Boundary Layer Problem and Quasineutral Limit of Compressible Euler-Poisson System
Shu Wang Chundi Liu
Communications on Pure & Applied Analysis 2017, 16(6): 2177-2199 doi: 10.3934/cpaa.2017108

We study the boundary layer problem and the quasineutral limit of the compressible Euler-Poisson system arising from plasma physics in a domain with boundary. The quasineutral regime is the incompressible Euler equations. Compared to the quasineutral limit of compressible Euler-Poisson equations in whole space or periodic domain, the key difficulty here is to deal with the singularity caused by the boundary layer. The proof of the result is based on a λ-weighted energy method and the matched asymptotic expansion method.

keywords: Euler-Poisson system incompressible Euler equations quasineutral limit boundary layer
Asymptotic expansions in two-fluid compressible Euler-Maxwell equations with small parameters
Yue-Jun Peng Shu Wang
Discrete & Continuous Dynamical Systems - A 2009, 23(1&2): 415-433 doi: 10.3934/dcds.2009.23.415
This work is concerned with the two-fluid Euler-Maxwell equations for plasmas with small parameters. We study, by means of asymptotic expansions, the zero-relaxation limit, the non-relativistic limit and the combined non-relativistic and quasi-neutral limit. For each limit with well-prepared initial data, we show the existence and uniqueness of an asymptotic expansion up to any order. For general data, an asymptotic expansion up to order 1 of the non-relativistic limit is constructed by taking into account the initial layers. Finally, we discuss the justification of the limits.
keywords: formal limits initial layer expansion Compressible Euler-Maxwell equations asymptotic expansion
Global existence for the 2D Navier-Stokes flow in the exterior of a moving or rotating obstacle
Shuguang Shao Shu Wang Wen-Qing Xu Bin Han
Kinetic & Related Models 2016, 9(4): 767-776 doi: 10.3934/krm.2016015
We consider the global existence of the two-dimensional Navier-Stokes flow in the exterior of a moving or rotating obstacle. Bogovski$\check{i}$ operator on a subset of $\mathbb{R}^2$ is used in this paper. One important thing is to show that the solution of the equations does not blow up in finite time in the sense of some $L^2$ norm. We also obtain the global existence for the 2D Navier-Stokes equations with linearly growing initial velocity.
keywords: Global existence Navier-Stokes flow Bogovski$\check{i}$ operator.
On the local C1, α solution of ideal magneto-hydrodynamical equations
Shu-Guang Shao Shu Wang Wen-Qing Xu Yu-Li Ge
Discrete & Continuous Dynamical Systems - A 2017, 37(4): 2103-2113 doi: 10.3934/dcds.2017090

This paper is devoted to the study of the two-dimensional andthree-dimensional ideal incompressible magneto-hydrodynamic (MHD)equations in which the Faraday law is inviscid. We consider thelocal existence and uniqueness of classical solutions for the MHDsystem in Hölder space when the general initial data belongs to$C^{1,α}(\mathbb{R}^n)$ for $n=2$ and $n=3$.

keywords: Ideal MHD equations local C1, α solution Hölder space
On one multidimensional compressible nonlocal model of the dissipative QG equations
Shu Wang Zhonglin Wu Linrui Li Shengtao Chen
Discrete & Continuous Dynamical Systems - S 2014, 7(5): 1111-1132 doi: 10.3934/dcdss.2014.7.1111
In this paper we study the Cauchy problem for one multidimensional compressible nonlocal model of the dissipative quasi-geostrophic equations and discuss the effect of the sign of initial data on the wellposedness of this model. First, we prove the existence and uniqueness of local smooth solutions for the Cauchy problem for the model with the nonnegative initial data, which seems to imply that whether the well-posedness of this model holds or not depends heavily upon the sign of the initial data even for the subcritical case. Secondly, for the sub-critical case $1<\alpha\leq 2$, we obtain the global existence and uniqueness results of the nonnegative smooth solution. Next, we prove the global existence of the weak solution for $0<\alpha\le 2$ and $\nu>0$. Finally, for the sub-critical case $1<\alpha\leq 2$, we establish $H^\beta(\beta\geq 0)$ and $L^p(p\geq 2)$ decay rates of the smooth solution as $t\to\infty$. A inequality for the Riesz transformation is also established.
keywords: dissipative quasi-geostrophic equations Riesz transformation Cauchy problem. sub-critical case Multidimensional compressible nonlocal model
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
Incompressible type euler as scaling limit of compressible Euler-Maxwell equations
Jianwei Yang Ruxu Lian Shu Wang
Communications on Pure & Applied Analysis 2013, 12(1): 503-518 doi: 10.3934/cpaa.2013.12.503
In this paper, we study the convergence of time-dependent Euler-Maxwell equations to incompressible type Euler equations in a torus via the combined quasi-neutral and non-relativistic limit. For well prepared initial data, the local existence of smooth solutions to the limit equations is proved by an iterative scheme. Moreover, the convergences of solutions of the former to the solutions of the latter are justified rigorously by an analysis of asymptotic expansions and the symmetric hyperbolic property of the systems.
keywords: Euler-Maxwell equations non-relativistic limit. incompressible type Euler equations quasineutral limit
Global regularity for a model of Navier-Stokes equations with logarithmic sub-dissipation
Shuguang Shao Shu Wang Wen-Qing Xu
Kinetic & Related Models 2018, 11(1): 179-190 doi: 10.3934/krm.2018009

In this paper, we study the global regularity to a three-dimensional logarithmic sub-dissipative Navier-Stokes model. This system takes the form of ${\partial _t}u +(\mathcal {D}^{-1/2}u)·\nabla u + \nabla p =-\mathcal {A}^2u$, where $\mathcal {D}$ and $\mathcal {A}$ are Fourier multipliers defined by $\mathcal {D}=|\nabla|$ and $\mathcal {A}= |\nabla|\ln^{-1/4}(e + λ \ln (e + |\nabla|)) $ with $λ≥q0$. The symbols of the $\mathcal {D}$ and $\mathcal {A}$ are $m(ξ) =\left| ξ \right|$ and $h(ξ) = \left| ξ \right| / g(ξ)$ respectively, where $g(ξ) = {\ln ^{{1 / 4}}}(e + λ \ln (e + |ξ|))$, $λ≥0$. It is clear that for the Navier-Stokes equations, global regularity is true under the assumption that $h(ξ) =|ξ|^α$ for $α≥q 5/4$. Here by changing the advection term we greatly weaken the dissipation to $ h(ξ)={{\left| ξ \right|} / g(ξ)}$. We prove the global well-posedness for any smooth initial data in $H^s(\mathbb{R}^3)$, $ s≥q3 $ by using the energy method.

keywords: Navier-Stokes equations global regularity sub-dissipation energy estimates
Quasineutral limit for the quantum Navier-Stokes-Poisson equations
Min Li Xueke Pu Shu Wang
Communications on Pure & Applied Analysis 2017, 16(1): 273-294 doi: 10.3934/cpaa.2017013

In this paper, we study the quasineutral limit and asymptotic behaviors for the quantum Navier-Stokes-Possion equation. We apply a formal expansion according to Debye length and derive the neutral incompressible Navier-Stokes equation. To establish this limit mathematically rigorously, we derive uniform (in Debye length) estimates for the remainders, for well-prepared initial data. It is demonstrated that the quantum effect do play important roles in the estimates and the norm introduced depends on the Planck constant $\hbar>0$.

keywords: Quantum Navier-Stokes-Possion system quasineutral limit formal expansion well-prepared initial data uniform energy estimates

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