Weak solutions for the stationary anisotropic and nonlocal compressible Navier-Stokes system

Abstract

In this paper, we prove the existence of weak solutions for the stationary compressible Navier-Stokes equations with an anisotropic and nonlocal viscous stress tensor in a periodic domain ${\mathbb{T}}^3$. This gives an answer to an open problem which is important for applications for instance in geophysics or in microfluidics. When dealing with weak solutions for such non-linear PDE system, the most delicate aspect is the stability analysis: Given a sequence of weak solutions for some well-chosen approximated systems, show that this sequence converges to a solution for the initial system. One of the key ingredients in the proof of stability is the adaptation of a new identity discovered by the authors [2] which was developed to study the quasi-stationary anisotropic compressible Brinkman system. This identity is used in order to recover strong convergence properties for the sequence of velocities and recover a posteriori strong convergence for the sequence of densities.

Résumé

Dans ce papier, nous montrons l'existence de solutions faibles pour les équations de Navier-Stokes compressible stationnaire avec un tenseur des contraintes visqueuses anisotrope et non local dans un domaine périodique ${\mathbb{T}}^3$. Ceci donne une réponse à un problème ouvert qui est important pour des applications notamment en géophysique et en microfluidique. La difficulté principale quand on s'intéresse aux solutions faibles de tels systèmes d'EDP non-linéaires est usuellement l'analyse de stabilité. Plus précisemment si l'on considère une suite de solutions faibles pour un système approché, on espère montrer qu'il existe une suite qui converge vers une solution faible du système initial. Une des clés dans la preuve de stabilité est l'adapataion d'une nouvelle identité découverte par les auteurs [2] qui a été développée pour étudier le système de Brinkman quasi-stationnaire anisotrope. Cette identité est utilisée afin de montrer, dans un premier temps, des propriétés de convergence forte sur une sous suite des champs de vitesse associés. Il faut ensuite, dans un deuxième temps, montrer la convergence forte pour la suite correspondante de densités.

Introduction

The stationary Navier-Stokes system for a barotropic compressible viscous fluid reads

$$\{\begin{array}{c}\mathrm{div}\left(\rho u\right)=0,\hfill \\ \mathrm{div}(\rho u\otimes u)-\mu \mathrm{\Delta }u-(\mu +\lambda )\mathrm{\nabla }\mathrm{div}u+\mathrm{\nabla }p\left(\rho \right)=\rho f+g,\hfill \end{array}$$

where μ and λ are given positive constants representing the shear respectively the bulk viscosities, $f,g\in {\mathbb{R}}^3$ are given exterior forces acting on the fluid, $\rho \ge 0$ is the density, $p\left(\rho \right)=a{\rho }^{\gamma }$ represents the pressure, where $a>0$ and $\gamma \ge 1$ are given constants while $u\in {\mathbb{R}}^3$ is the velocity field. The total mass of the fluid is given i.e. the above system should be considered along with the equation

$$\underset{\mathrm{\Omega }}{\int }\rho =M>0,$$

where M is given.

It is important to point out that all the known mathematical results regarding the existence of weak solutions for the stationary Navier-Stokes system strongly use the isotropic and local structure of the viscous stress tensor owing to the nice algebraic properties it induces for the so-called effective flux. Extending these results such as to take in account anisotropic or nonlocal viscous stress-tensors remained an open problem until now.

As explained in [16], one cannot expect that (1.1)-(1.2) with periodic boundary conditions to have a solution for any $f,g\in L^{\infty }$ because of the compatibility condition

$$\underset{{\mathbb{T}}^3}{\int }(\rho f+g)=0$$

which comes from integrating the momentum equation. Thus, if f and g have positive components this would imply that $\rho =0$ which clearly violates the total mass condition. One way to bypass this structural defect of the periodic case is to proceed as in [5] and consider forces f that posses a certain symmetry which ensures the validity of (1.3). Another way to bypass this problem was suggested by P.L. Lions in [16] and consists in introducing the term $B\times (B\times u)$ with $B\in L^{\infty }\left({\mathbb{T}}^3\right)$ a non-constant function in the momentum equation which can be interpreted as the effect of a magnetic field on the fluid. We claim that the ideas presented in the present paper can be adapted to handle both situations but in order to avoid extra technical difficulties we choose to treat the case where $f=0$. We propose here to investigate the problem of existence of weak solutions $(\rho ,u)$ for the following system:

$$\{\begin{array}{c}\mathrm{div}\left(\rho u\right)=0,\hfill \\ \mathrm{div}(\rho u\otimes u)-\mathcal{A}u+a\mathrm{\nabla }{\rho }^{\gamma }=g,\hfill \end{array}$$

with

$$\rho \ge 0,\underset{{\mathbb{T}}^3}{\int }\rho \left(x\right)dx=M>0,\underset{{\mathbb{T}}^3}{\int }u\left(x\right)dx=0,$$

where the viscous diffusion operator $\mathcal{A}$ is given by

$$\mathcal{A}\cdot =\underset{\text{classical part}}{\underbrace{\mu \mathrm{\Delta }\cdot +(\mu +\lambda )\mathrm{\nabla }\mathrm{div}\cdot }}+\underset{\text{anisotropic part}}{\underbrace{\mu \theta {\partial }_{33}\cdot }}+\underset{\text{nonlocal part}}{\underbrace{\eta ⁎\mathrm{\Delta }\cdot +\xi ⁎\mathrm{\nabla }\mathrm{div}\cdot }}.$$

We will assume the following hypothesis (H):

• A given total mass $M>0$ of the fluid.

• An adiabatic constant $\gamma >3$ and a positive constant $a>0$.

• A forcing term g such that

  $$g\in {(L^{\frac{3(\gamma -1)}{2\gamma -1}}\left({\mathbb{T}}^3\right))}^3\text{ with }\underset{{\mathbb{T}}^3}{\int }g=0.$$

• The constant μ, λ and θ such that

  $$\mu ,\mu +\lambda >0\text{ and }\theta >-1.$$

• The functions η and ξ satisfying

  $$\min \{1,1+\theta \}\mu -{\Vert \eta \Vert }_{L^1}-\frac{1}{3}{\Vert \xi \Vert }_{L^1}>0\text{ or }\hat{\eta }\left(k\right),\hat{\xi }\left(k\right)\in {\mathbb{R}}^{+}\text{ for all }k\in {\mathbb{Z}}^3$$

  and

  $$\mathrm{\nabla }\eta ,\mathrm{\nabla }\xi \in L^2\left({\mathbb{T}}^3\right).$$

The main objective of the paper is to prove existence of a weak solution à la Leray for the steady compressible barotropic Navier-Stokes system with anisotropic and nonlocal diffusion. Anisotropic diffusion is present for instance in geophysical flows, see [22], while nonlocal diffusion is considered when studying confined fluids or in microfluidics where fluids flow thought narrow vessels. In order to achieve this goal, one key ingredient is the identity that we proposed in [2] which allowed us to give a simple proof for the existence of global weak-solutions for the anisotropic quasi-stationary Stokes system (compressible Brinkman equations).

More precisely, in this paper, we prove the following existence result

Theorem 1

Let us assume Hypothesis (H) be satisfied. There exists a constant $c_0$ such that if

$$(1+|\theta |)\left|\theta \right|\mu \frac{2\lambda +\mu }{{(\lambda +\mu )}^2}\le c_0,$$

then there exists a pair $(\rho ,u)\in L^{3(\gamma -1)}\left({\mathbb{T}}^3\right)\times {(W^{1,\frac{3(\gamma -1)}{\gamma }}\left({\mathbb{T}}^3\right))}^3$ which is a weak-solution of the stationary Navier-Stokes system (1.4)–(1.5).

Remark 1.1

Motivated by physically relevant phenomena like anisotropy or thermodynamically unstable pressure state laws, D. Bresch and P.E. Jabin introduced in [3], see also [4], a new method for the identification of the pressure in the study of stability of solutions for the non-stationary compressible Navier-Stokes system. More precisely, if one considers a sequence of solutions generated by a sequence of initial data for which the corresponding sequence of initial densities is compact in $L^1$, then one is able to propagate this information for latter times via a nonlocal compactness criterion modulated with appropriate nonlinear weights. The idea in [3], propagation of compactness, is intimately related to the non-stationary transport equation and it does not seem to adapt to the stationary case.

Remark 1.2

The proof of Theorem 1 can be adapted to accommodate more general diffusion operators than (1.6). In particular, our method adapts to viscous stress tensors that include space-dependent coefficients or different convolution kernels for each component of u. In the opinion of the authors, the particular form of $\mathcal{A}$ proposed in (1.6), besides being physically relevant, see for instance [7] or [8], is also relatively easier to manipulate in computations.

To the authors's knowledge this is the first existence result of weak solutions taking in consideration anisotropic and nonlocal diffusion for the steady Navier-Stokes system. The first steps of the proof of Theorem 1 follow a rather well-known path: we consider an elliptic regularization for the system (1.4) to which classical theory can be applied and therefore we may construct a sequence of solutions parametrized by the regularization parameter. Of course, the more delicate part is to recover uniform estimates with respect to the regularization parameter and to show that the limiting object is a solution of the stationary Navier-Stokes system. The first key ingredient in the proof of stability is the new identity discovered by the authors in [2] in the context of the compressible Brinkman system. As it turns out, the $L^2$-integrability of the velocity field obtained via the basic energy estimate is not enough, better integrability is needed in order to justify rigorously the aforementioned identity. This is achieved by showing that it is possible to estimate the pressure ${\rho }^{\gamma }$ in a better space than $L^2$: the smallness Condition (1.7) is required at this level. We point out that a similar condition is imposed in [3] in order to treat the non-stationary compressible Navier-Stokes system. In particular, one can consider an arbitrary anisotropic amplitude if the bulk viscosity is large enough.

The rest of the paper is organized in the following way: in Section 2, we recall existing results concerning weak solutions for the steady compressible Navier–Stokes equations and we discuss energy dissipation properties for the diffusion operator (1.6). In Section 3 we prove a nonlinear weak stability result, see Theorem 2 bellow, for the stationary compressible Navier-Stokes equation with anisotropic and nonlocal diffusion operator. In particular, we show that it is possible to recover strong convergence of the sequence of the gradients of the velocities and we show how to combine this fact with the compactness properties of the anisotropic viscous flux in order to identify the pressure. This is the main idea in the paper. In Section 4, we propose an approximate system and prove the existence of solutions to such system. Such approximate system is based on two layers of regularization: one ensuring ellipticity while the other one providing positivity of the density. In the last Section 5, we prove our main result, Theorem 1 first by establishing uniform estimates with respect to the two parameter and secondly using the non-linear stability results established in Section 3.