The global classical solution to compressible Euler system with velocity alignment

In this paper, the compressible Euler system with velocity alignment and damping is considered, where the influence matrix of velocity alignment is not positive definite. Sound speed is used to reformulate the system into symmetric hyperbolic type. The global existence and uniqueness of smooth solution for small initial data is provided.


Introduction
In this paper, we study the following Cauchy problem ∂ t ρ + ∇ · (ρu) = 0, (1.1) in (0, ∞) × R N and the initial conditions where ρ and u are the unknown density and velocity, and the pressure p(ρ) = Aρ γ . The matrix is Γ(x) ∈ L 1 (R N ). The constants A, γ ≥ 1, τ > 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 Γ(x) = φ(|x|)I N×N , where φ(|x|) has a positive lower bound and I N×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 Γ = x |x| ⊗ x |x| as the weight of velocity alignment. In our model, the influence function of velocity alignment Γ 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 (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 (R N ) denotes the Sobolev space = W k,2 (R N ), and C k (I; E) is the space of k-times continuously differentiable functions from an interval I ⊂ R into a Banach space E. ∇ k denotes any partial derivative ∂ α with multi-index α, where |α| = k. For simplicity, we write f dx := R N f dx . 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.

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 γ > 1 in this paper and introduce the sound speed: κ(ρ) = p (ρ), whereκ = κ(ρ) is set to the sound speed at a background densityρ > 0. The symmetrization in the case of γ = 1 can be done similarly with a new variable ln ρ. Define Then the Eqs (1.1) and (1.2) are transformed into the following system: where the constants are a = γ −1/(γ−1) > 0. The initial condition (1.3) becomes . Note that as we did at the formal level, we can find the relation between the classical solutions (ρ, u) and (σ, 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].

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.
for some finite T > 0. Remark 1. Since the matrix Γ(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 2aκ ν Γ L 1 < 1 τ is therefore natural.

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.

Approximate solutions
We construct approximate solution by the following iterative method: • the zeroth approximation: (σ 0 , u 0 )(x, t) = (σ 0 , u 0 ); • Suppose that the kth approximation (σ k , u k )(x, t), k ≥ 1 is given. Then define the (k + 1)th approximation (σ k+1 , u k+1 )(x, t) as a solution of the linear system The local existence of the solutions (σ k+1 , u k+1 ) in Sobolev spaces can be obtained by applying the linear theory of the multi-dimensional hyperbolic equations in [1].

Priori estimates
We first set up several constants: and choose T 0 > 0 so that where C(M, ν,κ) is given in the proof of Lemma 3 below.
Lemma 3. Let (σ k , u k ) be a sequence of the approximate solutions generated by (3.1)-(3.2) together with the initial step (σ 0 , u 0 ) = (σ 0 , u 0 ). Then the following estimate holds where s > 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 (σ 0 , u 0 ) = (σ 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 sup 0≤t≤T 0 First, multiplying σ k+1 , u k+1 on both sides of (3.1), (3.2) respectively, summing up and integrating over 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 Next we will get the higher order estimate of (σ k+1 , u k+1 ). Taking ∇ r , 1 ≤ r ≤ s with respect to x on both sides of (3.1)-(3.2), and then multiplying the resulting identities by ∇ r σ k+1 , ∇ r u k+1 respectively, summing up and integrating over 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 ∇u k+1 L ∞ ≤ ∇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 (3.14) Applying the Sobolev embedding theorem and the inductive assumption (3.7), direct calculation shows where C(M, ν,κ) 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 ≤ r ≤ s and combine (3.9) to obtain This yields So, we obtain σ k+1 H s + u k+1 H s ≤ M (3.21) which completes the induction process.

Convergence in lower-order norm
In this subsection, we will show that the {σ k , u k } ∞ k=1 are convergent in some lower-order Sobolev spaces using the contraction mapping principle. Let Note that (σ k+1 , u k+1 ) and (σ k , u k ) satisfy It follows from (3.22) and (3.23) that Multiplying (3.25) and (3.26) by (σ k+1 − σ k ), (u k+1 − u k ) respectively, summing up and integrating over R N , similar to the estimate in subsection 3.2, we obtain where we use the following estimate, for a.e.(t, x) We can integrate (3.27) over (0, t) to obtain The we sum up for k = 1, 2, · · · together with the Gronwall's inequality to obtain This implies that σ k and u k are Cauchy sequences in C([0, T 0 ]; L 2 ) .

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.
So, we complete the proof of theorem.

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

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 s > N 2 + 1 and a sufficiently small δ > 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 σ L 2 .
Proof. We multiply (2.1) and (2.2) by σ and u respectively, sum up and integrate over R N , we obtain 1 2 We estimate I i item by item. Using Young's inequality, we have , (4.5) 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 ≤ r ≤ s and (4.1), (4.2) hold, then for s > N 2 + 1, we have 1 2 Proof. For 1 ≤ r ≤ s, we apply ∇ r to (2.1), (2.2), and multiply the resulting identities by ∇ r σ, ∇ r u respectively, sum up and integrating over R N to obtain Similar to the estimate of (3.12) and (3.13) in Section 3, by Hölder's inequality and Moser type inequality, we have Next, we estimate I 3 . Applying Moser type inequality and the Hölder inequality we have To deal with the dissipation of u, we need the following estimates. Similar to (3.28), we can get (4.14) Using the differential properties of the convolution and the Sobolev embedding theorem, we can compute and similar to (3.16), we have where C( σ L ∞ , ν,κ) is non-decreasing in σ L ∞ . 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 σ.

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 δ is small and conditions (4.1), we can deduce that there exists ε 1 > 0, C 1 > 0 such that