Zero Mach number limit of the compressible Euler–Korteweg equations

In this paper, we investigate the zero Mach number limit for the three-dimensional compressible Euler–Korteweg equations in the regime of smooth solutions. Based on the local existence theory of the compressible Euler–Korteweg equations, we establish a convergence-stability principle. Then we show that when the Mach number is sufficiently small, the initial-value problem of the compressible Euler–Korteweg equations has a unique smooth solution in the time interval where the corresponding incompressible Euler equations have a smooth solution. It is important to remark that when the incompressible Euler equations have a global smooth solution, the existence time of the solution for the compressible Euler–Korteweg equations tends to infinity as the Mach number goes to zero. Moreover, we obtain the convergence of smooth solutions for the compressible Euler–Korteweg equations towards those for the incompressible Euler equations with a convergence rate.


Introduction
In this paper, we are concerned with the three-dimensional compressible Euler-Korteweg system ⎧ ⎨ ⎩ ∂ t ρ + div(ρu) = 0, ∂ t (ρu) + div(ρu ⊗ u) + ∇p(ρ) = κρ∇ ρ, (1.1) for (x, t) ∈ Ω × [0, +∞). Throughout this paper, Ω is assumed to be the three-dimensional torus. Here, the unknown functions are the density ρ and the velocity u ∈ R 3 , p(ρ) is a given pressure function, and κ is the Weber number. This compressible Euler-Korteweg system results from a modification of the standard Euler equations governing the motion of compressible inviscid fluids through the adjunction of the Korteweg stress tensor, and arises as a mathematical model for a lot of phenomena in vortex dynamics, quantum hydrodynamics and hydrodynamics, e.g., flow of capillary fluids: liquid-vapor mixtures (for instance, highly pressurized and hot water in nuclear reactors cooling system), superfluids (for instance, helium near absolute zero), or even regular fluids at sufficiently small scales (for instance, ripples on shallow water or other thin films). We can see more details in [34] and [22] for the early developments of the theory of capillarity and, for instance, [21,35] for the derivation of the equations of motion. Note that when κ = 0, (1.1) reduces to the compressible Euler equations.
Recently, some results regarding the well-posedness of the compressible Euler-Korteweg system have been obtained. Benzoni-Gavage, Danchin and Descombes [6] addressed the well-posedness of the Cauchy problem for the Euler-Korteweg model in the one-dimensional case by reformulating the equations in Lagrangian coordinates. Benzoni-Gavage, Danchin and Descombes [7] also considered the multidimensional case in Eulerian formulation and established a blow-up criterion. Audiard [2] constructed a Kreiss symmetrizer and obtained the well-posedness of the Euler-Korteweg system in a halfspace. Audiard [3] obtained some dispersive smoothing effect of the Euler-Korteweg system both in one dimension and in higher dimensions. Audiard and Haspot [5] justified the global well-posedness of the multi-dimensional Euler-Korteweg equations for small irrotational initial data under a natural stability condition on the pressure. Giesselmann and Tzavaras [18] showed the weak-strong uniqueness, the large friction limit, and the vanishing capillarity limit of the Euler-Korteweg system under the relative energy framework. Audiard [4] obtained the existence of traveling waves for Euler-Korteweg equations with arbitrarily small energies in two dimensions and found that the standard for the linear instability of traveling waves implied nonlinear instability in one dimension.
Moreover, it is well known that the incompressible limit of compressible fluid dynamical equations is an important and challenging mathematical problem. Klainerman and Majda [23] first justified the convergence of the incompressible limit by using the partial differential equation method and singular limit approach of symmetric hyperbolic equations. Lin [26] proved the incompressible limit of the assumed weak solutions for the time-discretized compressible Navier-Stokes equations with big initial data by the uniform entropy-energy inequality. Hoff [19] proved that a compressible Navier-Stokes system with well-prepared initial data converged to an incompressible Navier-Stokes system as the Mach number goes to zero. Nevertheless, no smallness hypothesis is set on the external forces or on the initial data. Lions and Masmoudi [27] studied the incompressible limit of global weak solutions of compressible isentropic Navier-Stokes systems without size restrictions on the initial data. Desjardins and Grenier [13] researched the low Mach number limit for weak solutions of the compressible Navier-Stokes equations on the whole space by using a different method based on Strichartz's estimates for the linear wave equation, and gained better convergence results and simpler proof than former similar papers. Desjardins, Grenier, Lions and Masmoudi [14] investigated the limit of global weak solutions of the compressible isentropic Navier-Stokes equations in a bounded domain. They stated that the velocity of the compressible equations converged weakly to the global weak solution of the incompressible Navier-Stokes equations as the Mach number approached to 0, and the convergence became strong under certain geometrical assumptions on the domain. These results have been extended or improved by many others, e.g., the authors of Refs. [1,8,12,15,16,20,25,31,33].
To the best of our knowledge, no results about the incompressible limit of this Euler-Korteweg model can be found, except for Giesselmann [17], who gave a low Mach asymptotic-preserving scheme for the Euler-Korteweg model. In this paper, we analyze the incompressible limit of smooth solutions for the compressible Euler-Korteweg equations (1.1) with well-prepared initial data on the basis of the convergence-stability criterion, which was first formulated in [37]. Since then, this method has commonly been used in dealing with the singular limit of the partial differential equations. Yong [38] considered the zero Mach number limit of the smooth solution to the isentropic compressible Euler equations based on the convergence-stability principle. Li [24] presented the incompressible limit from the compressible MHD equations to ideal incompressible MHD equations under the framework of the convergence-stability principle. In fact, this approach is derived from singular perturbation theory [36], which is extensively used in the research of partial differential equations. For instance, Marin and Bhatti [29] studied the head-on collision model between capillary-gravity solitary waves using the singular perturbation method. This model was appropriate for shallow water waves and deep water waves and was investigated to find that the surface tension and the free parameter tended to remarkably decrease the solitary-wave profile. For similar methods, see [11,30].
The main difficulty in the analysis of this model, a third-order system of conservation laws, is the absence of dissipative regularization since the viscosity is neglected. To overcome this difficulty, we need more refined treatments, which is different from the compressible Navier-Stokes-Korteweg equation in [25]. See the treatments of the terms I 4 and H 3 in Sect. 5 for the difference. In addition, our approach in this paper is simpler than that adopted in previous work [23,28], and the requirements on the initial data and limit solution are fewer.
From a physical standpoint, when the density becomes almost constant, the velocity is very small, and we observe that at large time scales, the compressible fluid should act like the incompressible fluid. Therefore we introduce the following scaling: and assume that the capillarity coefficient κ is small and scaled as κ = εκ with ε ∈ (0, 1) a small parameter. With such scalings, the compressible Euler-Korteweg with the initial data Formally, letting ε → 0, we obtain from the momentum Eq. (1.2) 2 that ρ ε converges to a positive constant ρ * due to the periodic boundary conditions. Without loss of generality, let us assume that ρ * = 1. Then, passing to the limit in the mass conservation equation of (1.2), we obtain div u ε = 0. Therefore, by denoting the formal limits of ∇p(ρ ε )-κ ερ ε ∇ ρ ε ε 2 and u ε by ∇p 0 and u 0 , respectively, we can formally obtain the incompressible Euler equations: with the initial data To justify the above formal procedure, we first follow [25,38] and reformulate the compressible Euler-Korteweg equations (1.2) in terms of the pressure variable p ε = p(ρ ε ) and the velocity u ε . Assume that p(ρ ε ) is a smooth function with p (ρ ε ) > 0 for ρ ε > 0, then it has an inverse function ρ ε = ρ(p ε ). Set q(p ε ) = [ρ(p ε )p (ρ(p ε ))] -1 . Then the compressible Euler-Korteweg equations (1.2) for a smooth solution can be rewritten as with the initial data with p 0 = p(1) > 0. Further, we introducẽ Then (1.6) can be rewritten as with the initial datã

Main result
The main result can be stated as follows.
Moreover, there exists a constant K > 0, independent of ε but dependent on T 0 , such that In case T 0 = ∞, the maximal existence time T ε of (ρ ε , u ε )(x, t) tends to infinity as ε goes to zero.
can be relaxed as without changing our arguments. Here, we do not know whether the convergence rate in (2.1) is optimal, in particular, the velocity convergence rate. Using the arguments in [9], we will try to address this topic in the future. However, with the method here, we can obtain the sharp convergence rate (2.1), and no smallness condition on the initial data is required.
Remark 2.2 Here, we only consider the zero-Mach limit of the smooth solutions for the compressible Euler-Korteweg equations with well-prepared initial data. It is more interesting to consider the similar problem of the compressible Euler-Korteweg equations for general initial data (ill-prepared initial data). That is, we should take into account acoustic waves that propagate with the high speed 1 ε in the space domain, as in [23,28,33]. Moreover, we hope that similar results can be obtained for the limit of the compressible Euler-Korteweg equations in critical space. These issues are what our efforts should aim at in the future.
Recalling a local-in-time existence theory due to Benzoni-Gavage, Danchin and Descombes [7] for (1.2), we have the local-in-time existence of the classical solution to the compressible Euler-Korteweg equations (1.2) as follows.
From Theorem 2.2, we immediately have Theorem 2.1. In the following, we focus on the proof of Theorem 2.2.
Let us outline the idea of the proof as follows. On the basis of a local-in-time existence theory due to Benzoni-Gavage, Danchin and Descombes [7] for (1.2), we first establish a convergence-stability principle, which is similar to those developed in [36,37] for singular limit problems of symmetrizable hyperbolic systems. Thus, instead of deriving ε-uniform a priori estimates, we only need to make the error estimate (2.1) in the common time interval [0, min{T 0 , T ε }), where both solutions (ρ ε , u ε ) and (ρ 0 , u 0 ) are regular. Due to the third-order term and the absence of dissipative regularization in (1.1) or (1.2), deriving the error estimate requires some elaborated treatments. This is the difference from the compressible Navier-Stokes-Korteweg equation in [25]. See the treatments of the terms I 4 and H 3 in Sect. 5 for the difference.
The rest of this paper is organized as follows. In the next section, we make some preliminaries. That is, we give some notations and Moser-type calculus inequalities. Then, we prove the convergence-stability principle in Sect. 4. Finally, all required (error) estimates are obtained in Sect. 5.

Preliminaries
In this section, we mainly make some preliminaries.
Notation |U| denotes some norm of a vector or matrix U. For a nonnegative integer k, H k = H k (Ω) denotes the usual L 2 -type Sobolev space of order k. We write · k for the standard norm of H k and · for · 0 . When U is a function of another variable t as well as x ∈ Ω, we write U(·, t) to recall that the norm is taken with respect to x while t is viewed as a parameter. In addition, we denote by C([0, T], X) (resp. L 2 ([0, T], X)) the space of continuous (resp. square integrable) functions on [0, T] with values in a Banach space X.
In the subsequent proof process, we need the following Moser-type calculus inequalities in Sobolev spaces (refer to Proposition 2.1 in [28]).

(iii) Assume that g(u) is a smooth function on G, u(x) is a continuous function with
Here, | · | r,Ḡ 1 is the C r -norm on the setḠ 1 and C s is a generic constant depending only on s.

Convergence-stability principle
In fact, our proof of Theorem 2.2 is guided by the spirit of the convergence-stability principle developed in [36,37] for singular limit problems of symmetrizable hyperbolic systems. Fix ε > 0 in (1.8). According to Corollary 2.1, there is a time interval [0, T] such that the equations (1.8) with initial data (p,ū)(x, ε) have a unique solution (p ε ,ũ ε ) satisfying εp ε + p 0 > 0 for all (x, t) ∈ Ω × [0, T] and (Here, 2 can be replaced with any positive number larger than 1.) Namely, [0, T ε ) is the maximal time interval of H 5 × H 4 -existence. Note that T ε may tend to 0 as ε goes to 0.

Lemma 4.1 There exists T 0 ∈ (0, +∞) such that the IVP (4.2) of the incompressible Euler equation has a unique smooth solution
In the next section, we will prove the following theorem.
Having this theorem, we slightly modify the arguments in [25,36] to prove a theorem.

Theorem 4.2 Under the conditions of Theorem 2.1, there exists a constant
Proof Otherwise, there is a sequence {ε k } k≥1 such that lim k→∞ ε k = 0 and T ε k ≤ T 0 . Thanks to the error estimate in Theorem 4.1, (4.3) and Sobolev's inequality, there exists a k such that 4p ε k (x, t) ∈ (-3p 0 , 5p 0 ) for all x and t. Next, we deduce from  4 is bounded uniformly with respect to t ∈ [0, T ε k ). Now, we could apply Corollary 2.1, beginning at a time t less than T ε k (k is fixed here), to continue this solution beyond T ε k . This contradicts the definition of T ε k in (4.1). Finally, Theorem 2.2 is proved by combining Theorem 4.1 and Theorem 4.2.
We conclude this section with the following interesting remark, which is a by-product of our approach.
Let α be a multi-index with |α| ≤ 4. Differentiating the two sides of Eqs. (5.4) and (5.5) with ∂ α x and setting and Taking the inner product of (5.7) and (5.8) with q(p 0 + εp ε )P α and ρ(p 0 + εp ε )U α , respectively, and summing up the two resultant equalities gives Here and below, we often usẽ To estimate the I i , we first have the bounds of ρ (p 0 + εp ε ) and q (p 0 + εp ε ) as follows.
Proof Because p(ρ ε ) is a smooth function with p (ρ ε ) > 0 and has an inverse function ρ ε = ρ(p ε ), from the smoothness of ρ and 1 2 p 0 ≤ p 0 + εp ε ≤ 2p 0 , it is easy to see that there are positive constants c 1 and c 2 such that Moreover, from the definition of q, similarly, we have This completes the proof.
Now we turn to estimating the I i in (5.9). Using integration by parts and Lemma 5.1 and Lemma 5.2, we deduce that with the help of (1.8) 1 .
we have (5.14) Moreover, due to x , ρ (p 0 + εp ε )]∇P + ρ (p 0 + εp ε )∇P α , here and in the following, δ is a proper positive constant, which is determined. Moreover, with the help of (5.1) 1 and using Lemma 4.1, Lemma 3.1, and Lemma 5.1-5.2, we can obtain Similarly, using (5.4), Lemma 4.1, Lemma 3.1, and Lemma 5.1-5.2, we can obtain Therefore, substitution of the above inequalities and (5.15) into (5.12) yields Finally, from (5.6), we deduce that Hence, putting the estimates of I i (i = 1, 2, . . . , 5) into (5.9), we have To control the term with δ, we multiply (5.8) by ερ(p 0 + εp ε )∇P α and integrate the resultant equality by parts over Ω to obtain Here, we have used the fact that the initial data are in equilibrium. Furthermore, we apply Gronwall's inequality (refer to Theorem 6.2 in [10]) to (5.20) to obtain Applying the nonlinear Gronwall-type inequality (refer to Lemma 6.3 in [36]) to the last inequality yields