Compressible Euler equations interacting with incompressible flow

We investigate the global existence and large-time behavior of classical solutions to the compressible Euler equations coupled to the incompressible Navier-Stokes equations. The coupled hydrodynamic equations are rigorously derived in [1] as the hydrodynamic limit of the Vlasov/incompressible Navier-Stokes system with strong noise and local alignment. We prove the existence and uniqueness of global classical solutions of the coupled system under suitable assumptions. As a direct consequence of our result, we can conclude that the estimates of hydrodynamic limit studied in [1] hold for all time. For the large-time behavior of the classical solutions, we show that two fluid velocities will be aligned with each other exponentially fast as time evolves.

1. Introduction. In this paper, we are concerned with the global existence and large-time behavior of classical solutions to the compressible isothermal Euler equations coupled to the incompressible Navier-Stokes equations in the periodic domain T 3 . More precisely, let ρ(x, t) and u(x, t) be the density and the velocity of compressible fluid, respectively, and v(x, t) be the velocity of incompressible fluid at a domain (x, t) ∈ T 3 × R + . In this situation, our coupled hydrodynamic equations can be described as follows: Since the total mass is conserved in time, without loss of generality, we may assume that ρ is a probability density function, i.e., ρ(·, t) L 1 (T 3 ) = 1. We also assume that the viscosity coefficient µ = 1 for simplicity. In fact, a more general condition on the viscosity coefficient µ > 0 does not yield any difficulties for our analysis.

YOUNG-PIL CHOI
Recently, in [1], the author and his collaborators studied coupled kinetic-fluid equations with local alignment forces and rigorously derived the coupled hydrodynamic equations (1.1) when the noise and local alignment are strong enough. More specifically, let f = f (x, ξ, t) be the one-particle distribution function at a spatial periodic domain (x, ξ) ∈ T 3 × R 3 at time t and v = v(x, t) be the velocity of fluid, then the following kinetic-fluid equations are considered in [1]: where ρ ε := In [1], the authors showed that as long as there exists a unique classical solution to the system (1.1) the weak solutions (f ε , v ε ) to the system (1.3) satisfying a natural entropy inequality converge to (M ρ,u , v) where M ρ,u denotes the Maxwellian distribution with the density ρ and the velocity u as ε goes to zero, i.e., and v ε (x, t) → v(x, t) as ε → 0, where ρ and u are the ε → 0 limits of (1.4). Moreover, (ρ, u, v) solve the system (1.1). Without the interactions with the fluid, the rigorous hydrodynamic limit of the kinetic equations via relative entropy arguments and the global existence of classical solutions to the limiting system are investigated in [2,11]. For the coupling with other fluids without the local alignment force, global existence of weak solutions to Vlasov-Fokker-Planck/Navier-Stokes equations is treated in [14], and the global existence of classical solutions near equilibrium to Vlasov-Fokker-Planck/Euler equations is discussed in [3,7]. We also refer to [4,9,10,15] for the hydrodynamic limit of kinetic-fluid equations.
In this paper, we are interested in the drag forcing effect which comes from the coupling term ρ(u − v) in the system (1.1) on the regularity and large-time behavior of solutions. Without the interactions with the incompressible fluid, the system (1.1) 1 -(1.1) 2 becomes compressible Euler equations, and it is well-known that this system has the formation of singularities such as δ-shock in finite time no matter how smooth initial data are. Thus it is natural to consider the measure solutions for the global well-posedness. We refer the readers to [5] and references therein for the general survey of the Euler equations. The issue of development of the singularity presents new challenges to the global existence theory of classical solutions. For the global existence of the unique classical solution to the coupled system (1.1), we reinterpret the drag forcing term as the relative damping. To be more precise, we reduce the system (1.1) to a symmetric system (see (1.5) below), and show that the drag forcing term can prevent the formation of singularities in the compressible Euler equations if the initial data are small enough in an appropriate norm. Since the drag forcing term does not give the real damping effect, we carefully analyze the coupling term with the help of the viscous term in the incompressible Navier-Stokes equations. As a direct consequence of this result, we can conclude that the hydrodynamic limit studied in [1] holds for all time. For the large-time behavior, we employ a similar strategy which is recently proposed in [6] for the Vlasov/compressible Navier-Stokes equations. We construct a Lyapunov function measuring local variances of fluids around their local averages and the distance between local averaged velocities. We show the emergence of alignment between two fluid velocities as time evolves using our proposed Lyapunov function.
Before stating our main results on the global existence and large-time behavior of classical solutions, we introduce several simplified notations. For a function f (x), we denote by f L p the usual L p (T 3 )-norm. f g represents that there exists a positive constant C > 0 such that f ≤ Cg. We also denote by C a generic positive constant depending only on the norms of the data, but independent of t, and drop x-dependence of differential operators ∇ x , that is, ∇f := ∇ x f and ∆f := ∆ x f . For any nonnegative integer k, H k denotes the k-th order L 2 Sobolev space. C k ([0, T ]; E) is the set of k-times continuously differentiable functions from an interval [0, T ] ⊂ R into a Banach space E, and L p (0, T ; E) is the set of the L 2 functions from an interval (0, T ) to a Banach space E. ∇ k denotes any partial derivative ∂ α with multi-index α, |α| = k.
For the global existence of a unique classical solution, by setting n := ln ρ, we first reformulate the system (1.1) into the symmetric system as follows: (1.6) We notice that the Cauchy problem for the symmetric system (1.5) has a unique classical solution if and only if the system (1.1) has a unique classical solution (see Proposition 1 for details). From this observation, we prove the global existence of classical solutions to the system (1.5) under suitable assumptions on the initial data (1.6).
In order to state our second result on the large-time behavior of classical solutions, we define a total fluctuated energy function L(t) by where m c and v c are averaged quantities given by  (i) (ii) ρ ∈ [0,ρ] for someρ > 0, and u ∈ L ∞ (T 3 × R + ).
(iii) An initial total energy E 0 := Then we have where C is a positive constant independent of t.
Remark 1. 1. The smallness assumption on the initial data U 0 and v 0 is necessary. More precisely, we need the smallness of solution v in order to control the convection term in the incompressible Navier-Stokes equations (1.5) 3 -(1.5) 4 , and the smallness of solutions n and u are required to avoid the formation of singularity in compressible Euler equations (1. For the estimate of large-time behavior, we do not require that L ∞ (T 3 )-norms of ρ and u should be small, we need only the small total initial energy. We also notice that where we used |v c (0) · m c (0)| ≤ E 0 . Hence we have L 0 ≤ 3E 0 , and this implies that the initial total fluctuated energy function L 0 is also small.
3. The unique global classical solution obtained in Theorem 1.1 satisfies the assumptions in Theorem 1.2.
The rest of this paper is organized as follows. In Section 2, we provide a priori energy estimates for the system (1.1) and several useful lemmas. We also discuss the local existence of the unique classical solution and the relation between Cauchy problems (1.1)-(1.2) and (1.5)-(1.6). Section 3 is devoted to give the details of the proof for Theorem 1.1. We present the major energy estimates to extend the local existence to the global one. Finally in Section 4, we establish the largetime behavior of global classical solutions showing the alignment between two fluid velocities exponentially fast.

Preliminaries.
2.1. A priori energy estimates and useful inequalities. We first present basic energy estimates which show conservations of mass and total momentum, and a dissipation of total energy of the system (1.1). Lemma 2.1. Let (ρ, u, v) be any global classical solutions to the system (1.1)-(1.2). Then we have Proof. First we can easily obtain the estimates (i). For the estimate of (ii), we find We also notice that We next provide elementary estimates for the pressure of the compressible Euler equations.
Proof. A straightforward computation yields the desired result.
Then there exist positive constants c 1 , c 2 > 0 we have Then we can easily find that lim ρ→0 g(ρ) = 1 > 0 and lim Thus we deduce that g(ρ) is a continuous function on [0,ρ] with g(ρ) > 0, and this completes the proof.
We set In the following two lemmas, we recall Sobolev inequalities and provide an equivalence relation between ln f L 2 and f − 1 L 2 under suitable assumptions.
where c 3 and c 4 depend only on a and b, respectively.
Thus we combine these two estimates to obtain and this implies This completes the proof.

Local existence.
Theorem 2.6. Let s > 5 2 , and suppose . Then there exists a positive constant T 0 > 0 such that the system (1.5)-(1.6) admits a unique solution U ∈ C([0, Proof. Since local existence theories for a type of conservation laws have been well developed, we omit the proof here. We refer to [12,13] for the readers who are interested in it. The proof of the following proposition is straightforward, and the positivity of the density in the proposition below is obtained from the corresponding positivity of the initial density by using the method of characteristics. For more details, we refer to [16]. 3.1. A priori estimates. In this part, we present the a priori estimates for the global existence of the classical solutions to the system (1.5)-(1.6). For this, we first introduce a norm W m (f, g) for f, g ∈ H m (T 3 ) which is recursively defined by 11) for m ≥ 1. Then it is clear to get W m (f, g) ≈ f 2 H m + g 2 H m in the sense that there exists a positive constant C > 0 such that for some 0 < 1 1. Then we have where δ 0 is a positive constant.
This yields Similarly, we find where J i , i = 1, · · · , 4 are estimated by Here we used
Combining (3.19) and (3.20), we have and this completes the proof.
We next provide the estimates of U t and v t . Note that the regularities of U t and v t are different. More precisely, we will estimate U t H s and v t H s−1 in the lemma below.
Lemma 3.2. Let T > 0 be given and s > 5 2 . and (U, v) is a solution to the system (1.5)-(1.6). Furthermore we assume where δ 0 is the positive constant determined in Lemma 3.1.
Proof. By differentiating (1.5) with respect to t, we find (3.21) For the zeroth-order estimates, we multiply (3.21) 1 and (3.21) 2 by n t and u t , respectively, and integrating over T 3 to obtain where I i , i = 1, · · · , 5 are estimated by Similar fashion to the above, we also find (3.23) We now combine (3.22) and (3.23) to get For the high-order estimates, we take ∇ k -derivatives of the (3.21) for 1 ≤ k ≤ s to find (3.25) Then one can obtain that where J i , i = 1, · · · , 7 are estimated as follows.
This yields (3.26) For the estimate of v, it follows from (3.25) that We estimate K i , i = 1, · · · , 6 as follows.
where we used the following estimates for K 4 with the help of Lemma 2.4 and e n n t L ∞ n t L ∞ ≤ C 1 .

Thus we obtain
Combining (3.26) and (3.27), we get Then we again consider W s−1 (U t , v t ). Using the similar arguments in the proof of Lemma 3.1, we find from (3.24) and (3.28) that This concludes the desired result.
In the following lemma, we provide the relation between (n t , ∇n) H s and (u t , ∇u) H s .

Furthermore we have
Proof. It follows from (1.5) that Then we easily find that for 0 ≤ k ≤ s and this yields that Similarly, we also obtain We next show the estimates of upper bounds of (n, v) L 2 and v t L 2 .
Proof. (i) We first notice from Remark 3 that (3.31) We also find from Lemma 2.5 and Remark 3 that (3.32) Then we now combine (3.31) and (3.32) to conclude the proof of (i).
(ii) It follows from (1.5) 3 that This yields where C is a positive constant independent of t.
Proof. We combine the two differential inequalities obtained in Lemmas 3.1 and 3.2 to find where H(U, U t , v, v t ) is given by Note that