Zero dispersion and viscosity limits of invariant manifolds for focusing nonlinear Schrödinger equations

Zero dispersion and viscosity limits of invariant manifolds for focusing nonlinear Schrödinger equations (NLS) are studied. We start with spatially uniform and temporally periodic solutions (the so-called Stokes waves). We find that the spectra of the linear NLS at the Stokes waves often have surprising limits as dispersion or viscosity tends to zero. When dispersion (or viscosity) is set to zero, the size of invariant manifolds and/or Fenichel fibers approaches zero as viscosity (or dispersion) tends to zero. When dispersion (or viscosity) is nonzero, the size of invariant manifolds and/or Fenichel fibers approaches a nonzero limit as viscosity (or dispersion) tends to zero. When dispersion is nonzero, the center-stable manifold, as a function of viscosity, is not smooth at zero viscosity. A subset of the center-stable manifold is smooth at zero viscosity. The unstable Fenichel fiber is smooth at zero viscosity. When viscosity is nonzero, the stable Fenichel fiber is smooth at zero dispersion.     

Global regularity for the 2D Boussinesq equations with partial viscosity terms

In this paper, we prove the global in time regularity for the 2D Boussinesq system with either the zero diffusivity or the zero viscosity. We also prove that as diffusivity (viscosity) tends to zero, the solutions of the fully viscous equations converge strongly to those of zero diffusion (viscosity) equations. Our result for the zero diffusion system, in particular, solves the Problem no. 3 posed by Moffatt in [R.L. Ricca, (Ed.), Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001, pp. 3-10].     

The Asymptotic Limit for the 3D Boussinesq System

In this paper, we show the asymptotic limit for the 3D Boussinesq system with zero viscosity limit or zero diffusivity limit. By the classical energy method, we prove that as viscosity(or diffusivity) coefficient goes to zero the solutions of the fully viscous equations converges to those of zero viscosity(or zero diffusivity) equations, which extend the previous results on the asymptotic limit under the conditions of the zero parameter(zero viscosity ν = 0 or zero diffusivity η = 0) in 2D case separately.     

A variational approach to the inviscous limit of fractional conservation laws

We are concerned with a control problem related to the vanishing fractional viscosity approximation to scalar conservation laws. We investigate the Γ-convergence of the control cost functional, as the viscosity coefficient tends to zero.     

On the asymptotics of the motion of a nonlinear viscous fluid

Under study is the nonstationary problem of the motion of a nonlinear-viscous fluid in the case of low or high viscosity. We establish that the convergence of solutions to the corresponding limit solutions as the viscosity converges to zero or infinity.     

Boundary layers for parabolic regularizations of totally characteristic quasilinear parabolic equations

In this paper we study boundary layers of nonlinear characteristic parabolic equations as the viscosity goes to zero. We obtain and justify in small time a complete expansion of the solution with respect to the viscosity.     

On the asymptotics of motion of a viscous incompressible fluid for small viscosity

We study a nonstationary initial–boundary value problem on the motion of a viscous incompressible fluid in the case of small viscosity. We prove the convergence of solutions to the corresponding limit relations as the viscosity tends to zero.     

Uniform boundary stabilization of the finite difference space discretization of the 1−d wave equation

The energy of solutions of the wave equation with a suitable boundary dissipation decays exponentially to zero as time goes to infinity. We consider the finite-difference space semi-discretization scheme and we analyze whether the decay rate is independent of the mesh size. We focus on the one-dimensional case. First we show that the decay rate of the energy of the classical semi-discrete system in which the 1?d Laplacian is replaced by a three-point finite difference scheme is not uniform with respect to the net-spacing size h. Actually, the decay rate tends to zero as h goes to zero. Then we prove that adding a suitable vanishing numerical viscosity term leads to a uniform (with respect to the mesh size) exponential decay of the energy of solutions. This numerical viscosity term damps out the high frequency numerical spurious oscillations while the convergence of the scheme towards the original damped wave equation is kept. Our method of proof relies essentially on discrete multiplier techniques.     

黏性系数依赖密度型可压缩Navier-Stokes 方程的零耗散极限问题

本文考虑黏性系数依赖密度的可压缩Navier-Stokes 方程解的零耗散极限问题. 假定Euler 方程的稀疏波解一端被真空状态连接, 我们证明Navier-Stokes 方程存在一列(依赖黏性的) 整体解, 且随着粘性的消失, 此整体解逐渐稳定于Euler 方程对应的稀疏波解和真空状态; 并且得到了一致衰减率估计. 此结果推广了常黏性系数的情形.     

Wellposedness and zero microrotation viscosity limit of the 2D micropolar fluid equations

In this paper, we consider the 2D micropolar fluid equations in the whole space . We prove the global wellposedness of the system with rough initial data and show the vanishing microrotation viscosity limit in the case of zero kinematic viscosity or zero angular viscosity. Copyright © 2011 John Wiley & Sons, Ltd.     

A level set approach for computing discontinuous solutions of Hamilton-Jacobi equations

We introduce two types of finite difference methods to compute the L-solution and the proper viscosity solution recently proposed by the second author for semi-discontinuous solutions to a class of Hamilton-Jacobi equations. By regarding the graph of the solution as the zero level curve of a continuous function in one dimension higher, we can treat the corresponding level set equation using the viscosity theory introduced by Crandall and Lions. However, we need to pay special attention both analytically and numerically to prevent the zero level curve from overturning so that it can be interpreted as the graph of a function. We demonstrate our Lax-Friedrichs type numerical methods for computing the L-solution using its original level set formulation. In addition, we couple our numerical methods with a singular diffusive term which is essential to computing solutions to a more general class of HJ equations that includes conservation laws. With this singular viscosity, our numerical methods do not require the divergence structure of equations and do apply to more general equations developing shocks other than conservation laws. These numerical methods are generalized to higher order accuracy using weighted ENO local Lax-Friedrichs methods as developed recently by Jiang and Peng. We verify that our numerical solutions approximate the proper viscosity solutions obtained by the second author in a recent Hokkaido University preprint. Finally, since the solution of scalar conservation law equations can be constructed using existing numerical techniques, we use it to verify that our numerical solution approximates the entropy solution.

     

Blow-up criteria of smooth solutions to the 3D Boussinesq system with zero viscosity in a bounded domain

We prove that a smooth solution of the 3D Boussinesq system with zero viscosity in a bounded domain breaks down, if a certain norm of vorticity blows up at the same time. Here this norm is weaker than the bmo-norm.     

Vanishing viscosity and Debye-length limit to rarefaction wave with vacuum for the 1D bipolar Navier–Stokes–Poisson equation

In this paper, we consider the one-dimensional (1D) compressible bipolar Navier–Stokes–Poisson equations. We know that when the viscosity coefficient and Debye length are zero in the compressible bipolar Navier–Stokes–Poisson equations, we have the compressible Euler equations. Under the case that the compressible Euler equations have a rarefaction wave with one-side vacuum state, we can construct a sequence of the approximation solution to the one-dimensional bipolar Navier–Stokes–Poisson equations with well-prepared initial data, which converges to the above rarefaction wave with vacuum as the viscosity and the Debye length tend to zero. Moreover, we also obtain the uniform convergence rate. The results are proved by a scaling argument and elaborate energy estimate.     

Uniform Local Well-Posedness to the Density-Dependent Navier-Stokes-Maxwell System

In this paper we establish the uniform local-in-time existence and uniqueness of classical solutions to the density-dependent Navier-Stokes-Maxwell system. We then apply this uniform result to investigate the zero dielectric constant limit and the vanishing viscosity limit to Navier-Stokes-Maxwell system. We obtain the well-known density-dependent magnetohydrodynamic equations when the dielectric constant goes to zero.     

Global well-posedness for the 2D incompressible magneto-micropolar fluid system with partial viscosity

In this paper, we consider an initial-boundary value problem for the 2D incompressible magnetomicropolar fluid equations with zero magnetic diffusion and zero spin viscosity in the horizontally infinite flat layer with Navier-type boundary conditions. We establish the global well-posedness of strong solutions around the equilibrium(0, e_1, 0).     

Homogenization and penalization of functional first-order PDE

In this paper we study the asymptotic behavior of viscosity solutions for a functional partial differential equation with a small parameter as the parameter tends to zero. We study simultaneous effects of homogenization and penalization in functional first-order PDE. We establish a convergence theorem in which the limit equation is identified with some first order PDE.     

Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity

Summary. We consider the finite-difference space semi-discretization of a locally damped wave equation, the damping being supported in a suitable subset of the domain under consideration, so that the energy of solutions of the damped wave equation decays exponentially to zero as time goes to infinity. The decay rate of the semi-discrete systems turns out to depend on the mesh size h of the discretization and tends to zero as h goes to zero. We prove that adding a suitable vanishing numerical viscosity term leads to a uniform (with respect to the mesh size) exponential decay of the energy of solutions. This numerical viscosity term damps out the high frequency numerical spurious oscillations while the convergence of the scheme towards the original damped wave equation is kept. We discuss this problem in 1D and 2D in the interval and the square respectively. Our method of proof relies on discrete multiplier techniques. Mathematics Subject Classification (1991):65M06     

Bernoulli jets and the zero mean curvature equation

We consider an elliptic PDE problem related with fluid mechanics. We show that level sets of rescaled solutions satisfy the zero mean curvature equation in a suitable weak viscosity sense. In particular, such level sets cannot be touched from below (above) by a convex (concave) paraboloid in a suitably small neighborhood.     

Vanishing viscosity limit for a coupled Navier-Stokes/Allen-Cahn system

In this paper, we study the vanishing viscosity limit for a coupled Navier-Stokes/Allen-Cahn system in a bounded domain. We first show the local existence of smooth solutions of the Euler/Allen-Cahn equations by modified Galerkin method. Then using the boundary layer function to deal with the mismatch of the boundary conditions between Navier-Stokes and Euler equations, and assuming that the energy dissipation for Navier-Stokes equation in the boundary layer goes to zero as the viscosity tends to zero, we prove that the solutions of the Navier-Stokes/Allen-Cahn system converge to that of the Euler/Allen-Cahn system in a proper small time interval. In addition, for strong solutions of the Navier-Stokes/Allen-Cahn system in 2D, the convergence rate is cν1/2.     

On second grade fluids with vanishing viscosity

We consider in ℝⁿ (n = 2, 3) the equation of a second grade fluid with vanishing viscosity, also known as Camassa-Holm equation. We prove local existence and uniqueness of solutions for smooth initial data. We also give a blow-up condition which implies that the solution is global for n = 2. Finally, we prove the convergence of the solutions of second grade fluid equation to the solution of the Camassa-Holm equation as the viscosity tends to zero.