Global existence and convergence rates of smooth solutions for the compressible magnetohydrodynamic equations

In this paper, we are concerned with the global existence and convergence rates of the smooth solutions for the compressible magnetohydrodynamic equations in R³. We prove the global existence of the smooth solutions by the standard energy method under the condition that the initial data are close to the constant equilibrium state in H³-framework. Moreover, if additionally the initial data belong to L^p with , the optimal convergence rates of the solutions in L^q-norm with 2≤q≤6 and its spatial derivatives in L²-norm are obtained.     

Compactness of weak solutions to the three-dimensional compressible magnetohydrodynamic equations

The compactness of weak solutions to the magnetohydrodynamic equations for the viscous, compressible, heat conducting fluids is considered in both the three-dimensional space R³ and the three-dimensional periodic domains. The viscosities, the heat conductivity as well as the magnetic coefficient are allowed to depend on the density, and may vanish on the vacuum. This paper provides a different idea from [X. Hu, D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys. (2008), in press] to show the compactness of solutions of viscous, compressible, heat conducting magnetohydrodynamic flows, derives a new entropy identity, and shows that the limit of a sequence of weak solutions is still a weak solution to the compressible magnetohydrodynamic equations.     

Global existence of solutions to the compressible Navier-Stokes equation around parallel flows

The initial boundary value problem for the compressible Navier-Stokes equation is considered in an infinite layer of Rn. It is proved that if n?3, then strong solutions to the compressible Navier-Stokes equation around parallel flows exist globally in time for sufficiently small initial perturbations, provided that the Reynolds and Mach numbers are sufficiently small. The proof is given by a variant of the Matsumura-Nishida energy method based on a decomposition of solutions associated with a spectral property of the linearized operator.     

Global existence and asymptotic behavior of solutions to the nonisentropic bipolar hydrodynamic models

In this paper, we consider a one-dimensional bipolar nonisentropic hydrodynamical model from semiconductor devices. This system takes the nonisentropic Euler-Poisson form with electric field and frictional damping added to the momentum equations. First, we prove global existence of smooth solutions to the Cauchy problem. Next, we also discuss the asymptotic behavior of the smooth solutions. We find that in large time, the densities of electron and hole tend to the same nonlinear diffusive wave, the momentums tend to the Darcy's law, and the temperatures tend to the ambient device temperature. Finally, we can obtain the algebraic decay rate of the densities to the same nonlinear diffusive wave, the momentums to the Darcy's law and the temperatures to the ambient device temperature, and the exponential decay of their difference and the electric field to zero. We can show our results by precise energy methods.     

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 well-posedness of the hydrodynamic model for two-carrier plasmas

We prove results on the global well-posedness of the hydrodynamic model for two-carrier plasmas in whole space and periodic domain. We remove a technical condition which was first introduced by Alì and Jüngel [2] and developed in  and  to deal with the difficulty mainly arising from complicated coupling and cancellation between two carriers. The proofs depend on a result on continuity for compositions in Chemin–Lerner spaces and an elementary fact which indicates the connection between homogeneous and inhomogeneous Chemin–Lerner spaces.     

Regularity criteria for the 3D generalized MHD equations in terms of vorticity

We prove regularity criteria for the 3D generalized MHD equations. These criteria impose assumptions on the vorticity only. In addition, we also prove a result of global existence for smooth solution under some special conditions.     

On existence of global solutions to the Navier-Stokes equations for compressible and viscous flows on the surface of a sphere

Following tecniques proposed by A. V. Khazikhov and V. A. Weigant in 1995, we prove the global, with respect to time, existence and uniqueness of the solution to the Navier-Stokes equations for a compressible, viscous and barotropic fluid which moves on the surface of a sphere. In obtaining the main estimates we make use of the Hodge decomposition and the generalized potential theory due to K. Kodaira.

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Sunto Seguendo le tecniche proposte da A. V. Khazikhov and V. A. Weigant nel 1995, si prova l'esistenza ed unicità della soluzione per le equazioni di Navier-Stokes per un fluido comprimibile, viscoso e barotropico che si muova sulla superficie di una sfera. Nell'ottenere le principali stime si utilizzano la decomposizione di Hodge e la teoria del potenziale generalizzato, dovuta a K. Kodaira.

     

Local strong solution to the compressible viscoelastic flow with large data

The existence and uniqueness of local in time strong solution with large initial data for the three-dimensional compressible viscoelastic flow is established. The strong solution has weaker regularity than the classical solution. The Lax-Milgram theorem and the Schauder-Tychonoff fixed-point argument are applied.     

On weighted-norm estimates for nonstationary incompressible Navier-Stokes flows in a 3D exterior domain

We study the time-decay of weighted norms of weak and strong solutions to the Navier-Stokes equations in a 3D exterior domain. Moment estimates for weak solutions and weighted Lq-estimates for strong solutions are deduced, both of which seem to be optimal. The relation is discussed between the space-time decay and the vanishing of the total net force exerted by the fluid to the body. A class of initial data is given so that the total net force associated to the corresponding fluid flows does not vanish.     

On cylindrical symmetric flows through pipe-like domains

We prove existence of cylindrical symmetric solutions to the steady Navier-Stokes equations in bounded pipe-like domains in with the slip boundary conditions. The result is shown for any large flows across the boundary assuming only a geometrical constraint on the shape of the domain which is independent of data. The simply connectedness of the domain is not required. The technique is based on a reformulation of the original problem and delivers us a new type of estimates in the Hölder spaces for this class of the solutions.     

On the temporal decay for the Hall-magnetohydrodynamic equations

We establish temporal decay estimates for weak solutions to the Hall-magnetohydrodynamic equations. With these estimates in hand we obtain algebraic time decay for higher order Sobolev norms of small initial data solutions.     

A note on the regularity of flows with shear-dependent viscosity

We consider a non-Newtonian fluid governed by stationary, incompressible Navier–Stokes equations with shear-dependent viscosity. Using a fixed point argument in an appropriate functional setting, we establish the existence of a strong solution for small and suitably regular data. Uniqueness results are obtained under similar conditions.     

A blow-up criterion for 3D compressible viscoelastic flow with large initial data

In this paper, we consider a Cauchy problem for the three-dimensional compressible viscoelastic flow with large initial data. We establish a blow-up criterion for the strong solutions in terms of the gradient of velocity only, which is similar to the Beale-Kato-Majda criterion for ideal incompressible flow (cf. Beale et al. (1984) [20]) and the blow-up criterion for the compressible Navier-Stokes equations (cf. Huang et al. (2011) [21]).     

Global solutions of the super-critical 2D quasi-geostrophic equation in Besov spaces

In this paper we study the super-critical 2D dissipative quasi-geostrophic equation. We obtain some regularization effects allowing us to prove a global well-posedness result for small initial data lying in critical Besov spaces constructed over Lebesgue spaces Lp, with p∈[1,∞]. Local results for arbitrary initial data are also given.     

Low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data

The low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data is rigorously justified in the whole space R³${\mathbb{R}}^3$. First, the uniform-in-Mach-number estimates of the solutions in a Sobolev space are established on a finite time interval independent of the Mach number. Then the low Mach number limit is proved by combining these uniform estimate with a theorem due to Métivier and Schochet (2001) [45] for the Euler equations that gives the local energy decay of the acoustic wave equations.     

A blow-up criterion for compressible viscous heat-conductive flows

We study an initial boundary value problem for the three-dimensional Navier–Stokes equations of viscous heat-conductive fluids in a bounded smooth domain. We establish a blow-up criterion for the local strong solutions in terms of the temperature and the gradient of velocity only, similar to the Beale–Kato–Majda criterion for ideal incompressible flows.     

Dynamical behaviors for 1D compressible Navier-Stokes equations with density-dependent viscosity

The dynamical behaviors of vacuum states for one-dimensional compressible Navier-Stokes equations with density-dependent viscosity coefficient are considered. It is first shown that a unique strong solution to the free boundary value problem exists globally in time, the free boundary expands outwards at an algebraic rate in time, and the density is strictly positive in any finite time but decays pointwise to zero time-asymptotically. Then, it is proved that there exists a unique global weak solution to the initial boundary value problem when the initial data contains discontinuously a piece of continuous vacuum and is regular away from the vacuum. The solution is piecewise regular and contains a piece of continuous vacuum before the time T_∗>0, which is compressed at an algebraic rate and vanishes at the time T_∗, meanwhile the weak solution becomes either a strong solution or a piecewise strong one and tends to the equilibrium state exponentially.