Barotropic Navier-Stokes Equations Research Articles

Large-Time Behavior of Composite Waves of Viscous Shocks for the Barotropic Navier–Stokes Equations

Large-Time Behavior of Composite Waves of Viscous Shocks for the Barotropic Navier–Stokes Equations

Time-asymptotic stability of composite waves of viscous shock and rarefaction for barotropic Navier-Stokes equations

We prove the time-asymptotic stability of composite waves consisting of the superposition of a viscous shock and a rarefaction for the one-dimensional compressible barotropic Navier-Stokes equations. Our result solves a long-standing problem first mentioned in 1986 by Matsumura and Nishihara in [28]. The same authors introduced it officially as an open problem in 1992 in [29] and it was again described as very challenging open problem in 2018 in the survey paper [26]. The main difficulty is due to the incompatibility of the standard anti-derivative method, used to study the stability of viscous shocks, and the energy method used for the stability of rarefactions. Instead of the anti-derivative method, our proof uses the a-contraction with shifts theory recently developed by two of the authors. This method is energy based, and can seamlessly handle the superposition of waves of different kinds.

Liouville-type theorems for the stationary compressible barotropic and incompressible inhomogeneous Navier–Stokes equations

The present paper is dedicated to the Liouville-type problem for the three-dimensional stationary compressible barotropic Navier–Stokes equations and incompressible inhomogeneous Navier–Stokes equations. We show that velocity u is trivial under some suitable assumptions stated in terms of BMO−1 spaces.

Anisotropy and stratification effects in the dynamics of fast rotating compressible fluids

The primary goal of this paper is to develop robust methods to handle two ubiquitous features appearing in the modeling of geophysical flows: (i) the anisotropy of the viscous stress tensor and (ii) stratification effects.We focus on the barotropic Navier–Stokes equations with Coriolis and gravitational forces. Two results are the main contributions of the paper. Firstly, we establish a local well-posedness result for finite-energy solutions, via a maximal regularity approach. This method allows us to circumvent the use of the effective viscous flux, which plays a key role in the weak solutions theories of Lions–Feireisl and Hoff, but seems to be restricted to isotropic viscous stress tensors. Moreover, our approach is sturdy enough to take into account nonconstant reference density states; this is crucial when dealing with stratification effects. Secondly, we study the structure of the solutions to the previous model in the regime when the Rossby, Mach and Froude numbers are of the same order of magnitude. We prove an error estimate on the relative entropy between actual solutions and their approximation by a large-scale quasi-geostrophic flow supplemented with Ekman boundary layers. Our analysis holds for a large class of barotropic pressure laws.

Incompressible and fast rotation limit for barotropic Navier–Stokes equations at large Mach numbers

We study the incompressible and fast rotation limit for the barotropic Navier–Stokes equations with Coriolis force, in the case when the Mach number Ma is large with respect to the Rossby number Ro: namely, we focus on the regime Ro≪Ma. For this, we follow a recent approach in Danchin and Mucha (2017) and take also a large bulk viscosity coefficient. We prove that the limit dynamics is described by an incompressible Navier–Stokes type equation, recasted in the vorticity formulation, where however an additional unknown, linked to density oscillations around a fixed constant reference state, comes into play. The proof of the convergence is based on a compensated compactness argument and on the derivation of sharp decay estimates for solutions to a heat equation with fast diffusion in time.

Darcy’s law as low Mach and homogenization limit of a compressible fluid in perforated domains

We consider the homogenization limit of the compressible barotropic Navier–Stokes equations in a three-dimensional domain perforated by periodically distributed identical particles. We study the regime of particle sizes and distances such that the volume fraction of particles tends to zero but their resistance density tends to infinity. Assuming that the Mach number is decreasing with a certain rate, the rescaled velocity and pressure of the microscopic system converges to the solution of an effective equation which is given by Darcy’s law. The range of sizes of particles we consider is exactly the same which leads to Darcy’s law in the homogenization limit of incompressible fluids. Unlike previous results for the Darcy regime we estimate the deficit related to the pressure approximation via the Bogovskiĭ operator. This allows for more flexible estimates of the pressure in Lebesgue and Sobolev spaces and allows to proof convergence results for all barotropic exponents [Formula: see text].

Ergodic theory for energetically open compressible fluid flows

The ergodic hypothesis is examined for energetically open fluid systems represented by the barotropic Navier–Stokes equations with general inflow/outflow boundary conditions. We show that any globally bounded trajectory generates a stationary statistical solution, which is interpreted as a stochastic process with continuous trajectories supported by the family of weak solutions of the problem. The abstract Birkhoff–Khinchin theorem is applied to obtain convergence (in expectation and a.s.) of ergodic averages for any bounded Borel measurable function of state variables associated to any stationary solution. Finally, we show that validity of the ergodic hypothesis is determined by the behavior of entire solutions (i.e. a solution defined for any t∈R). In particular, the ergodic averages converge for any trajectory provided its ω-limit set in the trajectory space supports a unique (in law) stationary solution.

Low Mach number limit of some staggered schemes for compressible barotropic flows

In this paper, we study the behaviour at low Mach number of numerical schemes based on staggered discretizations for the barotropic Navier-Stokes equations. Three time discretizations are considered: the implicit-in-time scheme and two non-iterative pressure correction schemes. The last two schemes differ by the discretization of the convection term: linearly implicit for the first one, so the resulting scheme is unconditionnally stable, and explicit for the second one, so the scheme is stable under a CFL condition involving the material velocity only. We rigorously prove that these three variants are asymptotic preserving in the following sense: for a given mesh and a given time step, a sequence of solutions obtained with a sequence of vanishing Mach numbers tend to a solution of a standard scheme for incompressible flows. This convergence result is obtained by mimicking the proof already known in the continuous case.

Numerical investigations of the compressible Navier-Stokes system

In this paper we write, analyze and experimentally compare three different numerical schemes dedicated to the one dimensional barotropic Navier-Stokes equations: a staggered scheme based on the Rusanov one for the inviscid (Euler) system,a staggered pseudo-Lagrangian scheme in which the mesh “follows” the fluid,the Eulerian projection (on a fixed mesh) of the preceding scheme. All these schemes only involve the resolution of linear systems (all the nonlinear terms are solved in an explicit way). We propose numerical illustrations of their behaviors on particular solutions in which the density has discontinuities (hereafter called Hoff solutions). We show that the three schemes seem to converge to the same solutions, and we compare the evolution of the amplitude of the discontinuity of the numerical solution (with the pseudo-Lagrangian scheme) with the one predicted by Hoff and observe a good agreement.

Contraction property for large perturbations of shocks of the barotropic Navier–Stokes system

This paper is dedicated to the construction of a pseudo-norm for which small shockprofiles of the barotropic Navier–Stokes equations have a contraction property. This contraction property holds in the class of any large solutions to the barotropic Navier–Stokes equations. It implies a stability condition which is independent of the strength of the viscosity. The proof is based on the relative entropy method, and is related to the notion of a -contraction first introduced by the authors in the hyperbolic case.

Global Smooth Solutions for 1D Barotropic Navier–Stokes Equations with a Large Class of Degenerate Viscosities

We prove the global existence and uniqueness of smooth solutions to the one-dimensional barotropic Navier-Stokes system with degenerate viscosity $\mu(\rho)=\rho^\alpha$. We establish that the smooth solutions have possibly two different far-fields, and the initial density remains positive globally in time, for the initial data satisfying the same conditions. In addition, our result works for any $\alpha>0$, i.e., for a large class of degenerate viscosities. In particular, our models include the viscous shallow water equations. This extends the result of Constantin-Drivas-Nguyen-Pasqualotto \cite[Theorem 1.5]{CDNP} (on the case of periodic domain) to the case where smooth solutions connect possibly two different limits at the infinity on the whole space.

Global existence and exponential stability for the compressible Navier–Stokes equations with discontinuous data

The initial boundary value problem for the compressible barotropic Navier–Stokes equations is investigated in the case that the initial density has a jump discontinuity across an interior closed curve in two-dimensional bounded domain. If the initial data is a small perturbation of the constant state and the interior closed curve is near a circle inside the domain, the global existence and large time behavior of the piecewise strong solution is shown, in particular, the jump of the fluid density across the convecting curve decays exponentially in time.

Error estimates for a numerical method for the compressible Navier–Stokes system on sufficiently smooth domains

We derive an a priori error estimate for the numerical solution obtained by time and space discretization by the finite volume/finite element method of the barotropic Navier–Stokes equations. The numerical solution on a convenient polyhedral domain approximating a sufficiently smooth bounded domain is compared with an exact solution of the barotropic Navier–Stokes equations with a bounded density. The result is unconditional in the sense that there are no assumed bounds on the numerical solution. It is obtained by the combination of discrete relative energy inequality derived in [T. Gallouet, R. Herbin, D. Maltese and A. Novotný, IMA J. Numer. Anal. 36 (2016) 543–592.] and several recent results in the theory of compressible Navier–Stokes equations concerning blow up criterion established in [Y. Sun, C. Wang and Z. Zhang, J. Math. Pures Appl. 95 (2011) 36–47] and weak strong uniqueness principle established in [E. Feireisl, B.J. Jin and A. Novotný, J. Math. Fluid Mech. 14 (2012) 717–730].

Dimension reduction for compressible pipe flows including friction

We present the full derivation of a one dimensional Saint-Venant like equations for barotropic compressible pipe flows including friction. The one dimensional hyperbolic system is called γ−pressurized model where γ is the adiabatic constant. It is obtained through the three dimensional barotropic Navier-Stokes equations under thin layer assumptions as a first order approximation. Prescribing suitable boundary conditions, one can introduce a general friction law and then explicitly show its geometrical (w.r.t the hydraulic radius) and hydrodynamical (w.r.t the Oser number) dependencies in the reduced model. In particular, for linear pressure law (γ = 1), we justify the one dimensional pressurized model (called P-model) introduced by the author in the context of unsteady mixed flows in closed water pipes. For non linear pressure law (γ = 1), the γ−pressurized model describes the evolution of a compressible (almost) gravityless flow.

Error estimates for a numerical approximation to the compressible barotropic Navier–Stokes equations

We present here a general method based on the investigation of the relative energy of the system, that provides an unconditional error estimate for the approximate solution of the barotropic Navier Stokes equations obtained by time and space discretization. We use this methodology to derive an error estimate for a specific DG/finite element scheme for which the convergence was proved in [26].

Approximate controllability for linearized compressible barotropic Navier–Stokes system in one and two dimensions

In this paper we consider compressible barotropic Navier–Stokes equations in one and two dimensions linearized around a constant steady state with Dirichlet boundary conditions. We explore the controllability of this linearized system using a control only for the velocity equation. We prove that the system with homogeneous Dirichlet boundary conditions, is approximately controllable by a localized interior control when time is sufficiently large.

Incompressible limit of global strong solutions to 3-D barotropic Navier–Stokes equations with well-prepared initial data and Navier's slip boundary conditions

This paper studies the incompressible limit of global strong solutions to the three-dimensional barotropic Navier–Stokes equations associated with Navier's slip boundary conditions, provided that the initial data are sufficiently small, and “well-prepared” in the sense that the time derivatives, up to first order, of solutions are bounded initially. The main idea is to derive a differential inequality with decay, so that the estimates are bounded uniformly both in the Mach number ϵ∈(0,ϵ0] (for some ϵ0>0) and the time t∈[0,+∞).

A Lagrangian approach for the compressible Navier-Stokes equations

Here we investigate the Cauchy problem for the barotropic Navier-Stokes equations in ℝ n , in the critical Besov spaces setting. We improve recent results as regards the uniqueness condition: initial velocities in critical Besov spaces with (not too) negative indices generate a unique local solution. Apart from (critical) regularity, the initial density just has to be bounded away from 0 and to tend to some positive constant at infinity. Density-dependent viscosity coefficients may be considered. Using Lagrangian coordinates is the key to our statements as it enables us to solve the system by means of the basic contraction mapping theorem. As a consequence, conditions for uniqueness are the same as for existence, and Lipschitz continuity of the flow map (in Lagrangian coordinates) is established.

Staggered schemes for all speed flows

We review in this paper a class of schemes for the numerical simulation of compressible flows. In order to ensure the stability of the discretizations in a wide range of Mach numbers and introduce sufficient decoupling for the numerical resolution, we choose to implement and study pressure correction schemes on staggered meshes. The implicit version of the schemes is also considered for the theoretical study. We give both algorithms for the barotropic Navier-Stokes equations, for the full Navier-Stokes equations and for the Euler equations. In each case, we summarize the theoretical results that were recently obtained concerning the stability and consistency of the schemes and present some numerical results which confirm their good performance.

Global Weak Solutions to One-Dimensional Non-Conservative Viscous Compressible Two-Phase System

In this paper, we deal with global weak solutions of a non-conservative viscous compressible two-phase model in one space dimension. This work extends in some sense the previous work, [Bresch et al., Arch Rat Mech Anal 196:599–629, 2010], which provides the global existence of weak solutions in the multi-dimensional framework with 1 1. Then we prove that any possible vacuum state has to vanish within finite time after which densities are always away from vacuum. This allows to prove that at least one phase corresponding to the global weak solution is a locally in time and space (in a sense to be defined) strong solution after the vacuum states vanish. Our paper may be understood as a non-straightforward generalization to the two-phase flow system of a previous paper [Li et al., Commun Math Phys 281(2):401–444, 2008], which treated the usual compressible barotropic Navier-Stokes equations for mono-fluid with a degenerate viscosity. Various important mathematical difficulties occur when we want to generalize those results to the two-phase flows system since the corresponding model is non-conservative. Far from vacuum, it involves a strong coupling between a nonlinear algebraic system and a degenerate PDE system under constraint linked to fractions. Moreover, fractional densities may vanish if densities or fractions vanish: A difficulty is to find estimates on the densities from estimates on fractional densities using the algebraic system. Original approximate systems have also to be introduced compared to the works on the degenerate barotropic mono-fluid system. Note that even if our result concerns “only” the one-dimensional case, it points out possible global weak solutions (for such a non-conservative system) candidates to approach for instance shock structures and to define an appropriate a priori family of paths in the phase space (in numerical schemes) at the zero dissipation limit.