Shijin Ding

Landau-Lifshitz equations

Physical Background of Landau-Lifshitz Equations and Landau-Lifshitz-Maxwell Equations Existence of Weak Solutions to Landau-Lifshitz Equations and Landau-Lifshitz-Maxwell Equations in Multi-Dimensions Existence and Uniqueness of Smooth Solutions for Landau-Lifshitz Equations with or without Gilbert Damping in One Dimension Discovery of the Relations with Harmonic Map Heat Flows and Partial Regularity for Chen-Struwe Solutions on Two Dimensional Riemannian Manifolds Partial Regularity for Weak Solutions to Landau-Lifshitz Equations and Landau-Lifshitz-Maxwell Equations in Higher Dimensions Long Time Behaviors and Attractors.

Global Spherically Symmetric Classical Solution to Compressible Navier–Stokes Equations with Large Initial Data and Vacuum

In this paper, we obtain a result on the existence and uniqueness of global spherically symmetric classical solutions to the compressible isentropic Navier-Stokes equations with vacuum in a bounded domain or exterior domain {\Omega} of Rn(n >= 2). Here, the initial data could be large. Besides, the regularities of the solutions are better than those obtained in [H.J. Choe and H. Kim, Math. Methods Appl. Sci., 28 (2005), pp. 1-28; Y. Cho and H. Kim, Manuscripta Math., 120 (2006), pp. 91-129; S.J. Ding, H.Y.Wen, and C.J. Zhu, J. Differential Equations, 251 (2011), pp. 1696-1725]. The analysis is based on some new mathematical techniques and some new useful energy estimates. This is an extension of the work of Choe and Kim, Cho and Kim, and Ding, Wen, and Zhu, where the global radially symmetric strong solutions, the local classical solutions in three dimensions, and the global classical solutions in one dimension were obtained, respectively. This paper can be viewed as the first result on the existence of global classical solutions with large initial data and vacuum in higher dimension

On the initial boundary value problem for a shallow water equation

In this paper, we obtain the existence and uniqueness of the local strong solutions to the initial boundary problem for a one-dimensional shallow-water equation (Camassa–Holm equation) on the half-space {x>0} with initial data uo∈H2(R+)∩H01(R+). The solution is obtained as a limit of the solutions for a class of approximation problems. We also establish the global result of the corresponding solution, provided that the initial data u0 satisfies certain positivity condition.

Global existence of strong solutions for incompressible hydrodynamic flow of liquid crystals with vacuum

We consider the Cauchy problem for incompressible hydrodynamic flow of nematic liquid crystals in three dimensions. We prove the global existence and uniqueness of the strong solutions with nonnegative ρ0 and small initial data.

Global solutions to one-dimensional compressible Navier–Stokes–Poisson equations with density-dependent viscosity

In this paper, we prove the global existence of weak solutions to one-dimensional compressible isentropic Navier–Stokes–Poisson equations with density-dependent viscosity and free boundaries. The initial density ρ0∊W1,2n is bounded below by a positive constant, and the initial velocity u0∊L2n. In contrast to Jiang et al. [“Global weak solutions to 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity,” Methods Appl. Anal. 12, 239 (2005)], the Sobolev exponent n is less in this paper, and the viscosity coefficient μ=μ(ρ) is a general function of ρ including the cases μ(ρ)=c0ρθ (0

Measure-valued solution to the strongly degenerate compressible Heisenberg chain equations

In this paper we are concerned with the existence of solutions to the compressible Heisenberg chain equations. By the vanishing viscosity method we prove that this system admits at least one measure-valued solution.

Global classical solutions to the 2D compressible Navier-Stokes equations with vacuum

We investigate the global classical solutions to the Cauchy problem for the 2-D viscous compressible Navier-Stokes equations with vacuum and small initial density but possibly large initial energy, provided the shear viscosity is a positive constant and the bulk one is λ = ρβ for any β ∈ [0, 1]. This extends the earlier work on the well-posedness theory of the compressible Navier-Stokes equations with small initial energy [such as Huang et al., Commun. Pure Appl. Math. 65, 549–585 (2012); Li and Xin, e-print arXiv:1310.1673; and Fang and Guo, Z. Angew. Math. Phys. 67, 22 (2016), respectively].