On the Incompressible Limit for the Compressible Free-Boundary Euler Equations with Surface Tension in the Case of a Liquid

TL;DR

This paper proves the incompressible limit for the compressible free-boundary Euler equations with surface tension, addressing new technical challenges and developing a weighted energy approach that accounts for boundary geometry and non-irrotational flows.

Contribution

It introduces a novel weighted energy method to handle boundary terms with surface tension, enabling the passage to the incompressible limit without assuming irrotationality.

Findings

Established the incompressible limit with surface tension

Developed a weighted energy framework for boundary geometry

Achieved uniform estimates in sound speed

Abstract

In this paper we establish the incompressible limit for the compressible free-boundary Euler equations with surface tension in the case of a liquid. Compared to the case without surface tension treated recently, the presence of surface tension introduces severe new technical challenges, in that several boundary terms that automatically vanish when surface tension is absent now contribute at top order. Combined with the necessity of producing estimates uniform in the sound speed in order to pass to the limit, such difficulties imply that neither the techniques employed for the case without surface tension, nor estimates previously derived for a liquid with surface tension and fixed sound speed, are applicable here. In order to obtain our result, we devise a suitable weighted energy that takes into account the coupling of the fluid motion with the boundary geometry. Estimates are closed by exploiting the full non-linear structure of the Euler equations and invoking several geometric properties of the boundary in order to produce some remarkable cancellations. We stress that we do not assume the fluid to be irrotational.

Equations

$$\{\begin{array}{ll}\rho ({\partial }_{t}u+{\nabla }_{u}u)=-\nabla p,& \text{in}\mathcal{D},\\ {\partial }_t\rho +{\nabla }_u\rho +\rho \text{div}u=0,& \text{in}\mathcal{D},\\ p=p(\rho ),& \text{in}\mathcal{D}.\end{array}$$

$$\{\begin{array}{l}\{x:(0,x)\in \mathcal{D}\}={\mathcal{D}}_0,\\ u=u_0,\rho ={\rho }_0\text{in}\{0\}\times {\mathcal{D}}_0,\end{array}\{\begin{array}{l}({\partial }_t+{\nabla }_u){|}_{\partial \mathcal{D}}\in T(\partial \mathcal{D}),\\ p{|}_{\partial \mathcal{D}}=\sigma \mathcal{H},\end{array}$$

$$p=p(\rho ),p^{\prime }(\rho )>0.$$

$${\partial }_t\eta (t,y)$$

$$\eta (0,y)$$

$${\partial }_t\eta =v.$$

$${\Gamma }_0:={\mathbb{T}}^2\times \{x_3=0\},{\Gamma }_1:={\mathbb{T}}^2\times \{x_3=1\},$$

$$A=Ja,$$

$${\partial }_{\mu }{A}^{\mu \alpha }=0.$$

$$\{\begin{array}{ll}R{\partial }_t{v}^{\alpha }+a^{\mu \alpha }{\partial }_{\mu }q=0,& \text{in}[0,T]\times \Omega \\ {\partial }_{t}R+R{a}^{\mu \alpha }{\partial }_{\mu }{v}_{\alpha }=0,& \text{in}[0,T]\times \Omega \\ q=q(R),& \text{in}[0,T]\times \Omega \\ A^{\mu \alpha }{N}_{\mu }q+\sigma \sqrt{g}{\Delta }_g{\eta }^{\alpha }=0,& \text{on}[0,T]\times \Gamma ,\\ \eta (0,\cdot )=\mathrm{id},R(0,\cdot )=R_0(={\rho }_0),v(0,\cdot )=v_0,& \end{array}$$

$$g_{ij}={\partial }_i{\eta }^{\mu }{\partial }_j{\eta }_{\mu },{\Delta }_g(\cdot )=\frac{1}{\sqrt{g}}{\partial }_i(\sqrt{g}{g}^{ij}{\partial }_j(\cdot )),$$

$${\partial }_{t}J=J{a}^{\mu \nu }{\partial }_{\mu }{v}_{\nu },[0,T]\times \Omega .$$

$$RJ={\rho }_0,[0,T]\times \Omega ,$$

$${\rho }_0{\partial }_t{v}^{\alpha }+A^{\mu \alpha }{\partial }_{\mu }q=0,\text{in}[0,T]\times \Omega .$$

$$\{\begin{array}{ll}\beta {\mathfrak{v}}_t^{\alpha }+{\mathfrak{a}}^{\mu \alpha }{\partial }_{\mu }\mathfrak{q}=0,& \text{in}[0,T]\times \Omega \\ \text{div}\mathfrak{v}=0,& \text{in}[0,T]\times \Omega \\ {\mathfrak{A}}^{\mu \alpha }{N}_{\mu }\mathfrak{q}+\sigma \sqrt{\mathfrak{g}}{\Delta }_{\mathfrak{g}}{\tilde{\eta }}^{\alpha }=0,& \text{on}[0,T]\times \Gamma ,\end{array}$$

$$\kappa :=q_{\kappa }^{\prime }(R){|}_{R=\beta }.$$

$$q_{\kappa }(R)=c_{\gamma }\kappa (R^{\gamma }-{\beta }^{\gamma }),c_{\gamma }={\gamma }^{-1}>0,\beta >0,\gamma \ge 1.$$

$$\frac{1}{c_0}{\mathfrak{R}}_{\kappa }\le R_{\kappa }^{\prime }(q)\le c_0{\mathfrak{R}}_{\kappa },$$

$$|R_{\kappa }^{(k)}(q)|\le c_0,|R_{\kappa }^{(k)}(q)|\le c_0|R_{\kappa }^{\prime }(q){|}^k\le c_0|R_{\kappa }^{\prime }(q)|,$$

$$|q_{\kappa }^{(k)}(R)|\le c_0|q_{\kappa }^{\prime }(R)|,$$

Full text

1. 1 Introduction

2. 1.1 Lagrangian coordinate and the reference domain

3. 1.2 Background

4. 1.3 The sound speed

5. 1.4 The main results

1 Introduction

We consider the motion of a compressible liquid with free surface boundary in $\mathbb{R}^{3}$. We use the notation $\mathcal{D}_{t}$ to represent the bounded domain occupied by the fluid at each time $t$, whose boundary is advected by the fluid. The motion of the fluid is described by the compressible Euler equations

[TABLE]

Here, $\mathcal{D}=\cup_{0\leq t\leq T}\{t\}\times\mathcal{D}_{t}$, $u=u(t,x)$ is the velocity of the fluid, whereas $p=p(t,x)$ and $\rho=\rho(t,x)$ are the pressure and density, respectively. The density is bounded from below away from zero, i.e., $\rho\geq\text{constant}>0$. This condition on the density is what characterizes the fluid as a liquid. The initial and boundary conditions are

[TABLE]

where $\mathcal{H}$ is the mean curvature of $\partial\mathcal{D}_{t}$, $\sigma\geq 0$ is a constant, and $T(\partial\mathcal{D})$ is the tangent bundle of $\partial\mathcal{D}$ (the condition $(\partial_{t}+\nabla_{u})|_{\partial\mathcal{D}}\in T(\partial\mathcal{D})$ expresses the fact that the boundary moves with speed equal to the normal component of the velocity). Finally, the equation of state is assumed to be a strictly increasing function of the density, i.e.,

[TABLE]

We shall consider the specific equation of state given by (1.11) in this manuscript. The unknowns in (1.1)-(1.2) are $u,\rho$ and $\mathcal{D}_{t}$, and hence, $\mathcal{H}$ and $p$ are function of the unknowns, and therefore, are not known a priori.

Problem (1.1)-(1.2) behaves significantly different depending on whether $\sigma=0$ or $\sigma>0$. The former is known as the case without surface tension whereas the latter is the case with surface tension, which is the situation treated in this manuscript. Our goal is to show that, for $\sigma>0$, the motion of a free-boundary incompressible fluid with surface tension (corresponding to the idealized situation of a constant density fluid) is well-approximated by (1.1)-(1.2) when an appropriate notion of compressibility is very small. It is well-known that solutions to the incompressible equations, written in section 1.2 below, cannot be obtained by simply setting $\rho$ to a constant in (1.1)-(1.2) (see, e.g., [lindblad2018priori]). The correct way of setting the incompressible limit is via the fluid’s sound speed introduced in section 1.3.

The study of the incompressible limit has a long history in fluid dynamics, see section 1.2. For the case of a motion with free-boundary, the only results we are aware are the recent works [lindblad2018priori, luo2017motion] by Lindblad and the second author, both treating the case $\sigma=0$. In particular, to the best of our knowledge this is the first proof of the incompressible limit for the free-boundary compressible Euler equations with surface tension, i.e., $\sigma>0$. Despite many new difficulties introduced by the presence of surface tension, which are discussed in section LABEL:section_1.5, it is important to consider the case $\sigma>0$ because real fluids have surface tension. Thus, this feature has to be incorporated in the construction of more realistic models. We remark that we do not assume that the fluid is irrotational.

1.1 Lagrangian coordinate and the reference domain

We introduce Lagrangian coordinates, under which the moving domain becomes fixed. Let $\Omega$ be a bounded domain in $\mathbb{R}^{3}$. Denoting coordinates on $\Omega$ by $y=(y_{1},y_{2},y_{3})$, we define $\eta:[0,T]\times\Omega\to\mathcal{D}$ to be the flow of the velocity $u$, i.e.,

[TABLE]

We introduce the Lagrangian velocity, density and pressure, respectively, by $v(t,y):=u(t,\eta(t,y))$, $R(t,y):=\rho(t,\eta(t,y))$ and $q(t,y):=p(t,\eta(t,y))$. Therefore,

[TABLE]

For the sake of simplicity and clean notation, here we consider the model case when $\mathcal{D}_{0}=\Omega=\mathbb{T}^{2}\times(0,1)$. We set

[TABLE]

so that $\Gamma:=\partial\Omega=\Gamma_{0}\cup\Gamma_{1}$. Using a partition of unity, as in, e.g., [coutand2013LWP0.5, DKT], a general domain can be treated with the same tools we shall present. Choosing $\Omega$ as above, however, allows us to focus on the real issues of the problem without being distracted by the cumbersomeness of the partition of the unity. We also note that one might want to consider a situation more akin the finite-depth water waves problem, where the bottom boundary, $\Gamma_{0}$, remains fixed. This case requires only minor modifications from our presentation but, again, we believe that this would be a distraction from the main problem.

Let $\partial$ be the spatial derivative with respect to the spatial variable $y$. We introduce the matrix $a=(\partial\eta)^{-1}$. This is well-defined since $\eta(t,\cdot)$ is almost id (i.e., the identity diffeomorphism on $\Omega$) whenever $t$ is sufficiently small. Define the cofactor matrix

[TABLE]

where $J=\det(\partial\eta)$. Then, $A$ satisfies the Piola identity:

[TABLE]

Here, the summation convention is used for repeated upper and lower indices, and in above and throughout, we adopt the convention that the Greek indices range over $1,2,3$, while the Latin indices range over $1$ and $2$.

In terms of $v,R,q$ and $a$, the system (1.1)-(1.2) becomes

[TABLE]

where $N$ is the unit outward normal to $\Gamma$, and $\Delta_{g}$ is the Laplacian of the metric $g_{ij}$ induced on $\Gamma(t)=\eta(t,\Gamma)$ by the embedding $\eta$, i.e.,

[TABLE]

where $g=\det g$. Since $\eta(0,\cdot)=\operatorname{id}$, the initial Eulerian and Lagrangian velocities (i.e., $u_{0}$ and $v_{0}$) agree. In addition, we also have $a(0,\cdot)=I$, where $I$ is the identity matrix. Finally, $J=\text{det}(\partial\eta)$ satisfies

[TABLE]

This, together with the second equation of (1.4) imply

[TABLE]

and hence the first equation in (1.4) is equivalent to

[TABLE]

1.2 Background

The study of the motion of a fluid has a long history in mathematics. In particular, the study of free-boundary fluid problems has blossomed over the past decade or so. However, much of this activity has focused on the study of the incompressible free-boundary Euler equations, i.e.,

[TABLE]

where $\beta$ is a positive constant corresponding to the fluid’s constant density, $\mathfrak{v}$ and $\mathfrak{q}$ are the incompressible Lagrangian velocity and pressure, $\mathfrak{a}=(\partial\tilde{\eta})^{-1}$, $\mathfrak{A}=\det(\partial\tilde{\eta})\mathfrak{a}$, where $\tilde{\eta}$ is the Lagrangian map associated with $\mathfrak{v}$.

It is well-known that for the incompressible equations, $\mathfrak{q}$ is not determined by an equation of state. Rather, it is a Lagrange multiplier enforcing the constraint $\text{div}\,\mathfrak{v}=0$. The local well-posedness for the incompressible free-boundary Euler equations has been studied by many authors, see [bieri2017motion, christodoulou2000motion, coutand2007LWP, coutand2010LWP, disconzi2014limit, disconzi2016free, disconzi2017prioriI, ignatova2016local, kukavica2017local, lindblad2002, lindblad2005well, lindblad2009priori, nalimov1974cauchy, SchweizerFreeEuler, shatah2008geometry, shatah2008priori, shatah2011local, wu1997LWPww, wu1999LWPww, zhang2008free] and references therein. It is worth mentioning here that when $\mathcal{D}_{0}$ is unbounded (with finite or infinite depth) and the velocity $\mathfrak{v}_{0}$ is irrotational (i.e., $\text{curl}\,\mathfrak{v}_{0}=0$, a condition that is preserved by the evolution), this problems is called the water-waves problem, which has received a great deal of attention [alazard2013global, alazard2013sobolev, alazard2018morawetz, deng2016global, germain2012GWP, germain2015GWP, harrop2017finite, hunter2016two, ifrim2014two, ifrim2015two, ifrim2017lifespan, ionescu2018global, ionescu2014global, ionescu2015global, ionescu2016global, totz2012rigorous, wang2015global, wang20173, wang2018global, wu2009GWP, wu2011GWP].

However, the theory of the free-boundary compressible Euler equations is far less developed. It is known that for suitable initial data, the system (1.1) modeling a liquid admits a local (in time) solution, e.g., [coutand2013LWP0.5, disconzi2017prioriC, disconzi2018local, lindblad2003well, lindblad2005well', trakhinin2009local], and for the gas model, the existence of a local solution was obtained in [coutand2010priori, coutand2011LWP1D, coutand2012LWP, jang2009LWPgas, jang2015LWPgas, luo2014well].

In this paper we study how the solutions to (1.4) and (1.9) are related. Intuitively, one expects that the solution of (1.4) should converge to that of (1.9) when the “compressibility vanishes”. The proper way to define this problem is via the fluid’s sound speed (see (1.10) below), which corresponds to the speed of propagation of sound waves inside the fluid and captures the fluid’s compressibility in that stiffer fluids have larger sound speed111This is an experimental fact, see, e.g., [white1999fluid]..

The incompressible limit problem consists in proving that if a sequence $(v_{0,\kappa},R_{0,\kappa})$ of well-prepared initial data for (1.4) converges to $(\mathfrak{v}_{0},\beta)$, where $\mathfrak{v}_{0}$ is the initial data for the incompressible problem (1.9), and the sound speed at time zero diverges to infinity, then the respective solution $(v,R)$ of (1.1) converges to $(\mathfrak{v},\beta)$, where $\mathfrak{v}$ solves (1.9). Here, well-prepared initial data means that, in addition to satisfying the compatibility conditions, the initial data has to be tailored to the above limit (see Theorem LABEL:data).

The incompressible limit for the compressible Euler equations in a fixed domain (i.e., $\mathcal{D}_{t}=\mathcal{D}_{0}$ or the whole space) was established by several authors under different assumptions, see [alazard2006low, alazard2008minicourse, disconzi2017inL, ebin1977motion, ebin1982motion, klainerman1981singular, klainerman1982compressible, Metivier2011, Schochet1986The] and references therein. In addition, the incompressible limit for the compressible free-boundary Euler equations was solved by Lindblad and the second author in [lindblad2018priori] with $\sigma=0$ in a bounded domain, and by the second author [luo2017motion] in the same case but with unbounded domain. To our best knowledge, the aforementioned works [lindblad2018priori, luo2017motion] are the only known results in the study of the incompressible limit for equations (1.4). In particular, no result is available for the case with $\sigma>0$. We will establish a priori estimates for (1.4) that are uniform in the sound speed (see sections LABEL:section_3-LABEL:section_4). In addition, we will construct a sequence of well-prepared data for (1.4) which converges to that of (1.9) when the sound speed tends to infinity (see section LABEL:section_5). As a consequence, we conclude the convergence of the compressible solution to the incompressible one by an Arzelà-Ascoil-type theorem.

1.3 The sound speed

Physically, the sound speed is defined as $c=\sqrt{q^{\prime}\circ R}$. To set up the incompressible limit, it is conveninet to view the sound speed as a paramenter. As in [disconzi2017inL, ebin1982motion], we consider a family $\{q_{\kappa}(R)\}$ parametrized by $\kappa\in[0,\infty)$, where

[TABLE]

Here, ${}^{\prime}=\frac{d}{dR}$, and

[TABLE]

We slightly abuse terminology and call $\kappa$ the sound speed. In order to consider the incompressible limit, we view the density as a function of the pressure, i.e., $R_{\kappa}=R_{\kappa}(q)=[(c_{\gamma}\kappa)^{-1}q+\beta^{\gamma}]^{1/\gamma}$, and we see that $R_{\kappa}^{\prime}(q)$ satisfies

[TABLE]

for some fixed constant $c_{0}>0$, where $\mathfrak{R}_{\kappa}=(c_{\gamma}\kappa)^{-\frac{1}{\gamma}}$. Also, for $0\leq k\leq 4$, we have that:

[TABLE]

hold uniformly in $\kappa$.

1.4 The main results

Notations. All notations will be defined as they are introduced. In addition, a list of symbols is given at the end of this section for a quick reference.

Definition 1.1**.**

The $L^{2}$-based Sobolev spaces are denoted by $H^{s}(\Omega)$, with the corresponding norm denoted by $||\cdot||_{s}$; note that $||\cdot||_{0}=||\cdot||_{L^{2}(\Omega)}$. We denote by $H^{s}(\Gamma)$ the Sobolev space of functions defined on $\Gamma$, with norm $||\cdot||_{s,\Gamma}$.