On the low Mach number limit for Quantum Navier-Stokes equations

TL;DR

This paper studies the low Mach number limit of the 3-D quantum Navier-Stokes equations, proving strong convergence to incompressible flows using dispersive analysis and energy estimates, even for ill-prepared initial data.

Contribution

It introduces a dispersive analysis approach for acoustic parts and establishes strong convergence results for weak solutions in the quantum Navier-Stokes system.

Findings

Strong convergence of solutions to incompressible Navier-Stokes equations

Dispersive analysis based on Bogoliubov dispersion relation

Results hold for ill-prepared initial data

Abstract

We investigate the low Mach number limit for the 3-D quantum Navier-Stokes system. For general ill-prepared initial data, we prove strong convergence of finite energy weak solutions to weak solutions of the incompressible Navier-Stokes equations. Our approach relies on a quite accurate dispersive analysis for the acoustic part, governed by the well-known Bogoliubov dispersion relation for the elementary excitations of the weakly-interacting Bose gas. Once we have a control of the acoustic dispersion, the a priori bounds provided by the energy and Bresch-Desjardins entropy type estimates lead to the strong convergence. Moreover, for well-prepared data we show that the limit is a Leray weak solution, namely it satisfies the energy inequality. Solutions under consideration in this paper are not smooth enough to allow for the use of relative entropy techniques.

Equations

$$\{\begin{array}{ll}& {\partial }_t\rho +\mathrm{div}(\rho u)=0,\\ & {\partial }_t(\rho u)+\mathrm{div}\left(\rho u\otimes u\right)+\nabla P(\rho )=2\nu \mathrm{div}(\rho \mathbf{D}u)+2{\kappa }^2\rho \nabla \left(\frac{\Delta \sqrt{\rho }}{\sqrt{\rho }}\right),\end{array}$$

$$E(t)={\int }_{{\mathbb{R}}^3}\frac{1}{2}\rho |u{|}^2+2{\kappa }^2|\nabla \sqrt{\rho }{|}^2+\pi (\rho )\mathrm{d}x,$$

$$\pi =\pi (\rho )=\frac{{\rho }^{\gamma }-1-\gamma (\rho -1)}{\gamma (\gamma -1)}.$$

$$\rho \to 1\text{as}|x|\to \infty .$$

$$\{\begin{array}{ll}& {\partial }_t\rho +\mathrm{div}(\rho u)=0,\\ & {\partial }_t(\rho u)+\mathrm{div}(\rho u\otimes u)+\nabla P(\rho )=2\nu \mathrm{div}(\mathbb{S})+{\kappa }^2\mathrm{div}(\mathbb{K}),\end{array}$$

$$\mathbb{S}=h(\rho )\mathbf{D}u+g(\rho )\mathrm{div}u\mathbb{I},$$

$$\mathbb{K}=\left(\rho \mathrm{div}(k(\rho )\nabla \rho )-\frac{1}{2}(\rho k^{\prime }(\rho )-k(\rho )|\nabla \rho {|}^2)\right)\mathbb{I}-k(\rho )\nabla \rho \otimes \nabla \rho .$$

$$\{\begin{array}{ll}& {\partial }_t{\rho }_{\epsilon }+\mathrm{div}({\rho }_{\epsilon }{u}_{\epsilon })=0,\\ & {\partial }_t({\rho }_{\epsilon }{u}_{\epsilon })+\mathrm{div}\left({\rho }_{\epsilon }{u}_{\epsilon }\otimes u_{\epsilon }\right)+\frac{1}{{\epsilon }^2}\nabla P({\rho }_{\epsilon })=2\nu \mathrm{div}({\rho }_{\epsilon }\mathbf{D}{u}_{\epsilon })+2{\kappa }^2{\rho }_{\epsilon }\nabla \left(\frac{\Delta \sqrt{{\rho }_{\epsilon }}}{\sqrt{{\rho }_{\epsilon }}}\right),\end{array}$$

$${\rho }_{\epsilon }(0,x)$$

$$({\rho }_{\epsilon }{u}_{\epsilon })(0,x)$$

$${\pi }_{\epsilon }={\pi }_{\epsilon }({\rho }_{\epsilon })=\frac{{\rho }^{\gamma }-1-\gamma (\rho -1)}{{\epsilon }^2\gamma (\gamma -1)}.$$

$${\partial }_{t}u+u\cdot \nabla u+\nabla p=\nu \Delta u,\mathrm{div}u=0.$$

$$\Vert u{\Vert }_{L^p{L}^q}={\left({\int }_0^T{\left|{\int }_{{\mathbb{R}}^d}|u(t,x){|}^q\right|}^{\frac{p}{q}}\text{d}x\text{d}t\right)}^{\frac{1}{p}}.$$

$$L_2^p({\mathbb{R}}^d)=\left\{f\in L_{loc}^1({\mathbb{R}}^d):|f|{\chi }_{\{|f|\le \frac{1}{2}\}}\in L^2({\mathbb{R}}^d),|f|{\chi }_{\{|f|\ge \frac{1}{2}\}}\in L^p({\mathbb{R}}^d)\right\},$$

$$\mathbf{Q}=\nabla {\Delta }^{-1}\mathrm{div},\mathbf{P}=\mathbf{I}-\mathbf{Q}.$$

$$\frac{1}{p}=\frac{\theta }{p_1}+\frac{1-\theta }{p_2},s=\theta s_0+(1-\theta )s_1,$$

$$\Vert u{\Vert }_{L^p(0,T;W^{s,q}({\mathbb{R}}^d))}\le \Vert u{\Vert }_{L^{p_1}(0,T;W^{s_0,q}({\mathbb{R}}^d))}^{\theta }\Vert u{\Vert }_{L^{p_2}(0,T;W^{s_1,q}({\mathbb{R}}^d))}^{1-\theta }.$$

$$t\mapsto \epsilon t_{\epsilon },u\mapsto \epsilon u_{\epsilon }.$$

$$\nu \mapsto \epsilon {\nu }_{\epsilon },\kappa \mapsto \epsilon {\kappa }_{\epsilon },$$

$${\nu }_{\epsilon }\to \tilde{\nu }>0,{\kappa }_{\epsilon }\to \tilde{\kappa }>0,$$

$${\rho }_{\epsilon }\mathbf{D}{u}_{\epsilon }=\sqrt{{\rho }_{\epsilon }}{\mathbf{S}}_{\epsilon },$$

$$\sqrt{{\rho }_{\epsilon }}{\mathbf{T}}_{\epsilon }=\nabla ({\rho }_{\epsilon }{u}_{\epsilon })-2{\Lambda }_{\epsilon }\otimes \nabla \sqrt{{\rho }_{\epsilon }},$$

$$2\rho \nabla \left(\frac{\Delta \sqrt{\rho }}{\sqrt{\rho }}\right)=\mathrm{div}\left(\rho {\nabla }^2\log \rho \right)=\nabla \Delta \rho -4\mathrm{div}(\nabla \sqrt{\rho }\otimes \nabla \sqrt{\rho }).$$

$${\partial }_t({\rho }_{\epsilon }{u}_{\epsilon })+\mathrm{div}\left({\rho }_{\epsilon }{u}_{\epsilon }\otimes u_{\epsilon }+4{\kappa }^2\nabla \sqrt{{\rho }_{\epsilon }}\otimes \nabla \sqrt{{\rho }_{\epsilon }}\right)$$

$$=2\nu \mathrm{div}(\sqrt{{\rho }_{\epsilon }}{\mathbf{S}}_{\epsilon })+{\kappa }^2\nabla \Delta {\rho }_{\epsilon }.$$

$$\sqrt{{\rho }_{\epsilon }}\in L_{loc}^2((0,T)\times {\mathbb{R}}^3);$$

$$\nabla \sqrt{{\rho }_{\epsilon }}\in L_{loc}^2((0,T)\times {\mathbb{R}}^3);$$

$${\int }_{{\mathbb{R}}^3}{\rho }_{\epsilon }^0\varphi (0)+{\int }_0^T{\int }_{{\mathbb{R}}^3}{\rho }_{\epsilon }{\varphi }_t+\sqrt{{\rho }_{\epsilon }}\sqrt{{\rho }_{\epsilon }}{u}_{\epsilon }\nabla \varphi =0,$$

$${\int }_{{\mathbb{R}}^3}{\rho }_{\epsilon }^0{u}_{\epsilon }^0\psi (0)+{\int }_0^T{\int }_{{\mathbb{R}}^d}\sqrt{{\rho }_{\epsilon }}\sqrt{{\rho }_{\epsilon }}{u}_{\epsilon }{\psi }_t+(\sqrt{{\rho }_{\epsilon }}{u}_{\epsilon }\otimes \sqrt{{\rho }_{\epsilon }}{u}_{\epsilon })\nabla \psi +\frac{1}{{\epsilon }^2}{\rho }_{\epsilon }^{\gamma }\mathrm{div}\psi$$

$$-2\nu {\int }_0^T{\int }_{{\mathbb{R}}^3}\left(\sqrt{{\rho }_{\epsilon }}{u}_{\epsilon }\otimes \nabla \sqrt{{\rho }_{\epsilon }}\right)\nabla \psi -2\nu {\int }_0^T{\int }_{{\mathbb{R}}^3}\left(\nabla \sqrt{{\rho }_{\epsilon }}\otimes \sqrt{{\rho }_{\epsilon }}{u}_{\epsilon }\right)\nabla \psi$$

$$+\nu {\int }_0^T{\int }_{{\mathbb{R}}^3}\sqrt{{\rho }_{\epsilon }}\sqrt{{\rho }_{\epsilon }}{u}_{\epsilon }\Delta \psi +\nu {\int }_0^T{\int }_{{\mathbb{R}}^3}\sqrt{{\rho }_{\epsilon }}\sqrt{{\rho }_{\epsilon }}{u}_{\epsilon }\nabla \mathrm{div}\psi$$

$$-4{\kappa }^2{\int }_0^T{\int }_{{\mathbb{R}}^3}\left(\nabla \sqrt{{\rho }_{\epsilon }}\otimes \nabla \sqrt{{\rho }_{\epsilon }}\right)\nabla \psi +2{\kappa }^2{\int }_0^T{\int }_{{\mathbb{R}}^3}\sqrt{{\rho }_{\epsilon }}\nabla \sqrt{{\rho }_{\epsilon }}\nabla \mathrm{div}\psi =0,$$

$$E(t)={\int }_{{\mathbb{R}}^3}\frac{1}{2}{\left|\sqrt{{\rho }_{\epsilon }}{u}_{\epsilon }\right|}^2+2{\kappa }^2{\left|\nabla \sqrt{{\rho }_{\epsilon }}\right|}^2+{\pi }_{\epsilon }({\rho }_{\epsilon })\mathrm{d}x,$$

Full text

1. Introduction

In this paper, we study the low Mach number limit for a dispersive-diffusive fluid model, usually called the Quantum-Navier-Stokes equations (QNS). The system is posed on $(0,T)\times{\mathbb{R}}^{3}$,

[TABLE]

the unknowns are given by the mass density $\rho$ and the fluid velocity field $u$. We consider a pressure given by the usual $\gamma$-law, i.e. $P(\rho)=\frac{1}{\gamma}\rho^{\gamma}$, with $\gamma>1$, and where $\nu,\kappa>0$ denote the viscosity and capillarity coefficients respectively. The energy we consider for this system (1.5) is given by

[TABLE]

where the internal energy takes the form

[TABLE]

Thus, the finite energy assumption yields

[TABLE]

System (1.1) enters the more general class of Navier-Stokes-Korteweg systems and it is sometimes used as a model for dissipative quantum fluids particularly for numerical purposes, namely

[TABLE]

where the viscous stress tensor $\mathbb{S}=\mathbb{S}(\nabla u)$ is given by

[TABLE]

while the capillary (dispersive) term $\mathbb{K}=\mathbb{K}(\rho,\nabla\rho)$ reads

[TABLE]

System (1.1) is then recovered by choosing $h(\rho)=\rho$, $g(\rho)=0$ and $k(\rho)=\frac{1}{\rho}$. The QNS equations can also be derived from a Chapman-Enskog expansion for the Wigner equation with a BGK term [19, 37], see also [36] where several dissipative quantum fluid models are derived by means of a moment closure of (quantum) kinetic equations with appropriate choices of the collision terms. A more robust model, from the point of view of quantum physics, is given by the inviscid counterpart of (1.5), namely the classical Quantum Hydrodynamic system (QHD), see [6, 7, 8, 4], that arises in many different situations, for instance as a hydrodynamic model in superfluids [39] and Bose-Einstein condensates [48] or to describe the carrier transport in semiconductors [28].

After a suitable rescaling (see subsection 2.1), the system (1.1) reads,

[TABLE]

with initial data

[TABLE]

where $\varepsilon\ll 1$ is the scaled Mach number. Therefore, the scaled internal energy becomes

[TABLE]

In the low Mach number regime, i.e. in the limit as $\varepsilon\to 0$, the dynamics of (1.5) is formally governed by the incompressible Navier-Stokes equations,

[TABLE]

The aim of this paper is to rigorously study this limit in its full generality, i.e. by considering arbitrary finite energy initial data without imposing further regularity or smallness assumptions and in particular without being well-prepared. This class of initial data cannot provide in the limit smooth solutions to incompressible Navier-Stokes, for this reason we have to exclude the use of relative entropy methods [25], [26].

The QNS system entails some mathematical difficulties due to the possible appearance of vacuum regions. Indeed, the degenerate viscosity prevents a suitable control of the velocity field in the vacuum. In particular this yields some problems in establishing the necessary compactness estimates on the convective term $\rho_{\varepsilon}u_{\varepsilon}\otimes u_{\varepsilon}$. As a consequence, the mathematical analysis of fluid dynamical equations with degenerate viscosities differs substantially from the standard Feireisl-Lions theory [44, 27]. On the other hand, for certain compressible fluids with degenerate viscosity a further entropy estimate is available, first introduced by Bresch and Desjardins in [18] and later extended to a large class of Navier-Stokes-Korteweg systems [17]. For system (1.1), these estimates yield a regularizing effect for the mass density [35, 10], hence in our case they also allow for a better control of the far-field behavior compared to the classical compressible Navier-Stokes equations [45], see Lemmata 3.3 and 3.4. Finite energy weak solutions to (1.5) in the two and three dimensional torus were studied in [9, 41]. The Cauchy Problem on ${\mathbb{R}}^{d}$, $d=2,3$, with non-trivial far-field behavior has recently been addressed in [5]. The existence theory of finite energy weak solutions in [5] is based on the standard argument of invading domains, suitable truncations, and the available theory for ${\mathbb{T}}^{d}$ of [41]. Besides the mathematical difficulties originated from the degenerate viscosity, in this framework we also have to cope with the lack of integrability of the mass density due to non-trivial boundary conditions.

One of the main tools in this paper is provided by a class of suitable Strichartz estimates, that allow to capture more accurately the different dispersive scales involved in the propagation of the acoustic waves, as a consequence of the specific dispersion relation. Indeed, contrarily to the classical case where the fluctuations evolve accordingly to the classical wave equation [45, 22, 50], here in our problem the presence of the quantum term contributes in a non-trivial way to the dispersion relation, especially at high frequencies. The dispersion relation inferred here, see formula (4.1) below, is strictly related to the Bogoliubov spectrum describing excitations in a Bose-Einstein condensate, which predicts the superfluid behavior of the gas [16, 15, 49]. This is somehow reminiscent of the analysis of fluctuations done when studying the quasi-neutral limit for a class of Navier-Stokes-Korteweg systems [23, 24]. The analysis related to the dispersion relation (4.1) can be regarded as the $\varepsilon-$version of the results in [31]. We will present this analysis in the Appendix B. Here we remark that since the dispersion relation (4.1) is not homogeneous, we cannot obtain our estimates by a rescaling argument and we need to adapt the proof in [31]. On the other hand, it should be remarked that if we perform a frequency splitting as in [14], then the estimates (4.12), (4.13) deteriorate at low frequencies. A more detailed explanation can be found in Section 4 and Appendix B.

Furthermore, in the limit we recover a weak solution of the incompressible Navier-Stokes equation $u\in L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))\cap L^{2}(0,T,\dot{H}^{1}({\mathbb{R}}^{3}))$. We remark that we are able to obtain bounds on the gradient of the limiting solution to (1.7), even though at fixed $\varepsilon>0$ only a weak version of the energy inequality is available [9, 12, 41], see also the discussion in Section 2. However, this weak version of the energy inequality will anyway yield the aforementioned natural bounds on the gradient of the velocity field in the low Mach number limit. In fact, thanks to some uniform bounds satisfied by the momentum density, we can also infer further smoothing properties for the limiting solution to (1.7), see Theorem 2.4 and Proposition 5.4 for more details. Due to the presence of an initial layer which cannot be avoided for general ill-prepared data, the weak solution enters the Leray class only if further assumptions on the initial data are made. More precisely, only for well-prepared data it is possible to show that the solutions obtained in the limiting procedure satisfy the energy inequality.

The study of singular limits for fluid dynamical equations occupies a vast portion of mathematical literature, for a more comprehensive introduction to the topic we address the reader to the monograph [26] and the reviews [2, 46]. Our method shares some similarities with [22] which studies the compressible Navier-Stokes equations on the whole space. Indeed, there the authors exploit some Strichartz type estimates to analyse the acoustic waves. On the other hand, for the QNS system the dispersion relation is modified and reads as in formula (4.1); thus for high frequencies the fluctuations appearing in classical fluid dynamics and in system (1.1) differ considerably. Recently, the incompressible limit for a similar system has been investigated in [51] and later in [40]. In both papers the quantum Navier-Stokes system is augmented by adding a damping term in the momentum equation. This extra term allows to circumvent mathematical difficulties related to the lack of control of the velocity field in the vacuum. Moreover both papers deal with smooth local in time solutions for the limiting incompressible dynamics. By using this further regularity assumption it is then possible for them to exploit a relative entropy method.

Here, we tackle the problem from a different perspective, namely we retrieve global weak solutions in the limit rather than convergence to the unique local strong solution to the limiting system. Moreover, while in [40, 51] the fluctuations are studied by using a wave-like dispersion as for classical fluid dynamical systems, here we consider the full dispersion relation determined by the Bogoliubov spectrum (4.1) and obtain a better control on the fluctuations. This is achieved by carrying out a refined analysis on the dispersive properties of the acoustic waves that together with new uniform estimates enables us to study the low Mach number limit for general ill-prepared initial data without regularity or smallness assumptions and without damping. For the inviscid system, i.e. the QHD system, the low Mach number limit with ill-prepared data has been studied in [25] on the torus and on the plane in the forthcoming paper [3].

This paper is organized as follows, we introduce notations and preliminary results in Section 2. Subsequently, the needed a priori estimates are provided in Section 3. This is particularly relevant since finite energy weak solutions of (1.5) only obey a weak form of the energy inequality, for a detailed discussion see Appendix A. Section 4 is dedicated to the analysis of the acoustic waves. The strong convergence of finite energy weak solutions of (1.5) to weak solutions of (1.6) is achieved in Section 5 by means of an Aubin-Lions compactness argument. Furthermore, we investigate the regularity properties of the limit ${u}$ and show that $u$ lies in the class of Leray solutions under suitable additional assumptions. Appendix B is devoted to the proof of the dispersive estimates.

2. Preliminaries

Notations

We list the notations of function spaces and operators used in the following. We denote

• •

the symmetric part of the gradient by $\mathbf{D}u=\frac{1}{2}(\nabla u+(\nabla u)^{T})$ and the asymmetric part by $\mathbf{A}u=\frac{1}{2}(\nabla u-(\nabla u)^{T})$,

• •

by $\mathcal{D}({\mathbb{R}}_{+}\times{\mathbb{R}}^{3})$ the space of test functions $C_{c}^{\infty}({\mathbb{R}}_{+}\times{\mathbb{R}}^{3})$ an by $\mathcal{D^{\prime}}({\mathbb{R}}_{+}\times{\mathbb{R}}^{3})$ the space of distributions. The duality bracket between $\mathcal{D}$ and $\mathcal{D^{\prime}}$ is denoted by $\left\langle\cdot,\cdot\right\rangle$,

• •

by $L^{p}({\mathbb{R}}^{d})$ for $1\leq p\leq\infty$ the Lebesgue space with norm $\|\cdot\|_{L^{p}}$. We denote by $p^{\prime}$ the Hölder conjugate exponent of $p$, i.e. $1=\frac{1}{p}+\frac{1}{p^{\prime}}$, and for $0<T\leq\infty$ by $L^{p}(0,T;L^{q}({\mathbb{R}}^{d}))$ the space of functions $u:(0,T)\times{\mathbb{R}}^{d}\rightarrow{\mathbb{R}}^{n}$ with norm

[TABLE]

If $T=\infty$, we write $L^{p}({\mathbb{R}}_{+};L^{q}({\mathbb{R}}^{d}))$. Further, we denote by $L^{{p-}}(0,T;L^{q}({\mathbb{R}}^{d}))$, the functions $f$ such that $f\in L^{p_{0}}(0,T;L^{q}({\mathbb{R}}^{d}))$ for any $1\leq p_{0}<p$,

• •

The sum $L^{q_{1}}({\mathbb{R}}^{d})+L^{q_{2}}({\mathbb{R}}^{d})$ is a Banach space with norm $\|f\|_{L^{q_{1}}+L^{q_{2}}}=\inf\{\|g\|_{L^{q_{1}}}+\|h\|_{L^{q_{2}}}\,\,:\,f=g+h,\,g\in L^{q_{1}}({\mathbb{R}}^{d}),h\in L^{q_{2}}({\mathbb{R}}^{d})\}$,

• •

by $L_{2}^{p}({\mathbb{R}}^{d})$ the Orlicz space defined as

[TABLE]

we refer to [1, 44] for details.

• •

for $s\in{\mathbb{R}}$ and $p\in[1,\infty]$ the non-homogeneous Sobolev space by $W^{s,p}({\mathbb{R}}^{d})=(I-\Delta)^{-\frac{s}{2}}L^{p}({\mathbb{R}}^{d})$ and $H^{s}({\mathbb{R}}^{d})=W^{s,2}({\mathbb{R}}^{d})$. Its dual will be denoted by $W^{-s,p^{\prime}}({\mathbb{R}}^{d})$ with $p^{\prime}$ being the Hölder conjugate of $p$. The homogeneous spaces are denoted by $\dot{W}^{s,p}({\mathbb{R}}^{d})=(-\Delta)^{-\frac{k}{s}}L^{p}({\mathbb{R}}^{d})$ and $\dot{W}^{s,2}({\mathbb{R}}^{d})=\dot{H}^{s}({\mathbb{R}}^{d})$, and the dual space $\dot{W}^{-s,p^{\prime}}({\mathbb{R}}^{d})$. For $ps<d$ we denote the critical Sobolev exponent by $p^{\ast}=\frac{dp}{d-ps}$. We refer to Theorem 6.4.5 and Theorem 6.5.1 in [13], see also Chapter 4 in [1], for the classical embedding results for Sobolev and Lebesgue spaces.

• •

by $\mathbf{Q}$ and $\mathbf{P}$ the Helmholtz–Leray projectors on irrotational and divergence-free vector fields, respectively:

[TABLE]

For $f\in W^{k,p}({\mathbb{R}}^{d})$ with $1<p<\infty$ and $s\in{\mathbb{R}}$ the operators $\mathbf{P},\mathbf{Q}$ can be expressed as composition of Riesz multipliers and are bounded linear operators on $W^{s,p}({\mathbb{R}}^{d})$.

• •

the Fourier transform of $f$ by $\hat{f}:=\mathcal{F}(f)$ and the inverse Fourier transform by $f^{\vee}$,

• •

for $R>0$ the frequency cut-off $P_{R}(f)=(\phi_{R}(\xi)\hat{f})^{\vee}$, where $\phi$ is a smooth frequency cut-off compactly supported in $supp(\phi)\subset\{\frac{1}{2}R\leq|\xi|\leq 2R\}$. Similarly, by $P_{\leq R}(f)$ we denote the projection on frequencies of order $|\xi|\leq R$,

• •

by $\dot{B}_{q,r}^{s}({\mathbb{R}}^{d})$ the homogeneous Besov space. The dual space of $\dot{B}_{q,r}^{s}({\mathbb{R}}^{d})$ can be identified with $\dot{B}_{q^{\prime},r^{\prime}}^{-s}({\mathbb{R}}^{d})$, see Chapter 5 of [13].

In what follows $C$ will be any constant independent from $\varepsilon$.

For the convenience of the reader, we recall an interpolation result used several times throughout the paper.

Lemma 2.1** (Interpolation).**

Let $T>0$, $p_{1},p_{2},q\in(1,\infty)$ and $s_{0}<s_{1}$ real numbers. Further, let $u\in L^{p_{1}}(0,T;W^{s_{0},q}({\mathbb{R}}^{d}))\cap L^{p_{2}}(0,T;W^{s_{1},q}({\mathbb{R}}^{d}))$. Then, for all $(p,s)$ such that there exists $\theta\in(0,1)$ with

[TABLE]

it holds $u\in L^{p}(0,T;W^{s,q}({\mathbb{R}}^{d}))$ with

[TABLE]

The Lemma is a simplified statement of Theorem 5.1.2 in [13] applied with $A_{0}=W^{s_{0},q}({\mathbb{R}}^{d})$ and $A_{1}=W^{s_{1},q}({\mathbb{R}}^{d})$. The Lemma can also be proven by standard interpolation of Sobolev spaces in the space variables, see e.g. Paragraph 7.53 in [1], followed by Hölder’s inequality in the time variable.

2.1. Scaling

To recast the introduced scaling (1.5) of system (1.1), one starts writing the equations by re-scaling each length scale by its characteristic value (dimensionless scaling) and we assume the Mach number to be small. We expect the fluid to behave like an incompressible fluid on large time scales when the density is almost constant and the the velocity is small. Thus, we introduce the change of variable and unknowns,

[TABLE]

Moreover, the viscosity and capillarity coefficients scale as

[TABLE]

where

[TABLE]

as $\varepsilon$ goes to [math]. We refer the reader to the review papers [2, 46] for a more detailed discussion of suitable low Mach number scalings.

Weak solutions

As we already mentioned in the Introduction, the degenerate viscosity prevents the velocity field to be uniquely determined in the vacuum region; indeed system (1.5) lacks bounds for $u_{\varepsilon}$. Consequently, in this framework (see for example [9, 41]) it turns out that the problem is best studied in terms of the more suitable variables $\sqrt{\rho_{\varepsilon}}$ and $\Lambda_{\varepsilon}=\sqrt{\rho_{\varepsilon}}u_{\varepsilon}$. In fact, this occurs also when studying the QHD system [6] and the barotropic compressible Navier-Stokes equations with degenerate viscosities [43]. Mathematically speaking, this means that whenever the symbol $\rho_{\varepsilon}$ appears, it should be read as $\rho_{\varepsilon}=(\sqrt{\rho_{\varepsilon}})^{2}$ and similarly for the momentum density $m_{\varepsilon}=\rho_{\varepsilon}u_{\varepsilon}=\sqrt{\rho_{\varepsilon}}\Lambda_{\varepsilon}$. At no moment, the velocity field $u_{\varepsilon}$ or its gradient $\nabla u_{\varepsilon}$ are defined a.e. in ${\mathbb{R}}^{3}$. For those reasons, the viscous tensor should be rather thought as

[TABLE]

where $\mathbf{S}_{\varepsilon}$ is the symmetric part of the tensor $\mathbf{T}_{\varepsilon}$ defined through the following identity

[TABLE]

in $\mathcal{D}^{\prime}((0,T)\times{\mathbb{R}}^{3})$. In our paper, we do not use the notation $\Lambda_{\varepsilon}$ instead of $\sqrt{\rho_{\varepsilon}}u_{\varepsilon}$ for the sake of consistency with the literature regarding (quantum) Navier-Stokes equations. The definition of finite energy weak solutions will therefore be given in terms of the mathematical unknowns $\sqrt{\rho_{\varepsilon}}$ and $\sqrt{\rho_{\varepsilon}}u_{\varepsilon}$ instead of the physical unknowns of density $\rho_{\varepsilon}$ and momentum $m_{\varepsilon}$. We recall that under suitable assumptions on the mass density $\rho$ the quantum pressure term can be alternatively rewritten as

[TABLE]

For the sake of simplifying our exposition, from now on we will suppress the $\varepsilon$-dependence of $\nu$ and $\kappa$. In this way the equation for the momentum density in (1.5) reads

[TABLE]

Definition 2.2** (Finite energy weak solutions).**

A pair $(\rho_{\varepsilon},u_{\varepsilon})$ with $\rho_{\varepsilon}\geq 0$ is said to be a finite energy weak solution of the Cauchy Problem (1.5) if

1. (i)

integrability conditions

[TABLE] 2. (ii)

continuity equation

[TABLE]

for any $\phi\in C_{c}^{\infty}([0,T)\times{\mathbb{R}}^{3})$. 3. (iii)

momentum equation

[TABLE]

for any $\psi\in C_{c}^{\infty}([0,T)\times{\mathbb{R}}^{3};{\mathbb{R}}^{3})$. 4. (iv)

there exists a tensor $\mathbf{T}_{\varepsilon}\in L^{2}((0,T)\times{\mathbb{R}}^{3})$ satisfying identity (2.2) in $\mathcal{D}^{\prime}((0,T)\times{\mathbb{R}}^{3})$ such that for

[TABLE]

the following energy inequality holds for a.e. $t\in[0,T]$,

[TABLE]

where $\mathbf{S}_{\varepsilon}$ is the symmetric part of $\mathbf{T}_{\varepsilon}$, i.e. $\mathbf{S}_{\varepsilon}=\mathbf{T}_{\varepsilon}^{sym}$.

The definition of finite energy weak solutions including (2.4) is motivated by the degeneracy of the viscosity. While for smooth solutions of (1.5) with $\rho_{\varepsilon}>0$ the tensor $\mathbf{S}_{\varepsilon}$ is equivalent to $\sqrt{\rho_{\varepsilon}}\mathbf{D}u_{\varepsilon}$, this information may not be recovered for finite energy weak solutions. In fact, at present it is not clear whether arbitrary finite energy weak solutions to (1.5) satisfy the following energy inequality

[TABLE]

Therefore, we only assume the weaker version of the energy inequality (2.4). The aforementioned Bresch-Desjardins entropy further leads us to define the following notion of BD-entropic solutions.

Definition 2.3**.**

Let $(\rho_{\varepsilon},u_{\varepsilon})$ be a finite energy weak solution to (1.5) as in Definition 2.2, we say that $(\rho_{\varepsilon},u_{\varepsilon})$ is a BD-entropic weak solution if there exists $c>0$ such that defining the BD entropy as

[TABLE]

then the Bresch-Desjardins entropy inequality holds for a.e. $t\in[0,T]$,

[TABLE]

where $\mathbf{A}_{\varepsilon}=\mathbf{T}_{\varepsilon}^{asym}$, with $\mathbf{T}_{\varepsilon}$ satisfying identity (2.2).

Let us further comment on Definition 2.2 and 2.3 respectively. The existence of global in time BD-entropic weak solutions to equation (1.1), posed in ${\mathbb{R}}^{d}$, $d=2,3$ with non-trivial far-field behavior for the mass density, has recently been proved in [5], see also [9, 41, 11] for similar results on the periodic domain ${\mathbb{T}}^{d}$. More specifically, in [5] it has been proven that, for any finite energy initial data, there exists a global solution satisfying the Definitions 2.2 and 2.3 above. Their proof exploits the construction of a sequence of approximating solutions by solving the Cauchy problem on a sequence of invading domains. One of the key points in [5] is that the approximating solutions satisfy both the energy and BD entropy estimates, which allow to obtain the necessary compactness and to pass to the limit. Therefore, the solutions studied satisfy the bounds (2.4) and (2.6) by construction. Let us remark that in general it is not true that weak solutions to (1.1) satisfy the energy inequality (2.4) and the BD entropy estimate (2.6). Moreover, it is an open problem to determine the minimal assumptions on weak solutions to (1.1) in order to satisfy the two aforementioned inequalities. Presently, a physical interpretation of the BD type entropies is not yet clear, however this kind of estimates are naturally satisfied by a large class of weak solutions to compressible, viscous, fluid dynamic equations with a degenerate viscosity coefficient. At the formal level, they can be interpreted as the energy estimates associated to an equivalent system to (1.1), reformulated in terms of an effective velocity involving the density gradient [35], see also system (A.6). For the convenience of the reader, in Appendix A we provide a class of weak solutions to (1.1) which satisfies (2.4) and (2.6). More precisely, we give a direct proof that the weak solutions on ${\mathbb{T}}^{d}$ constructed in [9] as limit of smooth approximating solutions satisfy the aforementioned inequalities. Let us also mention that it is possible to adopt different approximation procedures to show the existence of weak solutions satisfying (2.4) and (2.6), for example a suitable strategy would be the one adopted in [5], see also [12] where a similar issue is dealt with for a Navier-Stokes-Korteweg type system and [20] where a further approximation method is used in the isothermal case. We stress that despite the fact that only (2.4) and (2.6) are available for the compressible system (1.5), we achieve convergence to a Leray weak solution for well-prepared initial data, see Proposition 2.6.

Main result

Let us specify the assumptions on the initial data for the system (1.5). We consider initial data $(\rho_{\varepsilon}^{0},u_{\varepsilon}^{0})$ of finite energy, namely such that

[TABLE]

where is $C$ independent on $\varepsilon>0$. Furthermore, we assume that

[TABLE]

With this definition at hand, we now state the main Theorem characterising the low Mach number regime for (1.5).

Theorem 2.4**.**

*Let $\gamma>1$ and let $(\rho_{\varepsilon},u_{\varepsilon})$ be a BD-entropic weak solution of (1.5) with initial data satisfying (2.7) and (2.8) and let $0<T<\infty$ be an arbitrary time. Then $\rho_{\varepsilon}-1$ converges strongly to [math] in $L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))\cap L^{4}(0,T;H^{s}({\mathbb{R}}^{3}))$ for any $0\leq s<1$. For any subsequence (not relabeled) $\sqrt{\rho_{\varepsilon}}u_{\varepsilon}$ converging weakly to ${u}$ in $L^{\infty}(0,T,L^{2}({\mathbb{R}}^{3}))$, then $u\in L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))\cap L^{2}(0,T;\dot{H}^{1}({\mathbb{R}}^{3}))$ is a global weak solution to the incompressible Navier-Stokes equation (1.7) with initial data {u}_{\big{|}t=0}=\mathbf{P}(u_{0}) and $\sqrt{\rho_{\varepsilon}}u_{\varepsilon}$ converges strongly to $u$ in $L^{2}(0,T;L_{loc}^{2}({\mathbb{R}}^{3}))$.

Moreover, $\mathbf{Q}(\rho_{\varepsilon}u_{\varepsilon})$ converges strongly to [math] in $L^{2}(0,T,L^{q}({\mathbb{R}}^{3}))$ for any $2<q<\frac{12}{5}$. Finally the limiting solution $u$ also satisfies $u\in L^{\frac{4}{1+4s}-}(0,T;H^{s}({\mathbb{R}}^{3}))$, for $0\leq s\leq\frac{1}{2}$.*

The existence of global in time BD-entropic results to (1.5) posed on ${\mathbb{R}}^{d}$, $d=2,3$ with far-field (1.4) has recently been introduced in [5], see also [9, 41] for related results on ${\mathbb{T}}^{d}$ with $d=2,3$.

Remark 2.5**.**

Let us remark that in order for the limiting function ${u}$ to satisfy the energy inequality, i.e. to be a Leray weak solution [42], stronger assumptions on the initial data $(\rho_{\varepsilon}^{0},u_{\varepsilon}^{0})$ are needed. Indeed the initial total energy for the compressible system in general does not converge, as $\varepsilon\to 0$, to the initial energy for (1.7), which would be given by $\frac{1}{2}\int|\mathbf{P}u_{0}|^{2}$. The excess energy determines an initial layer which cannot be avoided for ill-prepared data. On the other hand, if we require

[TABLE]

then the following Proposition holds true.

Proposition 2.6**.**

Under the same assumptions of Theorem 2.4, let $(\rho_{\varepsilon}^{0},u_{\varepsilon}^{0})$ further satisfy (2.9). Then the limiting solution $u$ to (1.7) satisfies the energy inequality

[TABLE]

for almost every $t\in[0,T]$.

3. Uniform estimates

In this Section, we start our analysis on the low Mach number limit by inferring some uniform estimates for finite energy weak solutions to (1.5). In our framework we need to take the non trivial boundary conditions for the mass density into account. For this reason, we provide some estimates on the quantities $\sqrt{\rho_{\varepsilon}}-1$ and $\rho_{\varepsilon}-1$. Furthermore, the lack of control for $\nabla u_{\varepsilon}$ in the vacuum will be compensated by the bounds inferred from (2.4) and (2.6).

Remark 3.1**.**

We shall use repeatedly the following observation, see for instance Theorem 4.5.9 in [34]: if $f\in\mathcal{D^{\prime}}({\mathbb{R}}^{d})$ with $\nabla f\in L^{p}({\mathbb{R}}^{d})$ for $p<d$, then there exists a constant $c$ such that $f-c\in L^{p^{\ast}}({\mathbb{R}}^{d})$, where $p^{\ast}=\frac{pd}{d-p}$. The condition $p<d$ is sharp.

If $(\rho_{\varepsilon},u_{\varepsilon})$ is of finite energy, i.e. $E(\rho_{\varepsilon},u_{\varepsilon})<\infty$, then

[TABLE]

This implies additional bounds which we summarize.

Lemma 3.2**.**

Let $C>0$ independent from $\varepsilon>0$ and $(\rho_{\varepsilon},u_{\varepsilon})$ be such that $E(\rho_{\varepsilon},u_{\varepsilon})\leq C$, then

1. (i)

$\rho_{\varepsilon}$* converges strongly to $1$ in $L^{2}({\mathbb{R}}^{3})$, more precisely*

[TABLE] 2. (ii)

$\sqrt{\rho_{\varepsilon}}-1\in H^{1}({\mathbb{R}}^{3})$* is uniformly bounded. Moreover,*

[TABLE] 3. (iii)

$\rho_{\varepsilon}u_{\varepsilon}\in L^{\frac{3}{2}}({\mathbb{R}}^{3})+L^{2}({\mathbb{R}}^{3})$* uniformly bounded. In particular $\rho_{\varepsilon}u_{\varepsilon}\in H^{-s}({\mathbb{R}}^{3})$ uniformly bounded for all $s\geq\frac{1}{2}$.*

We remark that the decay rate of (i) is not optimal but sufficient for our purpose.

Proof.

We recall the argument in [45] relating the integrability of the internal energy to the Orlicz spaces $L_{2}^{\gamma}({\mathbb{R}}^{3})$. Exploiting the convexity of

[TABLE]

for $\gamma>1$ one obtains from the uniform bound of the energy functional $E$ that

[TABLE]

where $\pi_{\varepsilon}(\rho_{\varepsilon})$ is defined in (1.6). It follows in particular that $\rho_{\varepsilon}-1\in L_{2}^{\gamma}({\mathbb{R}}^{3})$ uniformly bounded for $\gamma>1$ and

[TABLE]

provided $\gamma\geq 2$. Next, we show (ii). It follows from Remark 3.1 and $\nabla\sqrt{\rho_{\varepsilon}}\in L^{2}({\mathbb{R}}^{3})$ that there exists $c\in{\mathbb{R}}$ such that $\sqrt{\rho_{\varepsilon}}-c\in L^{6}({\mathbb{R}}^{3})$ uniformly bounded. We notice that

[TABLE]

and therefore (3.2) implies that

[TABLE]

It follows that $\sqrt{\rho_{\varepsilon}}-1\in L_{2}^{\gamma}({\mathbb{R}}^{3})$ uniformly bounded. In particular, for $\gamma\geq 2$ we have

[TABLE]

and $\sqrt{\rho_{\varepsilon}}-1\in H^{1}({\mathbb{R}}^{3})$ provided $\gamma\geq 2$. For $1<\gamma<2$, we infer the desired decay by interpolation. First, we notice that $\sqrt{\rho_{\varepsilon}}-1\in L^{6}({\mathbb{R}}^{3})$ uniformly, namely $c=1$. This easily follows from (3.3) and Markov inequality. Therefore, for $1<\gamma<2$ and $\theta=\frac{2\gamma}{6-\gamma}$ we obtain by interpolation that

[TABLE]

This estimate together with (3.3) yields

[TABLE]

We proceed to show (i) for $1<\gamma<2$. We observe that

[TABLE]

for $|z-1|\geq\frac{1}{2}$. Hence by interpolation between (3.4) and the uniform $L^{6}$-bound for $\sqrt{\rho_{\varepsilon}}-1$ of the previous step, we obtain

[TABLE]

It follows,

[TABLE]

Finally, to prove (iii) we notice that if $(\rho_{\varepsilon},u_{\varepsilon})$ is of finite energy then the bounds $\sqrt{\rho_{\varepsilon}}-1\in L^{6}({\mathbb{R}}^{3})$ from (ii) and $\sqrt{\rho_{\varepsilon}}u_{\varepsilon}\in L^{2}({\mathbb{R}}^{3})$ allow us to conclude

[TABLE]

As $L^{\frac{3}{2}}({\mathbb{R}}^{3})\hookrightarrow H^{-s}({\mathbb{R}}^{3})$ continuously for $s\geq\frac{1}{2}$, statement (iii) is proven. This completes the proof. ∎

3.1. Uniform estimates on the solution

By Definition 2.3, the BD-entropic weak solution $(\rho_{\varepsilon},u_{\varepsilon})$ we consider satisfies the energy inequality (2.4) and the BD entropy type inequality (2.6) that imply the following a priori estimates listed below.

Lemma 3.3**.**

If $(\rho_{\varepsilon},u_{\varepsilon})$ is a BD-entropic weak solution of (1.5), then for any $0<T<\infty$ there exists $C>0$ independent from $\varepsilon>0$ such that

1. (i)

$\|\rho_{\varepsilon}-1\|_{L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))}\leq C\varepsilon^{\alpha}$, for $\alpha(\gamma)$ defined as in (3.1) 2. (ii)

$\|\frac{1}{\varepsilon}\nabla\rho_{\varepsilon}^{\frac{\gamma}{2}}\|_{L^{2}(0,T;L^{2}({\mathbb{R}}^{3}))}\leq C$* ,* 3. (iii)

$\sqrt{\rho_{\varepsilon}}-1\in L^{\infty}(0,T;H^{1}({\mathbb{R}}^{3}))$* uniformly and in particular for $2\leq p<6$*

[TABLE] 4. (iv)

$\|\nabla^{2}\sqrt{\rho_{\varepsilon}}\|_{L^{2}({0,T};L^{2}({\mathbb{R}}^{3}))}\leq C$, 5. (v)

for any $0<s<2$ and $2\leq p<\frac{4}{s}$, there exists $0<\beta(p,s)<2$ such that $\|\sqrt{\rho_{\varepsilon}}-1\|_{L^{p}(0,T;H^{s}({\mathbb{R}}^{3}))}\leq C\varepsilon^{\beta}$. Moreover, for $1<s\leq 2$,

$\|\sqrt{\rho_{\varepsilon}}-1\|_{L^{\frac{2}{s-1}}(0,T;H^{s}({\mathbb{R}}^{3}))}\leq C$.* In particular, $\sqrt{\rho_{\varepsilon}}-1\in L^{4-}({0,T};L^{\infty}({\mathbb{R}}^{3}))$ uniformly bounded.* 6. (vi)

$\|\sqrt{\rho_{\varepsilon}}u_{\varepsilon}\|_{L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))}\leq C$,

Proof.

From (2.4), one has that $\sup_{t\geq 0}E(\rho_{\varepsilon}(t),u_{\varepsilon}(t))\leq C$, hence the bounds of Lemma 3.2 are uniform in $t\geq 0$. We notice that (i) and the first statement of (iii) are immediate consequences of (i) and (ii) of Lemma 3.2. The second part of (iii) follows by Sobolev embedding and interpolation with (ii) of Lemma 3.2 being valid uniformly in time. More precisely, for $p\in(2,6)$ and $\theta=3\left(\frac{1}{p}-\frac{1}{6}\right)$ one has

[TABLE]

with $\alpha$ as defined in (ii) of Lemma 3.2. The remaining statements except from (v) are direct consequences of inequalities (2.4) and (2.6). Statement (v) follows by interpolation of (iii) and (iv). We notice that (iii) and (iv) yield $\sqrt{\rho_{\varepsilon}}-1\in L^{2}(0,T;H^{2}({\mathbb{R}}^{3}))$ uniformly. We apply Lemma 2.1 to conclude (v). For $0<s<2$ and $0<\theta<1$ there exists $\beta>0$ such that

[TABLE]

where $s=2\theta$ and $p$ such that $p\leq\frac{2}{\theta}$. In particular if $s>\frac{3}{2}$ this yields for any $2\leq p<\frac{8}{3}$ that $\sqrt{\rho_{\varepsilon}}-1$ converges strongly to [math] in $L^{p}(0,T;L^{\infty}({\mathbb{R}}^{3}))$. By interpolation between the bounds $\sqrt{\rho_{\varepsilon}}-1\in L^{\infty}(0,T;H^{1}({\mathbb{R}}^{3}))$ and $\nabla^{2}(\sqrt{\rho_{\varepsilon}}-1)\in L^{2}(0,T;L^{2}({\mathbb{R}}^{3}))$, one may infer the slightly stronger bound $\sqrt{\rho_{\varepsilon}}-1\in L^{\frac{2}{s-1}}(0,T;H^{s}({\mathbb{R}}^{3}))$ for $1<s\leq 2$. ∎

3.2. Bounds on density fluctuations and momenta

First, we provide bounds on the density fluctuation $\sigma_{\varepsilon}:=\frac{\rho_{\varepsilon}-1}{\varepsilon}$.

Lemma 3.4**.**

If $(\rho_{\varepsilon},u_{\varepsilon})$ is a BD-entropic weak solution of (1.5), then for any $0<T<\infty$, the fluctuations $\sigma_{\varepsilon}$ satisfy the following

1. (i)

$\sigma_{\varepsilon}^{0}\in L_{2}^{\gamma}({\mathbb{R}}^{3})$* uniformly bounded and in particular $\sigma_{\varepsilon}^{0}\in L^{2}({\mathbb{R}}^{3})$ for $\gamma\geq 2$ and $\sigma_{\varepsilon}^{0}\in H^{-\frac{3}{2}}({\mathbb{R}}^{3})$ for $\gamma\in(1,2)$ uniformly bounded,* 2. (ii)

$\varepsilon\nabla\sigma_{\varepsilon}^{0}\in H^{-\frac{1}{2}}({\mathbb{R}}^{3})$* uniformly bounded,* 3. (iii)

$\sigma_{\varepsilon}\in L^{\infty}(0,T;L_{2}^{\gamma}({\mathbb{R}}^{3}))$* uniformly bounded, in particular $\sigma_{\varepsilon}\in L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))$ for $\gamma\geq 2$ and $\sigma_{\varepsilon}\in L^{\infty}(0,T;H^{-\frac{3}{2}}({\mathbb{R}}^{3}))$ for $\gamma\in(1,2)$ uniformly bounded,* 4. (iv)

$\varepsilon\sigma_{\varepsilon}\in L^{4-}(0,T;H^{1}({\mathbb{R}}^{3}))$* uniformly bounded,* 5. (v)

if $\gamma=2$, then $\sigma_{\varepsilon}\in L^{2}(0,T,H^{1}({\mathbb{R}}^{3}))$ uniformly bounded.

Proof.

For (i), we notice that $\sigma_{\varepsilon}^{0}=\frac{\rho_{\varepsilon}^{0}-1}{\varepsilon}$. It follows from Lemma 3.2 that $\sigma_{\varepsilon}^{0}\in L_{2}^{\gamma}({\mathbb{R}}^{3})$ uniformly bounded. If $\gamma\geq 2$, then $\sigma_{\varepsilon}^{0}\in L^{2}({\mathbb{R}}^{3})$ uniformly bounded. If $\gamma\in(1,2)$, we observe that $\sigma_{\varepsilon}^{0}\in L_{2}^{\gamma}({\mathbb{R}}^{3})$ uniformly bounded implies in particular $\sigma_{\varepsilon}^{0}\in L^{2}({\mathbb{R}}^{3})+L^{\gamma}({\mathbb{R}}^{3})$ uniformly. Since $L^{\gamma}({\mathbb{R}}^{3})\hookrightarrow H^{-\frac{3}{2}}({\mathbb{R}}^{3})$ for $\gamma\in(1,2]$, we obtain $\sigma_{\varepsilon}^{0}\in H^{-\frac{3}{2}}({\mathbb{R}}^{3})$. To show (ii), we observe that

[TABLE]

in virtue of (iii) of Lemma 3.3. Hence $\varepsilon\nabla\sigma_{\varepsilon}^{0}\in H^{-\frac{1}{2}}({\mathbb{R}}^{3})$. The first part of (iii) follows from (3.2). The second part follows from the Sobolev embedding upon observing that $L_{2}^{\gamma}({\mathbb{R}}^{3})\subset L^{2}({\mathbb{R}}^{3})$ for $\gamma\geq 2$ and $L_{2}^{\gamma}({\mathbb{R}}^{3})\subset L^{2}({\mathbb{R}}^{3})+L^{\gamma}({\mathbb{R}}^{3})\hookrightarrow H^{-\frac{3}{2}}({\mathbb{R}}^{3})$ otherwise. To show (iv), we observe that $\varepsilon\sigma_{\varepsilon}\in L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))$ from (i) of Lemma 3.3. By exploiting (iii) and (v) of Lemma 3.3, we obtain

[TABLE]

Moreover, for $\gamma=2$, the estimate $\frac{1}{\varepsilon}\nabla\rho_{\varepsilon}^{\frac{\gamma}{2}}\in L^{2}(0,T;L^{2}({\mathbb{R}}^{3}))$ provided by (2.6) allows us to conclude that $\sigma_{\varepsilon}\in L^{2}(0,T;H^{1}({\mathbb{R}}^{3}))$. ∎

Corollary 3.5**.**

If $(\rho_{\varepsilon},u_{\varepsilon})$ is a BD-entropic weak solution of (1.5), then for any $0<T<\infty$,

[TABLE]

Proof.

It is sufficient to write $\rho_{\varepsilon}u_{\varepsilon}=\sqrt{\rho_{\varepsilon}}u_{\varepsilon}+(\sqrt{\rho_{\varepsilon}}-1)\sqrt{\rho_{\varepsilon}}u_{\varepsilon}$ and to see that

[TABLE]

since $\sqrt{\rho_{\varepsilon}}-1\in L^{4-}(0,T;L^{\infty}({\mathbb{R}}^{3}))$ uniformly in view of (v) from Lemma 3.3. ∎

The Corollary 3.5 together with the a priori bounds on $\sqrt{\rho_{\varepsilon}}-1$ and $\mathbf{T_{\varepsilon}}$ allow us to prove a stronger estimate on $\rho_{\varepsilon}u_{\varepsilon}$.

Proposition 3.6**.**

If $(\rho_{\varepsilon},u_{\varepsilon})$ is a BD-entropic weak solution of (1.5), then for any $0<T<\infty$, $0\leq s\leq\frac{1}{2}$ and $1\leq p<\frac{4}{1+4s}$,

[TABLE]

*uniformly in $\varepsilon>0$.

In particular, for any $0\leq s_{1}<\frac{1}{4}$ one has $\rho_{\varepsilon}u_{\varepsilon}\in L^{2}(0,T;H^{s_{1}}({\mathbb{R}}^{3}))$.*

Proof.

From (2.2), one has

[TABLE]

in distributional sense where $F_{\varepsilon}:=F_{\varepsilon}^{1}+F_{\varepsilon}^{2}$ with

[TABLE]

We observe that $F_{\varepsilon}^{1}\in L^{2}(0,T;L^{\frac{3}{2}}({\mathbb{R}}^{3}))$ uniformly and $F_{\varepsilon}^{2}\in L^{\frac{4}{3}-}(0,T;L^{2}({\mathbb{R}}^{3}))$ uniformly in virtue of the uniform bounds of Lemma 3.3 and the inequalities (2.4) and (2.6). As $L^{\frac{3}{2}}({\mathbb{R}}^{3})\hookrightarrow\dot{H}^{-\frac{1}{2}}({\mathbb{R}}^{3})$ we have $F_{\varepsilon}^{1}\in L^{2}(0,T;\dot{H}^{-\frac{1}{2}}({\mathbb{R}}^{3}))$. It follows

[TABLE]

We deduce that

[TABLE]

Combining this bound with $\rho_{\varepsilon}u_{\varepsilon}\in L^{4-}(0,T;L^{2}({\mathbb{R}}^{3}))$ from Corollary 3.5 yields

[TABLE]

Hence, $\rho_{\varepsilon}u_{\varepsilon}\in L^{\frac{4}{3}-}(0,T;H^{\frac{1}{2}}({\mathbb{R}}^{3}))$ as $H^{s_{1}}({\mathbb{R}}^{3})\subset H^{s_{0}}({\mathbb{R}}^{3})$ for all real $s_{1}\geq s_{0}$. Finally, using again $\rho_{\varepsilon}u_{\varepsilon}\in L^{4-}(0,T;L^{2}({\mathbb{R}}^{3}))$ from Corollary 3.5 we conclude by applying Lemma 2.1 that

[TABLE]

for $1<p<\frac{4}{1+4s}$. ∎

4. Acoustic waves

This section is devoted to the analysis of the acoustic waves in the system. For highly subsonic flows they undergo rapid oscillations in time, so that one expects the acoustic waves to converge weakly to [math]. Furthermore, we will see that the dispersion relation satisfied by the fluctuations around the incompressible flow is not given by the classical waves but by the (scaled) Bogoliubov dispersion relation [16], which in our system reads

[TABLE]

see (4.10) below.

To perform this analysis we use identity (2.3) and rewrite system (1.5) as

[TABLE]

where we recall that the term $\rho_{\varepsilon}\mathbf{D}u_{\varepsilon}$ should be interpreted as in (2.1). We notice that, by using (1.6) we can write

[TABLE]

so that upon denoting the density fluctuations $\sigma_{\varepsilon}=\frac{\rho_{\varepsilon}-1}{\varepsilon}$ and momentum $m_{\varepsilon}=\rho_{\varepsilon}u_{\varepsilon}$ equation (4.2) reads

[TABLE]

and

[TABLE]

Projecting onto irrotational vector fields we obtain the system describing acoustic waves

[TABLE]

The initial datum for (4.5) is given by

[TABLE]

where we observe that from (i) of Lemma 3.4 and (iii) of Lemma 3.2,

[TABLE]

The main result of this section shows the strong convergence to [math] of the acoustic waves.

Theorem 4.1**.**

Let $(\rho_{\varepsilon},u_{\varepsilon})$ be a BD-entropic weak solution of (1.5). Then, for any $0<T<\infty$,

1. (i)

the density fluctuations $\rho_{\varepsilon}-1$ converge strongly to [math] in $C(0,T;L^{2}({\mathbb{R}}^{3}))$ and in $L^{4}(0,T;H^{s}({\mathbb{R}}^{3}))$ for any $s<1$, 2. (ii)

If $\gamma=2$, then $\sigma_{\varepsilon}$ converges strongly to [math] in $L^{2}(0,T;L^{q}({\mathbb{R}}^{3}))$ for any $2<q<6$, 3. (iii)

for any $2<q<\frac{12}{5}$ there exists $\delta>0$ such that

[TABLE]

In order to infer estimates on $(\sigma_{\varepsilon},\mathbf{Q}(m_{\varepsilon}))$ by studying (4.5), we derive Strichartz estimates for a symmetrization of the linearised system (4.3) that will ultimately imply the convergence of $(\sigma_{\varepsilon},\mathbf{Q}(m_{\varepsilon}))$. More precisely, we define

[TABLE]

and check that if $(\sigma_{\varepsilon},m_{\varepsilon})$ is a solution of (4.3) then $(\tilde{\sigma}_{\varepsilon},\tilde{m}_{\varepsilon})$ satisfies the symmetrised system

[TABLE]

where $\tilde{F}_{\varepsilon}=(-\Delta)^{-\frac{1}{2}}\mathop{\mathrm{div}}F_{\varepsilon}$. Hence, the linear evolution is characterised by the unitary semigroup $e^{-itH_{\varepsilon}}$, where

[TABLE]

is a self-adjoint operator with Fourier multiplier given by (4.1). In what follows, we are going to provide a class of Strichartz estimates for the linear propagator $e^{-itH_{\varepsilon}}$ which will yield a control of some mixed space-time norms of $(\tilde{\sigma_{\varepsilon}},\tilde{m_{\varepsilon}})$ in terms of the (scaled) Mach number. An interpolation argument exploiting the a priori estimates introduced in Section 3 gives the final result. For the sake of conciseness we postpone the proof of the Strichartz estimates to the appendix B.

Before stating the next Proposition, we recall that a pair of Lebesgue exponents $(p,q)$ is called Schrödinger admissible if $2\leq p,q\leq\infty$ and $\frac{2}{p}+\frac{3}{q}=\frac{3}{2}$. Given $q\in(2,6]$, we denote

[TABLE]

Proposition 4.2**.**

Let $\varepsilon>0$, $s\in{\mathbb{R}}$ and $(p,q)$, $(p_{1},q_{1})$ admissible pairs with $(p,q)\neq(\infty,2)$. Then for all $\alpha\in[0,\alpha_{0}]$ the following estimates hold true

[TABLE]

[TABLE]

Proposition 4.2 will be proved in Appendix B, in fact it will be a consequence of the more general Proposition B.12. We notice that Proposition B.12 yields that (4.12) is valid for all $\alpha\in[0,\alpha_{0}]$ with $\alpha_{0}$ as in (4.11). The non-homogeneous estimate (4.13) holds for $\alpha\in[0,\frac{d-2}{2}(1-\frac{1}{q}-\frac{1}{q_{1}})]$, we observe that $\alpha_{0}\leq\frac{d-2}{2}(1-\frac{1}{q}-\frac{1}{q_{1}})$ as $q_{1}\geq 2$. Let us remark that the case $\varepsilon=1$ was already studied in [31], where the authors infer dispersive estimates for the propagator $e^{itH_{1}}$ in order to study scattering properties for the Gross-Pitaevskii equation. In our case we need to keep track of the $\varepsilon-$dependence of the estimates, in order to show the convergence to zero of the acoustic part. However, since $H_{\varepsilon}=H_{\varepsilon}(\sqrt{-\Delta})$ is a non-homogeneous function of $\sqrt{-\Delta}$, it is not possible to obtain a decay in $\varepsilon$ by simply scaling the estimates in [31]. This is for example different from what happens for classical fluids [22] where the wave-like acoustic dispersion yields the convergence to zero by scaling the estimates and by considering the fast dynamics for the fluctuations.

On the other hand here we can exploit that the Strichartz estimates associated to the operator (4.10) are sligthly better than the ones for the Schrödinger operator close to the Fourier origin. This fact is also noticed in [31] for $H_{1}$. By exploiting this regularizing effect, Proposition 4.2 somehow improves a class of similar estimates inferred in [14] in another context (the linear wave regime for the Gross-Pitaevskii equation). Indeed the authors of [14] consider $H_{\varepsilon}$ in two different regimes: for low frequencies below the threshold $\frac{1}{\varepsilon}$ the operator behaves like the wave operator, while above the threshold it is Schrödinger-like. In this way the low frequency part experiences a derivative loss, due to the wave-type dispersive estimates inferred.

Here we do not split $H_{\varepsilon}$ in low and high frequencies, nevertheless we prove the convergence to zero of the acoustic part by only losing a small amount of derivatives.

In order to apply the estimates (4.12), (4.13) to system (4.5), we first need to bound $F_{\varepsilon}$ defined in (4.4) in suitable spaces.

Lemma 4.3**.**

If $(\rho_{\varepsilon},u_{\varepsilon})$ is a BD-entropic weak solution to (1.5), then one has,

1. (i)

$F_{\varepsilon}^{1}=\mathop{\mathrm{div}}\left(\rho_{\varepsilon}u_{\varepsilon}\otimes u_{\varepsilon}+4\kappa^{2}\nabla\sqrt{\rho_{\varepsilon}}\otimes\nabla\sqrt{\rho_{\varepsilon}}\right)+(\gamma-1)\nabla\pi_{\varepsilon})\in L^{\infty}(0,T;W^{-2,q^{\prime}}({\mathbb{R}}^{3}))$* for $q^{\prime}\in(1,\frac{3}{2})$,* 2. (ii)

$F_{\varepsilon}^{2}=2\nu\mathop{\mathrm{div}}\left(\rho_{\varepsilon}\mathbf{D}u_{\varepsilon}\right)\in L^{\frac{4}{3}-}(0,T;H^{-1}({\mathbb{R}}^{3}))$.

Proof.

We recall that $W^{1,p}({\mathbb{R}}^{3})\hookrightarrow L^{\infty}({\mathbb{R}}^{3})$ for any $p>3$. By duality for $p^{\prime}$, one has $L^{1}({\mathbb{R}}^{3})\hookrightarrow W^{-1,p^{\prime}}({\mathbb{R}}^{3})$. For the first statement, we observe that

[TABLE]

and thus $F_{\varepsilon}^{1}\in L^{\infty}(0,T;W^{-2,q^{\prime}}({\mathbb{R}}^{3}))$ for $q\in(3,\infty)$. Regarding the second statement, we observe that

[TABLE]

and thus $F_{\varepsilon}^{2}\in L^{\frac{4}{3}-}(0,T;H^{-1}({\mathbb{R}}^{3}))$. ∎

Remark 4.4**.**

Here, we need to use Strichartz estimates in non-homogeneous spaces. This is due to the fact that $L^{1}$ fails to embed in a homogeneous Sobolev space.

By combining the dispersive estimates of Proposition 4.2 and the bounds in Lemma 4.3 we can then infer the convergence to zero of $(\sigma_{\varepsilon},\mathbf{Q}(m_{\varepsilon}))$.

Proposition 4.5**.**

Let $(\sigma_{\varepsilon},m_{\varepsilon})$ be solution of (4.3) with initial data $(\sigma_{\varepsilon}^{0},m_{\varepsilon}^{0})$. For any $s\in{\mathbb{R}}$, $(p,q)$ admissible pair with $q>2$ and $\alpha_{0}$ as in (4.11), one has for any $\alpha\in[0,\alpha_{0}]$ and any admissible pairs $(p_{1},q_{1})$, $(p_{2},q_{2})$ that

[TABLE]

In particular, if $(\rho_{\varepsilon},u_{\varepsilon})$ is a BD-entropic solution to (1.5) with initial data $(\rho_{\varepsilon}^{0},u_{\varepsilon}^{0})$, $s\geq 2$ and $(p,q)$ any admissible pair with $q>2$ then

[TABLE]

for all $\alpha\in[0,\alpha_{0}]$.

The condition $s\geq 2$ is due to the low regularity of the nonlinearity in (4.5).

Proof.

First we observe that for any $s\in{\mathbb{R}}$, $q\in(1,\infty)$ and for $\tilde{\sigma_{\varepsilon}},\tilde{m_{\varepsilon}}$ as defined in (4.8) that

[TABLE]

and

[TABLE]

Indeed, to show (4.16) we define $T(f)=(1-\varepsilon^{2}\kappa^{2}\Delta)^{-\frac{1}{2}}f$. The symbol of $T$ is given by $m(\xi)=(1+\varepsilon^{2}\kappa^{2}|\xi|^{2})^{-\frac{1}{2}}$. It is straight forward to check that $T$ is a pseudo-differential operator of order [math] and thus $T:W^{s,q}\rightarrow W^{s,q}$ bounded. The inequality (4.17), follows from observing that the projection on the gradient part $\mathbf{Q}$ is given by a matrix valued Fourier multiplier $m(\xi)=\frac{\xi_{k}\xi_{j}}{|\xi|^{2}}$ while the change of variables $(-\Delta)^{-\frac{1}{2}}\mathop{\mathrm{div}}$ corresponds to the multiplier $\frac{\xi_{j}}{|\xi|}$. Inequality (4.17) follows. Second, we notice that (4.16), (4.17) combined with the Strichartz estimates (4.12) and (4.13) yield that for any $(p,q)$ admissible with $q>2$ there exists $\alpha_{0}>0$ such that for any $\alpha\in[0,\alpha_{0}]$ one has

[TABLE]

provided that $(p_{1},q_{1}),(p_{2},q_{2})$ are admissible. Finally, we have that

[TABLE]

as the operator $m_{\varepsilon}^{0}\mapsto\tilde{m}_{\varepsilon}^{0}$ with symbol $m(\xi)=\frac{\xi_{i}}{|\xi|}$ is of order [math]. Analogously, one derives the respective bounds for $\tilde{F}_{\varepsilon}^{i}$ with $i=1,2$. The operator $T^{-1}:{\sigma_{\varepsilon}}^{0}\mapsto(1-\varepsilon^{2}\kappa^{2}\Delta)^{\frac{1}{2}}\sigma_{\varepsilon}^{0}$ is characterised by the symbol $m(\xi)=(1+\varepsilon\kappa^{2}|\xi|^{2})^{\frac{1}{2}}$. One has that

[TABLE]

It follows,

[TABLE]

Inequality (4.14) now follows from (4.18) combined with (4.19) and (4.20). It remains to show (4.15). We recall that $\sigma_{\varepsilon}^{0}\in H^{-\frac{3}{2}}({\mathbb{R}}^{3})$ and $\varepsilon\nabla\sigma_{\varepsilon}^{0}\in H^{-\frac{1}{2}}({\mathbb{R}}^{3})$ uniformly in $\varepsilon$ in virtue of Lemma 3.4. Further, $m_{\varepsilon}^{0}\in H^{-\frac{1}{2}}({\mathbb{R}}^{3})$ from Lemma 3.2. Finally, for $(p_{1},q_{1})=(\frac{8}{3},4)$ and $(p_{2},q_{2})=(\infty,2)$ we have in virtue of Lemma 4.3 that

[TABLE]

Equation (4.14) now follows from (4.18) and the uniform bounds for the terms in the parenthesis on the right-hand side provided that $s\geq 2$. This completes the proof. ∎

Proof of Theorem 4.1.

Proof.

Lemma 3.3 (i) states that $\rho_{\varepsilon}-1$ converges strongly to [math] in $L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))$ with explicit convergence rate. Statement (iv) of Lemma 3.4 yields that $\rho_{\varepsilon}-1=\varepsilon\sigma_{\varepsilon}\in L^{4}(0,T;H^{1}({\mathbb{R}}^{3}))$ uniformly bounded. Interpolation of these bounds gives the desired strong convergence in $L^{4}(0,T;H^{s}({\mathbb{R}}^{3}))$ for $s\in[0,1)$.

Statement (ii) is inferred by observing that from (v) of Lemma 3.4 one has $\sigma_{\varepsilon}\in L^{2}(0,T;H^{1}({\mathbb{R}}^{3}))$ uniformly if $\gamma=2$. Provided that $q\in[2,6)$, one has $s=1-3(\frac{1}{2}-\frac{1}{q})>0$ and $\sigma_{\varepsilon}\in L^{2}(0,T;W^{s,q}({\mathbb{R}}^{3}))$ uniformly for $\gamma=2$. Inequality (4.15) yields that $\sigma_{\varepsilon}$ converges to [math] in $L^{p}(0,T;W^{-2-\alpha,q}({\mathbb{R}}^{3}))$ for $(p,q)$ admissible with $q\in(2,6)$ and $\alpha>0$ sufficiently small with convergence rate $\varepsilon^{\alpha}$. We obtain by applying Lemma 2.1 that

[TABLE]

where $\theta\in(0,1)$ and

[TABLE]

We notice that for $(p,q)$ admissible such that $q\in(2,6)$ one has $p\in(2,\infty)$ and $s>0$. Hence, for any admissible pair $(p,q)$ such that $q\in(2,6)$ there exists $s_{0}>0$ such that $\sigma_{\varepsilon}$ converges strongly to [math] in $L^{p}(0,T;W^{s_{0},q}({\mathbb{R}}^{3}))$. In particular, $\sigma_{\varepsilon}$ converges strongly to [math] in $L^{2}(0,T;L^{q}({\mathbb{R}}^{3}))$ for any $q\in(2,6)$ at convergence rate $\varepsilon^{\alpha\theta}$.

Finally, to obtain a bound on $\mathbf{Q}(m_{\varepsilon})$, we interpolate between the a priori bound (3.5) and the inequality (4.15). Proposition 4.5 yields that for $q>2$ such that $(p,q)$ is admissible there exists $\alpha_{0}=\alpha_{0}(q)>0$ such that

[TABLE]

for any $\alpha\in[0,\alpha_{0}]$. We notice that (3.5) implies that

[TABLE]

uniformly in $\varepsilon$. By Sobolev embedding, it follows for $q\in(2,\infty)$ that

[TABLE]

with $s_{1}=\frac{1}{4}-3(\frac{1}{2}-\frac{1}{q})$. We notice that $s_{1}>0$ provided that $q\in[2,\frac{12}{5})$. By applying Lemma 2.1 to interpolate between (4.15) and (4.22), we obtain that for $q\in(2,\frac{12}{5})$ with $(p,q)$ admissible

[TABLE]

where $(\theta,p_{0},s_{0})$ are such that

[TABLE]

We observe that provided $s_{1}>0$ there exists $\theta\in(0,1)$ such that $s_{0}>0$. As $(p,q)$ is an admissible pair with $q\in(2,\frac{12}{5})$ it follows in particular that $p\in(8,\infty)$. Hence, $p_{0}\geq 2$ for all $\theta\in(0,1)$. Therefore, for any $(p,q)$ admissible with $q\in(2,\frac{12}{5})$ there exist $\alpha\in[0,\alpha_{0}]$, $\theta\in(0,1)$, $s_{0}>0$ and $p_{0}\geq 2$ such that

[TABLE]

upon applying (4.15). The final statement follows by choosing $\delta=\min(s_{0},\alpha\theta)$. ∎

5. Convergence to the limiting system

In this section, we show strong compactness of $\sqrt{\rho_{\varepsilon}}u_{\varepsilon}$ in $L^{2}(0,T;L_{loc}^{2}({\mathbb{R}}^{3}))$ and conclude the proof of Theorem 2.4. To that end, we first prove strong compactness of the incompressible part $\mathbf{P}(\rho_{\varepsilon}u_{\varepsilon})$ that together with Theorem 4.1 will yield strong compactness of $\rho_{\varepsilon}u_{\varepsilon}$ in $L^{2}(0,T;L_{loc}^{2}({\mathbb{R}}^{3}))$. We notice that if a subsequence $\sqrt{\rho_{\varepsilon}}u_{\varepsilon}$ converges weakly to some $u$ in $L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))$ then $\rho_{\varepsilon}u_{\varepsilon}$ converges weakly to $u$ in $L^{2}(0,T;L^{2}({\mathbb{R}}^{3}))$. Indeed, $\rho_{\varepsilon}u_{\varepsilon}=\sqrt{\rho_{\varepsilon}}u_{\varepsilon}+(\sqrt{\rho_{\varepsilon}}-1)\sqrt{\rho_{\varepsilon}}u_{\varepsilon}$ where the second summand converges weakly to [math] as $\sqrt{\rho_{\varepsilon}}u_{\varepsilon}$ uniformly bounded and $\sqrt{\rho_{\varepsilon}}-1$ converges strongly to [math] in $L^{\frac{8}{3}}(0,T;L^{\infty}({\mathbb{R}}^{3}))$ in virtue (v) of Lemma 3.3.

Lemma 5.1**.**

Under the assumptions of Theorem 2.4, let $\rho_{\varepsilon}u_{\varepsilon}$ be a subsequence converging weakly to some $u\in L^{2}(0,T;L^{2}({\mathbb{R}}^{3}))$. Then $\rho_{\varepsilon}u_{\varepsilon}$ converges strongly to $u$ in $L^{2}(0,T;L_{loc}^{2}({\mathbb{R}}^{3}))$ as $\varepsilon$ goes to [math].

Proof.

We decompose $\rho_{\varepsilon}u_{\varepsilon}=\mathbf{Q}(\rho_{\varepsilon}u_{\varepsilon})+\mathbf{P}(\rho_{\varepsilon}u_{\varepsilon})$ by means of the Leray-Helmholtz projection operator. Theorem 4.1 states that $\mathbf{Q}(\rho_{\varepsilon}u_{\varepsilon})$ converges strongly to [math] in $L^{2}(0,T;W^{\delta,q}({\mathbb{R}}^{3}))$ for any $q\in(2,\frac{12}{5})$ and $\delta=\delta(q)>0$ sufficiently small. As $W^{\delta,q}({\mathbb{R}}^{3})$ is continuously embedded in $L^{q}({\mathbb{R}}^{3})$, it follows that $\mathbf{Q}(\rho_{\varepsilon}u_{\varepsilon})$ converges strongly to [math] in $L^{2}(0,T;L^{q}({\mathbb{R}}^{3}))$ for $q\in(2,\frac{12}{5})$ and therefore in particular in $L^{2}(0,T;L_{loc}^{2}({\mathbb{R}}^{3}))$. It remains to analyse the convergence of the incompressible part $\mathbf{P}(\rho_{\varepsilon}u_{\varepsilon})$. From (3.5), we have $\mathbf{P}(\rho_{\varepsilon}u_{\varepsilon})\in L^{p}(0,T;H^{s}({\mathbb{R}}^{3}))$ for $0\leq s\leq\frac{1}{2}$ and $1\leq p<\frac{4}{1+4s}$. Moreover, from

[TABLE]

with $\mathbf{S_{\varepsilon}}$ defined in (2.1), we conclude that $\partial_{t}\mathbf{P}(\rho_{\varepsilon}u_{\varepsilon})\in L^{2}(0,T;H^{-s}({\mathbb{R}}^{3})$ for any $s>\frac{5}{2}$. Indeed, it suffices to observe that from the energy bounds of Lemma 3.3 we have $\nabla\sqrt{\rho_{\varepsilon}}\in L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))$, $\sqrt{\rho_{\varepsilon}}-1\in L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))$, $\mathbf{S_{\varepsilon}}\in L^{2}(0,T;L^{2}({\mathbb{R}}^{3}))$ and $\sqrt{\rho_{\varepsilon}}u_{\varepsilon}\in L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))$. As the embedding $H^{s}({\mathbb{R}}^{3})\hookrightarrow L_{loc}^{2}({\mathbb{R}}^{3})$ is compact for any $s>0$, the Aubin-Lions Lemma yields strong compactness of $\mathbf{P}(\rho_{\varepsilon}u_{\varepsilon})$ in $L^{2}(0,T;L_{loc}^{2}({\mathbb{R}}^{3}))$. It follows that

[TABLE]

∎

By combing the strong compactness of $\rho_{\varepsilon}u_{\varepsilon}$ and the strong convergence of $\sqrt{\rho_{\varepsilon}}-1$ to [math], we infer strong compactness of $\sqrt{\rho_{\varepsilon}}u_{\varepsilon}$ and pass to the limit in (1.5).

Proposition 5.2**.**

Under the assumption of Theorem 2.4, let $\sqrt{\rho_{\varepsilon}}u_{\varepsilon}$ be a subsequence weakly converging to some $u\in L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))$. Then $\sqrt{\rho_{\varepsilon}}u_{\varepsilon}$ converges strongly to $u$ in $L^{2}(0,T;L_{loc}^{2}({\mathbb{R}}^{3}))$ and $u$ is a global weak solution to the incompressible Navier-Stokes equation with initial data u\big{|}_{t=0}=\mathbf{P}(u_{0}) with $u^{0}$ defined in (2.8).

We notice that the passage to the limit relies on the additional uniform bounds provided by the BD entropy inequality (2.6) in a crucial way: for both the strong compactness of $\sqrt{\rho_{\varepsilon}}u_{\varepsilon}$ and the convergence to [math] of the dispersive tensor.

Proof.

First, we show strong compactness of $\sqrt{\rho_{\varepsilon}}u_{\varepsilon}$. We notice that

[TABLE]

For any compact $K\subset{\mathbb{R}}^{n}$, one has

[TABLE]

for some $\beta>0$, where we used the convergence provided by (v) of Lemma 3.3 in the last step. Second, we carry out the $\varepsilon$-limit in the weak formulations of (1.5). The strong compactness of $\rho_{\varepsilon}u_{\varepsilon}$ provided by Lemma 5.1 and (3.1) applied to $\rho_{\varepsilon}$ and $\rho_{\varepsilon}^{0}$ allow us to pass to the limit in the weak formulation of the continuity equation. We recover,

[TABLE]

Next, we observe that the dispersive tensor satisfies

[TABLE]

We notice that $\nabla\sqrt{\rho_{\varepsilon}}$ converges strongly to [math] in $L^{2}(0,T;L^{2}({\mathbb{R}}^{3}))$. Indeed, one has that $\nabla^{2}\sqrt{\rho_{\varepsilon}}\in L^{2}(0,T;L^{2}({\mathbb{R}}^{3}))$ uniformly bounded stemming from the BD entropy inequality (2.6). It follows, that $\sqrt{\rho_{\varepsilon}}-1$ converges strongly to [math] in $L^{2}(0,T;H^{1}({\mathbb{R}}^{3}))$, see (v) of Lemma 3.3. Similarly, $\rho_{\varepsilon}-1$ converges strongly to [math] in $L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))$ from (i) of Lemma 3.2. Finally, we consider the weak formulation of the momentum equation projected onto divergence free vector fields. Let $\psi\in C_{c}^{\infty}([0,T)\times{\mathbb{R}}^{3};{\mathbb{R}}^{3})$ such that $\mathop{\mathrm{div}}\psi=0$, then the momentum equation reduces to

[TABLE]

The strong convergence of $\sqrt{\rho_{\varepsilon}}u_{\varepsilon}$ in $L^{2}(0,T;L_{loc}^{2}({\mathbb{R}}^{3}))$ together with Lemma 3.3 is sufficient to pass to the limit in (LABEL:eq:_weak_formulation). We conclude that (LABEL:eq:_weak_formulation) converges to

[TABLE]

where we used that $\rho_{\varepsilon}^{0}u_{\varepsilon}^{0}$ converges weakly to $u_{0}$ in $L_{loc}^{\frac{3}{2}-}({\mathbb{R}}^{3})$ as consequence of (2.8) and Lemma 3.2. Indeed, (ii) of Lemma 3.2 implies that $\sqrt{\rho_{\varepsilon}^{0}}-1$ converges strongly to [math] in $L^{\infty}(0,T;L^{q}({\mathbb{R}}^{3}))$ for any $q\in[2,6)$ by interpolation. We conclude by exploiting that $\psi$ is divergence free. Therefore, there exists a distribution $p$ defined on $(0,T)\times{\mathbb{R}}^{3}$ such that $u$ is solution of

[TABLE]

with initial data $\mathbf{P}(u_{0})$. ∎

As we already said, at fixed $\varepsilon>0$ the finite energy weak solutions $(\rho_{\varepsilon},u_{\varepsilon})$ to (1.5) satisfy a weak version of the energy inequality due to the degenerate viscosity, namely

[TABLE]

where $\mathbf{S_{\varepsilon}}$ is given by (2.1). We remark that in fact in the limit as $\varepsilon\to 0$ it is possible to recover the usual energy dissipation. More precisely, the uniform boundedness of $\mathbf{S_{\varepsilon}}\in L^{2}(0,T,L^{2}({\mathbb{R}}^{3}))$ only yields that $\mathbf{S_{\varepsilon}}\rightharpoonup{S}$ weakly in $L^{2}((0,T)\times{\mathbb{R}}^{3})$ up to subsequences. In the next Proposition we show that in fact we have ${S}=\frac{1}{2}\mathbf{D}u$. Moreover, by assuming the initial data to be well-prepared we obtain the convergence of the total energy at initial time and thus we can also show that the limit function $u$ obtained is indeed a Leray solution.

Proposition 5.3**.**

Under the assumptions of Theorem 2.4, let $\mathbf{S_{\varepsilon}}$ be as defined in (2.1) then

[TABLE]

Consequently, $u$ is a weak solution to (1.7) that satisfies $u\in L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))\cap L^{2}(0,T;\dot{H}^{1}({\mathbb{R}}^{3}))$. If additionally, $(\rho_{\varepsilon}^{0},u_{\varepsilon}^{0})$ satisfies (2.9), then $u$ is a Leray solution of (1.7), i.e. it satisfies (2.10).

Proof.

In virtue of Proposition 5.2, $u$ is a weak solution of (1.7) with initial data u\big{|}_{t=0}=\mathbf{P}(u_{0}). Next, we show that $\mathbf{S_{\varepsilon}}\rightharpoonup\mathbf{D}u$ in $L^{2}(0,T;L^{2}({\mathbb{R}}^{3}))$. From (2.4), one has that there exists $S\in L^{2}((0,T;L^{2}({\mathbb{R}}^{3}))$ such that $\mathbf{S_{\varepsilon}}\rightharpoonup S$ weakly in $L^{2}((0,T)\times{\mathbb{R}}^{3})$ up to passing to subsequences. Moreover, $\sqrt{\rho_{\varepsilon}}\mathbf{S_{\varepsilon}}\rightarrow S$ in $\mathcal{D}^{\prime}((0,T)\times{\mathbb{R}}^{3})$. Indeed, let us write $\sqrt{\rho_{\varepsilon}}\mathbf{S_{\varepsilon}}=\mathbf{S_{\varepsilon}}+(\sqrt{\rho_{\varepsilon}}-1)\mathbf{S_{\varepsilon}}$. The second term converges to [math] in $\mathcal{D}^{\prime}((0,T)\times{\mathbb{R}}^{3})$ since $\sqrt{\rho_{\varepsilon}}-1\rightarrow 0$ strongly in $L^{\infty}(0,T,L^{q}({\mathbb{R}}^{3}))$ for $2\leq q<6$ from Lemma 3.3. On the other hand, from (2.1) and (2.2) we infer that $\sqrt{\rho_{\varepsilon}}\mathbf{S_{\varepsilon}}\rightarrow\mathbf{D}u$ in $\mathcal{D}^{\prime}((0,T)\times{\mathbb{R}}^{3})$. Indeed, from Proposition 5.1, we have $\nabla(\rho_{\varepsilon}u_{\varepsilon})\rightarrow\nabla u$ in $\mathcal{D}^{\prime}((0,T)\times{\mathbb{R}}^{3})$ and from $\nabla\sqrt{\rho_{\varepsilon}}\rightarrow 0$ in $L^{2}(0,T;L^{2}({\mathbb{R}}^{3}))$ by Lemma 3.3, it follows $\sqrt{\rho_{\varepsilon}}u_{\varepsilon}\otimes\nabla\sqrt{\rho_{\varepsilon}}\rightarrow 0$ in $L^{2}(0,T,L^{1}({\mathbb{R}}^{3}))$. Thus $S=\mathbf{D}u\in L^{2}(0,T;L^{2}({\mathbb{R}}^{3}))$. We observe that for $u\in H^{1}({\mathbb{R}}^{3})$ such that $\mathop{\mathrm{div}}u=0$, one has

[TABLE]

Finally, by lower semi-continuity we conclude that

[TABLE]

Thus, $u\in L^{\infty}(0,T;L^{2}({\mathbb{R}}^{3}))$ and $\nabla u\in L^{2}(0,T;L^{2}({\mathbb{R}}^{3}))$. In order to conclude (2.10), it remains to show that,

[TABLE]

If the initial data is well-prepared, namely satisfies (2.9), the proof is complete. ∎

Finally, we stress that, since the bounds obtained in Proposition 3.6 are uniform in $\varepsilon>0$, they are also inherited by the solution to (1.7) obtained in the limit. The next Proposition proves the last statement of Theorem 2.4.

Proposition 5.4**.**

Let $u$ be the solution to (1.7) obtained in the limit. Then for any $0<T<\infty$ one has $u\in L^{p}(0,T;H^{s}({\mathbb{R}}^{3}))$ with $0\leq s\leq\frac{1}{2}$ and $1\leq p<\frac{1}{1+4s}$.

Acknowledgments

The first and the third author acknowledge partial support by PRIN-MIUR project 2015YCJY3A_003 Hyperbolic Systems of Conservation Laws and Fluid Dynamics: Analysis and Applications. The first and second author acknowledge partial support through the INdAM-GNAMPA project Esistenza, limiti singolari e comportamento asintotico per equazioni Eulero/Navier–Stokes–Korteweg.

Appendix A Energy and BD entropy inequality

In this Section we discuss the existence of BD-entropic weak solutions to (1.5) fulfilling the hypotheses of our main Theorem 2.4. More specifically, we focus on the existence of weak solutions satisfying the energy inequality (2.4) and the BD entropy inequality (2.6). Our specific interest stems from the fact that, for well-prepared initial data we want to show the convergence towards Leray weak solutions, see the Proposition 2.6, namely we want to recover the inequality (2.10) in the limit. For this purpose we need the weak solutions to (1.5) to satisfy the energy inequality in the form (2.4).

The QNS system has already been studied in the literature, see for example [9, 41]. however both papers consider the problem in the periodic domain ${\mathbb{T}}^{d}$ and more importantly the energy inequality satisfied there slightly differs from (2.4), see the Definition 1 in [9] or Theorem 1.1 in [41].

Let us emphasize that in this Appendix too, we focus on the periodic domain. Indeed, once (2.4) and (2.6) are proved on ${\mathbb{T}}^{d}$, then by exploiting an invading domain approach the same result can be proved also on ${\mathbb{R}}^{d}$. We address the interested reader to the recent preprint [5], where the Cauchy problem for the QNS system is studied with non-trivial far-field behavior. In [5] it is shown that, if we have a sequence of weak solutions satisfying (2.4) and (2.6) on larger and larger domains, then by a suitable truncation argument, it is possible to construct a weak solution satisfying the same entropy inequalities on ${\mathbb{R}}^{d}$.

As already said, one of the main problems in studying weak solutions to (1.1) is the lack of control for the velocity field and its gradient in the vacuum region. This fact immediately implies that in general weak solutions to (1.1) do not satisfy (2.4) and (2.6). Furthermore, it is still an open problem to determine the minimal assumptions for weak solutions such that (2.4) and (2.6) are valid. We address the interested reader to [47] for a recent result in this direction. On the other hand, by using a suitable approximation procedure, it is possible to show the existence of weak solutions to (1.1) for which both (2.4) and (2.6) are valid. This indeed reflects the commonly adopted strategy of showing the existence of weak solutions, which consists in constructing a sequence of (more regular) approximating solutions that also satisfy approximate versions of (2.4) and (2.6). By passing to the limit and by using the compactness provided by the a priori bounds, it is then possible to find a weak solution to the original system which additionally satisfies the estimates (2.4) and (2.6).

Let us stress that this strategy does not allow in any case to recover the energy inequality in the form (2.5). Indeed, while the a priori bounds imply the compactness of $\sqrt{\rho_{n}}u_{n}$ - which allows to pass to the limit in the convective term - the only information we have on the gradient of the velocity field is that $\rho_{n}|\nabla u_{n}|^{2}$ is uniformly bounded in $L^{1}$. Thus, in the limit we can only infer $\sqrt{\rho_{n}}\nabla u_{n}\rightharpoonup\mathbf{T}$ in $L^{2}$, where $\mathbf{T}$ is the tensor determined by the identity in (2.2).

In this Appendix we follow the approximation argument adopted in [9] to show that the weak solutions constructed in [9] indeed satisfy (2.4) and (2.6). In the following, we consider (1.5) on $[0,T)\times{\mathbb{T}}^{d}$. All domains of integration are hence adapted to ${\mathbb{T}}^{d}$ and the energy functional reads

[TABLE]

In particular, the internal energy does not need to be renormalized by considering $\pi(\rho)$ as in (1.2). We refer the reader to [5] for a more detailed discussion.

Theorem A.1**.**

Let $d=2,3$. Let $\nu,\kappa$ and $\gamma$ positive such that $\kappa<\nu$, $\gamma>1$ for $d=2$ and $\kappa^{2}<\nu^{2}<\frac{9}{8}\kappa^{2}$ and $1<\gamma<3$ for $d=3$. For any $0<T<\infty$ and initial data $(\rho^{0},u^{0})$ such that $E(\rho^{0},u^{0})<+\infty$ and in addition $u^{0}=0$ on $\{\rho^{0}=0\}$, $\sqrt{\rho^{0}}u^{0}\in L^{2}({\mathbb{T}}^{d})\cap L^{2+}({\mathbb{T}}^{d})$, there exists a BD-entropic weak solution $(\rho,u)$ of (1.1) on ${\mathbb{T}}^{d}$ with initial data $(\rho^{0},u^{0})$.

We stress that existence has been introduced in [9], here we show the validity of (2.4) and (2.6). The weak solutions constructed in [9] are obtained as limit of a sequence of approximating solutions $\{(\rho_{\delta},u_{\delta})\}_{\delta}$ satisfying the following system.

[TABLE]

with initial data

[TABLE]

We refer to [9] for the motivation of the regularizing terms, see also [43] where a similar regularizing approximations have been introduced for the compressible Navier-Stokes equations with density dependent viscosity. The approximating viscosity term is defined as

[TABLE]

with

[TABLE]

The approximating dispersive term reads

[TABLE]

The coefficient $\tilde{p}_{\delta}(\rho_{\delta})$ in the damping term is defined as

[TABLE]

The cold pressure is defined by $P_{\delta}(\rho_{\delta})=\left(\nu-\sqrt{\nu^{2}-\kappa^{2}}\right)\tilde{p}_{\delta}(\rho_{\delta})\frac{h_{\delta}^{\prime}(\rho_{\delta})}{\rho_{\delta}}$. The energy functional for the approximating system (A.1) reads

[TABLE]

where $f_{\delta}(\rho_{\delta})=\rho_{\delta}P_{\delta}^{\prime}(\rho_{\delta})-P_{\delta}(\rho_{\delta})$ that we observe to be non-negative and strictly convex, see Section 2 in [9]. In virtue of Theorem 6 in [9], there exists a global smooth solution to the Cauchy problem (A.1) for smooth initial data $(\rho_{\delta}^{0},u_{\delta}^{0})$. By means of direct computations one infers the energy equality for the system (A.1).

Lemma A.2** ([9]).**

Let $(\rho_{\delta},u_{\delta})$ be a global smooth solution of (A.1). Then for any $0\leq s<t\leq T$ one has

[TABLE]

In order to show that smooth solutions to (A.1) satisfy a Bresch-Desjardins entropy estimate, we introduce the effective velocity $v_{\delta}=u_{\delta}+c\nabla\phi_{\delta}(\rho_{\delta})$ with $\phi$ defined through $\rho_{\delta}\phi_{\delta}^{\prime}(\rho_{\delta})=h_{\delta}^{\prime}(\rho_{\delta})$ and for a suitable constant $c$ to be chosen below. Then $(\rho_{\delta},v_{\delta})$ is a smooth solution of the viscous Euler system

[TABLE]

where

[TABLE]

The Bresch-Desjardins entropy is then defined as energy functional associated to (A.6),

[TABLE]

The BD entropy equality arises as energy equality of (A.6).

Lemma A.3** ([9]).**

Let $(\rho_{\delta},u_{\delta})$ be a global smooth solution of (A.1). Given $c\in(0,\mu)$, the pair $(\rho_{\delta},v_{\delta})$ is a global smooth solution of (A.6) and the BD entropy equality is satisfied ,

[TABLE]

Next, we address existence of smooth solutions to (A.1) for a sequence of initial data given by smooth approximations of $(\rho^{0},u^{0})$ as specified in Theorem A.1. More precisely, we consider $(\rho_{\delta}^{0},u_{\delta}^{0})$ such that

[TABLE]

Again, Theorem 6 in [9] guarantees the existence of a sequence of global smooth solutions to the Cauchy problem (A.1) with initial data as specified by (A.9).

Proposition A.4** ([9]).**

Let $d=2,3$ and $\nu,\kappa>0$, $\gamma>1$ as in Theorem A.1. Let $(\rho_{\delta},u_{\delta})$ be a sequence of global smooth solutions to (A.1) with initial data (A.9), then $(\rho_{\delta},u_{\delta})$ converges to a global weak solution $(\rho,u)$ to (1.1) with initial data $(\rho^{0},u^{0})$. In particular, for any $T>0$ one has

[TABLE]

We are now in position to prove Theorem A.1.

Proof of Theorem A.1.

Given initial data $(\rho^{0},u^{0})$, we construct a sequence of smooth initial data $(\rho_{\delta}^{0},u_{\delta}^{0})$ satisfying (A.9) by mollification. Theorem 6 in [9] states the existence of a sequence of smooth solutions $(\rho_{\delta},u_{\delta})$ to (A.1). Proposition A.4 yields convergence towards a weak solution $(\rho,u)$ of (1.1). It remains to pass to the limit in (A.6) and (A.8) to show (2.4) and (2.6) respectively. First, we show (2.4). To that end, we notice that

[TABLE]

For all $0\leq t\leq T$, one has

[TABLE]

In virtue of (A.5), we conclude that $\sqrt{\rho_{\delta}}\mathbf{D}u_{\delta}\in L^{2}(0,T;L^{2}({\mathbb{T}}^{d}))$ uniformly bounded. Hence, there exists $\mathbf{S}\in L^{2}(0,T;L^{2}({\mathbb{T}}^{d}))$ such that $\sqrt{\rho_{\delta}}\mathbf{D}u_{\delta}\rightharpoonup\mathbf{S}$ in $L^{2}(0,T;L^{2}({\mathbb{T}}^{d}))$, up to passing to subsequences. By lower semi-continuity of norms, we obtain that

[TABLE]

Similarly,

[TABLE]

where we exploit that $|\nabla\sqrt{\rho_{\delta}}|\leq|h_{\delta}^{\prime}(\rho_{\delta})\nabla\sqrt{\rho_{\delta}}|$ and that $f(\rho_{\delta})$ is non-negative for $\delta$ sufficiently small. Finally, summing up (A.10), (A.11) and applying (A.5) yields

[TABLE]

where we used that $\tilde{p}_{\delta}(\rho_{\delta})$ is non-negative for the first inequality and (A.9) in the last identity. Next, we check that $\mathbf{S}$ satisfies (2.1). As $\sqrt{\rho_{\delta}}$ converges strongly to $\sqrt{\rho}$ in $L^{2}(0,T;L^{2}({\mathbb{T}}^{d}))$ in virtue of Proposition A.4 and $\sqrt{\rho_{\delta}}\mathbf{D}u_{\delta}$ converges weakly to $\mathbf{S}$ in $L^{\infty}(0,T;L^{2}({\mathbb{T}}^{d}))$, one has that $\rho_{\delta}\mathbf{D}u_{\delta}$ converges to $\sqrt{\rho}\mathbf{S}$ in $\mathcal{D}^{\prime}([0,T)\times{\mathbb{T}}^{d})$. Let $\phi\in\mathcal{D}((0,T)\times{\mathbb{T}}^{d})$ and consider

[TABLE]

As $\sqrt{\rho_{\delta}}\rightarrow\sqrt{\rho}$ and $\nabla\sqrt{\rho_{\delta}}\rightarrow\nabla\sqrt{\rho}$ both strongly in $L^{2}(0,T;L^{2}({\mathbb{T}}^{d}))$ and $\sqrt{\rho_{\delta}}u_{\delta}\rightharpoonup\sqrt{\rho}u$ weakly-$\ast$ in $L^{\infty}(0,T;L^{2}({\mathbb{T}}^{d}))$ we conclude that $\mathbf{S}$ satisfies (2.1). Second, we show (2.6) by proceeding similarly. As $h_{\delta}(\rho_{\delta})\geq\rho_{\delta}$, we infer from (A.8) that $\sqrt{\rho_{\delta}}\mathbf{A}v_{\delta}\in L^{2}(0,T;L^{2}({\mathbb{T}}^{d}))$ uniformly bounded. We notice that $\mathbf{A}v_{\delta}=\mathbf{A}u_{\delta}$. Hence, there exists $\mathbf{A}$ such that $\sqrt{\rho_{\delta}}\mathbf{A}u_{\delta}\rightharpoonup\mathbf{A}$ up to subsequences and

[TABLE]

In particular, proceeding as for the identification of $\mathbf{S}$, we obtain that $\mathbf{T}=\mathbf{S}+\mathbf{A}$ satisfies (2.2). Further, we observe that

[TABLE]

Thus, from Lemma 4 in [9] we obtain that there exists $C>0$ such that

[TABLE]

It follows that, up to passing to subsequences, $\nabla^{2}\sqrt{\rho_{\delta}}$ converges weakly to $\nabla^{2}\sqrt{\rho}$ in $L^{2}(0,T;L^{2}({\mathbb{T}}^{d}))$. Moreover, (A.8) together with

[TABLE]

implies that $\nabla\rho_{\delta}^{\frac{\gamma}{2}}\in L^{2}(0,T;L^{2}({\mathbb{T}}^{d}))$ uniformly bounded, hence up to passing to subsequences

[TABLE]

Summing up and exploiting lower semi-continuity of norms we have

[TABLE]

By exploiting lower-semicontinuity of norms, (A.8), (A.12), as well as $B_{\delta}(0)\rightarrow B(0)$ due to (A.9) we infer (2.6). For that purpose, we notice that $\tilde{p}_{\delta}(\rho_{\delta})$ and $f_{\delta}^{\prime\prime}(\rho_{\delta})$ are non-negative for $\delta$ sufficiently small by construction. ∎

Appendix B Strichartz estimates for the acoustic wave system

The main purpose of this appendix is to give a proof of Proposition 4.2 (see Proposition B.12 below), that is we want to study the dispersive properties satisfied by solutions to system (4.9). Even if the paper only studies the three dimensional setting for the sake of completeness the whole analysis is carried out in the general $d$-dimensional setting ($d\geq 2)$. As already mentioned, for $\varepsilon=1$, the dispersive analysis associated to the operator $H_{1}$ has been carried out in [31, 32, 33]. In this paper, we need to carefully track down the $\varepsilon$-dependence on the estimates as the (scaled) Mach number $\varepsilon$ not only determines a time scale but also a frequency threshold such that the operator behaves differently. This is due to the non-homogeneity of the dispersion relation and is opposite to the analysis of low Mach number limit in classical fluid dynamics where the Mach number $\varepsilon$ only determines the time scale. The dispersive analysis for non-homogeneous symbols has been investigated in more general framework also in [30, 21, 29]. Homogeneous Besov spaces suit best for the purpose of proving the refined Strichartz estimates while simultaneously tracking the $\varepsilon$-dependence.

We structure the exposition of this Appendix as follows. In Proposition B.3 we recall the stationary phase estimate proved in [31]. The scaled dispersive estimate associated to the operator $H_{\varepsilon}$ in (4.10) is then given in Corollary B.6. By using the dispersive estimate it is then possible to derive the frequency localized Strichartz estimates of Lemma B.10. By summing over all frequencies we then obtain Proposition B.11. For the convenience of the reader, the final estimates used in the body of this paper, namely Section 4, are stated in terms of Sobolev spaces, see Proposition B.12.

B.1. Besov spaces and embeddings

We recall the embeddings relating homogeneous Besov spaces to Lebesgue space $L^{q}$ and homogeneous Sobolev spaces $\dot{W}^{s,q}$.

Lemma B.1**.**

Let $d\geq 1$.

1. (i)

For any $s\in{\mathbb{R}}$, we have $\dot{B}_{2,2}^{s}({\mathbb{R}}^{d})=\dot{H}^{s}({\mathbb{R}}^{d})$. 2. (ii)

For $q\in[2,\infty)$, the homogeneous Besov space $\dot{B}_{q,2}^{0}({\mathbb{R}}^{d})$ is continuously embedded in $L^{q}({\mathbb{R}}^{d})$ and $L^{q^{\prime}}({\mathbb{R}}^{d})$ embeds continuously in $\dot{B}_{q^{\prime},2}^{0}({\mathbb{R}}^{d})$. 3. (iii)

For $s\in{\mathbb{R}}$ and $q\in[2,\infty)$, one has

[TABLE]

The proof of statement (iii) can be found e.g. in [13], see Theorem 6.4.4. The first and second statements are direct consequences of the third.

We state a useful property for space-time Besov spaces.

Lemma B.2**.**

Let $d\geq 1$, $T>0$ and $p,q,r\in[1,\infty)$. Then, for all $p\geq r$ one has

[TABLE]

while for $p\leq r$,

[TABLE]

The statements follow upon applying Minkowski’s integral inequality.

B.2. Dispersive estimate

In what follows we are going to prove the $L^{\infty}-L^{1}$ dispersive estimate associated for the semigroup $e^{itH_{\varepsilon}}$. For the convenience of the reader, we recall the stationary phase estimate in [31]. Here and below, we adopt the notation $r=|\xi|$.

Proposition B.3** ([31]).**

Let $\phi(r)\in C^{\infty}(0,\infty)$ satisfy the following.

1. (i)

$\phi^{\prime}(r),\phi^{\prime\prime}(r)>0$* for all $r>0$,* 2. (ii)

$\phi^{\prime}(r)\sim\phi^{\prime}(s)$* and $\phi^{\prime\prime}(r)\sim\phi^{\prime\prime}(s)$ for all $0<s<r<2s$,* 3. (iii)

$\left|\phi^{(k+1)}(r)\right|\lesssim\frac{\phi^{\prime}(r)}{r^{k}}$* for all $r>0$ and $k\in{\mathbb{N}}$.*

Let $\chi(r)$ be a dyadic cut-off function with support around $r\sim R$ and that satisfies

[TABLE]

These estimates are supposed to hold uniformly for $r$ and $R$, but may depend on $k$. Then if

[TABLE]

we have

[TABLE]

Several observations are in order. We define

[TABLE]

and exploiting that $\phi$ is a radial function we compute

[TABLE]

so that the right hand side of (B.1) involves $h(R)^{-1/2}$. This is consistent with the general theory for stationary phase estimates, see for instance Theorem 7.7.6 in [34]. Furthermore, from Proposition 2 in [21] it follows that the dispersive estimate (B.1) is sharp in the sense that there exists $t_{0}$ and $R_{0}$ such that for all $|t|>|t_{0}|$ and $R>R_{0}$ there exists $C>0$ such that

[TABLE]

In [31], the estimate (B.1) has been applied to the pseudo-differential operator $H_{1}$, i.e. $\phi(r)=r\sqrt{1+\kappa^{2}r^{2}}$. We remark that the dispersive estimate for the symbol $\omega$ defined by the Bogliubov dispersion relation (4.1) can be obtained from estimate (B.1) by defining

[TABLE]

Indeed, we notice that $\omega(r)=\phi_{\varepsilon}(r)$ and after a computation that

[TABLE]

as well as

[TABLE]

by rescaling. We remark that the scaling affects the support of frequencies. Finally, to track down the $\varepsilon$-dependence in the dispersive estimate it is enough to study the properties of the Hessian matrix of $\phi_{\varepsilon}$ in terms of its determinant $h(\varepsilon r)$.

Lemma B.4**.**

Let $h$ be defined as in (B.2). There exists $C>0$ such that for any $\lambda\in[0,\infty]$,

[TABLE]

For $d=2$, there exists $C>0$ such that for any $\lambda\in[0,\infty)$,

[TABLE]

We shall use these estimates for $\lambda=\varepsilon|\xi|$. We notice that the estimates blow up as $\kappa$ goes to [math]. Here, we consider $\kappa>0$ fixed, see also Remark B.7.

Proof.

This follows from immediate computations. ∎

The information on the (scaled) function $h$ allows to derive the dispersive estimate for the symbol $\phi_{\varepsilon}$.

Corollary B.5**.**

Let $\phi_{\varepsilon}(r)=\frac{1}{\varepsilon}r\sqrt{1+(\varepsilon\kappa)^{2}r^{2}}$, $R>0$ be given and let $\chi(r)\in C_{c}(0,\infty)$ be as in Proposition B.3. Then there exists a constant $C>0$ such that

[TABLE]

In particular, this implies there exists $C>0$ independent from $\varepsilon$ such that,

[TABLE]

Proof.

For fixed $\varepsilon>0$, we may exploit the dispersive estimate of Proposition B.3 upon using (B.5). More precisely, we obtain

[TABLE]

Estimate (B.7) then follows by applying (B.4). To conclude the second estimate, it is enough to observe that for $d\geq 2$ there exists $C>0$ such that $h(r)^{-\frac{1}{2}}\leq C$ uniformly on $(0,\infty)$ as consequence of Lemma B.4. ∎

The estimate in (B.8) implies that the operator $H_{\varepsilon}$ has the same dispersive properties as the Schrödinger operator. As a consequence (B.8) would yield Schrödinger type dispersive estimates for frequency localized functions. However, from (B.6) we shall infer that in fact, for $d>2$, we can derive better estimates, due to the regularizing effect of $\frac{(\varepsilon r)}{\sqrt{1+(\varepsilon r)^{2}}}$ when $\varepsilon r$ is small. This has already been pointed out in [31] for the operator $e^{itH_{1}}$ and is explained by a different curvature of the geometric surface $|\xi|\sqrt{1+|\xi|^{2}}$ with respect to $|\xi|^{2}$. We reformulate this observation in the next Corollary.

Corollary B.6**.**

Let $d\geq 2$, $\phi_{\varepsilon}(r)=\frac{1}{\varepsilon}r\sqrt{1+(\varepsilon\kappa)^{2}r^{2}}$, $R>0$ be given and let $\chi(r)\in C_{c}(0,\infty)$ be as in Proposition B.3. Then there exists a constant $C>0$ such that

[TABLE]

for any $0\leq\delta\leq\frac{d-2}{2}$.

Remark B.7**.**

We emphasize that the RHS in the estimate (B.9) blows up as $\kappa$ goes to [math]. This reflects the contribution of the quantum pressure term to the dispersion relation. In the absence of the quantum pressure, i.e. $\kappa=0$ one recovers a linear dispersion relation for which wave-type dispersive estimates are to be expected and (B.9) being of Schrödinger type does not hold. However, here we consider the system (1.5) for fixed and bounded $\kappa$ and thus we set $\kappa=1$ in the following, see also the scaling chosen in Section 2.1.

This motivates to define the pseudo-differential operator $U_{\varepsilon}$ corresponding to the Fourier multiplier

[TABLE]

In particular, this allows for the gain of the factor $\varepsilon^{\delta}$ in the estimate at the expense of a factor $R^{\delta}$ corresponding to a loss of derivatives, see inequality (B.11).

B.3. Strichartz estimates

Next, we infer the needed Strichartz estimates from the dispersive estimate (B.9). First, we recall the definition of admissible exponents.

Definition B.8**.**

We say the pair of exponents $(p,q)$ is Schrödinger admissible if $2\leq p,q\leq\infty$,

[TABLE]

and $(p,q,d)\neq(2,\infty,2)$.

The first step consists in showing a pointwise in time estimate.

Lemma B.9** (Pointwise estimate).**

For fixed $\varepsilon>0$ and $R>0$, let $f\in L^{1}({\mathbb{R}}^{d})$ such that $supp(\hat{f})\subset\{\xi\in{\mathbb{R}}^{d}\,:\,\frac{1}{2}R\leq|\xi|\leq 2R\}$. The following estimate holds for any $2\leq q\leq\infty$ and $0\leq\delta\leq\frac{d-2}{2}$:

[TABLE]

and consequently

[TABLE]

Proof.

The operator $e^{itH_{\varepsilon}}$ is unitary on $L^{2}$ therefore

[TABLE]

Furthermore, Corollary B.6 guarantees that there exists $C_{2}>0$ not depending on $\varepsilon,R$

[TABLE]

By a standard interpolation argument we conclude the proof. Estimate (B.11) follows from

[TABLE]

∎

Next, we show Strichartz estimates localized in frequencies on dyadic blocks.

Lemma B.10**.**

For $d\geq 2$, $\varepsilon,R>0$ and $0<\delta\leq\frac{d-2}{2}$, let $f\in L^{2}({\mathbb{R}}^{d})$ and $F\in L^{p^{\prime}}(0,T;L^{q^{\prime}}({\mathbb{R}}^{d}))$ such that $supp(\hat{f}),supp(\hat{F}(t))\subset\{\xi\in{\mathbb{R}}^{d}\,:\,\frac{1}{2}R\leq|\xi|\leq 2R\}$ Then there exists a constant $C>0$ independent from $T,\varepsilon$ such that for any $(p,q)$, $(p_{1},q_{1})$ admissible pairs,

[TABLE]

[TABLE]

Moreover,

[TABLE]

Proof.

Given (B.9) and considering the fact that $e^{itH_{\varepsilon}}$ is an isometry on $L^{2}({\mathbb{R}}^{d})$, we observe that Theorem 1 of [38] applies. We notice that the constants in the estimates (B.12) are identical as coming from an abstract duality argument. ∎

We remark that for $\varepsilon=1$, we recover the Strichartz estimates provided by Theorem 2.1 in [31].

Proposition B.11**.**

Let $d\geq 2$, $\varepsilon>0$, $0\leq\delta\leq\frac{d-2}{2}$. Then there exists a constant $C>0$ independent from $T,\varepsilon$ such that for any $(p,q)$, $(p_{1},q_{1})$ admissible pairs,

[TABLE]

and

[TABLE]

Proof.

Fix $R>0$, by scaling $t^{\prime}=\frac{t}{R^{2}}$ and $x^{\prime}=\frac{x}{R}$, we achieve that the projection $P_{R}(e^{it^{\prime}H_{\varepsilon}}f)(t^{\prime},x^{\prime})$ is spectrally supported in the annulus $\{\xi\in{\mathbb{R}}^{d}\,:\,\frac{1}{2}\leq|\xi|\leq 2\}$. In the following we use the subscript $L_{\mathbf{\cdot}}^{p}$ for Lebesgue spaces to indicate the variable w.r.t which the norm is computed. We infer from (B.12) that

[TABLE]

for any admissible pair $(p,q)$, namely such that $\frac{2}{p}+\frac{d}{q}=\frac{d}{2}$. Similarly, the bound (B.14) implies that for admissible pairs $(p,q)$ and $(p_{1},q_{1})$ , we have

[TABLE]

Hence, given an admissible pair $(p,q)$ and setting $R=2^{k}$, we compute

[TABLE]

where we have used the inequalities of Lemma B.2 in the first and third inequality respectively and (B.12) in the second. Similarly, we proceed for (B.13). Indeed,

[TABLE]

∎

The final estimates follow upon observing that the presence of the operator $U_{\varepsilon}$ may be exploited to gain a factor $\varepsilon$ as shown in (B.11). For the purpose of Section 4, it is sufficient to state the estimates in non-homogeneous Sobolev spaces rather than in the more general framework of Besov spaces.

Proposition B.12**.**

Let $d\geq 2$, fix $\varepsilon>0$ and $s\in{\mathbb{R}}$. There exists a constant $C>0$ independent from $T,\varepsilon$ such that for any $(p,q)$ admissible pair and any $\alpha_{0}\in[0,\frac{d-2}{2}(\frac{1}{2}-\frac{1}{q})]$, the following hold true,

[TABLE]

Moreover for any $(p_{1},q_{1})$ admissible and $\alpha_{1}\in[0,\frac{d-2}{2}(1-\frac{1}{q}-\frac{1}{q_{1}})]$ it holds

[TABLE]

We observe that $\alpha_{0}\leq\alpha_{1}$ as $q_{1}\geq 2$.

Proof.

We notice that $U_{\varepsilon}$ defined by (2.9) is such that its symbol satisfies

[TABLE]

It follows that for any $q\in(1,\infty)$, $\alpha\geq 0$ that

[TABLE]

Next we recall from Lemma B.1 that $\dot{B}_{q,2}^{0}({\mathbb{R}}^{d})\hookrightarrow L^{q}({\mathbb{R}}^{d})$ for any $q\in[2,\infty)$ and that $\dot{B}_{2,2}^{-s}({\mathbb{R}}^{d})=\dot{H}^{-s}({\mathbb{R}}^{d})$. Hence, (B.15) yields

[TABLE]

for any $\alpha_{0}\in[0,\frac{d-2}{2}(\frac{1}{2}-\frac{1}{q})]$. Given $s\geq 0$, applying (B.20) to $\tilde{f}=(1-\Delta)^{-\frac{s+\alpha_{0}}{2}}f$ we obtain the estimate

[TABLE]

for any admissible pair $(p,q)$ and $\alpha_{0}\in[0,\frac{d-2}{2}(\frac{1}{2}-\frac{1}{q})]$. It remains to show (B.18). Lemma B.1 provides the embedding $\dot{W}^{s,q^{\prime}}({\mathbb{R}}^{d})\hookrightarrow\dot{B}_{q^{\prime},2}^{s}({\mathbb{R}}^{d})$ for $s\in{\mathbb{R}}$ and $q\in[2,\infty)$, we conclude from (B.16) that for any $(p_{1},q_{1})$ admissible, we have

[TABLE]

provided $\alpha_{1}\in[0,\frac{d-2}{2}(1-\frac{1}{q}-\frac{1}{q_{1}})]$. Applying (B.21) to $\tilde{F}=(1-\Delta)^{-\frac{s+\alpha_{1}}{2}}F$ we infer the estimate

[TABLE]

for $s\in{\mathbb{R}}$ and $\alpha_{1}\in[0,\frac{d-2}{2}(1-\frac{1}{q}-\frac{1}{q_{1}})]$. This completes the proof. ∎