CPAA
Global solutions to the incompressible magnetohydrodynamic equations
Xiaoli Li Dehua Wang
Communications on Pure & Applied Analysis 2012, 11(2): 763-783 doi: 10.3934/cpaa.2012.11.763
An initial-boundary value problem of the three-dimensional incompressible magnetohydrodynamic (MHD) equations is considered in a bounded domain. The homogeneous Dirichlet boundary condition is prescribed on the velocity, and the perfectly conducting wall condition is prescribed on the magnetic field. The existence and uniqueness is established for both the local strong solution with large initial data and the global strong solution with small initial data. Furthermore, the weak-strong uniqueness of solutions is also proved, which shows that the weak solution is equal to the strong solution with certain initial data.
keywords: incompressible flow strong solutions Magnetohydrodynamic (MHD) existence and uniqueness weak-strong uniqueness.
PROC
Global existence and dynamical properties of large solutions for combustion flows
Dehua Wang
Conference Publications 2003, 2003(Special): 888-897 doi: 10.3934/proc.2003.2003.888
A mathematical model for viscous compressible realistic reactive flows without species diffusion in dynamic combustion is investigated. The initial-boundary value problem with Dirichlet-Neumann mixed boundaries in a finite domain is studied. The existence, uniqueness, and regularity of global solutions are established with general large initial data in $H^1$. It is proved that, although the solutions have large oscillations and the chemical reaction generates heat, there is no shock wave, turbulence, vacuum, mass or heat concentration developed in a finite time.
keywords: real flow reacting fluid Combustion existence a-priori estimates. uniqueness global solutions
CPAA
Global solution for the mixture of real compressible reacting flows in combustion
Dehua Wang
Communications on Pure & Applied Analysis 2004, 3(4): 775-790 doi: 10.3934/cpaa.2004.3.775
The equations for viscous, compressible, heat-conductive, real reactive flows in dynamic combustion are considered, where the equations of state are nonlinear in temperature unlike the linear dependence for perfect gases. The initial-boundary value problem with Dirichlet-Neumann mixed boundaries in a finite domain is studied. The existence, uniqueness, and regularity of global solutions are established with general large initial data in $H^1$. It is proved that, although the solutions have large oscillations, there is no shock wave, turbulence, vacuum, mass concentration, or extremely hot spot developed in any finite time.
keywords: global solutions existence real flows reacting fluids uniqueness Combustion a-priori estimates.
DCDS
A multidimensional piston problem for the Euler equations for compressible flow
Shuxing Chen Gui-Qiang Chen Zejun Wang Dehua Wang
Discrete & Continuous Dynamical Systems - A 2005, 13(2): 361-383 doi: 10.3934/dcds.2005.13.361
A multidimensional piston problem for the Euler equations for compressible isentropic flow is analyzed. Thepiston initially locates at the origin and experiences compressiveand expansive motions with spherical symmetry. The initialsingularity at the origin is one of the difficulties for thisspherically symmetric piston problem. A local shock front solutionfor the compressive motion is constructed based on thelinearization at an approximate solution and the Newton iteration. A global entropy solution for the piston problem is constructed byusing a shock capturing approach and the method of compensatedcompactness.
keywords: entropy compressive motion origin. expansive motion piston problem initial singularity local solutions Euler equations global solutions
DCDS
The initial-boundary value problem for the compressible viscoelastic flows
Xianpeng Hu Dehua Wang
Discrete & Continuous Dynamical Systems - A 2015, 35(3): 917-934 doi: 10.3934/dcds.2015.35.917
The initial-boundary value problem for the equations of compressible viscoelastic flows is considered in a bounded domain of three-dimensional spatial dimensions. The global existence of strong solution near equilibrium is established. Uniform estimates in $W^{1,q}$ with $q>3$ on the density and deformation gradient are also obtained.
keywords: strong solution Compressible viscoelastic flows existence. initial-boundary value problem
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

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