Jishan Fan

Global regularity for the 2D liquid crystal model with mixed partial viscosity

This paper proves the global existence of strong solutions of the 2D liquid crystal model when ν1 = k2 = 0, ν2 = k1 = 1 or ν1 = k2 = 1, ν2 = k1 = 0. We also prove some regularity criteria when ν1 = k1 = 1, ν2 = k2 = 0 or ν1 = k1 = 0, ν2 = k2 = 1.

Inverse problems for the Boussinesq system

We obtain two results on inverse problems for a 2D Boussinesq system. One is that we prove the Lipschitz stability for the inverse source problem of identifying a time-independent external force in the system with observation data in an arbitrary sub-domain over a time interval of the velocity and the data of velocity and temperature at a fixed positive time t0 > 0 over the whole spatial domain. The other one is that we prove a conditional stability estimate for an inverse problem of identifying the two initial conditions with a single observation on a sub-domain.

Well-posedness of an inverse problem of Navier–Stokes equations with the final overdetermination

Abstract This paper proves the existence, uniqueness and stability of solutions of an inverse problem for the 2-D Navier–Stokes equations with the final overdetermination with L 2 initial data and sufficiently large viscosity.

Global Strong Solution to the Density-Dependent 2-D Liquid Crystal Flows

The initial-boundary value problem for the density-dependent flow of nematic crystals is studied in a 2-D bounded smooth domain. For the initial density away from vacuum, the existence and uniqueness is proved for the global strong solution with the large initial velocity and small . We also give a regularity criterion of the problem with the Dirichlet boundary condition , on .

Local solvability of an inverse problem to the density-dependent Navier–Stokes equations

In this article, we prove the local solvability of an inverse problem to the density-dependent Navier–Stokes equations in the case of integral overdetermination.

REGULARITY CRITERIA FOR THE p-HARMONIC AND OSTWALD-DE WAELE FLOWS

Abstract. This paper considers regularity for the p-harmonic andOstwald-de Waele flows. Some Serrin’s type regularity criteria are es-tablished for 1 n ≥ 3, Fardoun-Regbaoui [12] showed the global well-posednessof strong solutions for large data. Hungerbu¨hler [14] established existence ofglobal weak solutions of the p-harmonic flow between Riemannian manifoldsM and N for arbitrary initial data having finite p-energy in the case when thetarget N is a homogeneous space with a left invariant metric when 2 < p < n.Chen-Hong-Hungerbu¨hler [8] proved existence of global weak solutions whenp ≥ 2.When 1 < p < 2, Misawa [18] proved that the problem (1.1)-(1.3) has aglobal weak solution satisfying(1.4)1pZ|∇u|

Regularity criteria and uniform estimates for the Boussinesq system with temperature-dependent viscosity and thermal diffusivity

In this paper, we establish some regularity criteria for the 3D Boussinesq system with the temperature-dependent viscosity and thermal diffusivity. We also obtain some uniform estimates for the corresponding 2D case when the fluid viscosity coefficient is a positive constant.

Regularity criteria for the incompressible magnetohydrodynamic equations with partial viscosity

We prove some regularity criteria for the incompressible magnetohydrodynamic equations with partial viscosity. We study three cases: incompressible magnetohydrodynamic equations with zero viscosity in a bounded domain, incompressible magnetohydrodynamic equations with zero resistivity in a bounded domain, and the density-dependent magnetohydrodynamic equations with zero heat conductivity and zero resistivity in the whole space ℝ3. Our results extend and improve some known results.

Uniform global solutions of the 3D compressible MHD system in a bounded domain

Abstract In this paper we consider the 3D compressible MHD system in a bounded domain. We first prove a regularity criterion and then use it and the bootstrap argument to show the uniform-in- η global small solutions. Here η is the resistivity coefficient.