arXiv subjects

Jiang Xu

Publications and source records attributed to Jiang Xu.

60 records · Page 4Linked to original sources

arXiv2012

Global classical solutions for partially dissipative hyperbolic system of balance laws

This work is concerned with ($N$-component) hyperbolic system of balance laws in arbitrary space dimensions. Under entropy dissipative assumption and the Shizuta-Kawashima algebraic condition, a general theory on the well-posedness of classical solutions in the framework of Chemin-Lerner's spaces with critical regularity is established. To do this, we first explore the functional space theory and develop an elementary fact that indicates the relation between homogeneous and inhomogeneous Chemin-Lerner's spaces. Then this fact allows to prove the local well-posedness for general data and global well-posedness for small data by using the Fourier frequency-localization argument. Finally, we apply the new existence theory to a specific fluid model-the compressible Euler equations with damping, and obtain the corresponding results in critical spaces.

math.AP↗

arXiv2011

Global classical solutions to the compressible Euler-Maxwell equations

In this paper, we consider the compressible Euler-Maxwell equations arising in semiconductor physics, which take the form of Euler equations for the conservation laws of mass density and current density for electrons, coupled to Maxwell's equations for self-consistent electromagnetic field. We study the global well-posedness in critical spaces and the limit to zero of some physical parameters in the scaled Euler-Maxwell equations. More precisely, using high- and low-frequency decomposition methods, we first construct uniform (global) classical solutions (around constant equilibrium) to the Cauchy problem of Euler-Maxwell equations in Chemin-Lerner's spaces with critical regularity. Furthermore, based on Aubin-Lions compactness lemma, it is justified that the (scaled) classical solutions converge globally in time to the solutions of compressible Euler-Poisson equations in the process of non-relativistic limit and to that of drift-diffusion equations under the relaxation limit or the combined non-relativistic and relaxation limits.

math.AP↗

arXiv2011

Local well-posedness for Euler-Poisson fluids with non-zero heat conduction

We consider the multidimensional Euler-Poisson equations with non-zero heat conduction, which consist of a coupled hyperbolic-parabolic-elliptic system of balance laws. We make a deep analysis on the coupling effects and establish a local well-posedness of classical solutions to the Cauchy problem pertaining to data in the critical Besov space. Proof mainly relies on a standard iteration argument. To achieve it, a new Moser-type inequality is developed by the Bony' decomposition.

math.AP↗

arXiv2011

Relaxation Limit in Besov Spaces for Compressible Euler Equations

The relaxation limit in critical Besov spaces for the multidimensional compressible Euler equations is considered. As the first step of this justification, the uniform (global) classical solutions to the Cauchy problem with initial data close to an equilibrium state are constructed in the Chemin-Lerner's spaces with critical regularity. Furthermore, it is shown that the density converges towards the solution to the porous medium equation, as the relaxation time tends to zero. Several important estimates are achieved, including a crucial estimate of commutator.

math.AP↗

arXiv2007

Global exponential stability of classical solutions to the hydrodynamic model for semiconductors

In this paper, the global well-posedness and stability of classical solutions to the multidimensional hydrodynamic model for semiconductors on the framework of Besov space are considered. We weaken the regularity requirement of the initial data, and improve some known results in Sobolev space. The local existence of classical solutions to the Cauchy problem is obtained by the regularized means and compactness argument. Using the high- and low- frequency decomposition method, we prove the global exponential stability of classical solutions (close to equilibrium). Furthermore, it is also shown that the vorticity decays to zero exponentially in the 2D and 3D space. The main analytic tools are the Littlewood-Paley decomposition and Bony's para-product formula.

math.AP↗

arXiv2007

Existence and asymptotic behavior of $C^1$ solutions to the multidimensional compressible Euler equations with damping

In this paper, the existence and asymptotic behavior of $C^1$ solutions to the multidimensional compressible Euler equations with damping on the framework of Besov space are considered. We weaken the regularity requirement of the initial data, and improve the well-posedness results of Sideris-Thomases-Wang (Comm.P.D.E. 28 (2003) 953). The global existence lies on a crucial a-priori estimate which is proved by the spectral localization method. The main analytic tools are the Littlewood-Paley decomposition and Bony's para-product formula.

math.AP↗