Barotropic Navier-Stokes Equations Research Articles

Global solutions of compressible barotropic Navier–Stokes equations with a density-dependent viscosity coefficient

In this paper, we consider the global existence and uniqueness of the classical (weak) solution for the 2D or 3D compressible Navier–Stokes equations with a density-dependent viscosity coefficient (λ = λ(ρ)). Initial data and solutions are only small in the energy-norm. We also give a description of the large time behavior of the solution. Then, we study the propagation of singularities in solutions. We obtain that if there is a vacuum domain initially, then the vacuum domain will exist for all time, and vanishes as time goes to infinity.

CAUCHY PROBLEM FOR VISCOUS SHALLOW WATER EQUATIONS WITH A TERM OF CAPILLARITY

In this paper, we consider the compressible Navier–Stokes equation with density-dependent viscosity coefficients and a term of capillarity introduced formally by van der Waals in Ref. 51. This model includes at the same time the barotropic Navier–Stokes equations with variable viscosity coefficients, shallow-water system and the model introduced by Rohde in Ref. 46.We first study the well-posedness of the model in critical regularity spaces with respect to the scaling of the associated equations. In a functional setting as close as possible to the physical energy spaces, we prove global existence of solutions close to a stable equilibrium, and local in time existence of solutions with general initial data. Uniqueness is also obtained.

Nonexistence results for a compressible non-Newtonian fluid with magnetic effects in the whole space

We consider a generalization of the compressible barotropic Navier–Stokes equations to the case of non-Newtonian fluid in the whole space. The viscosity tensor is assumed to be coercive with an exponent q > 1 . We prove that if the total mass and momentum of the system are conserved, then one can find a constant q γ > 1 depending on the dimension of space n and the heat ratio γ such that for q ∈ [ q γ , n ) there exists no global in time smooth solution to the Cauchy problem. We prove also an analogous result for solutions to equations of magnetohydrodynamic non-Newtonian fluid in 3D space.

Global Weak Solutions to a Generic Two-Fluid Model

This paper deals with mathematical properties of a generic two-fluid flow model commonly used in industrial applications. More precisely, we address the question of whether available mathematical results in the case of a single-fluid governed by the compressible barotropic Navier–Stokes equations may be extended to such a two-phase model. We focus on existence of global weak solutions, linear theory and determination of eigenvalues and invariant regions.

EXISTENCE AND UNIQUENESS OF STRONG SOLUTIONS FOR A COMPRESSIBLE MULTIPHASE NAVIER–STOKES MISCIBLE FLUID-FLOW PROBLEM IN DIMENSION n = 1

We prove the global existence and uniqueness of strong solutions for a compressible multifluid described by the barotropic Navier–Stokes equations in dim = 1. The result holds when the diffusion coefficient depends on the pressure. It relies on a global control in time of the L2norm of the space derivative of the density, via a new kind of entropy.

An unconditionally stable pressure correction scheme for the compressible barotropic Navier-Stokes equations

We present in this paper a pressure correction scheme for the barotropic compressible Navier-Stokes equations, which enjoys an unconditional stability property, in the sense that the energy and maximum-principle-based ap rioriestimates of the continuous problem also hold for the discrete solution. The stability proof is based on two independent results for general finite volume discretiza- tions, both interesting for their own sake: the L 2 -stability of the discrete advection operator provided it is consistent, in some sense, with the mass balance and the estimate of the pressure work by means of the time derivative of the elastic potential. The proposed scheme is built in order to match these theoretical results, and combines a fractional-step time discretization of pressure-correction type with a space discretization associating low order non-conforming mixed finite elements and finite volumes. Numerical tests with an exact smooth solution show the convergence of the scheme.

Open Boundary Control Problem for Navier-Stokes Equations Including a Free Surface: Data Assimilation

This paper develops the data-assimilation procedure in order to allow for the assimilation of measurements of currents and free-surface elevations into an unsteady flow solution governed by the free-surface barotropic Navier-Stokes equations. The flow is considered in a 2D vertical section in which horizontal and vertical components of velocity are represented as well as the elevation of the free surface. Since a possible application is to the construction of a coastal (limited area) circulation model, the open boundary control problem is the main scope of the paper. The assimilation algorithm is built on the limited memory quasi-Newton LBFGS method guided by the adjoint sensitivities. The analytical step search, which is based on the solution of the tangent linear model, is used. We process the gradients to regularize the solution. In numerical experiments we consider different wave patterns with a purpose to specify a set of incomplete measurements, which could be sufficient for boundary-control identification. As a result of these experiments we formulate some important practical conclusions.

On the construction of approximate solutions for the 2D viscous shallow water model and for compressible Navier–Stokes models

The purpose of this paper is to build sequences of suitably smooth approximate solutions to the Saint-Venant model that preserve the mathematical structure discovered in [D. Bresch, B. Desjardins, Comm. Math. Phys. 238 (1–2) (2003) 211–223]. The stability arguments in this paper then apply to such sequences of approximate solutions, which leads to the global existence of weak solutions for this model. Extension of this mollifying procedure to the case of compressible Navier–Stokes equations is also provided. Using the recent paper written by the authors, this provides global existence results of weak solutions for the barotropic Navier–Stokes equations and for compressible Navier–Stokes equations with heat conduction using a particular cold pressure term close to vacuum.