Asymptotic stability of contact waves for the compressible Navier–Stokes equations

In this article, we study the large-time asymptotic behaviour of contact wave for the Cauchy problem of one-dimensional compressible Navier–Stokes equations with zero viscosity. When the Riemann problem for the Euler system admits a contact discontinuity solution, we can construct a contact wave, which approximates the contact discontinuity on any finite-time interval for small heat conduction and then runs away from it for large time, and prove that it is nonlinearly stable provided that the strength of contact discontinuity and the perturbation of the initial data are suitably small.     

Existence of global weak solutions for a 3D Navier–Stokes–Poisson–Korteweg equations

The purpose of this work is to study the global-in-time existence of weak solutions of a viscous capillary model of plasma expressed as a so-called Navier–Stokes–Poisson–Korteweg model for large data in three-dimensional space. Using the compactness argument, we prove the existence of global weak solutions in the classical sense to such system with a cold pressure.     

A remark on the global existence of weak solutions to the compressible quantum Navier–Stokes equations

In this paper, we obtain the global existence of weak solutions to the compressible quantum Navier–Stokes equations. By virtue of a useful identity and an interesting estimate, we solve the critical case that the viscosity equals the dispersive coefficient. This result removes the restrictions on the coefficients and improves the recent work of Antonell and Spirito (2017) in some senses.     

Global well-posedness of classical solutions to the compressible Navier–Stokes equations in a half-space

In this paper, we establish the global well-posedness of classical solutions to the half-space problem with the boundary condition proposed by Navier for the isentropic compressible Navier–Stokes equations in three spatial dimensions. Initial data are of small energy but possibly large oscillations.     

Blow up of classical solutions to the isentropic compressible Navier–Stokes equations

In this work we consider the isentropic compressible Navier–Stokes equations in three space dimensions. Blow up result will be established, assuming the gradient of the velocity satisfies some decay constraint and the initial total momentum does not vanish. We prove the main result by a contradiction argument, based on the conservation of the total mass and the total momentum.     

A convergent FEM-DG method for the compressible Navier–Stokes equations

This paper presents a new numerical method for the compressible Navier–Stokes equations governing the flow of an ideal isentropic gas. To approximate the continuity equation, the method utilizes a discontinuous Galerkin discretization on piecewise constants and a basic upwind flux. For the momentum equation, the method is a new combined discontinuous Galerkin and finite element method approximating the velocity in the Crouzeix–Raviart finite element space. While the diffusion operator is discretized in a standard fashion, the convection and time-derivative are discretized using discontinuous Galerkin on the element average velocity and a Lax–Friedrich type flux. Our main result is convergence of the method to a global weak solution as discretization parameters go to zero. The convergence analysis constitutes a numerical version of the existence analysis of Lions and Feireisl.     

Asymptotic stability for the Navier–Stokes equations in L n

Considering the Navier–Stokes equations in , we prove the asymptotic stability for weak solutions in the marginal class u ∈ C _B (0, ∞; L ⁿ ) with arbitrary initial and external perturbations.      

Quasi-neutral and zero-viscosity limits of Navier–Stokes–Poisson equations in the half-space

The present paper is concerned with the quasi-neutral and zero-viscosity limits of Navier–Stokes–Poisson equations in the half-space. We consider the Navier-slip boundary condition for velocity and Dirichlet boundary condition for electric potential. By means of asymptotic analysis with multiple scales, we construct an approximate solution of the Navier–Stokes–Poisson equations involving two different kinds of boundary layer, and establish the linear stability of the boundary layer approximations by conormal energy estimate.     

Axisymmetric solutions of the 3D Navier–Stokes equations for compressible isentropic fluids

We prove the existence of global weak solutions to the Navier–Stokes equations for compressible isentropic fluids for any γ>1 when the Cauchy data are axisymmetric, where γ is the specific heat ratio. Moreover, we obtain a new integrability estimate of the density in any neighborhood of the symmetric axis (the singularity axis).     

Blowup of solutions for the compressible Navier–Stokes equations with density-dependent viscosity coefficients

We study the blowup phenomena of solutions to the compressible Navier–Stokes equations with density-dependent viscosity coefficients in arbitrary dimensions. By constructing a family of self-similar analytical solutions with spherical symmetry, some interesting information including the blowup and expanding properties are shown. In addition, the case of constant viscosity coefficients is also considered. The approach is based on the phase plane method.     

Asymptotic behavior of stochastic two-dimensional Navier–Stokes equations with delays

The paper proves the L ²-exponential stability of weak solutions of two-dimensional stochastic Navier?CStokes equations in the presence of delays. The results extend some of the existing results.