Energy dissipation for weak solutions of incompressible MHD equations

In this article, we mainly study the local equation of energy for weak solutions of 3D MHD equations. We define a dissipation term D(u, B) that stems from an eventual lack of smoothness in the solution, and then obtain a local equation of energy for weak solutions of 3D MHD equations. Finally, we consider the 2D case at the end of this article.     

On the Chern-Simons limit for a Maxwell-Chern-Simons model on bounded domains

This paper concerns the Chern-Simons limit for the Abelian Maxwell-Chern-Simons model on bounded domains with vanishing gauge fields. We prove that every sequence of solutions of the Maxwell-Chern-Simons equations has a subsequence converging to a solution of the Chern-Simons equation in any Ck norms. We also show that the Maxwell-Chern-Simons equations with the nontopological type boundary condition do not admit any nontrivial solutions on star-shaped domains.     

On regularity criterion to the 3D axisymmetric incompressible MHD equations

This paper is concerned with the regularity criterion for a class of axisymmetric solutions to 3D incompressible magnetohydrodynamic equations. More precisely, for the solutions that have the form of u = u_re_r+u_θe_θ+u_ze_z and b = b_θe_θ, we prove that if |ru(x,t)|≤C holds for ?1≤t < 0, then (u,b) is regular at time zero. This result can be thought as a generalization of recent results in for the 3D incompressible Navier‐Stokes equations. Copyright © 2016 John Wiley & Sons, Ltd.     

Blowup analysis for integral equations on bounded domains

Consider the integral equation

$f^{q?1}(x)=\underset{\mathrm{\Omega }}{\int }\frac{f(y)}{|x?y{|}^{n?\alpha }}dy,f(x)>0,x\in \stackrel{￣}{\mathrm{\Omega }},$

where $\mathrm{\Omega }?{\mathbb{R}}^n$ is a smooth bounded domain. For $1<\alpha <n$, the existence of energy maximizing positive solution in the subcritical case $2<q<\frac{2n}{n+\alpha }$, and nonexistence of energy maximizing positive solution in the critical case $q=\frac{2n}{n+\alpha }$ are proved in [6]. For $\alpha >n$, the existence of energy minimizing positive solution in the subcritical case $0<q<\frac{2n}{n+\alpha }$, and nonexistence of energy minimizing positive solution in the critical case $q=\frac{2n}{n+\alpha }$ are also proved in [4]. Based on these, in this paper, the blowup behaviour of energy maximizing positive solution as $q\to {(\frac{2n}{n+\alpha })}^{+}$ (in the case of $1<\alpha <n$), and the blowup behaviour of energy minimizing positive solution as $q\to {(\frac{2n}{n+\alpha })}^{?}$ (in the case of $\alpha >n$) are analyzed. We see that for $1<\alpha <n$ the blowup behaviour obtained is quite similar to that of the elliptic equation involving the subcritical Sobolev exponent. But for $\alpha >n$, different phenomena appear.     

A note on incompressible limit for compressible Euler equations

This paper presents a simple justification of the classical low Mach number limit in critical Besov spaces for compressible Euler equations with prepared initial data. As the first step of this justification, we formulate a continuation principle for general hyperbolic singular limit problems in the framework of critical Besov spaces. With this principle, it is also shown that, for the Mach number sufficiently small, the smooth compressible flows exist on the (finite) time interval where the incompressible Euler equations have smooth solutions, and the definite convergence orders are obtained. Copyright © 2011 John Wiley & Sons, Ltd.     

Global regularity to the 2D incompressible MHD with mixed partial dissipation and magnetic diffusion in a bounded domain

This article considers the global regularity to the initial–boundary value problem for the 2D incompressible MHD with mixed partial dissipation and magnetic diffusion.To overcome the difficulty caused by the vanishing viscosities,we first establish the elliptic system for uxand by,which are estimated by▽×u_x and▽×b_y,respectively.Then,we establish the global estimates for▽×u and▽×b.     

Global well-posedness for the density-dependent incompressible magnetohydrodynamic flows in bounded domains

In this paper, we study the three-dimensional incompressible magnetohydrodynamic equations in a smooth bounded domains, in which the viscosity of the fluid and the magnetic diffusivity are concerned with density. The existence of global strong solutions is established in vacuum cases, provided the assumption that(|| ?μ(ρ0)|| Lp +|| ?ν(ρ0) ||Lq + ||b0|| L3+ ||ρ0|| L∞)(p, q 3) is small enough, there is not any smallness condition on the velocity.     

On symmetry properties of parabolic equations in bounded domains

We study symmetry properties of nonnegative bounded solutions of fully nonlinear parabolic equations on bounded domains with Dirichlet boundary conditions. We propose sufficient conditions on the equation and domain, which guarantee asymptotic symmetry of solutions.     

Hardy-Sobolev-Maz'ya type equations in bounded domains

We study the regularity, Palais-Smale characterization and existence/nonexistence of solutions of the Hardy-Sobolev-Maz'ya equation in a bounded domain in RN where x∈RN is denoted as x=(y,z)∈Rk×RN−k and . We show different behaviors of PS sequences depending on t=0 or t>0.     

On the vanishing viscosity limit for two-dimensional Navier–Stokes equations with singlular initial data

We study the solutions of the Navier–Stokes equations when the initial vorticity is concentrated in small disjoint regions of diameter ?. We prove that they converge, uniformily in ?. for vanishing viscosity to the corresponding solutions of the Euler equations and they are connected to the vortex model.     

On the effects of nonlinear boundary conditions in diffusive logistic equations on bounded domains

We study a diffusive logistic equation with nonlinear boundary conditions. The equation arises as a model for a population that grows logistically inside a patch and crosses the patch boundary at a rate that depends on the population density. Specifically, the rate at which the population crosses the boundary is assumed to decrease as the density of the population increases. The model is motivated by empirical work on the Glanville fritillary butterfly. We derive local and global bifurcation results which show that the model can have multiple equilibria and in some parameter ranges can support Allee effects. The analysis leads to eigenvalue problems with nonstandard boundary conditions.     

Low-frequency asymptotics for dissipative Maxwell's equations in bounded domains

Consider a bounded domain Ω surrounded by a perfect conductor and containing a conducting cavity D. The behaviour of the solutions of the time harmonic Maxwell problem as frequency tends to 0 is analysed in this situation. Necessary and sufficient conditions on the excitations are given which guarantee the existence of a limit. This limit turns out to be the solution of some static problem.     

Global well-posedness for the incompressible MHD equations with density-dependent viscosity and resistivity coefficients

This paper concerns an initial–boundary value problem of the inhomogeneous incompressible MHD equations in a smooth bounded domain. The viscosity and resistivity coefficients are density-dependent. The global well-posedness of strong solutions is established, provided the initial norms of velocity and magnetic field are suitably small in some sense, or the lower bound of the transport coefficients are large enough. More importantly, there is not any smallness condition on the density and its gradient.     

Parabolic equations and Feynman-Kac formula on general bounded domains

Parabolic equations on general bounded domains are studied. Using the refined maximum principle, existence and the semigroup property of solutions are obtained. It is also shown that the solution obtained by PDE’s method has the Feynmann-Kac representation for any bounded domains.     

Convergence of an unconditionally stable semi-implicit scheme for linear incompressible MHD equations

Semi-implicit methods have been introduced by Harned et al. to solve magneto-hydrodynamic equations (MHD) with numerical schemes which are unconditionally stable with respect to fast and shear Alfven modes. They prove the stability of their scheme for linear ideal MHD equations with periodic boundary conditions, and with some technical assumptions. In this paper, we prove convergence of the numerical approximation (time discretization), under the same hypothesis, but looking for solutions on any regular bounded open set of R³ with appropriate boundary conditions, and introducing finite resistivity and viscosity.     

Extremal equilibria for reaction-diffusion equations in bounded domains and applications

We show the existence of two special equilibria, the extremal ones, for a wide class of reaction-diffusion equations in bounded domains with several boundary conditions, including non-linear ones. They give bounds for the asymptotic dynamics and so for the attractor. Some results on the existence and/or uniqueness of positive solutions are also obtained. As a consequence, several well-known results on the existence and/or uniqueness of solutions for elliptic equations are revisited in a unified way obtaining, in addition, information on the dynamics of the associated parabolic problem. Finally, we ilustrate the use of the general results by applying them to the case of logistic equations. In fact, we obtain a detailed picture of the positive dynamics depending on the parameters appearing in the equation.     

On the vanishing viscosity limit in a disk

We say that a solution of the Navier–Stokes equations converges in the vanishing viscosity limit to a solution of the Euler equations if their velocities converge in the energy (L ²) norm uniformly in time as the viscosity ν vanishes. We show that a necessary and sufficient condition for the vanishing viscosity limit to hold in a disk is that the space–time energy density of the solution to the Navier–Stokes equations in a boundary layer of width proportional to ν vanish with ν, and that one need only consider spatial variations whose frequencies in the radial or tangential direction lie in a band centered around 1/ν. The author was supported in part by NSF grant DMS-0705586 during the period of this work.