THE INVISCID AND NON-RESISTIVE LIMIT IN THE CAUCHY PROBLEM FOR 3-D NONHOMOGENEOUS INCOMPRESSIBLE MAGNETO-HYDRODYNAMICS

In this paper,the inviscid and non-resistive limit is justified for the local-in-time solutions to the equations of nonhomogeneous incompressible magneto-hydrodynamics （MHD）in R3.We prove that as the viscosity and resistivity go to zero,the solution of the Cauchy problem for the nonhomogeneous incompressible MHD system converges to the solution of the ideal MHD system.The convergence rate is also obtained simultaneously.

THE COMBINED INVISCID AND NON-RESISTIVE LIMIT FOR THE NONHOMOGENEOUS INCOMPRESSIBLE MAGNETOHYDRODYNAMIC EQUATIONS WITH NAVIER BOUNDARY CONDITIONS

In this paper, we establish the existence of the global weak solutions for the non-homogeneous incompressible magnetohydrodynamic equations with Navier boundary condi-tions for the velocity field and the magnetic field in a bounded domain Ω R^3. Furthermore,we prove that as the viscosity and resistivity coefficients go to zero simultaneously, these weaksolutions converge to the strong one of the ideal nonhomogeneous incompressible magneto-hydrodynamic equations in energy space.

On the inviscid limit of the compressible Navier-Stokes equations near Onsager's regularity in bounded domains

The viscous dissipation limit of weak solutions is considered for the Navier-Stokes equations of compressible isentropic flows confined in a bounded domain.We establish a Kato-type criterion for the validity of the inviscid limit for the weak solutions of the Navier-Stokes equations in a function space with the regularity index close to Onsager’s critical threshold.In particular,we prove that under such a regularity assumption,if the viscous energy dissipation rate vanishes in a boundary layer of thickness in the order of the viscosity,then the weak solutions of the Navier-Stokes equations converge to a weak admissible solution of the Euler equations.Our approach is based on the commutator estimates and a subtle foliation technique near the boundary of the domain.

求解气体动力学方程组的高效差分格式

给出了求解多维无粘可压Euler方程组的二阶半离散中心迎风格式。因考虑到了非线性波在Riemann扇内传播的局部速度,从而能更加准确地估计出局部Riemann扇的宽度,最终既回避了计算网格的交错,又降低了格式的数值粘性,建立了介于迎风格式和中心格式之间的高分辨率的半离散中心迎风格式。同时,该格式利用Tadmor等人的耗散型MinMod限制器和Harten等人的压缩型UNO限制器的凸组合来重构分片线性多项式,不仅能快速求解多维无粘可压Euler方程组,还可有效地防止数值解产生伪振荡。

On the Inviscid Limit of the 3D Navier–Stokes Equations with Generalized Navier-Slip Boundary Conditions

In this paper,we investigate the vanishing viscosity limit problem for the 3-dimensional(3D)incompressible Navier-Stokes equations in a general bounded smooth domain of R^3 with the generalized Navier-slip boundary conditions u^ε·n=0,n×(ω^ε)=[Bu^ε]τon∂Ω.Some uniform estimates on rates of convergence in C([0,T],L2(Ω))and C([0,T],H^1(Ω))of the solutions to the corresponding solutions of the ideal Euler equations with the standard slip boundary condition are obtained.

THE NAVIER-STOKES EQUATIONS WITH THE KINEMATIC AND VORTICITY BOUNDARY CONDITIONS ON NON-FLAT BOUNDARIES

We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in R^n with compact and smooth boundary, subject to the kinematic and vorticity boundary conditions on the non-flat boundary. We observe that, under the nonhomogeneous boundary conditions, the pressure p can be still recovered by solving the Neumann problem for the Poisson equation. Then we establish the well-posedness of the unsteady Stokes equations and employ the solution to reduce our initial-boundary value problem into an initial-boundary value problem with absolute boundary conditions. Based on this, we first establish the well-posedness for an appropriate local linearized problem with the absolute boundary conditions and the initial condition （without the incompressibility condition）, which establishes a velocity mapping. Then we develop apriori estimates for the velocity mapping, especially involving the Sobolev norm for the time-derivative of the mapping to deal with the complicated boundary conditions, which leads to the existence of the fixed point of the mapping and the existence of solutions to our initial-boundary value problem. Finally, we establish that, when the viscosity coefficient tends zero, the strong solutions of the initial-boundary value problem in R^n（n ≥ 3） with nonhomogeneous vorticity boundary condition converge in L^2 to the corresponding Euler equations satisfying the kinematic condition.

非牛顿流无粘极限的进一步研究

扩展了在不同范数意义下,当粘性系数趋于零时非牛顿流的解收敛到Navier-Stokes方程解的结论,将研究结果扩展为3<p<11/3,得到了在L2范数、H1范数及L!范数意义下解的收敛.

三维非齐次不可压MHD方程组在Slip边界条件下的无粘无电阻极限

在一般光滑有界区域下,研究了三维非齐次不可压MHD方程组在slip边界条件下的无粘无电阻极限问题,并且得到了在C([0,T],H^2(Ω))意义下强收敛结果.

THE RELATION OF TWO-DIMENSIONAL VISCOUS CAMASSA-HOLM EQUATIONS AND THE NAVIER-STOKES EQUATIONS

In this article, the authors show the existence of global solution of two-dimensional viscous Camassa-Holm (Navier-Stokes-alpha) (NS-α) equations. The authors also prove that the solution of the NS-α equations converges to the solution of the 2D NS equations in the inviscid limit and give the convergence rate of the difference of the solution.

Uniform local well-posedness and inviscid limit for the Benjamin-Ono-Burgers equation

In this paper,we study the Cauchy problem for the Benjamin-Ono-Burgers equation ∂_(t)u−ϵ∂^(2)/_(x)u+H∂^(2)_(x)u+uu_(x)=0,where H denotes the Hilbert transform operator.We obtain that it is uniformly locally well-posed for small data in the refined Sobolev space H~σ(R)(σ■0),which is a subspace of L2(ℝ).It is worth noting that the low-frequency part of H~σ(R)is scaling critical,and thus the small data is necessary.The high-frequency part of H~σ(R)is equal to the Sobolev space Hσ(ℝ)(σ■0)and reduces to L2(ℝ).Furthermore,we also obtain its inviscid limit behavior in H~σ(R)(σ■0).

Inviscid Limit for Scalar Viscous Conservation Laws in Presence of Strong Shocks and Boundary Layers

In this paper, we study the inviscid limit problem for the scalar viscous conservation laws on half plane. We prove that if the solution of the corresponding inviscid equation on half plane is piecewise smooth with a single shock satisfying the entropy condition, then there exist solutions to the viscous conservation laws which converge to the inviscid solution away from the shock discontinuity and the boundary at a rate of ε1 as the viscosity ε tends to zero.

The inviscid limit for the Landau-Lifshitz-Gilbert equation in the critical Besov space

We prove that in dimensions three and higher the Landau-Lifshitz-Gilbert equation with small initial data in the critical Besov space is globally well-posed in a uniform way with respect to the Gilbert damping parameter. Then we show that the global solution converges to that of the Schr¨odinger maps in the natural space as the Gilbert damping term vanishes. The proof is based on some studies on the derivative Ginzburg-Landau equations.

THE RELATION OF TWO-DIMENSIONAL VISCOUS CAMASSA-HOLM EQUATIONS AND THE NAVIER-STOKES EQUATIONS

In this article, the authors show the existence of global solution of two-dimensional viscous Camassa-Holm (Navier-Stokes-alpha) (NS-α) equations. The authors also prove that the solution of the NS-α equations converges to the solution of the 2D NS equations in the inviscid limit and give the convergence rate of the difference of the solution.

Vlasov-Maxwell-Fokker-Planck方程组的极限问题

本文研究Vlasov-Maxwell-Fokker-Planck方程组的拟中性和粘性消失复合极限,证明了Vlasov-Maxwell-Fokker-Planck的重整化解到电磁流体方程组的强解的收敛性.主要结果的证明基于弱收敛紧性方法和相对熵方法.

THE EULER EQUATION AND ONSAGER CONJECTURE

In this paper, we introduce the progress of the Euler equation and Onsager conjecture. We also introduce the Euler＇s life, the researches about the incompressible Euler equation, and the Onsager conjecture.

Examples of Boundary Layers Associated with the Incompressible Navier-Stokes Equations

The author surveys a few examples of boundary layers for which the Prandtl boundary layer theory can be rigorously validated.All of them are associated with the incompressible Navier-Stokes equations for Newtonian fluids equipped with various Dirichlet boundary conditions(specified velocity).These examples include a family of(nonlinear 3D) plane parallel flows,a family of(nonlinear) parallel pipe flows,as well as flows with uniform injection and suction at the boundary.We also identify a key ingredient in establishing the validity of the Prandtl type theory,i.e.,a spectral constraint on the approximate solution to the Navier-Stokes system constructed by combining the inviscid solution and the solution to the Prandtl type system.This is an additional difficulty besides the wellknown issue related to the well-posedness of the Prandtl type system.It seems that the main obstruction to the verification of the spectral constraint condition is the possible separation of boundary layers.A common theme of these examples is the inhibition of separation of boundary layers either via suppressing the velocity normal to the boundary or by injection and suction at the boundary so that the spectral constraint can be verified.A meta theorem is then presented which covers all the cases considered here.

Global Steady Prandtl Expansion over a Moving Boundary I

This is the first of three papers in which we prove that steady,incompressible Navier-Stokes flows posed over the moving boundary,y=0,can be decomposed into Euler and Prandtl flows in the inviscid limit globally in[1,∞)×[0,∞),assum-ing a sufficiently small velocity mismatch.In this part,sharp decay rates and self-similar asymptotics are extracted for both Prandtl and Eulerian layers.