Some limit analysis of a three dimensional viscous compressible capillary model for plasma

In this study, we discuss some limit analysis of a viscous capillary model of plasma, which is expressed as a so‐called the compressible Navier‐Stokes‐Poisson‐Korteweg equation. First, the existence of global smooth solutions for the initial value problem to the compressible Navier‐Stokes‐Poisson‐Korteweg equation with a given Debye length λ and a given capillary coefficient κ is obtained. We also show the uniform estimates of global smooth solutions with respect to the Debye length λ and the capillary coefficient κ. Then, from Aubin lemma, we show that the unique smooth solution of the 3‐dimensional Navier‐Stokes‐Poisson‐Korteweg equations converges globally in time to the strong solution of the corresponding limit equations, as λ tends to zero, κ tends to zero, and λ and κ simultaneously tend to zero. Moreover, we also give the convergence rates of these limits for any given positive time one by one.     

Existence of global weak solutions for Navier‐Stokes‐Poisson equations with quantum effect and convergence to incompressible Navier‐Stokes equations

In this paper, we consider a three dimensional quantum Navier‐Stokes‐Poisson equations. Existence of global weak solutions is obtained, and convergence toward the classical solution of the incompressible Navier‐Stokes equation is rigorously proven for well prepared initial data. Furthermore, the associated convergence rates are also obtained. Copyright © 2014 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.     

Semi‐invariant solutions of the Navier‐Stokes equations

We present new exact solutions and reduced differential systems of the Navier‐Stokes equations of incompressible viscous fluid flow. We apply the method of semi‐invariant manifolds, introduced earlier as a modification of the Lie invariance method. We show that many known solutions of the Navier‐Stokes equations are, in fact, semi‐invariant and that the reduced differential systems we derive using semi‐invariant manifolds generalize previously obtained results that used ad hoc methods. Many of our semi‐invariant solutions solve decoupled systems in triangular form that are effectively linear. We also obtain several new reductions of Navier‐Stokes to a single nonlinear partial differential equation. In some cases, we can solve reduced systems and generate new analytic solutions of the Navier‐Stokes equations or find their approximations, and physical interpretation.     

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.     

Incompressible limit of all-time solutions to 3-D full Navier–Stokes equations for perfect gas with well-prepared initial condition

In this paper, we prove the incompressible limit of all-time strong solutions to the three-dimensional full compressible Navier–Stokes equations. Here the velocity field and temperature satisfy the Dirichlet boundary condition and convective boundary condition, respectively. The uniform estimates in both the Mach number \({\epsilon\in(0,\overline{\epsilon}]}\) and time \({t\in[0,\infty)}\) are established by deriving a differential inequality with decay property, where \({\overline{\epsilon}\in(0,1]}\) is a constant. Based on these uniform estimates, the global solution of full compressible Navier–Stokes equations with “well-prepared” initial conditions converges to the one of isentropic incompressible Navier–Stokes equations as the Mach number goes to zero.     

Stability and convergence of efficient Navier‐Stokes solvers via a commutator estimate

For strong solutions of the incompressible Navier‐Stokes equations in bounded domains with velocity specified at the boundary, we establish the unconditional stability and convergence of discretization schemes that decouple the updates of pressure and velocity through explicit time stepping for pressure. These schemes require no solution of stationary Stokes systems, nor any compatibility between velocity and pressure spaces to ensure an inf‐sup condition, and are representative of a class of highly efficient computational methods that have recently emerged. The proofs are simple, based upon a new, sharp estimate for the commutator of the Laplacian and Helmholtz projection operators. This allows us to treat an unconstrained formulation of the Navier‐Stokes equations as a perturbed diffusion equation. © 2007 Wiley Periodicals, Inc.     

Vanishing viscosity limit of the Navier‐Stokes equations to the euler equations for compressible fluid flow

We establish the vanishing viscosity limit of the Navier‐Stokes equations to the isentropic Euler equations for one‐dimensional compressible fluid flow. For the Navier‐Stokes equations, there exist no natural invariant regions for the equations with the real physical viscosity term so that the uniform sup‐norm of solutions with respect to the physical viscosity coefficient may not be directly controllable. Furthermore, convex entropy‐entropy flux pairs may not produce signed entropy dissipation measures. To overcome these difficulties, we first develop uniform energy‐type estimates with respect to the viscosity coefficient for solutions of the Navier‐Stokes equations and establish the existence of measure‐valued solutions of the isentropic Euler equations generated by the Navier‐Stokes equations. Based on the uniform energy‐type estimates and the features of the isentropic Euler equations, we establish that the entropy dissipation measures of the solutions of the Navier‐Stokes equations for weak entropy‐entropy flux pairs, generated by compactly supported C² test functions, are confined in a compact set in H^?1, which leads to the existence of measure‐valued solutions that are confined by the Tartar‐Murat commutator relation. A careful characterization of the unbounded support of the measure‐valued solution confined by the commutator relation yields the reduction of the measurevalued solution to a Dirac mass, which leads to the convergence of solutions of the Navier‐Stokes equations to a finite‐energy entropy solution of the isentropic Euler equations with finite‐energy initial data, relative to the different end‐states at infinity. © 2010 Wiley Periodicals, Inc.     

A two‐level finite element method for the stationary Navier‐Stokes equations based on a stabilized local projection

In this article, we propose a two‐level finite element method to analyze the approximate solutions of the stationary Navier‐Stokes equations based on a stabilized local projection. The local projection allows to circumvent the Babuska‐Brezzi condition by using equal‐order finite element pairs. The local projection can be used to stabilize high equal‐order finite element pairs. The proposed method combines the local projection stabilization method and the two‐level method under the assumption of the uniqueness condition. The two‐level method consists of solving a nonlinear equation on the coarse mesh and solving a linear equation on fine mesh. The nonlinear equation is solved by the one‐step Newtonian iteration method. In the rest of this article, we show the error analysis of the lowest equal‐order finite element pair and provide convergence rate of approximate solutions. Furthermore, the numerical illustrations coincide with the theoretical analysis expectations. From the view of computational time, the results show that the two‐level method is effective to solve the stationary Navier‐Stokes equations. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Stability of the rarefaction wave for a coupled compressible Navier‐Stokes/Allen‐Cahn system

In this paper, we are concerned with the large time behavior of solutions to the Cauchy problem for the one dimensional Navier‐Stokes/Allen‐Cahn system. Motivated by the relationship between the Navier‐Stokes/Allen‐Cahn system and the Navier‐Stokes system, we can prove that the solutions to the one‐dimensional compressible Navier‐Stokes/Allen‐Cahn system tend time‐asymptotically to the rarefaction wave, where the strength of the rarefaction wave is not required to be small. The proof is mainly based on a basic energy method.     

Low Mach number limit for a model of radiative flow

We consider a simplified model of compressible Navier–Stokes–Fourier coupled to the radiative transfer equation introduced by Seaïd, Teleaga and al., and we study its low Mach number limit. We prove the convergence toward the incompressible Navier–Stokes system coupled to a system of two stationary transport equations.     

Vanishing viscosity and Debye-length limit to rarefaction wave with vacuum for the 1D bipolar Navier–Stokes–Poisson equation

In this paper, we consider the one-dimensional (1D) compressible bipolar Navier–Stokes–Poisson equations. We know that when the viscosity coefficient and Debye length are zero in the compressible bipolar Navier–Stokes–Poisson equations, we have the compressible Euler equations. Under the case that the compressible Euler equations have a rarefaction wave with one-side vacuum state, we can construct a sequence of the approximation solution to the one-dimensional bipolar Navier–Stokes–Poisson equations with well-prepared initial data, which converges to the above rarefaction wave with vacuum as the viscosity and the Debye length tend to zero. Moreover, we also obtain the uniform convergence rate. The results are proved by a scaling argument and elaborate energy estimate.     

Zero‐viscosity limit of the linearized Navier‐Stokes equations for a compressible viscous fluid in the half‐plane

The zero‐viscosity limit for an initial boundary value problem of the linearized Navier‐Stokes equations of a compressible viscous fluid in the half‐plane is studied. By means of the asymptotic analysis with multiple scales, we first construct an approximate solution of the linearized problem of the Navier‐Stokes equations as the combination of inner and boundary expansions. Next, by carefully using the technique on energy methods, we show the pointwise estimates of the error term of the approximate solution, which readily yield the uniform stability result for the linearized Navier‐Stokes solution in the zero‐viscosity limit. © 1999 John Wiley & Sons, Inc.     

A Liouville theorem for the compressible Navier‐Stokes equations

This paper is concerned with 3‐dimensional steady compressible Navier‐Stokes equations. A Liouville‐type theorem is proved when some suitable conditions are satisfied.     

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.     

A regularity condition of strong solutions to the two‐dimensional equations of compressible nematic liquid crystal flows

This paper is concerned with the short time strong solutions for Cauchy problem to a simplified Ericksen–Leslie system of compressible nematic liquid crystals in two dimensions with vacuum as far field density. We establish a blow‐up criterion for possible breakdown of such solutions at a finite time, which is analogous to the well‐known Serrin's blow‐up criterion for the incompressible Navier–Stokes equations. Copyright © 2016 John Wiley & Sons, Ltd.     

Incompressible limit of the degenerate quantum compressible Navier–Stokes equations with general initial data

In this paper we study the incompressible limit of the degenerate quantum compressible Navier–Stokes equations in a periodic domain ${\mathbb{T}}^3$ and the whole space ${\mathbb{R}}^3$ with general initial data. In the periodic case, by applying the refined relative entropy method and carrying out the detailed analysis on the oscillations of velocity, we prove rigorously that the gradient part of the weak solutions (velocity) of the degenerate quantum compressible Navier–Stokes equations converge to the strong solution of the incompressible Navier–Stokes equations. Our results improve considerably the ones obtained by Yang, Ju and Yang [25] where only the well-prepared initial data case is considered. While for the whole space case, thanks to the Strichartz's estimates of linear wave equations, we can obtain the convergence of the weak solutions of the degenerate quantum compressible Navier–Stokes equations to the strong solution of the incompressible Navier–Stokes/Euler equations with a linear damping term. Moreover, the convergence rates are also given.     

Analysis of nonlocal model of compressible fluid in 1‐D

We study a nonlocal modification of the compressible Navier–Stokes equations in mono‐dimensional case with a boundary condition characteristic for the free boundaries problem. From the formal point of view, our system is an intermediate between the Euler and Navier–Stokes equations. Under certain assumptions, imposed on initial data and viscosity coefficient, we obtain the local and global existence of solutions. Particularly, we show the uniform in time bound on the density of fluid. Copyright © 2010 John Wiley & Sons, Ltd.     

Local well‐posedness and blow‐up criterion for a compressible Navier‐Stokes‐Fourier‐P1 approximate model arising in radiation hydrodynamics

We establish a local well‐posedness and a blow‐up criterion of strong solutions for the compressible Navier‐Stokes‐Fourier‐P1 approximate model arising in radiation hydrodynamics. For the local well‐posedness result, we do not need the assumption on the positivity of the initial density and it may vanish in an open subset of the domain.     

Two‐level Newton iterative method for the 2D/3D steady Navier‐Stokes equations

A combination method of the Newton iteration and two‐level finite element algorithm is applied for solving numerically the steady Navier‐Stokes equations under the strong uniqueness condition. This algorithm is motivated by applying the m Newton iterations for solving the Navier‐Stokes problem on a coarse grid and computing the Stokes problem on a fine grid. Then, the uniform stability and convergence with respect to ν of the two‐level Newton iterative solution are analyzed for the large m and small H and h << H. Finally, some numerical tests are made to demonstrate the effectiveness of the method. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012