On the regularity of solutions to a parabolic system related to Maxwell's equations

The goal of this paper is to establish Hölder estimates for the solutions of a certain parabolic system related to Maxwell's equations arising in a quasi-stationary electromagnetic field.     

Remarks on regularity criteria for axially symmetric weak solutions to the Navier–Stokes equations

We examine the conditional regularity of the solutions to the Navier–Stokes equations in the entire three‐dimensional space under the assumption that the data are axially symmetric. We show that if a radial or angular component of velocity satisfies a weighted Serrin condition, then the solution is regular. Copyright © 2011 John Wiley & Sons, Ltd.     

On the regularity criterion for the solutions of 3D Navier–Stokes equations in weak multiplier spaces

In this work, we improved the regularity criterion on the Cauchy problem for the Navier–Stokes equations in multiplier space in terms of the two partial derivatives of velocity fields, ?₁u₁ and ?₂u₂. Copyright © 2011 John Wiley & Sons, Ltd.     

Partial regularity of weak solutions for Landau-Lifshitz system with potential

In this note, we prove the partial regularity of stationary weak solutions for the Landau-Lifshitz system with general potential in four or three dimensional space. As well known, in general, since the constraint of the methods, in order to get the partial regularity of stationary weak solution of the Landau-Lifshitz system with potential, we need to add some very strongly conditions on the potential. The main difficulty caused by potential is how to find the equation satisfied by the scaling function, which breaks down the blow-up processing. We estimate directly Morrey’s energy to avoid the difficulties by blowing up. This work was supported by National Natural Science Foundation of China (Grant Nos. 10631020, 60850005) and the Natural Science Foundation of Zhejiang Province (Grant No. D7080080)     

Sufficient conditions for the regularity of the solutions of the Navier–Stokes equations

In this paper we find sufficient conditions, involving only the pressure, that ensure the regularity of weak solutions to the Navier–Stokes equations. Conditions involving only the pressure were previously obtained in [7,4]. Following a remark in this last reference we improve, in particular, Kaniel's result [7]. Our condition can be seen at the light of the heuristic idea that the pressure behaves similarly to the modulus squared of the velocity. Copyright © 1999 John Wiley & Sons, Ltd.     

On singular solutions of the initial boundary value problem for the Stokes system in a power cusp domain

The initial boundary value problem for the non-steady Stokes system is considered in bounded domains with the boundary having a peak-type singularity (power cusp singularity). The case of the boundary value with a nonzero time-dependent flow rate is studied. The formal asymptotic expansion of the solution near the singular point is constructed. This expansion contains both the outer asymptotic expansion and the boundary-layer-in-time corrector with the ‘fast time’ variable depending on the distance to the cusp point. The solution of the problem is constructed as the sum of the asymptotic expansion and the term with finite energy.     

Endpoint regularity criterion for weak solutions of the 3D incompressible liquid crystals system

In this paper, we consider the endpoint case regularity for the 3D liquid crystals system. We prove that if , then weak solution (v,d) is smooth, and our main observation is that the condition is not necessary in this situation. The proof is based on the blow‐up analysis and backward uniqueness for the parabolic operator developed by Escauriaza‐Seregin‐S̆verák.     

Partial regularity for weak solutions of stationary navier-stokes systems

This article is concerned with the partial regularity for the weak solutions of stationary Navier-Stokes system under the controllable growth condition.By A-harmonic approximation technique,the optimal regularity is obtained.     

On interior regularity criteria for weak solutions of the navier-stokes equations

We are concerned with the behavior of weak solutions of the Navier-Stokes equations near possible singularities. We shall show that if a weak solution is in some Lebesgue space or small in some Lorentz space locally, it does not blowup there. Our basic idea is to estimate integral formulas for vorticity which satisfies parabolic equations.     

Remark on the regularity for weak solutions to the magnetohydrodynamic equations

We study the regularity criteria for weak solutions to the incompressible magnetohydrodynamic (MHD) equations. Some regularity criteria, which are related only with u+B or u?B, are obtained for weak solutions to the MHD equations. Copyright © 2008 John Wiley & Sons, Ltd.     

Global existence of weak solutions for the anisotropic compressible Stokes system

In this paper, we study the problem of global existence of weak solutions for the quasi-stationary compressible Stokes equations with an anisotropic viscous tensor. The key idea is a new identity that we obtain by comparing the limit of the equations of the energies associated to a sequence of weak-solutions with the energy equation associated to the system verified by the limit of the sequence of weak-solutions. In the context of stability of weak solutions, this allows us to construct a defect measure which is used to prove compactness for the density and therefore allowing us to identify the pressure in the limiting model. By doing so we avoid the use of the so-called effective flux. Using this new tool, we solve an open problem namely global existence of solutions à la Leray for such a system without assuming any restriction on the anisotropy amplitude. This provides a flexible and natural method to treat compressible quasilinear Stokes systems which are important for instance in biology, porous media, supra-conductivity or other applications in the low Reynolds number regime.     

Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations

In this paper, we study the local behavior of the solutions to the three-dimensional magnetohydrodynamic equations. we are interested in both the uniform gradient estimates for smooth solutions and regularity of weak solutions. It is shown that, in some neighborhood of (x₀,t₀), the gradients of the velocity field u and the magnetic field B are locally uniformly bounded in L^∞ norm as long as that either the scaled local L²-norm of the gradient or the scaled local total energy of the velocity field is small, and the scaled local total energy of the magnetic field is uniformly bounded. These estimates indicate that the velocity field plays a more dominant role than that of the magnetic field in the regularity theory. As an immediately corollary we can derive an estimates of Hausdorff dimension on the possible singular set of a suitable weak solution as in the case of pure fluid. Various partial regularity results are obtained as consequences of our blow-up estimates.     

On the existence of finite energy weak solutions to the Navier–Stokes equations in irregular domains

This paper studies the existence of weak solutions of the Navier–Stokes system defined on a certain class of domains in ?³ that may contain cusps. The concept of such a domain and weak energy solution for the system is defined and its existence is proved. However, thinness of cusps must be related to the adiabatic constant appearing in the pressure law. Copyright © 2008 John Wiley & Sons, Ltd.     

On regularity of a weak solution to the Navier–Stokes equation with generalized impermeability boundary conditions

We consider the 3D Navier–Stokes equation with generalized impermeability boundary conditions. As auxiliary results, we prove the local in time existence of a strong solution (‘strong’ in a limited sense) and a theorem on structure. Then, taking advantage of the boundary conditions, we formulate sufficient conditions for regularity up to the boundary of a weak solution by means of requirements on one of the eigenvalues of the rate of deformation tensor. Finally, we apply these general results to the case of an axially symmetric flow with zero angular velocity.     

On the boundary regularity of weak solutions to the MHD system

The partial regularity of boundary suitable weak solutions to the MHD system near the plane part of the boundary is proved. Bibliography: 16 titles.