Global properties of solutions to 1D-viscous compressible barotropic fluid equations with density dependent viscosity

The Navier-Stokes equations for compressible barotropic fluid in 1D with the mass force under zero velocity boundary conditions are studied. We prove the uniform upper and lower bounds for the density as well as the uniform in time L ²()-estimates for _x and u _x (u is the velocity). Moreover, a collection of the decay rate estimates for - (with being the stationary density) and u in ²()-norm and H ¹()-norm as time t are established. The results are given for general state function p() (but mainly monotone) and viscosity coefficient µ() of arbitrarily fast (or slow) growth as well as for the large data.     

Global weak solutions and asymptotic behavior to 1D compressible Navier-Stokes equations with density-dependent viscosity and vacuum

This paper is concerned with existence of global weak solutions to a class of compressible Navier-Stokes equations with density-dependent viscosity and vacuum. When the viscosity coefficient μ is proportional to ρθ with , a global existence result is obtained which improves the previous results in Fang and Zhang (2004) [4], Vong et al. (2003) [27], Yang and Zhu (2002) [30]. Here ρ is the density. Moreover, we prove that the domain, where fluid is located on, expands outwards into vacuum at an algebraic rate as the time grows up due to the dispersion effect of total pressure. It is worth pointing out that our result covers the interesting case of the Saint-Venant model for shallow water (i.e., θ=1, γ=2).     

Global weak solutions to a generalized Benjamin-Bona-Mahony-Burgers equation

The existence of global weak solutions for a generalized Benjamin-Bona-MahonyBurgers equation is established in the space C([0, ∞) × R) ∩ L~∞([0, ∞); H1(R)) under the condition that its initial value belongs to the space H1(R). A one-sided super bound estimate and a space-time higher-norm estimate on the first order derivatives of the solution with respect to the space variable are derived to prove the existence.     

Global existence of weak and classical solutions for the Navier-Stokes-Vlasov-Fokker-Planck equations

We consider a system coupling the incompressible Navier-Stokes equations to the Vlasov-Fokker-Planck equation. The coupling arises from a drag force exerted by each other. We establish existence of global weak solutions for the system in two and three dimensions. Furthermore, we obtain the existence and uniqueness result of global smooth solutions for dimension two. In case of three dimensions, we also prove that strong solutions exist globally in time for the Vlasov-Stokes system.     

Global variational solutions to the compressible magnetohydrodynamic equations

We prove the existence of globally defined variational solutions to the compressible magnetohydrodynamic (MHD) equations with the coefficients depending on the temperature. As a by-product, we give a simple proof for the nonexistence of nontrivial weak time-periodic solutions by the entropy principle of Clausius–Duhem and a new Poincaré-type inequality.     

Global smooth solutions for a one-dimensional nonisentropic hydrodynamic model with non-constant lattice temperature

In this paper, a one-dimensional nonisentropic hydrodynamic model for semiconductors with non-constant lattice temperature is studied. The model is self-consistent in the sense that the electric field, which forms a forcing term in the momentum equation, is determined by the coupled Poisson equation. The existence and uniqueness of the corresponding stationary solutions are investigated carefully under proper conditions. Then, global existence of the smooth solutions for the Cauchy problem with initial data, which are perturbations of stationary solutions, is established. It is shown that these smooth solutions tend to the stationary solutions exponentially fast as t → ∞.      

Behaviour of weak solutions of compressible Navier-Stokes equations for isothermal fluids with a nonlinear stress tensor

The aim of this paper is to study the behaviour of a weak solution to Navier-Stokes equations for isothermal fluids with a nonlinear stress tensor for time going to infinity. In an analogous way as in [18], we construct a suitable function which approximates the density for time going to infinity. Using properties of this function, we can prove the strong convergence of the density to its limit state. The behaviour of the velocity field and kinetic energy is mentioned as well.     

Global solution to the one-dimensional equations for a self-gravitating viscous radiative and reactive gas

In this paper we consider a system of equations describing a motion of a self-gravitating one-dimensional gaseous medium in the presence of radiation and reacting process. By introducing Lagrangian mass coordinate, this free-boundary problem is reduced to the problem in a fixed domain with an explicit gravitational term. Based on the fundamental local existence result and a priori estimates, we can construct a classical unique global solution.     

Formation of singularities in solutions to the compressible radiation hydrodynamics equations with vacuum

We study the Cauchy problem for multi-dimensional compressible radiation hydrodynamics equations with vacuum. First, we present some sufficient conditions on the blow-up of smooth solutions in multi-dimensional space. Then, we obtain the invariance of the support of density for the smooth solutions with compactly supported initial mass density by the property of the system under the vacuum state. Based on the above-mentioned results, we prove that we cannot get a global classical solution, no matter how small the initial data are, as long as the initial mass density is of compact support. Finally, we will see that some of the results that we obtained are still valid for the isentropic flows with degenerate viscosity coefficients as well as for one-dimensional case.     

Convergence from an electromagnetic fluid system to the full compressible MHD equations

We are concerned with the zero dielectric constant limit for the full electro-magneto-fluid dynamics in this article. This singular limit is justified rigorously for global smooth solution for both well-prepared and ill-prepared initial data. The explicit convergence rate is also obtained by a elaborate energy estimate. Moreover, we show that for the well-prepared initial data, there is no initial layer, and the electric field always converges strongly to the limit function. While for the ill-prepared data case, there will be an initial layer near t=0. The strong convergence results only hold outside the initial layer.     

Global existence of solutions for one-dimensional compressible navier-stokes equations in the half space

We prove the existence of global solutions to the initial-boundary-value problem on the half space R＋ for a one-dimensional viscous ideal polytropic gas. Some suitable assumptions are made to guarantee the existence of smooth solutions. Employing the L2- energy estimate, we prove that the impermeable problem has a unique global solutionis.     

Global existence and asymptotic convergence of weak solutions for the one-dimensional Navier-Stokes equations with capillarity and nonmonotonic pressure

We construct global weak solution of the Navier-Stokes equations with capillarity and nonmonotonic pressure. The volume variable v₀ is initially assumed to be in H¹ and the velocity variable u₀ to be in L² on a finite interval [0,1]. We show that both variables become smooth in positive time and that asymptotically in time u→0 strongly in L²([0,1]) and v approaches the set of stationary solutions in H¹([0,1]).     

Partial regularity of suitable weak solutions to the four‐dimensional incompressible magneto‐hydrodynamic equations

In this paper, we study the partial regularity of suitable weak solutions to the incompressible magneto‐hydrodynamic equations in dimension four by borrowing and improving the arguments given by Caffarelli, Kohn, and Nirenberg for incompressible Navier–Stokes equations. The so‐called ε‐regularity criteria are established for suitable weak solutions. As an application, an estimate on Hausdorff dimension of the possible singular points set for a suitable weak solution is given. Finally, we present further information on distribution of the possible singular points if the given initial data decay sufficiently rapidly or are not too singular at the origin, in some sense. Copyright © 2012 John Wiley & Sons, Ltd.     

Trivial stationary solutions to the Kuramoto-Sivashinsky and certain nonlinear elliptic equations

We show that the only locally integrable stationary solutions to the integrated Kuramoto-Sivashinsky equation in R and R² are the trivial constant solutions. We extend our technique and prove similar results to other nonlinear elliptic problems in RN.     

Global entropy solutions to the relativistic Euler equations for a class of large initial data

We are concerned with global entropy solutions to the relativistic Euler equations for a class of large initial data which involve the interaction of shock waves and rarefaction waves. We first carefully analyze the global behavior of the shock curves, the rarefaction wave curves, and their corresponding inverse curves in the phase plane. Based on these analyses, we use the Glimm scheme to construct global entropy solutions to the relativistic Euler equations for the class of large discontinuous initial data.Received: May 23, 2004     

Local regularity criteria of a suitable weak solution to MHD equations

We present a regularity condition of a suitable weak solution to the MHD equations in three dimensional space with slip boundary conditions for a velocity and magnetic vector fields. More precisely, we prove a suitable weak solution are H¨older continuous near boundary provided that the scaled mixed L_(x,t)~(p,q) -norm of the velocity vector field with 3/p + 2/q ≤ 2,2 q ∞ is sufficiently small near the boundary. Also, we will investigate that for this solution u ∈ L_(x,t)~(p,q) with 1≤3/p+2/q≤3/2, 3 p ∞, the Hausdorff dimension of its singular set is no greater than max{p, q}(3/p+2/q-1).     

Asymptotic stability of a composite wave of two viscous shock waves for the one-dimensional radiative Euler equations

This paper is devoted to the study of the wellposedness of the radiative Euler equations. By employing the anti-derivative method, we show the unique global-in-time existence and the asymptotic stability of the solutions of the radiative Euler equations for the composite wave of two viscous shock waves with small strength. This method developed here is also helpful to other related problems with similar analytical difficulties.     

Extension criterion on regularity for weak solutions to the 3D MHD equations

In this paper, we consider the regularity criteria for weak solutions to the 3D incompressible magnetohydrodynamic equations and prove some regularity criteria which are related only with u+B or u?B. This is an improvement of the result given by He and Wang (J. Differential Equations 2007; 238:1–17; Math. Meth. Appl. Sci. 2008; 31:1667–1684) and He and Xin (J. Differential Equations 2005; 213(2):235–254). Copyright © 2010 John Wiley & Sons, Ltd.