Incompressible limit of Euler equations with damping

1.   Introduction

We are concerned with the 3D compressible Euler equations with frictional damping

Here, $x\in \mathbb{T}^3$, $t > 0$, the unknown functions $\rho$ and $u$ denote the density and velocity of the fluid respectively; the pressure-density function $p$ is given by

the constant $A > 0$ is the friction coefficient and $\varepsilon$ is the Mach number. The system can be used to model the compressible fluid flow through a porous media.

The compressible Euler equations with damping can be used to simulate the motion for the compressible gas flow through a porous medium. The medium induces a friction force, proportional to the linear momentum in the opposite direction. Usually, this model has many variations. For example, the friction coefficient $A$ may depend on time ^[1] and, more generally, one can consider the nonlinear damping ^[2]. Here, we only study the constant-coefficient linear damping case.

It is well-known that the compressible Euler equations will develop singularity in finite time for general initial data ^[3]. However, the damping effect will prevent the formation of singularities in small amplitude flows, but large solutions may still break down ^[31]. Due to strong physical background and significant mathematical challenge, system (1.1) with fixed Mach number has been studied by many researchers. The readers are referred to ^[30] for BV solutions and ^[4,5] for $L^\infty$ solution. For the existence of weak solutions, we refer to ^[6,7]. While for global classical solution, see ^{[8,9,10,11,12]}. We also refer to ^{[13,14,15,16,17,18]} and references therein for further studies in this direction and related problems.

In particular, the global-in-time existence and asymptotic behavior of the classical solution were obtained by Wang and Yang ^[11]. By employing the Green's function method and energy estimate, they proved that if the suitable Sobolev norm of the initial data is small, then the global existence of classical solution can be obtained, and the $L^2$ norm of the solution decays at the rate of $t^{-3/4}$ in the whole space.

Physically, the Mach number represents the ratio of the typical speed of the flow to the speed of sound. In practical applications, the incompressible equations are often used when the Mach number is sufficiently small. So it is natural to think, under appropriate conditions on the initial data, that solutions of the compressible system converge to the solution of the incompressible system when the Mach number goes to zero. This is the low Mach number limit problem in fluid mechanics.

Over the past four decades, many results have been obtained about the incompressible limit of fluid dynamic equations, which is a special case of the low Mach number limit. In their ground-breaking works ^[19,20], Klainerman and Majda setup a general framework for the study of singular limit of hyperbolic PDEs. With the general theory, they proved the incompressible limit of isentropic Euler equations in the well-prepared initial data case. For general initial data, Ukai ^[21] obtained the convergence of the solutions in the whole space. Finally by using the filtering method, Schochet ^[22] solved the incompressible limit problem for Euler equations in the torus case. Concerning the low Mach number limit of the non-isentropic Euler equations, we refer to ^[23,24,25].

Since these pioneering works, the mathematical analysis of the low Mach number limit to the isentropic Navier-Stokes equations also attracted a lot of attention. In the framework of weak solutions, the incompressible limit of Navier-Stokes equations in the periodic case and bounded domain was studied in ^[26] and ^[27] respectively. For global-in-time regular solutions, Bessaih ^[28] established the uniform estimates of the solutions with almost incompressible initial data and proved the convergence to the solution of incompressible system with no-slip boundary conditions. Ou ^[29] further extended this result to the slip-type boundary conditions.

For the local classical solutions, the incompressible limit of the system (1.1) can be established in a similar fashion as the compressible Euler equations. If the damping effect is considered, it is possible to study the incompressible limit of the system (1.1) for all time since we have global small solutions. To the best of our knowledge, there are no results studying the global incompressible limit of Euler equations with damping. Incompressible fluid flow differs from compressible fluid flow in that the continuity equation is replaced by the divergence-free condition on the velocity field. Mathematically, the low Mach number limit attempts to bridge the gap between those two different descriptions and, in some sense, it is relatively easy to study the incompressible system instead of the compressible equations.

In this paper, based on the previous global existence results of the Euler system with damping, we are going to study the global incompressible limit to system (1.1) in the framework of small amplitude solutions. Namely, we will show that as the Mach number goes to zero, the solutions to Eqs (1.1) will converge to the solution of the following system

where $\bar{\rho}$ is some constant and $\pi$ can be formally obtained by applying $\nabla\cdot$ to the first equation.

The rest of the paper is arranged as follows. In Section 2, we reformulate the Cauchy problem for Eq (1.1) into a symmetric hyperbolic system and state the global existence of classical solutions to the system with fixed Mach number. In Section 3, the global uniform estimates of the solution are obtained. Finally with the uniform estimates, we will prove the convergence of the solutions of the original equations (1.1) to the solution of the limit system in Section 4.

Notation. Throughout the paper $C$ will denote a positive constant whose value may be different in each occurrence, that may depend PDEs, domain, and Sobolev index, but are independent of the small parameter $\varepsilon$. The small letter $c$ and its variants denote similar constants whose value is fixed. $\partial_x^\alpha$ or simply $\partial^\alpha$ with multi-index $\alpha$ stands for the usual spatial derivatives. For any integer $s\geq0$, $H^s$ denote the inhomogeneous Sobolev space $H^s(\mathbb{T}^3)$ with the norm $\|\cdot\|_{H^s}$. We denote $\|\cdot\| = \|\cdot\|_{L^2}$ for simplicity. Generally, the solution is dependent on the small parameter $\varepsilon$ in this paper, so it is better to write the solution $u^\varepsilon(t, x)$ rather than $u(t, x)$ for example. But for simplicity, we always omit the superscript $\varepsilon$ and use $u(t, x)$ instead when there is no confusion.

2.   Reformulation of the problem

In this short section, we will give a reformulation of the problem. To simplify the presentation, we introduce

Using $h$ and $u$ as the new unknown functions, we get from the original equation (1.1) that

where

is a smooth functions of $h > 0$.

In this paper, we will consider the perturbative solutions of the above equations. Thus we choose the constant equilibrium state $(h^\ast, 0)$ with $h^\ast > 0$ and set

The system satisfied by $q$ and $u$ is

Initial data is given by

For fixed $\varepsilon > 0$, it has been proved by many authors, c.f. ^[11,31], that (2.3)–(2.4) admits a global classical solution provided that the certain Sobolev norm of the initial data is sufficiently small. Namely, we have the following global existence theorem.

Theorem 2.1. For any fixed $\varepsilon > 0$, suppose that the initial data $(q_0(x), u_0(x))\in H^s(\mathbb{T}^3)$ with $s\geq3$ and $\|(q_0(x), u_0(x))\|_{H^s(\mathbb{T}^3)}$ is sufficiently small. Then there exists a unique, global, classical solution $(q^\varepsilon(x, t), u^\varepsilon(x, t))$ to the Cauchy problem of (2.3)–(2.4).

The aim of the present paper is to show that the sequences of the solution $(q^\varepsilon(x, t), u^\varepsilon(x, t))$ actually converge to the solution of a limit system for all time. The key step is to establish the uniform estimates both in Mach number and time.

3.   Uniform estimates

In this part, we are going to prove the uniform estimates of the solution. To this end, we define a weighted norm

First, we have the following estimate.

Lemma 3.1. If $U = (q, u)\in C([0, T]; H^3)$ is a solution of the system (2.3)–(2.4) for any given $T > 0$, then we have

Proof. The proof is divided into two parts. First, we will show that

Second, we shall prove that

Thus, by the definition of the functional $E(U)$, it is obvious that (3.2) holds.

Part Ⅰ: For $0\leq|\alpha|\leq3$, applying $\partial^\alpha_x$ to (2.3) and testing by $\partial^\alpha_xU$ gives

where the singular terms are cancelled by integration by parts and the commutators $\mathcal{C}_1$ and $\mathcal{C}_2$ are given by

By direct calculation, we get

and

Combining the above two equalities with (3.5), one finds

Clearly, the first three terms on the right-hand side of (3.6) can be controlled by

where standard Sobolev embedding inequalities are used here. The commutator estimates is given by

We only consider the case $|\alpha| = 3$, because other cases are relatively easy to deal with. When $|\beta| = 1, |\gamma| = 2$, we have

When $|\beta| = 2, |\gamma| = 1$,

When $|\beta| = 3, |\gamma| = 0$,

Collecting (3.8)–(3.10), we get the following estimate

In a similar fashion, one can show that

Substituting the commutator estimates and (3.7) into (3.6) and taking summation over $0\leq|\alpha|\leq3$, we get

This completes the proof of the first part.

Part Ⅱ: For $1\leq k\leq3$ and $k+|\alpha|\leq3$, applying $\partial^k_t\partial^\alpha_x$ to (2.3) and testing by $\varepsilon^{2k-2}\partial^k_t\partial^\alpha_xU$ gives

where

By integrating by parts, we still have

It remains to give the estimates of the commutators. Basically, the commutator estimates is similar to (3.8)–(3.10) except that we need to guarantee there are enough powers of $\varepsilon$ to balance the norms of the unknowns. We should be careful when we encountered with higher time derivatives. For example, the estimate of the first term in $\mathcal{C}_3$ will involve

with $|\alpha| = 2$. The definition of $E(U)$ suggests that there should be a $\varepsilon$ to balance $\partial^2_tq$. Fortunately, since $h = h^\ast+\varepsilon q$, taking derivative to $e(h)$ will give us an additional $\varepsilon$. The other terms in the commutator is similar, we just omit the details for the sake of simplicity.

Using the strategy we explained above to estimate the commutators, we get

Combined with (3.13), one has

By taking summation over $1\leq k\leq3$ and $k+|\alpha|\leq3$, we get the desired results (3.4). This completes the proof of Lemma 3.1.

Next, we have the following lemma.

Lemma 3.2. If $U = (q, u)\in C([0, T]; H^3)$ is a solution of the system (2.3)–(2.4), then

and

Proof. For $0\leq|\alpha|\leq 2$, taking $\partial_x^\alpha$ to the second equation of (2.3) yields

This further gives

Taking summation over $0\leq|\alpha|\leq 2$ gives

where we have used the Poincaré inequality to control $\|q\|_{L^2}$.

Next, for $0\leq k+|\alpha|\leq2$, taking $\partial_t^k\partial_x^\alpha$ to the first equation of (2.3) gives

where the commutator $\mathcal{C}_5$ is given by

Then, multiplying the above equation by $\partial_{t}^{k+1}\partial_x^\alpha q$ and integrating over $\mathbb{T}^3$, we have

Applying $\partial_{t}^{k+1}\partial_x^\alpha$ to the second equation of (2.3) gives

with

Multiplying the above equation by $\partial_{t}^{k}\partial_x^\alpha u$ and integrating over $\mathbb{T}^3$, one has

Adding (3.22) to (3.25), then multiplying by $\varepsilon^{2k}$ yields

Each term on the right-hand side of the above equality need to be estimated now. For $I_1$, we have

The second term $I_2$ can be reformulated into

Similarly, we have

While for $I_4$, it is easy to find that

For the estimates of the commutators, we mention that there are enough powers of $\varepsilon$ to balance the time derivatives of the solution. Thus, $I_5$ and $I_6$ can be handled in a similar way as Lemma 3.1. For simplicity, we omit the details of the estimates and give

Finally, putting the above estimates of $I_j$ into (3.26) and taking summation over$0\leq k+|\alpha|\leq2$, we get

This completes the proof of Lemma 3.2.

Now, following the two lemmas above, we are ready to give the uniform estimates of the solutions. Multiplying (3.16) and (3.17) by a sufficiently small constant $\kappa$, adding them onto (3.2), we get

Notice that the first term $C\kappa E(u)$ on the right-hand side of (3.28) can be absorbed by $AE(u)$ due to the smallness of $\kappa$, thus one has

where

Now, assume a priori that

with $\delta$ being sufficiently small. Then by the smallness of $\varepsilon$ and $\kappa$, we find that $\tilde{E}(U)$ is equivalent to $E(U)$. Namely, there exist positive constant $c_3$ and $c_4$ such that

Moreover, since $\delta$ is small, we find all the terms on the right-hand side of (3.29) can be absorbed by the $c_1E(U)$. This gives

Thus if we assume that the initial data is sufficiently small such that

we can get from (3.30) and (3.31) that

This closes the a priori assumption and we get the uniform estimates of the solution. In a word, we have proved:

Theorem 3.3. Suppose that the initial data $(q_0(x), u_0(x))\in H^3(\mathbb{T}^3)$. Then there exist positive constants $\delta_0$ and $\varepsilon_0$ such that if

and $0 < \varepsilon < \varepsilon_0$ hold, system (2.3)–(2.4) admits a global classical solution $(q(x, t), u(x, t))$ satisfying

for all $t\in \mathbb{R}^+$.

4.   Convergence

With the uniform estimates established in previous section, we shall study the convergence of the solution in this section.

Actually, from the uniform estimates (3.32) and the Aubin-Lions compactness lemma, we can find a limit function

with $\eta > 0$ such that, by passing to a subsequence,

Let $\mathbb{P}$ be the orthogonal projection of $(L^2(\Omega)\big)^3$ onto the subspace

Applying $\mathbb{P}$ to the second equation of (2.3) yields

Letting $\varepsilon$ goes to zero in the above equations and using (4.1), (4.2), one has

By the properties of $\mathbb{P}$, there exists a function $\pi^0\in L^\infty(\mathbb{R}^+; H^3(\mathbb{T}^3))$ such that

On the other hand, the first equation in (2.3) gives

The uniform estimates (3.32) enables us to take $\varepsilon\rightarrow0$ in the above equation to get

Equations (4.3) and (4.4) constitute the limit system

where the initial data $u^0_0(x)$ is given by $\|u^\varepsilon_0-u^0_0(x)\|_{H^3}\rightarrow0$ as $\varepsilon\rightarrow0$.

Conflict of interest

The author declares there is no conflict of interest.