DCDS
Low Mach number limit for the compressible magnetohydrodynamic equations in a periodic domain
Fucai Li Yanmin Mu
Discrete & Continuous Dynamical Systems - A 2018, 38(4): 1669-1705 doi: 10.3934/dcds.2018069

This paper studies the convergence of the compressible isentropic magnetohydrodynamic equations to the corresponding incompressiblemagnetohydrodynamic equations with ill-preparedinitial data in a periodic domain.We prove that the solution to the compressible isentropic magnetohydrodynamic equations with small Mach number exists uniformly in the time interval as long as that to the incompressible one does. Furthermore,we obtain the convergence result for the solutions filtered by the group of acoustics.

keywords: Isentropic compressible magnetohydrodynamic equations incompressible magnetohydrodynamic equations periodic domain low Mach number limit
DCDS
Zero viscosity-resistivity limit for the 3D incompressible magnetohydrodynamic equations in Gevrey class
Fucai Li Zhipeng Zhang
Discrete & Continuous Dynamical Systems - A 2018, 38(9): 4279-4304 doi: 10.3934/dcds.2018187

We study the zero viscosity-resistivity limit for the 3D incompressible magnetohydrodynamic (MHD) equations in a periodic domain in the framework of Gevrey class. We first prove that there exists an interval of time, independent of the viscosity coefficient and the resistivity coefficient, for the solutions to the viscous incompressible MHD equations. Then, based on these uniform estimates, we show that the solutions of the viscous incompressible MHD equations converge to that of the ideal incompressible MHD equations as the viscosity and resistivity coefficients go to zero. Moreover, the convergence rate is also given.

keywords: Incompressible MHD equations Gevrey class zero viscosity-resistivity limit convergence rate
CPAA
Combined quasineutral and inviscid limit of the Vlasov-Poisson-Fokker-Planck system
Ling Hsiao Fucai Li Shu Wang
Communications on Pure & Applied Analysis 2008, 7(3): 579-589 doi: 10.3934/cpaa.2008.7.579
The combined quasineutral and inviscid limit for the Vlasov-Poisson-Fokker-Planck (VPFP) system is rigorously derived in this paper. It is shown that the solution of VPFP system converges to the solution of incompressible Euler equations with damping. The proof of convergence result is based on compactness arguments and the so-called relative-entropy method.
keywords: incompressible Euler equations relative-entropy method. Vlasov-Poisson-Fokker-Planck system
PROC
Global existence and low Mach number limit to the 3D compressible magnetohydrodynamic equations in a bounded domain
Jishan Fan Fucai Li Gen Nakamura
Conference Publications 2015, 2015(special): 387-394 doi: 10.3934/proc.2015.0387
In this paper we establish the global existence of strong solutions to the three-dimensional compressible magnetohydrodynamic equations in a bounded domain with small initial data. Moreover, we study the low Mach number limit to the corresponding problem.
keywords: low Mach number limit. Compressible magnetohydrodynamic equations global existence
KRM
Global strong solutions to the planar compressible magnetohydrodynamic equations with large initial data and vacuum
Jishan Fan Shuxiang Huang Fucai Li
Kinetic & Related Models 2017, 10(4): 1035-1053 doi: 10.3934/krm.2017041
This paper considers the initial boundary problem to the planar compressible magnetohydrodynamic equations with large initial data and vacuum. The global existence and uniqueness of large strong solutions are established when the heat conductivity coefficient
$κ(θ)$
satisfies
$C_{1}(1+\theta^q)\leq \kappa(\theta)\leq C_2(1+\theta^q)$
for some constants
$q>0$
, and
$C_1,C_2>0$
.
keywords: Planar compressible magnetohydrodynamic equations large initial data vacuum global well-posedness
KRM
Local well-posedness and low Mach number limit of the compressible magnetohydrodynamic equations in critical spaces
Fucai Li Yanmin Mu Dehua Wang
Kinetic & Related Models 2017, 10(3): 741-784 doi: 10.3934/krm.2017030

The local well-posedness and low Mach number limit are considered for the multi-dimensional isentropic compressible viscous magnetohydrodynamic equations in critical spaces. First the local well-posedness of solution to the viscous magnetohydrodynamic equations with large initial data is established. Then the low Mach number limit is studied for general large data and it is proved that the solution of the compressible magnetohydrodynamic equations converges to that of the incompressible magnetohydrodynamic equations as the Mach number tends to zero. Moreover, the convergence rates are obtained.

keywords: Isentropic compressible magnetohydrodynamic equations incompressible magnetohydrodynamic equations local well-posedness low Mach number limit critical spaces
KRM
Asymptotic limit of nonlinear Schrödinger-Poisson system with general initial data
Qiangchang Ju Fucai Li Hailiang Li
Kinetic & Related Models 2011, 4(3): 767-783 doi: 10.3934/krm.2011.4.767
The asymptotic limit of the nonlinear Schrödinger-Poisson system with general WKB initial data is studied in this paper. It is proved that the current, defined by the smooth solution of the nonlinear Schrödinger-Poisson system, converges to the strong solution of the incompressible Euler equations plus a term of fast singular oscillating gradient vector fields when both the Planck constant $\hbar$ and the Debye length $\lambda$ tend to zero. The proof involves homogenization techniques, theories of symmetric quasilinear hyperbolic system and elliptic estimates, and the key point is to establish the uniformly bounded estimates with respect to both the Planck constant and the Debye length.
keywords: semi-classical limit incompressible Euler equations. quasi-neutral limit Nonlinear Schrödinger-Poisson System
DCDS-B
A regularity criterion for the 3D full compressible magnetohydrodynamic equations with zero heat conductivity
Jishan Fan Fucai Li Gen Nakamura
Discrete & Continuous Dynamical Systems - B 2018, 23(4): 1757-1766 doi: 10.3934/dcdsb.2018079

We establish a regularity criterion for the 3D full compressible magnetohydrodynamic equations with zero heat conductivity and vacuum in a bounded domain.

keywords: Compressible magnetohydrodynamic equations zero heat conductivity regularity criterion
KRM
Convergence of the full compressible Navier-Stokes-Maxwell system to the incompressible magnetohydrodynamic equations in a bounded domain
Jishan Fan Fucai Li Gen Nakamura
Kinetic & Related Models 2016, 9(3): 443-453 doi: 10.3934/krm.2016002
In this paper we establish the uniform estimates of strong solutions with respect to the Mach number and the dielectric constant to the full compressible Navier-Stokes-Maxwell system in a bounded domain. Based on these uniform estimates, we obtain the convergence of the full compressible Navier-Stokes-Maxwell system to the incompressible magnetohydrodynamic equations for well-prepared data.
keywords: Full compressible Navier-Stokes-Maxwell system incompressible magnetohydrodynamic equations bounded domain. zero Mach number limit zero dielectric constant limit
CPAA
Low Mach number limit of the full compressible Hall-MHD system
Jishan Fan Fucai Li Gen Nakamura
Communications on Pure & Applied Analysis 2017, 16(5): 1731-1740 doi: 10.3934/cpaa.2017084

In this paper we study the low Mach number limit of the full compressible Hall-magnetohydrodynamic (Hall-MHD) system in $\mathbb{T}^3$. We prove that, as the Mach number tends to zero, the strong solution of the full compressible Hall-MHD system converges to that of the incompressible Hall-MHD system.

keywords: Full compressible Hall-MHD system incompressible Hall-MHD system low Mach number limit

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