Global regularity for the fractional Euler alignment system

TL;DR

This paper proves that a fractional Euler alignment system with nonlinear, density-dependent alignment and fractional dissipation has globally regular solutions, highlighting a novel non-local nonlinear regularization mechanism.

Contribution

It demonstrates for the first time that nonlinear, density-dependent alignment enhances dissipation, ensuring global regularity in a fractional Euler system.

Findings

Solutions are globally regular for all fractional orders in (0, 1)

Non-local nonlinear modulation of dissipation prevents shock formation

Alignment term enhances dissipation leading to regularity

Abstract

We study a pressureless Euler system with a nonlinear density-dependent alignment term, originating in the Cucker-Smale swarming models. The alignment term is dissipative in the sense that it tends to equilibrate the velocities. Its density dependence is natural: the alignment rate increases in the areas of high density due to species discomfort. The diffusive term has the order of a fractional Laplacian $(-\partial_{xx})^{\alpha/2}$, $\alpha \in (0, 1)$. The corresponding Burgers equation with a linear dissipation of this type develops shocks in a finite time. We show that the alignment nonlinearity enhances the dissipation, and the solutions are globally regular for all $\alpha \in (0, 1)$. To the best of our knowledge, this is the first example of such regularization due to the non-local nonlinear modulation of dissipation.

Equations

$${\theta }_i(t+1)=\frac{1}{|{\mathcal{N}}_i(t)|}\sum _{j\in {\mathcal{N}}_i(t)}{\theta }_j(t)+\eta \Delta \theta .$$

$${\dot{x}}_i=v_i,{\dot{v}}_i=\frac{1}{N}\sum _{j=1}^N\varphi (|x_i-x_j|)(v_j-v_i).$$

$${\dot{x}}_i=v_i,{\dot{v}}_i=\frac{\lambda }{{\Phi }_i}\sum _{j=1}^N\varphi (|x_i-x_j|)(v_j-v_i),{\Phi }_i=\sum _{k=1}^N\varphi (|x_i-x_k|),$$

$$f_t+v\cdot {\nabla }_{x}f+{\nabla }_v\cdot (L[f]f)=0,$$

$$L[f](x,v,t)={\int }_{{\mathbb{R}}^{2d}}\varphi (x-y)(v^{\prime }-v)f(y,v^{\prime },t)d{v}^{\prime }dy.$$

$$S(t)=\sup \{|x-y|:(x,v),(y,v^{\prime })\in \text{supp}(f(\cdot ,\cdot ,t))\}$$

$$V(t)=\sup \{|v-v^{\prime }|:(x,v),(y,v^{\prime })\in \text{supp}(f(\cdot ,\cdot ,t))\}\to 0\; ast\to +\infty ,$$

$$f_t+v\cdot {\nabla }_{x}f+{\nabla }_v\cdot (L[f]f)+\lambda {\nabla }_v\cdot ((u(x,t)-v)f)={\Delta }_{v}f,$$

$$u(x,t)=\frac{1}{\rho (t,x)}{\int }_{{\mathbb{R}}^d}vf(x,v,t)dv,\rho (x,t)={\int }_{{\mathbb{R}}^d}f(x,v,t)dv.$$

$$f(x,v,t)=\rho (t,x)\delta (v-u(x,t)).$$

$${\partial }_t\rho +{\partial }_x(\rho u)=0,$$

$${\partial }_t(\rho u)+{\partial }_x(\rho u^2)={\int }_{\mathbb{R}}\varphi (x-y)(u(y,t)-u(t,x))\rho (y,t)\rho (x,t)dy.$$

$${\partial }_t\rho +{\partial }_x(\rho u)=0,$$

$${\partial }_t(\rho u)+{\partial }_x(\rho u^2)+{\partial }_x\rho ={\int }_{\mathbb{R}}\varphi (x-y)(u(y,t)-u(x,t))\rho (y,t)\rho (x,t)dy.$$

$$f(x,v,t)=\rho(x,t)\exp\Big{(}-\frac{(v-u(x,t))^{2}}{2}\Big{)},$$

$${\partial }_t\rho +{\partial }_x(\rho u)=0,$$

$${\partial }_t(\rho u)+{\partial }_x(\rho u^2)+{\partial }_x\rho =\alpha \rho u-\beta \rho |u{|}^{2}u-{\int }_{\mathbb{R}}\nabla V(x-y)\rho (y,t)\rho (x,t)dy.$$

$$u_t+u{u}_x=0.$$

$$z_t+(zu{)}_x=0,$$

$$f_t+v{f}_x=0,$$

$$f(x,v,t)=f_0(x-vt,v)=\delta (v+x-vt),$$

$$u_t+u{u}_x=0,$$

$$\underset{x,y\in S_t}{\sup }|u(x,t)-u(y,t)|\to 0\; ast\to +\infty .$$

$${\partial }_x{u}_0(x)\ge -(\varphi \star {\rho }_0)(x)for\; allx\in \mathbb{R},$$

$${\partial }_x{u}_0(x)<-(\varphi \star {\rho }_0)(x).$$

$${\varphi }_{\alpha }(x)=\frac{c_{\alpha }}{|x{|}^{1+\alpha }},$$

$${\Lambda }^{\alpha }f=c_{\alpha }{\int }_{\mathbb{R}}\frac{f(x)-f(y)}{|x-y{|}^{1+\alpha }}dy,\Lambda =(-{\partial }_{xx}{)}^{1/2}.$$

$${\partial }_t\rho +{\partial }_x(\rho u)=0$$

$${\partial }_{t}u+u{\partial }_{x}u=c_{\alpha }{\int }_{\mathbb{R}}\frac{u(y,t)-u(x,t)}{|y-x{|}^{1+\alpha }}\rho (y,t)dy.$$

$${\partial }_{t}u+u{\partial }_{x}u=-{\Lambda }^{\alpha }u,$$

$${\partial }_x{u}_0+\varphi \star {\rho }_0\ge 0,$$

$${\partial }_{x}u+\varphi \star \rho \ge 0$$

Full text

1 Introduction

The Cucker-Smale model

Modeling of the self-organized collective behavior, or swarming, has attracted a large amount of attention over the last few years. Even an attempt at a brief review of this field is well beyond the scope of this introduction, and we refer to the recent reviews [13, 16, 39]. A remarkable phenomenon commonly observed in biological systems is flocking, or velocity alignment by near-by individuals. One of the early flocking models, discrete in time and two-dimensional, is commonly referred to as the Vicsek model: the angle $\theta_{i}(t)$ of the velocity of $i$-th particle satisfies

[TABLE]

Here, ${\cal N}_{i}(t)=\{j:~{}|x_{i}(t)-x_{j}(t)|\leq r\}$, with some $r>0$ fixed, $\Delta\theta$ is a uniformly distributed random variable in $[-1,1]$, and $\eta>0$ is a parameter measuring the strength of the noise. This model preserves the modulus of the particle velocity and only affects its direction. First via numerical simulations and then by mathematical tools, it has been shown that this model has a rich behavior, ranging from flocking when $\eta$ is small, to a completely chaotic motion for large $\eta$, with a phase transition at a certain critical value $\eta_{c}$.

A natural generalization of the Vicsek model was introduced by F. Cucker and S. Smale [19]:

[TABLE]

Here, $\{x_{i},v_{i}\}_{i=1}^{N}$ represent, respectively, the locations and the velocities of the agents. Individuals align their velocity to the neighbors, with the interaction strength characterized by a non-negative influence function $\phi(x)\geq 0$. The relative influence is typically taken as a decreasing function of the distance between individuals. An important flexibility of the Cucker-Smale model is that it both does not impose the constraint on the velocity magnitude and allows to analyze the behavior based on the decay properties of the kernel $\phi(r)$. One of the main results of the Cucker-Smale paper was that, roughly, provided that $\phi(r)$ decays slower than $r^{-1}$ as $r\to+\infty$, then all velocities $v_{i}(t)$ converge to a common limit $\bar{v}(t)$, and the relative particle positions $x_{i}(t)-x_{j}(t)\to\bar{x}_{ij}$ also have a common limit – the particles form a swarm moving with a uniform velocity. This is what we would call a global flocking: all particles move with nearly identical velocities.

One potential shortcoming of the Cucker-Smale model is that an ”isolated clump” of particles may be more affected by ”far away” large mass than by its own neighbors. Essentially, the dynamics inside a small clump would be suppressed by the presence of a large group of particles ”far away”. This can be balanced by a different kind of averaging, rather than simple division by $N$ in (1.2), as was done by S. Motsch and E. Tadmor in [33]:

[TABLE]

with some $\lambda>0$. This modification reinforces the local alignment over the long distance interactions.

A kinetic Cucker-Smale model

Kinetic models are also commonly used to describe the collective behavior when the number of particles is large, in terms of the particle density $f(x,v,t)$, with $x\in{\mathbb{R}}^{d}$, $v\in{\mathbb{R}}^{d}$. A kinetic limit of the Cucker-Smale model was obtained by S.-Y. Ha and E. Tadmor in [25], as a nonlinear and non-local kinetic equation

[TABLE]

with

[TABLE]

Together, (1.4)-(1.5) give a nonlinear kinetic version of the Cucker-Smale system.

It was shown in [14] that its solutions exhibit global flocking, in the sense that the size of the support in $x$

[TABLE]

remains uniformly bounded in time, and the support in $v$ shrinks:

[TABLE]

under the assumption that $\phi(r)$ decays slower than $r^{-1}$ as $r\to+\infty$. A similar result was obtained in [38] for the kinetic Motsch-Tadmor system.

A kinetic model that combines the features of the Cucker-Smale and Motsch-Tadmor models was proposed in a paper by T. Karper, A. Mellet and K. Trivisa [26]:

[TABLE]

with $L[f]$ as in (1.5), $\lambda>0$, and the local average velocity $u(t,x)$ defined as

[TABLE]

The Laplacian in the right side of (1.7) takes into account the possible Brownian noise in the velocity.

One should also mention a large body of literature on the kinetic versions of the Vicsek model and its modifications, and their hydrodynamic limits: see [10, 20, 21, 22, 23] and references therein.

An Euler alignment model

The kinetic Cucker-Smale model can be further ”macroscopized” as a hydrodynamic model for the local density $\rho(t,x)$ and local average velocity $u(t,x)$ defined in (1.8). The standard formal derivation of the hydrodynamic limit for nonlinear kinetic equations often relies on a (often hard to justify) moment closure procedure. An alternative is to consider the ”monokinetic” solutions of (1.4)-(1.5) of the form

[TABLE]

In a sense, this is a ”local alignment” (as opposed to global flocking) ansatz – the particles move locally with just a single velocity but the velocity does vary in space. Inserting this expression into (1.4)-(1.5) gives the Euler alignment system, which we write in one dimension as

[TABLE]

The presence of the density $\rho$ under the integral in the right side of (1.11) has a very reasonable biological interpretation: the alignment effect between the individual agents becomes stronger where the density is high (assuming that the interaction kernel $\phi$ is localized). As far as a rigorous derivation of the hydrodynamic limit is concerned, the aforementioned paper [26] derives the hydrodynamic limit starting from the ”combined” Cucker-Smale-Motsch-Tadmor kinetic system (1.7):

[TABLE]

This system has an extra term $\partial_{x}\rho$ in the left side of (1.11) that can be thought of as pressure, with the constitutive law $p(\rho)=\rho$. The pressure appears as a result of the balance between the local interaction term in the left side of (1.7) and the Laplacian in the right side. In particular, the starting point of the derivation is not the single local velocity ansatz (1.9) but its smooth Maxwellian version (setting $\lambda=1$ in (1.7) for convenience)

[TABLE]

together with the assumption that the interaction is weak: $\phi\to\varepsilon\phi$, and a large time-space rescaling $(t,x)\to t/\varepsilon,x/\varepsilon$.

Another version of the Euler equations as a model for swarming has been proposed in [32], and formally justified in [17]:

[TABLE]

The key difference between models like (1.15) and the ones we consider here is the absence of the regularizing term $u(t,y)-u(t,x)$ in the right side, so one does not expect the regularizing effect of the interactions that we will observe here.

The Euler alignment system for Lipschitz interaction kernels

When particles do not interact, that is, $\phi(x)\equiv 0$, the system (1.10)-(1.11) is simply the pressure-less Euler equations. In particular, in that case, (1.11) is the Burgers equation:

[TABLE]

Its solutions develop a shock singularity in a finite time if the initial condition $u_{0}(x)$ has a point where $\partial_{x}u_{0}(x)<0$. In particular, if $u_{0}(x)$ is periodic and not identically equal to a constant, then $u(x,t)$ becomes discontinuous in a finite time. The function $z(x,t)=-u_{x}(x,t)$ satisfies the continuity equation

[TABLE]

and becomes infinite at the shock location.

The singularity in the Burgers equation does not mean that there is a singularity in the solution of kinetic equation: it only means that the ansatz (1.9) breaks down, and we can not associate a single velocity to a given position. This is a version of ”a shock implies no local alignment”. To illustrate this point, consider the solution of free transport equation

[TABLE]

with the initial condition $f_{0}(x)=\delta(v+x)$. The solution of the kinetic equation is

[TABLE]

hence the ansatz (1.9) fails at $t=1$. This is the time when the corresponding Euler equation

[TABLE]

with the initial condition $u(0,x)=-x$, develops a shock: $u(x,t)=-x/(1-t)$.

The integral term in the right side of (1.11) has a dissipative nature when $\phi\not\equiv 0$: it tries to regularize the velocity discontinuity. When the function $\phi(x)$ is Lipschitz, this system has been investigated in [12] and [37] that show two results. First, a version of global flocking: if $\phi$ decays slower than $|x|^{-1}$ at infinity, and the solution remains smooth for all $t\geq 0$ and the initial density $\rho_{0}$ is compactly supported, then the support $S_{t}$ of $\rho(t,\cdot)$ remains uniformly bounded in time, and

[TABLE]

An improvement in global regularity compared to the Burgers equation (1.16) was also obtained in [12] and [37]. As we have mentioned, solutions of the latter become discontinuous in a finite time provided there is a point $x\in{\mathbb{R}}$ where the initial condition $u_{0}(x)$ has a negative derivative: $\partial_{x}u_{0}(x)<0$. On the other hand, solutions of the Euler alignment equations remain regular for initial data such that

[TABLE]

while the solution blows up in a finite time if there exists $x_{0}\in{\mathbb{R}}$ such that

[TABLE]

Thus, the presence of the dissipative term in (1.11) leads to global regularity for some initial data that blows up for the Burgers equation: the right side of (1.22) may be negative. However, a Lipschitz interaction kernel $\phi(x)$ arrests the shock singularity for the Euler alignment equations only for some initial conditions.

Singular alignment kernels

Our interest is in singular interaction kernels of the form $\phi(x)=|x|^{-\beta}$, with $\beta>0$. One reason to consider such kernels is to strengthen the effect of the local interactions compared to the effect of ”far-away” particles, in the spirit of the Motsch-Tadmor correction. The well-posedness of the finite number of particles Cucker-Smale system with such interactions is a delicate issue – the difficulty is in either ruling out the possibility of particle collisions, or understanding the behavior of the system at and after a collision. This problem was addressed in [35, 36] for $\beta\in(0,1)$ – it was shown that in this range, particles may get stuck together but a weak solution of the ODE system can still be defined. When $\beta\geq 1$, a set of initial conditions that has no particle collisions was described in [1]. The absence of collisions was proved very recently for general initial configurations in [11]. As far as flocking is concerned, unconditional flocking was proved in [24] for $\beta\in(0,1)$, while for $\beta\geq 1$ there are initial configurations that do not lead to global flocking – the long distance interaction is too weak. The well-posedness of the kinetic Cucker-Smale system for $\beta\in(0,1/2)$ was established in [34].

We consider here the alignment kernels $\phi(x)$ with $\beta>1$:

[TABLE]

with $\alpha>0$. In particular, the decay of $\phi(x)$ at large $|x|$ is faster than the $1/|x|$ decay required for the Cucker-Smale and other proofs of flocking. It is compensated by a very strong alignment for $|x|\to 0$. The constant $c_{\alpha}$ is chosen so that

[TABLE]

Then the strong form of the Euler alignment system is

[TABLE]

Let us first compare the Euler alignment system (1.25)-(1.26) to the Burgers equation with a fractional dissipation

[TABLE]

obtained by formally setting $\rho(t,x)\equiv 1$ in (1.26) and dropping (1.25) altogether. This neglects the nonlinear mechanism of the dissipation. Global regularity of the solutions of the fractional Burgers equation has been studied in [30]. One can distinguish three regimes: first, when $\alpha>1$, the dissipative term in the right side has a higher order derivative than the nonlinear term in the left side. This is the sub-critical regime: the dissipation dominates the nonlinearity, and global existence of the strong solutions can be shown in a reasonably straightforward manner using the energy methods. On the other hand, when $0<\alpha<1$, the dissipation is too weak to compete with the nonlinear term, which has a higher derivative, and solutions with smooth initial conditions may develop a shock, as in the inviscid case. The critical case is $\alpha=1$ when the dissipation and the nonlinearity contain derivatives of the same order. One may expect that then the nonlinearity may win over the dissipation for some large data. This, however, is not the case: solutions with smooth initial conditions remain regular globally in time. The proof of the global regularity when $\alpha=1$ is much less straightforward than for $\alpha>1$ and does not rely solely on the energy methods.

One may hope that the nonlinearity in the dissipative term in the right side of (1.26) is actually beneficial, compared to the fractional Burgers equation (1.27). Indeed, on the qualitative level, as the shock would form, the density $\rho$ would be expected to increase near the point of the shock. This, in turn, would increase the dissipation in (1.26), moving the problem from ”like a super-critical Burgers” to ”like a sub-critical Burgers”. This intuition, however, may be slightly misleading: for instance, as we will see, strengthening the dissipation by increasing $\alpha$ does not appear to make the problem any easier, or change its critical character. The competition between the Burgers nonlinearity in the left side of (1.26) and the nonlinear dissipation in the right side is rather delicate.

The aforementioned results of [12, 37] may lead to a conjecture that a dissipation term involving the convolution kernel $\phi\notin L^{1}$, as in (1.26), should lead to global regularity. However, this is far from obvious. The global regularity argument of [12, 37] uses two ingredients: first, if initially

[TABLE]

for all $x\in{\mathbb{R}}$ then

[TABLE]

for all $x\in{\mathbb{R}}$ and $t\geq 0$. Second, an $L^{\infty}$-bound on $\rho$ is established. When $\phi$ is an $L^{1}$-function, one deduces a lower bound $\partial_{x}u>-C_{0}$, which is crucial for global regularity. One may combine an argument of [12] with the Constantin-Vicol nonlinear maximum principle to establish the $L^{\infty}$-bound for $\rho$ in our case, as well. However, in our case, the analogous inequality to (1.29) is

[TABLE]

This fails to give the required lower bound on $\partial_{x}u$ based on just the $L^{\infty}$ control of $\rho$, and the global regularity does not follow easily from the uniform bound on the density. Instead, we have to deploy a much subtler argument involving both upper and lower bounds on the density and a non-trivial modification of the modulus of continuity technique of [31].

The main result

We consider here the Euler alignment system (1.25)-(1.26) on the torus $\mathbb{T}$, for $\alpha\in(0,1)$. In particular, this range of $\alpha$ corresponds to the supercritical case for the fractional Burgers equation (1.27). We prove that the nonlinear, density modulated dissipation qualitatively changes the behavior of the solutions: instead of blowing up in a finite time, solutions are globally regular.

Theorem 1.1**.**

For $\alpha\in(0,1)$, the Euler alignment system (1.25)-(1.26) with periodic smooth initial data $(\rho_{0},u_{0})$ such that $\rho_{0}(x)>0$ for all $x\in\mathbb{T}$, has a unique global smooth solution.

The regularizing effect of a non-linear diffusion has been observed before, for instance, in the chemotaxis problems with a nonlinear diffusion – see [5, 6, 7, 8, 9]. The main novelties here are that the nonlinearity is non-local, and that, as we will see, increasing $\alpha$ does not, contrary to a naive intuition, and unlike what happens in the fractional Burgers equation, strengthen the regularization effect.

To explain the ideas behind the result and its proof, it is convenient to reformulate the Euler alignment system (1.25)-(1.26) as the following system for $\rho$ and $G=\partial_{x}u-\Lambda^{\alpha}\rho$:

[TABLE]

with the velocity $u$ related to $\rho$ and $G$ via

[TABLE]

We show in Section 2 that (1.25)-(1.26) and (1.31)-(1.33) are, indeed, equivalent for regular solutions. Note that (1.33) only defines $u$ up to its mean, which is determined from the conservation of the momentum:

[TABLE]

Somewhat paradoxically, (1.33) seems to indicate that increasing the dissipation $\alpha$ makes the velocity more singular in terms of the density rather than more regular.

The solutions of (1.31)-(1.32) with the initial conditions $\rho_{0}(x)$, $u_{0}(x)$ such that

[TABLE]

preserve the constraint $G=0$ for all $t>0$, and (1.31)-(1.32) then reduces to a single equation

[TABLE]

that is simpler to analyze. Note that (1.36) defines $u(x,t)$ only up to its spatial average – we assume that it has mean-zero for all $t>0$. The model (1.36) is interesting in its own right. When $\alpha=1$, so that the velocity is the Hilbert transform of the density, it was introduced as a 1D vortex sheet model in [4], and has been extensively studied in [15] as a 1D model of the 2D quasi-geostrophic equation. In particular, the global existence of the solution if $\rho_{0}>0$ is proved in [15] using the algebraic properties of the Hilbert transform. Our results in this paper can be directly applied to (1.36), and show the global regularity of the solutions for all $\alpha\in(0,1)$. The strategy of the regularity proof here is very different from that in [15]. A quintessential feature of (1.36) is that increasing $\alpha$ does not help the dissipation in its competition with the Burgers nonlinearity. Indeed, the toy model (1.36) can be written as

[TABLE]

Thus, the scalings of the dissipation in the right side and of the nonlinear transport term in the left side are exactly the same, both in $\rho$ and in $x$, no matter what $\alpha\in(0,1)$ is. While the proof of global regularity for (1.37) is inspired by the nonlocal maximum principle arguments of [31, 30], the nonlinear nature of dissipative term necessitates significant changes and new estimates. The upgrade of the proof from global regularity of the model equation to the full system is also highly non-trivial and requires new ideas.

We note that our results can be applied to the case $\alpha\in(1,2)$, where the global behavior is the same as for the fractional Burgers equation. One can also extend our results to influence kernels of the form

[TABLE]

with a non-negative smooth compactly supported function $\chi(r)$. This is the analog of the kernels in (1.24) for $\alpha=0$. We expect that as soon as the influence kernel is not integrable, solutions remain regular. The proofs of these extensions require some nontrivial adjustments and further technicalities compared to the arguments in this paper, and will be presented elsewhere.

Our results also lead to global flocking behavior for (1.25)-(1.26). The periodized influence function

[TABLE]

has a positive lower bound for all $x\in\mathbb{T}$. Since the solution is smooth, one can use the argument in [37] to obtain asymptotic flocking behavior in the sense that

[TABLE]

This paper is organized as follows. In Section 2 we prove an a priori $L^{\infty}$-bound on $\rho$, that is the key estimate for the regularity of the solutions, as well as lower bound on $\rho$. The local well-posedness of the solutions is proved in Section 3. Section 4 contains the proof of our main result, Theorem 1.1. Apppendix A contains the proof of an auxiliary technical estimate. Throughout the paper we denote by $C$, $C^{\prime}$, etc. various universal constants, and by $C_{0}$, $C_{0}^{\prime}$ etc. constants that depend only on the initial conditions.

Acknowledgment. This work was was partially supported by the NSF grants DMS-1412023 and DMS-1311903.

2 Bounds on the density

In this section, we prove the upper and lower bounds on the density $\rho(t,x)$. The upper bound is uniform in time, and is crucial for the global regularity. The lower bound will deteriorate in time but will be sufficient for our purposes.

2.1 The reformulation of the Euler alignment system

We first explain how the Euler alignment system (1.25)-(1.26) is reformulated as  (1.31)-(1.32), as we will mostly use the latter. We only need to obtain (1.32) for $G$ defined in (1.33). The idea comes from [12]. We apply the operator $\Lambda^{\alpha}$ to (1.25), and use the identity

[TABLE]

to obtain

[TABLE]

On the other hand, applying $\partial_{x}$ to (1.26), we get

[TABLE]

Subtracting (2.1) from (2.2) gives an equation for the function $G=\partial_{x}u-\Lambda^{\alpha}\rho$:

[TABLE]

which is (1.32).

Let us comment on how to recover $u$ from (1.33). Let us denote by

[TABLE]

the average of $\rho$ in $\mathbb{T}$, which is preserved in time by (1.31), at least as long as $\rho$ remains smooth. Note that $G(x,t)$ has mean zero automatically:

[TABLE]

We also define

[TABLE]

so that

[TABLE]

Thus, the primitive functions of $\theta(x,t)$ and $G(x,t)$ are periodic. We denote by $(\varphi,\psi)$ the mean-zero primitive functions of $(\theta,G)$, respectively:

[TABLE]

and

[TABLE]

Then, $u$ can be written as

[TABLE]

To determine $I_{0}(t)$, we use the conservation of the momentum. Note that the conservation law form of (1.26) is

[TABLE]

Integrating (2.9) gives

[TABLE]

thus

[TABLE]

Together with (2.8), $u$ is now uniquely defined, with $I_{0}(t)$ given by

[TABLE]

Note that we have

[TABLE]

thus

[TABLE]

In particular, $I_{0}(t)$ is time-independent in the special case $G\equiv 0$, that leads to (1.36), and then we have

[TABLE]

2.2 The upper bound on the density

We now prove an a priori $L^{\infty}$ bound on $\rho$.

Theorem 2.1**.**

Let $\rho(x,t),u(x,t)$ be a strong solution to (1.25)-(1.26) for $0\leq t\leq T$, with smooth periodic initial conditions $\rho_{0}(x)$, $u_{0}(x)$ such that $\rho_{0}(x)>0$ on $\mathbb{T}$. Then, there exists a constant $C_{0}>0$ that depends on $\rho_{0}$ and $u_{0}$ but not on $T$, so that $\|\rho(\cdot,t)\|_{L^{\infty}}\leq C_{0}$ for all $t\geq 0$.

This bound already indicates that the Euler alignment system behaves not as the fractional Burgers equation. Indeed, if we couple fractional Burgers equation with (1.25), the density may blow up for $\alpha\in(0,1)$ for suitable smooth initial conditions.

The proof of Theorem 2.1

As the functions $\rho$ and $G$ obey the same continuity equation, their ratio $F=G/\rho$ satisfies

[TABLE]

It follows that $F$ is uniformly bounded:

[TABLE]

as $\rho_{0}$ and $u_{0}$ are smooth, and $\rho_{0}$ is strictly positive.

In order to prove the upper bound on $\rho$, for a fixed $t\geq 0$, let $\bar{x}$ be such that

[TABLE]

It follows from (1.31) that

[TABLE]

Thus, to obtain an a priori upper bound on $\rho$, it suffices to show that there exists $C_{0}$ that depends on the initial conditions $\rho_{0}$ and $u_{0}$ so that if $\rho(\bar{x},t)>C_{0}$, then

[TABLE]

To obtain (2.18), note that

[TABLE]

In order to bound $\Lambda^{\alpha}\rho$ in the right side of (2.19) from below, we use the nonlinear maximum principle for the fractional Laplacian, see [18, Theorem 2.3]:

[TABLE]

Here, the constant $c>0$ only depends on $\alpha$. Recall that we denote by $\theta(x,t)$ the mean-zero shift of $\rho(x,t)$, as in (2.3) and (2.5), and by $\varphi(x,t)$ the mean-zero primitive of $\theta(x,t)$, as in (2.6). Note that $\|\varphi(\cdot,t)\|_{L^{\infty}}$ is uniformly bounded:

[TABLE]

Therefore, if

[TABLE]

with a sufficiently large $C$, which depends only on $\rho_{0}$ and $u_{0}$, then

[TABLE]

and the second possibility in (2.20) can not hold. Thus, as soon as (2.22) holds, we have

[TABLE]

with a constant $C_{0}$ that depends on the initial condition $\rho_{0}$. Going back to (2.19), this implies

[TABLE]

Thus, (2.18) indeed holds if $\rho(\bar{x},t)>C_{0}^{\prime}$, where $C_{0}^{\prime}$ is a constant that depends only on $\rho_{0}$ and $u_{0}$, and the proof of Theorem 2.1 is complete. $\Box$

One immediate consequence of Theorem 2.1 is that $I_{0}(t)$ in (2.13) is uniformly bounded for all time. Indeed, it suffices to bound

[TABLE]

while

[TABLE]

where $C$ is a universal constant independent of $t$. Summarizing, we have

[TABLE]

with a constant $C_{0}$ that depends only on $\rho_{0}$ and $u_{0}$.

Thus, we have the following a priori bound on $\|u\|_{L^{2}}$.

Corollary 2.2**.**

Let $\rho(x,t),u(x,t)$ be a strong solution to (1.25)-(1.26) for $0\leq t\leq T$, with smooth periodic initial conditions $\rho_{0}(x)$, $u_{0}(x)$ such that $\rho_{0}(x)>0$ on $\mathbb{T}$. There exists a constant $C_{0}$ that depends only on $\rho_{0}$ and $u_{0}$ but not not on $T$ so that $\|u(\cdot,t)\|_{L^{2}}\leq C_{0}$ for all $0\leq t\leq T$.

Proof. This follows immediately from the bound

[TABLE]

together with the bound

[TABLE]

and (2.24)-(2.25). $\Box$

The uniform upper bound on the density also implies a uniformly Lipschitz bound on $F$.

Lemma 2.3**.**

The function $F=G/\rho$ is Lipschitz, and the Lipschitz bound is uniform in time.

Proof. Recall that $F$ satisfies (2.15), thus $p=\partial_{x}F$ satisfies the same continuity equation as $\rho$:

[TABLE]

and $w=p/\rho$ is a solution of

[TABLE]

It follows that $\|w(\cdot,t)\|_{L^{\infty}}=\|w_{0}\|_{L^{\infty}}$, and therefore,

[TABLE]

Theorem 2.1 implies now that $F$ is Lipschitz, with a time-independent Lipschitz bound. $\Box$

2.3 A lower bound on the density

A uniform lower bound on $\rho$ plays an important role as it keeps the dissipation active. The following lemma ensures no creation of vacuum in finite time.

Lemma 2.4**.**

Let $\rho(x,t),u(x,t)$ be a strong solution to (1.25)-(1.26) for $0\leq t\leq T$, with smooth periodic initial conditions $\rho_{0}(x)$, $u_{0}(x)$ such that $\rho_{0}(x)>0$ on $\mathbb{T}$. There exists a positive constant $C_{0}>0$ that depends on $\rho_{0}$ and $u_{0}$ but not on $T$, so that

[TABLE]

Proof. Fix some $t>0$ and let $\underline{x}$ be such that

[TABLE]

Then we have

[TABLE]

and thus

[TABLE]

satisfies

[TABLE]

If the minimum is achieved at more than one point, we just need to take a minimum over all of them in the above estimate, which leads to the same bound. Notice that $\rho_{m}(t)$ is Lipschitz in time, so the estimate is valid for a.e. $t$, and ${d\rho_{m}}/{dt}$ determines $\rho_{m}(t).$ Integrating this differential inequality, we get

[TABLE]

finishing the proof. $\Box$

In particular, in the special case $G\equiv 0$, that is, for (1.36) we have the following.

Corollary 2.5**.**

Let $\rho(x,t)$ be the solution of (1.36). Then, we have

[TABLE]

3 The local wellposedness

The a priori bounds on $\rho$ established in the previous section rule out some kinds of finite time blow up, but do not imply that there is no finite time shock formation. This remains to be shown. To proceed further, we first establish a local well-posedness theory for solutions of the Euler alignment system with smooth initial conditions.

Theorem 3.1**.**

Let $\alpha\in(0,1).$ Assume that the initial conditions $\rho_{0}$ and $u_{0}$ satisfy

[TABLE]

with a sufficiently large even integer $s>0$. Then, there exists $T_{0}>0$ such that the system (1.25)-(1.26) has a unique strong solution $\rho(x,t),u(x,t)$ on $[0,T_{0}]$, with

[TABLE]

Moreover, a necessary and sufficient condition for the solution to exist on a time interval $[0,T]$ is

[TABLE]

Condition (3.3) is a Beale-Kato-Majda type criterion. It indicates that the solution is globally regular if $\partial_{x}\rho$ is uniformly bounded in the $L^{\infty}$ norm. We will show that such bound actually does hold in Section 4, using the modulus of continuity method.

3.1 The commutator estimates

We will need some commutator estimates for the local well-posedness theory. We will use the following notation:

[TABLE]

Lemma 3.2**.**

*The following commutator estimates hold:

(i) for any $n\geq 1$, we have*

[TABLE]

(ii) for any $\gamma\in(0,1)$ and $\epsilon>0$, we have

[TABLE]

(iii) for any $\gamma>0$, we have

[TABLE]

Let us comment briefly on the proof of these estimates. Estimate (3.4) can be obtained by the standard Gagliardo-Nirenberg interpolation inequality. As $\Lambda^{2}=-\partial^{2}_{xx}$, this estimate holds if we replace the operator $\partial_{x}^{n}$ by $\Lambda^{s}$ with an even integer $s$.

A version of (3.5) is discussed in [28, Theorem A.8]. We sketch the proof in Appendix A. Finally, estimate (3.6) is due to Kato and Ponce [27]. The proof is similar to that of (3.5).

3.2 The proof of the local well-posedness

It will be convenient to use the variables $(\theta,G)$, so that equations (1.31)-(1.32) take the form

[TABLE]

Here $\kappa$ is the constant in time mean of $\rho$, as in (2.3).

Let us fix $T>0$ and take a sufficiently large even integer $s>0$. We will aim to obtain a differential inequality on

[TABLE]

that will have bounded solutions on a time interval $[0,T_{0}]$, with a sufficiently small $T_{0}$ depending on the initial conditions. To this end, we apply the operator $\Lambda^{s}$ to the equation for $\theta$ in (3.7), multiply the result by $\Lambda^{s}\theta$ and integrate in $x$:

[TABLE]

The second term in the right side produces the dissipation. We shall use it to control the other two terms.

We split the first term in the right side of (3.10) into three pieces:

[TABLE]

Let us start with $I$:

[TABLE]

For $I_{1}$, we have, using (3.8):

[TABLE]

The term $I_{11}$ is controlled by the dissipation in the right side of (3.10): set

[TABLE]

Note that $\rho_{m}(t)>0$ by Lemma 2.4. Then we have, using Lemma 2.4:

[TABLE]

To bound $I_{12}$ we use the Hölder inequality:

[TABLE]

In order to control the term $I_{2}$ in (3.2), we, once again, use (3.8), and the Hölder inequality:

[TABLE]

The contribution of $I_{3}$ in (3.2) is bounded using the commutator estimate (3.5):

[TABLE]

Next, we estimate the term $II$ in (3.11), integrating by parts

[TABLE]

For the term $III$ in (3.11), we apply the commutator estimate (3.4) and get

[TABLE]

To estimate $\|u\|_{H^{s}}$ in the right side, we apply Corollary 2.2 to get

[TABLE]

We also have, using the uniform bound on the density:

[TABLE]

with a constant $C_{0}$ that depends on $\rho_{0}$ and $u_{0}$. Therefore, $\partial_{x}u$ satisfies

[TABLE]

Together, (3.18)-(3.21) give

[TABLE]

The third term in the right side of (3.10) can be estimated as

[TABLE]

Putting the above estimates together, we end up with the following inequality:

[TABLE]

In order to close the estimate, and obtain a bound on $Y(t)$ defined in (3.9), we write:

[TABLE]

The term $IV$ can be treated as $II$ via integration by parts, together with (3.20):

[TABLE]

To bound $V$, we apply the commutator estimate (3.6), as well as (3.21):

[TABLE]

Now, estimates (3.23)-(3.2), together with the uniform bound on $\|G\|_{L^{\infty}}$, yield an inequality

[TABLE]

For $s>{3}/2$, $H^{s}$ is embedded in $W^{1,\infty}$. This, together with Lemma 2.4, implies

[TABLE]

and the local in time well-posedness for solutions with $H^{s}$ initial data follows. Moreover, it follows from (3.27) that

[TABLE]

For all finite $T>0$, if the Beale-Kato-Majda criterion (3.3) is satisfied, the right side of (3.29) is finite, whence

[TABLE]

and thus $\rho\in C([0,T],H^{s}(\mathbb{T}))$. Furthermore, integrating (3.27) in $[0,T]$, we see that if (3.3) holds, then

[TABLE]

thus $\rho\in L^{2}([0,T],H^{s+\frac{\alpha}{2}})$. To recover the conditions on $u$ in (3.2), we apply Corollary 2.2 and get

[TABLE]

This ends the proof of Theorem 3.1.

4 The global regularity

In this section, we derive a uniform $L^{\infty}$-bound on $\partial_{x}\rho$, using a variant of the modulus of continuity method. Together with the Beale-Kato-Majda type criterion (3.3), this will imply the global well-posedness of the Euler alignment system (1.25)-(1.26), and prove Theorem 1.1. We will first consider the special case $G\equiv 0$, that is, the system (1.36). The nonlinear diffusive term makes the problem subtler than in the SQG or Burgers equation case. Finally, we prove the result to the general Euler alignment system, using a combination of an appropriate scaling argument, estimate on the minimum of $\rho,$ and additional regularity estimates. In this case, the bound on $\partial_{x}\rho$ will depend on time and may grow, but remains finite for every $t>0.$

For convenience, we work on $\mathbb{R}$, and extend $\rho$ and $u$ periodically in space.

4.1 The modulus of continuity

We say that a function $f$ obeys modulus of continuity $\omega$ if

[TABLE]

We will work with the following modulus of continuity for the density $\rho$:

[TABLE]

so that $\omega$ is continuous at $\xi=\delta$. The parameters $\delta$ and $\gamma$ are sufficiently small positive numbers to be specified later. The modulus $\omega$ is continuous, piecewise differentiable, increasing and concave, and satisfies

[TABLE]

The following proposition describes the only possible modulus breakthrough scenario for evolution equations.

Proposition 4.1** ([31]).**

Suppose $\rho_{0}$ obeys a modulus of continuity $\omega$ that satisfies (4.2). If the solution $\rho(x,t)$ violates $\omega$ at some positive time, then there must exist $t_{1}>0$ and $x_{1}\neq y_{1}$ such that

[TABLE]

Thus, to prove that $\rho$ obeys a modulus of continuity $\omega$ for all times $t>0$, it is sufficient to prove that if (4.3) holds, then

[TABLE]

As a remark on the notation, we will again use $C$ as a notation for various universal constants that do not depend on $T,\delta$ and $\gamma$.

4.2 The global regularity for the special system with $G\equiv 0$

Let us first consider the special case $G\equiv 0$, or, equivalently, the system (1.36):

[TABLE]

As the mean of $u$ is preserved by the evolution – see (2.14), we may assume without loss of generality that

[TABLE]

for otherwise we would simply consider (4.5) in a frame moving the speed equal to the mean of $u_{0}$. Thus, we have

[TABLE]

Here, $\varphi(x,t)$ is the mean-zero primitive of $\theta(x,t)$, as in (2.6). We will prove the following result.

Theorem 4.2**.**

The system (4.5) with a smooth periodic initial condition $\rho_{0}$ such that $\rho_{0}(x)>0$ for all $x\in\mathbb{T}$ has a unique global smooth solution.

The key step in the proof is

Lemma 4.3**.**

Suppose that $m={\rm min}_{x\in\mathbb{T}}\rho_{0}(x)>0.$ Then there exist $\delta_{m}$ and $\gamma_{m},$ independent of the period of the initial data, such that if $\rho_{0}(x)$ obeys the modulus of continuity $\omega$ given by (4.1), then $\rho(x,t)$ obeys $\omega$ for all $t>0$.

Theorem 4.2 is a consequence of Lemma 4.3. Indeed, suppose that Lemma 4.3 is true. Notice that the equation (4.5) has a scaling invariance: if $\rho(x,t)$ is a solution, then so is

[TABLE]

for any $\lambda>0$. From the properties of the modulus of continuity $\omega$ given by (4.1) (in particular its growth at infinity) it follows that we can find $\lambda>0$ sufficiently small such that $\rho_{\lambda}^{0}(x)=\rho_{0}(\lambda x)$ obeys $\omega$ with $\delta=\delta_{m}$, $\gamma=\gamma_{m}$ provided by Lemma 4.3. Note that the rescaling (4.8) does not change the minimum of $\rho$. As $\delta_{m}$ and $\gamma_{m}$ do not depend on the period, Lemma 4.3 shows that $\rho_{\lambda}(x,t)$ obeys $\omega$ for all $t>0$. In particular, it follows that

[TABLE]

As we have mentioned, (4.9) together with the Beale-Kato-Majda type criterion (3.3), implies that $\rho_{\lambda}(t,x)$ is a global in time solution of (4.5), and thus so is $\rho(t,x)$.

Therefore, we only need to prove Lemma 4.3. Our strategy is as follows. Let us assume that a modulus of continuity $\omega$, with some $\delta$ and $\gamma$ is broken at a time $t_{1}$, in the sense that (4.3) holds for some $x_{1},y_{1}\in\mathbb{T}$. We denote

[TABLE]

and, for simplicity, drop the time variable $t_{1}$ in the notation. We compute:

[TABLE]

We will obtain the following estimates for the three terms in the right side of (4.11). To bound the first term we note that if $\Omega(\xi)$ is a modulus of continuity for $u$, then it follows from [31] that

[TABLE]

The modulus $\Omega(\xi)$ for $u$ is given by the following.

Lemma 4.4**.**

Let $\rho$ obey the modulus of continuity $\omega$ as in (4.1). There exists a universal constant $C>0$ so that then $u(x)$ obeys a modulus of continuity

[TABLE]

We will prove Lemma 4.4 later in this section.

As $\omega^{\prime}(\xi)\leq 1$ for $0\leq\xi<\delta$, and $\omega^{\prime}(\xi)=\gamma/\xi$ for $\xi>\delta$, we conclude that

[TABLE]

again, with the constant $C>0$ that does not depend on $\rho_{0}$.

To bound the last two terms in the right side of (4.11) purely in terms of $\xi=|x_{1}-y_{1}|$ we will use the following lemma.

Lemma 4.5**.**

Let $\rho$ obey the modulus of continuity $\omega$ as in (4.1), and let $x_{1}$, $y_{1}$ be the breakthrough points as in (4.3). There exists a constant $C>0$ that may only depend on $\alpha$ such that

[TABLE]

and

[TABLE]

The first estimate in the above lemma gives a bound for the second term in (4.11):

[TABLE]

while (4.16) leads to:

[TABLE]

Here, $m$ is the minimum of $\rho_{0}$ and is preserved in time: see Corollary 2.5. Putting (4.12), (4.17) and (4.18) together, we obtain

[TABLE]

For $0\leq\xi<\delta$, using (4.13), (4.15) and (4.16), as well as the inequalities

[TABLE]

we see that

[TABLE]

provided that

[TABLE]

On the other hand, for $\xi\geq\delta$, the above bounds tell us

[TABLE]

if

[TABLE]

Therefore, for $\delta$ and $\gamma$ sufficiently small, we have

[TABLE]

which is a contradiction to the assumption that $t_{1}$ is the first breakthrough tine. Thus, $\omega$ can never be broken, and the proof of Lemma 4.3 is complete, except for the proof of Lemmas 4.4 and 4.5. $\Box$

4.2.1 The dissipation bound in Lemma 4.5

We first prove the dissipation bound (4.16) in Lemma 4.5. It was shown in [29] that

[TABLE]

with

[TABLE]

Both terms in the right side are positive due to the concavity of $\omega$.

To obtain a lower bound for $D(\xi)$, we consider two cases. For $\xi\leq\delta$, we only keep the first term. Note that

[TABLE]

due to the concavity of $\omega$, and

[TABLE]

due to the second order Taylor formula and the monotone growth of

[TABLE]

This gives

[TABLE]

which is the first bound in (4.16).

For $\xi>\delta$, we only keep the second term in (4.27). Due to the concavity of $\omega$, we have

[TABLE]

if

[TABLE]

In that case, we have, using (4.29):

[TABLE]

and the proof of (4.16) is complete.

4.2.2 A lower bound on $\Lambda^{\alpha}\rho$ in Lemma 4.5

The next step is obtain the lower bound (4.15) for $\Lambda^{\alpha}\rho(x_{1},t_{1})$. As $\omega$ is a modulus of $\rho$, we have for any $z\in\mathbb{R}$

[TABLE]

while

[TABLE]

This implies a lower bound

[TABLE]

Our goal is to bound $A(\xi)$ from above. Let us decompose the integral in the second line of (4.2.2) as

[TABLE]

We claim that $A_{2}$ and $A_{3}$ are positive, so that their contribution to $A(\xi)$ is negative. Indeed, we can estimate $A_{2}$ using the concavity of $\omega$:

[TABLE]

In addition, $A_{3}\geq 0$ simply due to the monotonicity of $\omega$, which implies

[TABLE]

It remains to bound $A_{1}$ and $A_{4}$ from below. We first consider $0\leq\xi\leq\delta$. In this region, we can estimate $A_{4}$ as follows:

[TABLE]

Thus, if we choose $\delta<1$ and $\gamma<\delta$, as in (4.30), we obtain

[TABLE]

The term $A_{1}$ can be estimated similarly for $0\leq\xi\leq\delta$. Indeed, for $\xi<{\delta}/{2}$, we have

[TABLE]

provided that $\gamma$ satisfies (4.30). On the other hand, for $\delta/2\leq\xi\leq\delta$, we have

[TABLE]

Summing up the above computation, we conclude that

[TABLE]

On the other hand, if $\xi>\delta$, we have the following estimates on $A_{1}$ and $A_{4}$:

[TABLE]

and

[TABLE]

Thus, we have the bound

[TABLE]

finishing the proof of (4.15), as well as of Lemma 4.5. $\Box$

4.2.3 The proof of Lemma 4.4

Next, we find a modulus of continuity $\Omega$ for $u$, if $\rho$ obeys $\omega$ given by (4.1). We start with (4.7):

[TABLE]

The first term in the right side can evaluated explicitly:

[TABLE]

The second term in the right side of (4.44) can be re-written using integration by parts as

[TABLE]

As $\theta\in L^{\infty}$, so that $\varphi$ is uniformly Lipschitz, we can combine (4.45) and (4.46), pass to the limit $\varepsilon\downarrow 0$, and obtain

[TABLE]

Let us note that, since $\theta(x)$ is a periodic mean-zero function, the integral in the right side of (4.47) converges as $|y|\to+\infty$, and

[TABLE]

Using an argument similar to that in the appendix of [31], one can show that, as long as $\rho(x)$ obeys a modulus of continuity $\omega$, the function $u(x)$ given by (4.48) obeys the modulus of continuity

[TABLE]

with a universal constant $C>0$.

Thus, for $0\leq\xi\leq\delta$, we get

[TABLE]

as long as we take $\gamma<\delta$. This is the first inequality in (4.13).

For $\xi>\delta$, we use (4.49) to write

[TABLE]

finishing the proof of Lemma 4.4.

4.3 The global regularity for the full system

We now consider the full system (1.31)-(1.33)

[TABLE]

without the extra assumption $G\equiv 0$. Let us recall representation (2.8):

[TABLE]

Here, $\phi(x)$ and $\psi(x)$ are the mean-zero primitives of $\theta$ and $G$, respectively, as in (2.6)-(2.7), and $I_{0}$ is given by (2.13).

Note that if $\rho(x,t)$ and $G(x,t)$ are solutions of (4.52)-(4.54), with the corresponding velocity $u(x,t)$, then

[TABLE]

are also solutions, with the corresponding velocity

[TABLE]

and

[TABLE]

Note that if $\rho_{\lambda}(x,t)$ obeys a modulus of continuity $\omega$, then $\rho(x,t)$ obeys the modulus of continuity

[TABLE]

The proof of the global regularity for the solutions of (4.52)-(4.54) is based on the following lemma.

Lemma 4.6**.**

Let $\omega$ and $\omega_{\lambda}$ be as in (4.1) and (4.59), respectively. Given a smooth periodic initial condition $(\rho_{0},u_{0})$ for (4.52)-(4.54) and $T>0$, there exist $\delta>0$, $\gamma>0$ and $\lambda>0$ so that $\rho(x,t)$ obeys the modulus of continuity $\omega_{\lambda}(\xi)$ for all $0\leq t\leq T$. The parameters $\delta$, $\gamma$ and $\lambda$ may depend on $\alpha,$ $\rho_{0},$ $u_{0},$ and $T$.

This will imply a uniform bound on $\|\partial_{x}\rho\|_{L^{\infty}}$ on $0\leq t\leq T$. As $T$ is arbitrary, this is sufficient for the global regularity of the solutions, according to (3.3). Note that $\rho(x,t)$ obeys $\omega_{\lambda}$ until a time $T$ if and only if $\rho_{\lambda}(x,t)$ obeys the modulus of continuity $\omega$ until the time $T_{\lambda}=\lambda^{-\alpha}T$, and this is what we will show. That is, given $\rho_{0}$ and $u_{0}$, and $T>0$, we will find $\lambda>0$, $\delta>0$ and $\gamma>0$ sufficiently small, so that (i) $\rho_{\lambda}(0,x)=\rho_{0}(\lambda x)$ obeys $\omega$, and (ii) $\rho_{\lambda}(x,t)$ obeys $\omega$ at least until the time $\lambda^{-\alpha}T$. The a priori bounds on $\rho(x,t)$ and $F(x,t)$ will play a crucial role in the proof.

As in the case $G\equiv 0$ considered above, we assume that a modulus of continuity $\omega$ of the form  (4.1), with some $\delta$ and $\gamma$, is broken by $\rho_{\lambda}$ at a time $t_{1}$, at some $x_{1},y_{1}\in{\mathbb{R}}$, in the sense of (4.3). If $\mathbb{T}=[0,L]$, then $\rho_{\lambda}$ is $\lambda^{-1}L$-periodic, and we can restrict our attention to $x_{1},y_{1}\in\mathbb{T}_{\lambda}:=\lambda^{-1}\mathbb{T}$. We also set

[TABLE]

and drop the time variable $t_{1}$ in the notation. We decompose as in (4.11):

[TABLE]

with the terms $R_{1}$ and $R_{2}$ coming from the contributions of $u_{\lambda}^{(1)}$ and $u_{\lambda}^{(2)}$ in (4.55). We treat $R_{1}$ as before:

[TABLE]

Note that $I$ and $II$ can be estimated exactly as before: first, as in (4.14), we have

[TABLE]

with a constant $C>0$ that does not depend on $\rho_{0}$ or $u_{0}$. The term $II$ can be bounded as in (4.17):

[TABLE]

with $A(\xi)$ defined in (4.15). The term $III$ is bounded slightly differently from (4.18)

[TABLE]

Here, $\rho_{m}^{(\lambda)}(T)$ is the minimum of $\rho_{\lambda}(x,t)$ over $0\leq t\leq\lambda^{-\alpha}T$, and $D_{1}(\xi)$ is defined in (4.16). The lower bound (2.30) in Lemma 2.4 implies that

[TABLE]

as follows from (4.58). That is, even though now, unlike in the special case $G\equiv 0$, the function $\rho(x,t)$ does not necessarily obey the minimum principle, and $\rho_{m}(t)$ may decrease in time, the value of $\rho_{m}^{(\lambda)}(t)$ does not depend on $\lambda>0$. Thus, we may first choose the parameters $\delta$ and $\gamma$ in the definition (4.1) of the modulus of continuity $\omega$ so that (4.22) and (4.24) hold with $m$ replaced by $\bar{\rho}_{m}(T)$, and, in addition, they satisfy (4.30). Next, we choose $\lambda$ sufficiently small, so that $\rho_{\lambda}^{0}(x)=\rho_{0}(\lambda x)$ obeys the modulus of continuity $\omega$ with the above choice of $\delta$ and $\gamma$.

It remains to take into account the contribution of $u_{\lambda}^{(2)}$ to the right side of (4.61). The goal is to control the corresponding terms in (4.11) by the dissipation, namely, to show that

[TABLE]

Note that the flow $u_{\lambda}^{(2)}(x)$ is Lipschitz, as

[TABLE]

with a constant $C_{0}$ that depends on the initial conditions $\rho_{0}$ and $u_{0}$ but not on $\lambda>0$. Therefore, $u_{\lambda}^{(2)}$ obeys the modulus of continuity

[TABLE]

and the first term in (4.3) can be bounded by

[TABLE]

Let us recall from (4.1) and (4.16) that

[TABLE]

hence, we have

[TABLE]

provided that $\delta$ and $\lambda$ are sufficiently small. On the other hand, we see from (4.1) and (4.16) again that

[TABLE]

It is also straightforward to check that $D_{1}(\xi)$ is decreasing for $\xi>\delta$, provided that

[TABLE]

with a sufficiently small constant $c>0$ that depends only on $\alpha$. We also have

[TABLE]

Hence, taking $\lambda$ sufficiently small, depending on $L$ as well, we have the inequality

[TABLE]

for $\delta\leq\xi\leq L\lambda^{-1}$. Together, (4.72) and (4.76) show that

[TABLE]

For the second term in (4.3), we write

[TABLE]

with a constant $C_{0}$ that depends only on the initial condition $\rho_{0}$ and $u_{0}$. Then, for $\lambda$ sufficiently small, we have, once again using the fact that $\omega(\xi)/\xi^{\alpha}$ is decreasing for $\xi>\delta$ and (4.75):

[TABLE]

To bound $R_{22}$ in the region $0\leq\xi\leq\delta$, we write

[TABLE]

Lemma 2.3 guarantees that $F$ is Lipschitz, and the Lipschitz bound is uniform in time, thus (4.58) implies

[TABLE]

with a constant $C_{0}$ that depends only on the initial conditions. In addition, it follows from (4.58) that

[TABLE]

Inserting the last two bounds in (4.80), together with the expression for $D_{1}(\xi)$ in (4.71), gives

[TABLE]

Here the constant $C_{0}$ depends only on the initial conditions $\rho_{0}$ and $u_{0}$, and the second inequality holds provided that $\delta$ and $\lambda$ are sufficiently small. This proves (4.3), and finishes the proof of Lemma 4.6.

Let us recap the order in which we choose the parameters. The value of $\alpha$ is fixed throughout the argument. Given the initial data, we also fix its period, $L.$ We can also assume that $\lambda$ does not exceed one. Next we choose $\delta$ sufficiently small so that (4.22) (with $m$ replaced by $\bar{\rho}_{m}(T)$), (4.72), and (4.81) hold. Then we choose $\gamma$ so that (4.24) (with $m$ replaced by $\bar{\rho}_{m}(T)$), (4.30) and (4.74) hold. Finally, we choose $\lambda$ so that $\rho_{\lambda}(0,x)$ obeys $\omega$ with the above choice of $\delta,$ $\gamma$ and so that (4.76) and (4.79) hold. The proof of Theorem 1.1 is now complete. $\Box$

Appendix A The proof of a commutator estimate

In this section, we prove the commutator estimate (3.5),

[TABLE]

The proof is for $x\in\mathbb{R}^{n}$, though it can be easily adapted to periodic case. Let $(\chi,\eta)$ be smooth functions such that $\chi$ is supported in a ball $\{\xi~{}:~{}|\xi|\leq 4/3\}$, $\eta$ is supported in an annulus $\{\xi~{}:~{}3/4\leq|\xi|\leq 8/3\}$, and

[TABLE]

It is standard to take

[TABLE]

which we will assume. Denote the Littlewood-Paley decomposition of $f$ as $\sum_{q=-1}^{\infty}\Delta_{q}f$, where $\Delta_{q}f=(\eta(2^{-q}\xi)\hat{f}(\xi))^{\vee}$ for $q\geq 0$, and $\Delta_{-1}f=(\chi(\xi)\hat{f}(\xi))^{\vee}$. The Besov norm is defined as [3]

[TABLE]

Let the partial sum $S_{q}f=\sum_{p\leq q-1}\Delta_{p}f$. The Bony decomposition states

[TABLE]

where

[TABLE]

Proof of the commutator estimate.

First, we observe

[TABLE]

Therefore, it suffies to prove

[TABLE]

We apply the Bony decomposition to both terms, to get

[TABLE]

The terms $\text{I}_{1},\text{II}_{1},\text{I}_{3},\text{II}_{3}$ can be estimated with standard paraproduct calculus, sketched as follows.

[TABLE]

as $C^{\gamma+\epsilon}$ is embedded in $B^{\gamma}_{\infty,2}$. These terms are nicely controlled.

The commutator structure is mainly used to estimate $\text{I}_{2}-\text{II}_{2}$. Let us denote the difference as III. Given any $q\in\mathbb{N}$,

[TABLE]

Note that $\text{III}_{p}\equiv 0$ for $|p-q|\geq 5$. Therefore, it is a finite sum. We discuss $\text{III}_{q}$ and the other terms can be treated similarly.

Followed from [28], we estimate $\text{III}_{q}$ in the Fourier side,

[TABLE]

Define a multiplier $m(\xi,\zeta)$ as

[TABLE]

It is easy to check that $m$ is uniformly bounded, compactly supported and $C^{\infty}$. Let $m_{q}(\xi,\zeta)=m(2^{-q}\xi,2^{-q}\zeta)$, then

[TABLE]

where

[TABLE]

Compute

[TABLE]

where the last integral is bounded due to smoothness of $m$, and the constant $C$ does not depend on $q$. Then, applying Young’s inequality, we get

[TABLE]

We collect all modes and conclude

[TABLE]

∎