On partial regularity for weak solutions to the Navier-Stokes equations

In this paper, we study the partial regularity of the general weak solution u∈L∞(0,T;L2(Ω))∩L2(0,T;H1(Ω)) to the Navier-Stokes equations, which include the well-known Leray-Hopf weak solutions. It is shown that there is a absolute constant ε such that for the weak solution u, if either the scaled local L^q(1?q?2) norm of the gradient of the solution, or the scaled local ) norm of u is less than ε, then u is locally bounded. This implies that the one-dimensional Hausdorff measure is zero for the possible singular point set, which extends the corresponding result due to Caffarelli et al. (Comm. Pure Appl. Math. 35 (1982) 717) to more general weak solution.     

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.     

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 weak solutions for two-dimensional semilinear wave equations with strong damping in an exterior domain

We consider two-dimensional mixed problems in an exterior domain for a semilinear strongly damped wave equation with a power-type nonlinearity |u|^p${\left|u\right|}^p$. If the initial data have a small weighted energy, we shall derive a global existence and energy decay results in the case when the power p$p$ of the nonlinear term satisfies p>6$p>6$.     

Asymptotic energy concentration in the phase space of the weak solutions to the Navier-Stokes equations

We study the asymptotic behavior of the energy of weak solutions of Navier-Stokes equations as t→∞. We characterize the space of the initial data which causes a concentration of the kinetic energy in the phase space. Moreover, an explicit convergence rate is obtained.     

Uniform boundedness in time and large time behavior of weak solutions to a kind of dissipative system

This paper deals with the uniform boundedness (as well as the existence) and large time behavior of the weak entropy solutions to a kind of compressible Euler equation with dissipation effect. The existence and uniform boundedness in time of weak solutions are proved by using the Lax-Friedrichs scheme and compensate compactness. Time asymptotically, the density is showed to satisfy a kind of nonlinear Fokker-Planck equation and the momentum obeys to the Darcy’s law. As a by product, the exponentially decay rate is obtained.     

Existence of global weak solutions for Navier-Stokes equations with large flux

Global existence of weak solutions to the Navier-Stokes equations in a cylindrical domain under boundary slip conditions and with inflow and outflow is proved. To prove the energy estimate, crucial for the proof, we use the Hopf function. This makes it possible to derive an estimate such that the inflow and outflow need not vanish as t→∞. The proof requires estimates in weighted Sobolev spaces for solutions to the Poisson equation. Our result is the first step towards proving the existence of global regular special solutions to the Navier-Stokes equations with inflow and outflow.     

On the existence of local strong solutions for the Navier-Stokes equations in completely general domains

There are only very few results on the existence of unique local in time strong solutions of the Navier-Stokes equations for completely general domains Ω⊆R³, although domains with edges and corners, bounded or unbounded, are very important in applications. The reason is that the L^q-theory for the Stokes operator A is available in general only in the Hilbert space setting, i.e., with q=2. Our main result for a general domain Ω is optimal in a certain sense: Consider an initial value and a zero external force. Then the condition is sufficient and necessary for the existence of a unique local strong solution u∈L⁸(0,T;L⁴(Ω)) in some interval [0,T), 0<T≤∞, with u(0)=u₀, satisfying Serrin’s condition . Note that Fujita-Kato’s sufficient condition u₀∈D(A^1/4) is strictly stronger and therefore not optimal.     

On the regularity criteria for weak solutions to the magnetohydrodynamic equations

We study the regularity criteria for weak solutions to the incompressible magnetohydrodynamic equations. Some regularity criteria are obtained for weak solutions to the magnetohydrodynamic equations, which generalize the results in [C. He, Z. Xin, On the regularity of solutions to the magneto-hydrodynamic equations, J. Differential Equations 213 (2) (2005) 235-254]. Our results reveal that the velocity field of the fluid plays a more dominant role than the magnetic field does on the regularity of solutions to the magnetohydrodynamic equations.     

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.     

Very weak, generalized and strong solutions to the Stokes system in the half-space

In this paper, we study the Stokes system in the half-space , with N?2. We give existence and uniqueness results in weighted Sobolev spaces. After the central case of the generalized solutions, we are interested in strong solutions and symmetrically in very weak solutions by means of a duality argument.     

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.     

On the regularity of weak solutions to the magnetohydrodynamic equations

In this paper, we study the regularity of weak solution to the incompressible magnetohydrodynamic equations. We obtain some sufficient conditions for regularity of weak solutions to the magnetohydrodynamic equations, which is similar to that of incompressible Navier-Stokes equations. Moreover, our results demonstrate that the velocity field of the fluid plays a more dominant role than the magnetic field does on the regularity of solution to the magneto-hydrodynamic equations.     

Smooth or singular solutions to the Navier-Stokes system ?

The existence of singular solutions of the incompressible Navier-Stokes system with singular external forces, the existence of regular solutions for more regular forces as well as the asymptotic stability of small solutions (including stationary ones), and a pointwise loss of smoothness for solutions are proved in the same function space of pseudomeasure type.     

On a theorem by Sohr for the Navier-Stokes equations

In this paper we study some criteria for the full (space-time) regularity of weak solutions to the Navier-Stokes equations. In particular, we generalize some classical and very recent criteria involving the velocity, or its derivatives. More specifically, we show with elementary tools that if a weak solution, or its vorticity, is small in appropriate Marcinkiewicz spaces, then it is regular.     

On the uniqueness of weak solutions for the 3D viscous magneto-hydrodynamic equations

A uniqueness result of weak solutions for the 3D viscous magneto-hydrodynamic equations is proved by means of the Fourier localization technique and Bony’s paraproduct decomposition.