Asymptotic stability of stationary solutions to the compressible bipolar Navier–Stokes–Poisson equations

In this paper, we consider the compressible bipolar Navier–Stokes–Poisson equations with a non‐flat doping profile in three‐dimensional space. The existence and uniqueness of the non‐constant stationary solutions are established when the doping profile is a small perturbation of a positive constant state. Then under the smallness assumption of the initial perturbation, we show the global existence of smooth solutions to the Cauchy problem near the stationary state. Finally, the convergence rates are obtained by combining the energy estimates for the nonlinear system and the L²‐decay estimates for the linearized equations. Copyright © 2017 John Wiley & Sons, Ltd.     

Blow‐up of smooth solutions to the Navier–Stokes–Poisson equations

The main purpose of this paper is concerned with blow‐up smooth solutions to Navier–Stokes–Poisson (N‐S‐P) equations. First, we present a sufficient condition on the blow up of smooth solutions to the N‐S‐P system. Then we construct a family of analytical solutions that blow up in finite time. Copyright © 2010 John Wiley & Sons, Ltd.     

Local strong solution of Navier–Stokes–Poisson equations with degenerated viscosity coefficient

In this paper, we investigate the vacuum free boundary problem of a compressible Navier–Stokes–Poisson system with density‐dependent viscosity. By introducing Eulerian and Lagrange energy, we obtain a local in time well‐posedness of the strong solution to the Navier–Stokes–Poisson system in a spherically symmetric case. Copyright © 2015 John Wiley & Sons, Ltd.     

Large time behavior for the non‐isentropic Navier–Stokes–Maxwell system

In this paper, we are concerned with the system of the non‐isentropic compressible Navier–Stokes equations coupled with the Maxwell equations through the Lorentz force in three space dimensions. The global existence of solutions near constant steady states is established, and the time‐decay rates of perturbed solutions are obtained. The proof for existence is due to the classical energy method, and the investigation of large‐time behavior is based on the linearized analysis of the non‐isentropic Navier–Stokes–Poisson equations and the electromagnetic part for the linearized isentropic Navier–Stokes–Maxwell equations. In the meantime, the time‐decay rates obtained by Zhang, Li, and Zhu [J. Differential Equations, 250(2011), 866‐891] for the linearized non‐isentropic Navier–Stokes–Poisson equations are improved. Copyright © 2016 John Wiley & Sons, Ltd.     

Zero‐viscosity‐capillarity limit to rarefaction waves for the 1D compressible Navier–Stokes–Korteweg equations

In this paper, we study the zero viscosity and capillarity limit problem for the one‐dimensional compressible isentropic Navier–Stokes–Korteweg equations when the corresponding Euler equations have rarefaction wave solutions. In the case that either the effects of initial layer are ignored or the rarefaction waves are smooth, we prove that the solutions of the Navier–Stokes–Korteweg equation with centered rarefaction wave data exist for all time and converge to the centered rarefaction waves as the viscosity and capillarity number vanish, and we also obtain a rate of convergence, which is valid uniformly for all time. These results are showed by a scaling argument and elementary energy analysis. Copyright © 2016 John Wiley & Sons, Ltd.     

On the global existence and time decay estimates in critical spaces for the Navier–Stokes–Poisson system

We are concerned with the study of the Cauchy problem for the Navier–Stokes–Poisson system in the critical regularity framework. In the case of a repulsive potential, we first establish the unique global solvability in any dimension for small perturbations of a linearly stable constant state. Next, under a suitable additional condition involving only the low frequencies of the data and in the L²‐critical framework (for simplicity), we exhibit optimal decay estimates for the constructed global solutions, which are similar to those of the barotropic compressible Navier–Stokes system. Our results rely on new a priori estimates for the linearized Navier–Stokes–Poisson system about a stable constant equilibrium, and on a refined time‐weighted energy functional.     

On the existence of solutions to the Navier–Stokes–Poisson equations of a two‐dimensional compressible flow

In this paper, we consider the Navier–Stokes–Poisson equations for compressible, barotropic flow in two space dimensions. We introduce useful tools from the theory of Orlicz spaces. Then we prove the existence of globally defined finite energy weak solutions for the pressure satisfying p(?)=a?log^d (?) for large ?. Here d>1 and a>0. Copyright © 2006 John Wiley & Sons, Ltd.     

A Navier–Stokes–Voight model with memory

In this article, we consider a three‐dimensional Navier–Stokes–Voight model with memory where relaxation effects are described through a distributed delay. We prove the existence of uniform global attractors , where ? ∈ (0,1) is the scaling parameter in the memory kernel. Furthermore, we prove that the model converges to the classical three‐dimensional Navier–Stokes–Voight system in an appropriate sense as ? → 0. In particular, we construct a family of exponential attractors Ξ_? that is robust as ? → 0. Copyright © 2013 John Wiley & Sons, Ltd.     

On compactness of the velocity field in the incompressible limit of the full Navier–Stokes–Fourier system on large domains

The incompressible limit for the full Navier–Stokes–Fourier system is studied on a family of domains containing balls of the radius growing with a speed that dominates the inverse of the Mach number. It is shown that the velocity field converges strongly to its limit locally in space, in particular, the effect of the sound waves is eliminated by means of the local decay estimates for the acoustic wave equation. Copyright © 2008 John Wiley & Sons, Ltd.     

The stability of rarefaction wave for Navier–Stokes equations in the half‐line

This paper studies the stability of the rarefaction wave for Navier–Stokes equations in the half‐line without any smallness condition. When the boundary value is given for velocity u∥_{x = 0} = u_? and the initial data have the state (v₊, u₊) at x→ + ∞, if u_?<u₊, it is excepted that there exists a solution of Navier–Stokes equations in the half‐line, which behaves as a 2‐rarefaction wave as t→ + ∞. Matsumura–Nishihara have proved it for barotropic viscous flow (Quart. Appl. Math. 2000; 58:69–83). Here, we generalize it to the isentropic flow with more general pressure. Copyright © 2011 John Wiley & Sons, Ltd.     

Stationary solutions of Euler–Poisson equations for non‐isentropic gaseous stars

The motion of the self‐gravitational gaseous stars can be described by the Euler–Poisson equations. For some velocity fields and entropy functions that solve the conservation of mass and energy, we consider the existence of stationary solutions of Euler–Poisson equations. Under various restriction to the strength of velocity field, different assumptions on the isentropic function and adiabatic exponent, we get the existence, multiplicity and uniqueness of the stationary solutions to the Euler–Poisson system, respectively. Copyright © 2012 John Wiley & Sons, Ltd.     

Energy stable numerical scheme for the viscous Cahn–Hilliard–Navier–Stokes equations with moving contact line

In the present article we study the numerics of the viscous Cahn–Hilliard–Navier–Stokes model, endowed with dynamic boundary conditions which allow us to take into account the interaction between the fluids interface and the moving walls of the physical domain. In what follows, we propose an energy stable temporal scheme for the problem and we prove the stability and the unconditional solvability of the discretization proposed. We also propose a fully discrete scheme for which we prove the stability and the unconditional solvability. Numerical simulations are presented to illustrate the theoretical results.     

Remark on stability of rarefaction waves to the one‐dimensional compressible Navier–Stokes equations with density‐dependent viscosity coefficient

In this paper, we study one‐dimensional compressible isentropic Navier–Stokes equations with density‐dependent viscosity. We can obtain the asymptotic stability of rarefaction waves for the compressible isentropic Navier–Stokes equations when the power of viscosity coefficient , which enlarge the range of α in the article [Jiu Q, Wang Y, Xin ZP, Communication in Partial Differential Equations 2011; 36: 602‐634]. Copyright © 2013 John Wiley & Sons, Ltd.     

Large time behavior of isentropic compressible Navier–Stokes system in ℝ3

We consider the long‐time behavior and optimal decay rates of global strong solution to three‐dimensional isentropic compressible Navier–Stokes (CNS) system in the present paper. When the regular initial data also belong to some Sobolev space with l?4 and s∈[0, 1], we show that the global solution to the CNS system converges to the equilibrium state at a faster decay rate in time. In particular, the density and momentum converge to the equilibrium state at the rates (1 + t)^?3/4?s/2 in the L²‐norm or (1 + t)^?3/2?s/2 in the L^∞‐norm, respectively, which are shown to be optimal for the CNS system. Copyright © 2010 John Wiley & Sons, Ltd.     

Global weak solutions to the Cauchy problem of compressible Navier–Stokes–Vlasov–Fokker–Planck equations

A fluid–particles system of the compressible Navier‐Stokes equations and Vlasov‐Fokker‐Planck equation (including the case of Vlasov equation) in three‐dimensional space is considered in this paper. The coupling arises from a drag force exerted by the fluid onto the particles. We study a Cauchy problem with large data, and establish the existence of global weak solutions through an approximation scheme, energy estimates, and weak convergence. Copyright © 2016 John Wiley & Sons, Ltd.     

Nonlinear stability of viscous contact wave for the one‐dimensional compressible fluid models of Korteweg type

This paper is concerned with the large time behavior of solutions of the Cauchy problem to the one‐dimensional compressible fluid models of Korteweg type, which governs the motions of the compressible fluids with internal capillarity. When the corresponding Riemann problem for the Euler system admits a contact discontinuity wave, it is shown that the viscous contact wave corresponding to the contact discontinuity is asymptotically stable provided that the strength of contact discontinuity and the initial perturbation are suitably small. The analysis is based on the elementary L²‐energy method together with continuation argument. Copyright © 2013 John Wiley & Sons, Ltd.     

The stationary three‐dimensional Navier–Stokes equations with a non‐zero constant velocity at infinity

This paper is devoted to some mathematical questions related to the three‐dimensional stationary Navier–Stokes equations. Our approach is based on a combination of properties of Oseen problems in ?³. Copyright © 2008 John Wiley & Sons, Ltd.     

Stokes and Navier–Stokes equations with nonhomogeneous boundary conditions

In this paper, we study the existence and regularity of solutions to the Stokes and Oseen equations with nonhomogeneous Dirichlet boundary conditions with low regularity. We consider boundary conditions for which the normal component is not equal to zero. We rewrite the Stokes and the Oseen equations in the form of a system of two equations. The first one is an evolution equation satisfied by Pu, the projection of the solution on the Stokes space – the space of divergence free vector fields with a normal trace equal to zero – and the second one is a quasi-stationary elliptic equation satisfied by (I−P)u, the projection of the solution on the orthogonal complement of the Stokes space. We establish optimal regularity results for Pu and (I−P)u. We also study the existence of weak solutions to the three-dimensional instationary Navier–Stokes equations for more regular data, but without any smallness assumption on the initial and boundary conditions.     

Moment estimates for weak solutions to the Navier–Stokes equations in half‐space

We establish the moment estimates for a class of global weak solutions to the Navier–Stokes equations in the half‐space. Copyright © 2009 John Wiley & Sons, Ltd.     

Mathematical analysis of the motion of a rigid body in a compressible Navier–Stokes–Fourier fluid

We study an initial and boundary value problem modelling the motion of a rigid body in a heat conducting gas. The solid is supposed to be a perfect thermal insulator. The gas is described by the compressible Navier–Stokes–Fourier equations, whereas the motion of the solid is governed by Newton's laws. The main results assert the existence of strong solutions, in an ‐ setting, both locally in time and globally in time for small data. The proof is essentially using the maximal regularity property of associated linear systems. This property is checked by proving the ‐sectoriality of the corresponding operators, which in turn is obtained by a perturbation method.