The global classical solution to compressible Euler system with velocity alignment

1.   Introduction

In this paper, we study the following Cauchy problem

in $(0, \infty)\times \mathbb{R}^N$ and the initial conditions

where $\rho$ and $u$ are the unknown density and velocity, and the pressure $p(\rho) = A\rho^{\gamma}.$ The matrix is ${\mathbf \Gamma}(x)\in L^1(\mathbb{R}^N)$. The constants $A, \gamma\geq1, \tau > 0$ are given. For simplicity, it is assumed that $A = 1$. This compressible Euler type system can be formally derived, for example in ^[11], from mean field type interacting many particle system where the particle velocities are also involved in the interacting force. In the intermediate step, from mesoscopic kinetic model to Fluid dynamic model, one can choose different closure Ansatz of the probability density to obtain the system with or without pressure term. For example, by taking the density to be a localized Gaussian function, one obtains a linear pressure in ^[7]. In this paper, we consider the case with nonlinear pressure, which can be obtained by taking the probability density as an indicator function in the velocity space.

For quasi-linear hyperbolic systems, intensive studies have been carried out in numerous literature, such as ^{[17,19,21,23]}, and etc. It has been proved in general that for symmetrizable hyperbolic systems, smooth solutions exist locally in time and shock waves formation will breakdown the smoothness of the solutions in finite time even for the scalar case, see ^[21]. However, if some damping terms are taken into account, shock waves can be avoided for small perturbation of the diffusion waves ^[10,15]. Moreover, with the help of velocity damping, the existence and uniqueness of the global classical solution can be obtained, see for example ^{[16,24,25,26,27]}.

The pressureless Euler system with non-local forces has been studied recently and very limited results have been done. The local existence of classical solutions of the complex material flow dynamics which has been derived in ^[11], under structural condition for the interaction force has been obtained in ^[6]. In ^[3], for the 1-D model with damping and non-local interaction, a critical threshold for the existence of classical solution by using the characteristic method is presented. 1-D entropy weak solution for Cucker-Smale type interaction has been obtained with the help of the compensated compactness argument in ^[12]. The global existence of smooth solutions with small initial data for the model with velocity alignment can be found in ^{[2,9,13,14,18]}. In these references, the influence function of velocity alignment is $\mathbf \Gamma (x) = \phi(|x|)\mathbf I_{N\times N}$, where $\phi(|x|)$ has a positive lower bound and $\mathbf I_{N\times N}$ is the identity matrix, with which the formation of shock can be prevented. In case that additional pressure and viscosity are added, for restrictive interaction potentials, the global weak solution and its long time behavior are obtained in ^[5]. By adapting the convex integration method, it has been shown in ^[4] that infinitely many weak solutions exist. Recently, the global existence of classical solutions for the hydrodynamic model with linear pressure term and non-local velocity alignment was given in ^[7], where the shock wave was prevented by velocity alignment.

In many models, the communication weight matrices have different structures. Many of them do not need to be positive definite, as for example in the material flow model that has been proposed in ^[11], the interaction force includes $\mathbf\Gamma = \frac{x}{|x|}\otimes\frac{x}{|x|}$ as the weight of velocity alignment.

In our model, the influence function of velocity alignment $\mathbf \Gamma$ is a matrix which corresponds to a linear projection of the velocity field. Furthermore it is non-constant and not positive definite, which reflects the anisotropic non-local interaction within the system. Therefore, the velocity alignment alone can not prevent the formation of shock wave. In order to obtain the global existence of smooth solutions, the additional damping effect in the system is necessary. Additionally, the symmetry of the coefficients plays an important role in the analysis of the existence of smooth solutions when using the method of standard energy estimates. We will use sound speed to reconstruct Eqs $(1.1)$ and $(1.2)$ into symmetric hyperbolic equations. It should be pointed out that this method will make the term of velocity alignment more complicated. After a detailed analysis of the relationship between velocity alignment and damping, the anisotropic non-local interaction is overcome by using damping, and the existence of the global classical solution of problem $(1.1)$–$(1.2)$ is obtained.

Here, we introduce several notations used throughout the paper. For a function $u = u(x)$, $\|u\|_{L^p}$ denotes the usual $L^p(\mathbb{R}^N)$ -norm. We also set $C$ as a generic positive constant independent of $t$. For any non-negative integer $k, \ H^k : = H^k(\mathbb{R}^N)$ denotes the Sobolev space $= W^{k, 2}(\mathbb{R}^N)$, and $C^k(I; E)$ is the space of $k$-times continuously differentiable functions from an interval $I \subset \mathbb{R }$ into a Banach space $E$. $\nabla^k$ denotes any partial derivative $\partial^\alpha$ with multi-index $\alpha$, where $|\alpha| = k$. For simplicity, we write $\int f\mathrm{d}x: = \int_{\mathbb{R}^N} f\mathrm{d}x$.

This paper is structured as follows: In Section 2, we reformulate the Cauchy problem $(1.1)$–$(1.3)$ into a symmetric hyperbolic system and present our main result. In Section 3, we demonstrate the local existence under uniqueness of the classical solution for the reconstructed system. Finally, we establish the priori estimates to prove the global existence result.

2.   Reformulation and main result

2.1.   Reformulation of the problem

In this subsection, we will reformulate the Cauchy problem of the compressible Euler system $(1.1)$–$(1.3)$ as in ^[24]. The main point is to obtain a symmetric system. We consider the case $\gamma > 1$ in this paper and introduce the sound speed:

where $\bar{\kappa} = \kappa(\bar{\rho})$ is set to the sound speed at a background density $\bar{\rho} > 0$. The symmetrization in the case of $\gamma = 1$ can be done similarly with a new variable $\ln \rho$.

Define

Then the Eqs $(1.1)$ and $(1.2)$ are transformed into the following system:

where the constants are $a = \gamma^{-1/{(\gamma-1)}} > 0$. The initial condition (1.3) becomes

with $\sigma_0 = \nu(\kappa(\rho_0)-\bar{\kappa})$. Note that as we did at the formal level, we can find the relation between the classical solutions $(\rho, u)$ and $(\sigma, u)$ to the systems $(1.1)$–$(1.2)$ and (2.1)–(2.2), respectively, in the following two lemmas. The proofs can be obtained by taking the similar strategy as in ^[24].

Lemma 1. For any $T > 0$, if $(\rho, u)\in C^1(\mathbb{R}^N\times [0, T])$ is a solution of system (1.1)–(1.2) with $\rho > 0$, then $(\sigma, u))\in C^1(\mathbb{R}^N\times [0, T])$ is a solution for the system (2.1)–(2.2) with $(\frac{1}{\nu} \sigma+\bar{\kappa})^\nu > 0$. Conversely, if $(\sigma, u))\in C^1(\mathbb{R}^N\times [0, T])$ is a solution for the system (2.1)–(2.2) with $(\frac{1}{\nu} \sigma+\bar{\kappa})^\nu > 0$, then $(\rho, u)\in C^1(\mathbb{R}^N\times [0, T])$ is a solution of system (1.1)–(1.2) with $\rho > 0$.

Lemma 2. For any $T > 0$, if $(\rho, u)\in C^1(\mathbb{R}^N\times [0, T])$ is a uniformly bounded solution of system (1.1)–(1.2) with $\rho_0 > 0$, then $\rho > 0$ on $\mathbb{R}^N\times [0, T]$. Conversely, if $(\sigma, u))\in C^1(\mathbb{R}^N\times [0, T])$ is a uniformly bounded solution of system (2.1)–(2.2) with $(\frac{1}{\nu} \sigma_0+\bar{\kappa})^\nu > 0$, then $(\frac{1}{\nu} \sigma+\bar{\kappa})^\nu > 0$ on $\mathbb{R}^N\times [0, T]$.

2.2.   Main result

In the subsection 2.1, we have presented the equivalent reconstruction system (2.1)–(2.2) of the problem (1.1)–(1.2) and the equivalence relation between them. Next, we will study the reconstructed system (2.1)–(2.2) and provide the following results.

Theorem 1. (Local-in-time exitence) For $s > \frac{N}{2}+1$, assume the initial values $(\sigma_0(x), u_0(x))\in H^{s}(\mathbb{R}^N).$ Then there exist a unique classical solution $(\sigma, u)$ of the Cauchy problem (2.1)–(2.3) satisfying

for some finite $T > 0$.

Theorem 2. (Global-in-time exitence) Suppose background sound speed $\bar{\kappa}$ satisfying $2a\bar{\kappa}^\nu\|\mathbf \Gamma\|_{L^1} < \frac{1}{\tau}$. If $\|\sigma_0\|_{H^s}+\|u_0\|_{H^s}\leq \delta_0$ with sufficiently small $\delta_0 > 0$, then the Cauchy problem (2.1)–(2.3) has a unique global classical solution.

Remark 1. Since the matrix $\mathbf \Gamma(x)$ is not positive definite, the damping coefficient needs to be large enough to make the damping term restrain the self-acceleration effect caused by velocity alignment to get the global well-posedness. The definition of condition $2a\bar{\kappa}^\nu\|\mathbf \Gamma\|_{L^1} < \frac{1}{\tau}$ is therefore natural.

3.   Local existence and uniqueness

In this section, we demonstrate the local existence and uniqueness of the classical solutions to (2.1)–(2.3). We will present a successive iteration scheme to construct approximate solutions and to obtain the energy estimates. Then we show that approximate solutions are convergent in Sobolev spaces using the contraction mapping principle and prove that the limit function is the local solution.

3.1.   Approximate solutions

We construct approximate solution by the following iterative method:

● the zeroth approximation: $(\sigma^0, \ u^0)(x, \ t) = (\sigma_0, \ u_0)$;

● Suppose that the $k$th approximation $(\sigma^k, \ u^k)(x, \ t), \ k\geq1$ is given. Then define the $(k+1)$th approximation $(\sigma^{k+1}, \ u^{k+1})(x, \ t)$ as a solution of the linear system

with the initial data

The local existence of the solutions $(\sigma^{k+1}, u^{k+1})$ in Sobolev spaces can be obtained by applying the linear theory of the multi-dimensional hyperbolic equations in ^[1].

3.2.   Priori estimates

We first set up several constants:

and choose $T_0 > 0$ so that

where $C(M, \nu, \bar{\kappa})$ is given in the proof of Lemma 3 below.

Lemma 3. Let ${(\sigma^k, \ u^k)}$ be a sequence of the approximate solutions generated by (3.1)–(3.2) together with the initial step $(\sigma^0, u^0) = (\sigma_0, u_0)$. Then the following estimate holds

where $s > \frac{N}{2}+1$, $M$ and $T_0$ are given in $(3.4)$ and $(3.5)$.

Proof. We use the method of induction to prove the Lemma.

Step 1. (Initial step) Because we choose $(\sigma^0, u^0) = (\sigma_0, u_0)$, together with the choice of $M, \ T_0 > 0$ in $(3.4)$ and $(3.5)$, it is easy to check that

Step 2. (Inductive step) Suppose that

where $T_0, \ M$ are positive constants determined in $(3.4)$ and $(3.5)$. We will prove that

First, multiplying $\sigma^{k+1}, \ u^{k+1}$ on both sides of (3.1), (3.2) respectively, summing up and integrating over $\mathbb{R}^N$, we obtain

We shall estimate the terms on the right-hand side of (3.8). Thanks to the Sobolev embedding theorem and the inductive assumption (3.7), using integration by parts, we obtain

Combining the estimate of $I_i, \ i = 1, \ 2, \ 3$, we obtain

Next we will get the higher order estimate of $(\sigma^{k+1}, u^{k+1})$.

Taking $\nabla^{r}, 1\leq r\leq s$ with respect to $x$ on both sides of $(3.1)$–$(3.2)$, and then multiplying the resulting identities by $\nabla^r \sigma^{k+1}, \ \nabla^r u^{k+1}$ respectively, summing up and integrating over $\mathbb{R}^N$, we obtain

In the following we will estimate $I_i$ term by term. Using the Sobolev embedding theorem and Moser type inequality, we obtain

where we have used $\|\nabla u^{k+1}\|_{L^{\infty}}\leq\|\nabla u^{k+1}\|_{H^{s-1}}$ and the inductive assumption (3.7).

Next, we estimate the $I_3$. Using Young's inequality and Moser type inequality, we have

Applying the Sobolev embedding theorem and the inductive assumption (3.7), direct calculation shows

where $C(M, \nu, \bar{\kappa})$ is non-decreasing in $M$.

Then, we obtain that

Finally, we provide the estimate of $I_4$. By applying the Moser type inequality and Young's inequality, we have

Here, we used (3.16) and the inductive assumption (3.7).

Collecting all estimates of $I_i$ from 1 to 4, we obtain that

We can sum (3.19) over $1\leq r\leq s$ and combine (3.9) to obtain

This yields

By the choise of $M$ and $T_0$ as in $(3.4)$ and $(3.5)$, we can easily check that

So, we obtain

which completes the induction process.

3.3.   Convergence in lower-order norm

In this subsection, we will show that the $\{\sigma^k, \ u^k\}_{k = 1}^{\infty}$ are convergent in some lower-order Sobolev spaces using the contraction mapping principle.

Let

Note that $(\sigma^{k+1}, \ u^{k+1})$ and $(\sigma^{k}, \ u^{k})$ satisfy

as well as

subject to the same initial data

It follows from $(3.22)$ and $(3.23)$ that

Multiplying (3.25) and (3.26) by $(\sigma^{k+1}- \sigma^k)$, $(u^{k+1}- u^k)$ respectively, summing up and integrating over $\mathbb{R}^N$, similar to the estimate in subsection 3.2, we obtain

where we use the following estimate, for $a.e.(t, x)\in (0, +\infty)\times \mathbb{R}^N$

We can integrate (3.27) over $(0, \ t)$ to obtain

The we sum up for $k = 1, \ 2, \ \cdots$ together with the Gronwall's inequality to obtain

This implies that ${ \sigma^k}$ and ${u^k}$ are Cauchy sequences in $C([0, T_0];\ L^2)$.

3.4.   The proof of Theorem 1

In this subsection, we will prove the local well-posedness of the system (2.1)–(2.2) given in Theorem 2.1. First, we prove the existence of classical solutions.

By the Gagliardo-Nirenberg inequality together with the uniform bound of $(\sigma^k, \ u^k)$ and the convergence result (3.30), we can conclude that, for $s > \frac{N}{2}+1$,

It easily follows from (3.31) that limit function $(\sigma, \ u)$ is a solution to (2.1)–(2.2) in a distributional sense. Using a similar argument as in ^[13], we can obtain the regularity of $(\sigma, \ u)$:

Applying Sobolev's embedding theorem, we prove $(\sigma, \ u)\in C^{1}([0, \ T_0]\times \mathbb{R}^N)$ is a classical solution.

Next, we prove the uniqueness. Let $(\sigma, \ u)$ and $(\widetilde{ \sigma}, \ \widetilde{u})$ be the two classical solutions of (2.1)–(2.2) corresponding the same initial data $(\sigma_0, \ u_0)$. We set

Then, by the same argument as in subsection 4.1, $U(t)$ satisfies Gronwall's inequality:

This yields that

So, we complete the proof of theorem.

4.   Global existence of classical solution

In this section, we discuss the global existence of the classical solution on the basis of the local existence results in Section 3. According to Remark 1, we assume that the background density and the bottom viscous damping satisfy

4.1.   A priori esimates

In this subsection, we will provide the a priori estimates for the Cauchy problem (2.1)–(2.3). Hence, we assume a priori assumption that for $\mathbf{s>}\frac{\mathbf{N}}{\mathbf{2}}\mathbf{+1}$ and a sufficiently small $\delta > 0$,

We show the $L^2$-norm estimates which contains the dissipation estimate for $u$. It should be noticed that there is no dissipation estimate of $\|\sigma\|_{L^2}$.

Lemma 4. Assume $(\sigma, u)$ are classical solution of (2.1)–(2.2) and $(4.1), (4.2)$ hold, then we have

Proof. We multiply (2.1) and (2.2) by $\sigma$ and $u$ respectively, sum up and integrate over $\mathbb{R}^N$, we obtain

We estimate $I_i$ item by item. Using Young's inequality, we have

Similar to (3.28), we can get

with the help of the Sobolev embedding theorem. Then, we obtain that

Collecting estimates (4.5)–(4.8) into (4.4), we obtain (4.3).

Next, we provide the high order energy estimates which contains the dissipation estimate for $u$.

Lemma 5. Assume $1\leq r \leq s$ and $(4.1), (4.2)$ hold, then for $\mathbf{s>}\frac{\mathbf{N}}{\mathbf{2}}\mathbf{+1}$, we have

Proof. For $1\leq r\leq s$, we apply $\nabla^r$ to (2.1), (2.2), and multiply the resulting identities by $\nabla^r \sigma, \ \nabla^r u$ respectively, sum up and integrating over $\mathbb{R}^N$ to obtain

Similar to the estimate of (3.12) and (3.13) in Section 3, by H${\rm\ddot{o}}$lder's inequality and Moser type inequality, we have

Next, we estimate $I_3$. Applying Moser type inequality and the H${\rm \ddot{o}}$lder inequality we have

To deal with the dissipation of $u$, we need the following estimates. Similar to (3.28), we can get

Using the differential properties of the convolution and the Sobolev embedding theorem, we can compute

and similar to (3.16), we have

where $C(\|\sigma\|_{L^{\infty}}, \nu, \bar{\kappa})$ is non-decreasing in $\|\sigma\|_{L^{\infty}}$.

Substituting (4.14)–(4.16) for (4.13), we obtain that

where the Sobolev embedding theorem is used.

Similar to estimate of $I_3$, we can deduce that

Collecting estimates (4.11), (4.12), (4.17), (4.18) and put them into (4.10), we obtain that

Now, we will bring forward the dissipation estimate for $\sigma$.

Lemma 6. For $1\leq r\leq s$,

Proof. First, we can directly calculate to obtain

Next, we will estimate the right-hand two terms of the upper equation. Let $1\leq r\leq s$, applying $\nabla^{r}$ to (2.1), multiplying it by $\nabla^{r-1}u$ and integrating over $\mathbb{R}^N$ we obtain

In order to get the estimate of (4.22), we estimate $\|\nabla^{r-1}(u\cdot\nabla \sigma)\|_{L^2}$ and $\|\nabla^{r-1}(\sigma\cdot\nabla u)\|_{L^2}$. By Moser type inequality and Sobolev embedding theorem, we have

Then, substituting (4.23) and (4.24) for (4.22) and applying Young's inequality, we can deduce that

Now, we estimate the second item on the right side of (4.21). Let $1\leq r\leq s$, applying $\nabla^{r-1}$ to (2.2), multiplying it by $\nabla^r\sigma$ and integrating over $\mathbb{R}^N$ we obtain

Using Young's and Holder's inequality, we have

By a method similar to the estimate for $\|\nabla^{r-1}(u\cdot\nabla \sigma)\|_{L^2}$ and $\|\nabla^{r-1}(\sigma\cdot\nabla u)\|_{L^2}$, we have

Next, we estimate $I_4$. Similar to (3.15) and (3.16), using Young's inequality and Moser type inequality, we have

Applying the same method as $I_4$, we can get

Collecting all estimates of $I_i$, we have

Combining (4.25) and (4.32) to obtain

4.2.   The proof of global existence

In this subsection, we construct the global-in-time solution by combining the local existence theory.

We sum up the estimate (4.9) in Lemma 5 form $r = 1$ to $s$, and then add the estimate (4.3) in Lemma 4, since $\delta$ is small and conditions (4.1), we can deduce that there exists $\varepsilon_1 > 0, C_1 > 0$ such that

Summing up the estimates (4.20) in Lemma 6 from $r = 1$ to $s$, for sufficient small $\delta$, there exist $\varepsilon_2 > 0, C_2 > 0$, such that

Multiplying (4.35) by $\dfrac{2C_1\delta}{\varepsilon_2}$, adding it to (4.34), since $\delta$ is small, then there exits a constant $\varepsilon_3 > 0$ such that

Note that there exist constant $C_3 > 0$ such that

Integrating directly in time, with help of (4.37), we obtain

Finally, we can use Theorem 1 and (4.38) to prove the global existence of classical solutions for (2.1)–(2.2). Applying the local existence of the classical solution in Theorem 1, we deduce that for any positive constant $\delta_0 < \delta$, there is a positive constant $T_0$ depending only on $\delta_0, \ \delta$ such that if $\|\sigma_0\|_{H^s}+\|u_0\|_{H^s} < \delta_0$, then the solution of the Cauchy problem (2.1)–(2.3) satisfies

Then, we choose that

where $\delta$ and $C_3^2$ are given in (4.2) and (4.38), respectively. Let us define the maximal existence time $T_{max} > 0$ of the system (2.1)–(2.2) by

Suppose $T_{max} < \infty$, then we can use the continuation argument and (4.38) to get

This is a contradiction, hence, we can conclude that $T_{max} = \infty$.

In summary, we have completed the proof of Theorem 2.2.

Acknowledgments

The publication of this article was funded by the Ministry of Science, Research and the Arts Baden-Württemberg and the University of Mannheim.

Lining Tong and Shu Wang would like to thank the University of Mannheim for the hospitality during their scientific visiting, where the main part of this article has been obtained.

Lining Tong is supported by NSFC(No. 11771274, 11901379). Simone Göttlich is supported by the German Research Foundation, DFG grant GO 1920/7-1. Shu Wang is supported by NSFC(No. 11831003, 11771031, 11531010).

Conflict of interest

The authors declare no conflict of interest.

Appendix

In the appendix, we present several lemmas used in the existence proof in Sections 3 and 4.

Lemma 7.(Moser type inequality) Let $s\geq 1$ and $1\leq r\leq s$, then for any pair of functions $f, \ g\in H^s\cap L^\infty$, we have

Furthermore if $\nabla f\in L^{\infty}(\mathbb{R}^N)$ we have

Proof. See Lemma 3.4 in ^[22].

Lemma 8. (Young's inequality) Let $p, q, r\geq1$ and $1/p+1/q+1/r = 2$. Let $f\in L^{p}, g\in L^{q}$ and $h\in L^{r}$. Then

Proof. See Theorem 4.2 in ^[20].

Lemma 9. (Gagliardo-Nirenberg inequality) Let $f\in W_{0}^{1, q}\cap L^r$ for some $r\leq1$. There exists a constant $C$ depending upon $N, P, r$ such that

where $\theta\in[0, 1]$ and $p, q\leq1$ are linked by

Proof. See Theorem 1.1 of the Chapter 10 in ^[8].