黏性系数依赖密度型可压缩Navier-Stokes 方程的零耗散极限问题

本文考虑黏性系数依赖密度的可压缩Navier-Stokes 方程解的零耗散极限问题. 假定Euler 方程的稀疏波解一端被真空状态连接, 我们证明Navier-Stokes 方程存在一列(依赖黏性的) 整体解, 且随着粘性的消失, 此整体解逐渐稳定于Euler 方程对应的稀疏波解和真空状态; 并且得到了一致衰减率估计. 此结果推广了常黏性系数的情形.     

Zero dissipation limit to strong contact discontinuity for the 1-D compressible Navier-Stokes equations

In this paper, we study the zero dissipation limit problem for the one-dimensional compressible Navier-Stokes equations. We prove that if the solution of the inviscid Euler equations is piecewise constants with a contact discontinuity, then there exist smooth solutions to the Navier-Stokes equations which converge to the inviscid solution away from the contact discontinuity at a rate of as the heat-conductivity coefficient κ tends to zero, provided that the viscosity μ is of higher order than the heat-conductivity κ. Without loss of generality, we set μ≡0. Here we have no need to restrict the strength of the contact discontinuity to be small.     

Global regularity for d-dimensional micropolar equations with fractional dissipation

This paper is devoted to the global in time existence of classical solutions to the d-Dimensional (dD) micropolar equations with fractional dissipation. Micropolar equations model a class of fluids with nonsymmetric stress tensor such as fluids consisting of particles suspended in a viscous medium. It remains unknown whether or not smooth solutions of the classical 3D micropolar equations can develop finite-time singularities. The purpose here is to explore the global regularity of solutions for dD micropolar equations under the smallest amount of dissipation. We establish the global regularity for two important fractional dissipation cases. Direct energy estimates are not sufficient to obtain the desired global a priori bounds in each case. To overcome the difficulties, we employ the Besov space techniques.     

Zero dissipation limit to rarefaction waves for the one-dimensional navier-stokes equations of compressible isentropic gases

We study the zero dissipation limit problem for the one-dimensional Navier-Stokes equations of compressible, isentropic gases in the case that the corresponding Euler equations have rarefaction wave solutions. We prove that the solutions of the Navier-Stokes equations with centered rarefaction wave data exist for all time, and converge to the centered rarefaction waves as the viscosity vanishes, uniformly away from the initial discontinuities. In the case that either the effects of initial layers are ignored or the rarefaction waves are smooth, we then obtain a rate of convergence which is valid uniformly for all time. Our method of proof consists of a scaling argument and elementary energy analysis, based on the underlying wave structure. © 1993 John Wiley & Sons, Inc.     

Zero relaxation limit to centered rarefaction waves for Jin-Xin relaxation system

In this paper, we study the zero relaxation limit problem for the following Jin-Xin relaxation system

(E)     

Zero dissipation limit of the compressible heat-conducting navier-stokes equations in the presence of the shock

The zero dissipation limit of the compressible heat-conducting Navier–Stokes equations in the presence of the shock is investigated. It is shown that when the heat conduction coefficient κ and the viscosity coefficient ε satisfy κ = O（ε）, κ/ε≥ c 〉 0, as ε→ 0 （see （1.3））, if the solution of the corresponding Euler equations is piecewise smooth with shock wave satisfying the Lax entropy condition, then there exists a smooth solution to the Navier–Stokes equations, which converges to the piecewise smooth shock solution of the Euler equations away from the shock discontinuity at a rate of ε. The proof is given by a combination of the energy estimates and the matched asymptotic analysis introduced in [3].     

Stability of boundary layer and rarefaction wave to an outflow problem for compressible Navier-Stokes equations under large perturbation

In this paper, we investigate the large-time behavior of solutions to an outflow problem for compressible Navier-Stokes equations. In 2003, Kawashima, Nishibata and Zhu [S. Kawashima, S. Nishibata, P. Zhu, Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space, Comm. Math. Phys. 240 (2003) 483-500] showed there exists a boundary layer (i.e., stationary solution) to the outflow problem and the boundary layer is nonlinearly stable under small initial perturbation. In the present paper, we show that not only the boundary layer above but also the superposition of a boundary layer and a rarefaction wave are stable under large initial perturbation. The proofs are given by an elementary energy method.     

The diffusive relaxation limit of non-isentropic Euler-Maxwell equations for plasmas

This paper concerns the non-isentropic Euler-Maxwell equations for plasmas with short momentum relaxation time. With the help of the Maxwell-type iteration, it is obtained that, as the relaxation time tends to zero, periodic initial-value problem of certain scaled non-isentropic Euler-Maxwell equations has unique smooth solutions existing in the time interval where the corresponding classical drift-diffusion model has smooth solutions. Meanwhile, we justify a formal derivation of the corresponding drift-diffusion model from the non-isentropic Euler-Maxwell equations.     

Delta-shocks and vacuum states in the vanishing pressure limit of solutions to the isentropic Euler equations for modified Chaplygin gas

The phenomena of concentration and cavitation and the formation of δ-shocks and vacuum states in solutions to the isentropic Euler equations for a modified Chaplygin gas are analyzed as the double parameter pressure vanishes. Firstly, the Riemann problem of the isentropic Euler equations for a modified Chaplygin gas is solved analytically. Secondly, it is rigorously shown that, as the pressure vanishes, any two-shock Riemann solution to the isentropic Euler equations for a modified Chaplygin gas tends to a δ-shock solution to the transport equations, and the intermediate density between the two shocks tends to a weighted δ-measure that forms the δ-shock; any two-rarefaction-wave Riemann solution to the isentropic Euler equations for a modified Chaplygin gas tends to a two-contact-discontinuity solution to the transport equations, the nonvacuum intermediate state between the two rarefaction waves tends to a vacuum state. Finally, some numerical results exhibiting the formation of δ-shocks and vacuum states are presented as the pressure decreases.     

The 2D magneto‐micropolar equations with partial dissipation

We study the global existence and regularity of classical solutions to the 2D incompressible magneto‐micropolar equations with partial dissipation. We establish the global regularity for one partial dissipation case. The proofs of our main results rely on anisotropic Sobolev type inequalities and suitable combination and cancellation of terms.     

Solution representations for a wave equation with weak dissipation

We consider the Cauchy problem for the weakly dissipative wave equation □v+μ/1+t v_t=0, x∈?ⁿ, t≥0 parameterized by μ>0, and prove a representation theorem for its solutions using the theory of special functions. This representation is used to obtain L_p–L_q estimates for the solution and for the energy operator corresponding to this Cauchy problem. Especially for the L₂ energy estimate we determine the part of the phase space which is responsible for the decay rate. It will be shown that the situation depends strongly on the value of μand that μ=2 is critical. Copyright © 2004 John Wiley & Sons, Ltd.     

Low Mach number limit for the non-isentropic Navier-Stokes equations

We study the low Mach number limit of the local in time solutions to the compressible Navier-Stokes equations with zero heat conductivity coefficient as the Mach number tends to zero. A uniform existence result for the one-dimensional initial-boundary value problem is proved provided that the initial data are “well-prepared” in the sense that the temporal derivatives up to order two are bounded initially.     

On potential wells and vacuum isolating of solutions for semilinear wave equations

In this paper, we study the initial boundary value problem of semilinear wave equations:

     

A logarithmically improved blow-up criterion for smooth solutions to the 3D micropolar fluid equations

In this paper, the Cauchy problem for the 3D micropolar fluid equations is investigated. A new logarithmically improved blow-up criterion for the 3D micropolar fluid equations in an appropriate homogeneous Besov space is established.     

Zero Dissipation Limit to Rarefaction Waves for the ρ-System

We study the zero-dissipation problem for a one-dimensional model system for the isentropic flow of a compressible viscous gas, the so-called p-system with viscosity. When the solution of the inviscid problem is a rarefaction wave with finite strength, there exists unique solution to the viscous problem with the same initial data which converges to the given inviscid solution as c goes to zero. The proof consists of a scaling argument and elementary energy analysis, based on the underlying wave structure.     

Stability of the planar rarefaction wave to two-dimensional Navier-Stokes-Korteweg equations of compressible fluids

This study is concerned with the large time behavior of the two-dimensional compressible Navier-Stokes-Korteweg equations, which are used to model compressible fluids with internal capillarity. Based on the fact that the rarefaction wave, one of the basic wave patterns to the hyperbolic conservation laws is nonlinearly stable to the one-dimensional compressible Navier-Stokes-Korteweg equations, the planar rarefaction wave to the two-dimensional compressible Navier-Stokes-Korteweg equations is first derived. Then, it is shown that the planar rarefaction wave is asymptotically stable in the case that the initial data are suitably small perturbations of the planar rarefaction wave. The proof is based on the delicate energy method. This is the first stability result of the planar rarefaction wave to the multi-dimensional viscous fluids with internal capillarity.     

Inverse problem for a class of one-dimensional wave equations with piecewise constant coefficients

We consider the problem of reconstructing the piecewise constant coefficient of a one-dimensional wave equation on the halfline from the knowledge of the displacement on the boundary caused by an impulse at time zero. This problem is formulated as a nonlinear optimization problem. The objective function of this optimization problem has several special features that have been exploited in building an ad hoc optimization method. The optimization method is based on the solution of a nonlinear system of equations by an algorithm consisting of the evaluation of the unknowns one by one.The research of the third author has been made possible through the support and sponsorship of the Italian Government through the Ministero Pubblica Istruzione under Contract M.P.I. 60% 1987 at the Università di Roma—La Sapienza.