Zero-electron-mass limit of Euler-Poisson equations
Jiang Xu Ting Zhang
Discrete & Continuous Dynamical Systems - A 2013, 33(10): 4743-4768 doi: 10.3934/dcds.2013.33.4743
We study the limit of vanishing ratio of the electron mass to the ion mass (zero-electron-mass limit) in the scaled Euler-Poisson equations. As the first step of this justification, we construct the uniform global classical solutions in critical Besov spaces with the aid of ``Shizuta-Kawashima" skew-symmetry. Then we establish frequency-localization estimates of Strichartz-type for the equation of acoustics according to the semigroup formulation. Finally, it is shown that the uniform classical solutions converge towards that of the incompressible Euler equations (for ill-preparedinitial data) in a refined way as the scaled electron-mass tends to zero. In comparison with the classical zero-mach-number limit in [7,23], we obtain different dispersive estimates due to the coupled electric field.
keywords: Zero-electron-mass limits Euler-Poisson equations Strichartz-type estimate. critical Besov spaces skew-symmetry
Well-posedness and stability of classical solutions to the multidimensional full hydrodynamic model for semiconductors
Jiang Xu
Communications on Pure & Applied Analysis 2009, 8(3): 1073-1092 doi: 10.3934/cpaa.2009.8.1073
This paper is concerned with the global well-posedness and stability of classical solutions to the Cauchy problem for the multidimensional full hydrodynamic model in semiconductors on the framework of Besov space. By using the high- and low- frequency decomposition method, we obtain the exponential decay of classical solutions (close to equilibrium). Moreover, it is also shown that the vorticity decays to zero exponentially in the 2D and 3D space. The work weakens the regularity requirement of the initial data and improves some known results in Sobolev space.
keywords: Exponential stability classical solutions hydrodynamic model.
Zero-relaxation limit of non-isentropic hydrodynamic models for semiconductors
Jiang Xu Wen-An Yong
Discrete & Continuous Dynamical Systems - A 2009, 25(4): 1319-1332 doi: 10.3934/dcds.2009.25.1319
This paper deals with non-isentropic hydrodynamic models for semiconductors with short momentum and energy relaxation times. With the help of the Maxwell iteration, we construct a new approximation and show that periodic initial-value problems of certain scaled non-isentropic hydrodynamic models have unique smooth solutions in a time interval independent of the two relaxation times. Furthermore, it is proved that as the two relaxation times both tend to zero, the smooth solutions converge to solutions of the corresponding semilinear drift-diffusion models.
keywords: relaxation limit continuation principle Non-isentropic hydrodynamic model energy estimates. Maxwell iteration
Global existence and optimal decay estimates of the compressible viscoelastic flows in $ L^p $ critical spaces
Xinghong Pan Jiang Xu
Discrete & Continuous Dynamical Systems - A 2019, 39(4): 2021-2057 doi: 10.3934/dcds.2019085

In this paper, we are concerned with the compressible viscoelastic flows in whole space $ \mathbb{R}^n $ with $ n\geq2 $. We aim at extending the global existence in energy spaces (see [18] by Hu & Wang and [30] by Qian & Zhang) such that it holds in more general $ L^p $ critical spaces, which allows to the case of large highly oscillating initial velocity. Precisely, We define "two effective velocities" which are used to eliminate the coupling between the density, velocity and deformation tensor. Consequently, the global existence in the $ L^p $ critical framework is constructed by elementary energy approaches. In addition, the optimal time-decay estimates of strong solutions are firstly shown in the $ L^p $ framework, which improve recent decay efforts for compressible viscoelastic flows.

keywords: Compressible viscoelastic flows critical Besov space global existence optimal decay

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