Vanishing Viscosity Limit for Riemann Solutions to a Class of Non-Strictly Hyperbolic Systems

By the vanishing viscosity approach, a class of non-strictly hyperbolic systems of conservation laws that contain the equations of geometrical optics as a prototype are studied. The existence, uniqueness and stability of solutions involving delta shock waves and generalized vacuum states are discussed completely.     

Remarks on the Emergence of Weak Euler Solutions in the Vanishing Viscosity Limit

Journal of Nonlinear Science - We prove that if the local second-order structure function exponents in the inertial range remain positive uniformly in viscosity, then any spacetime $$L^2$$ weak...     

Global Well‐Posedness of Incompressible Elastodynamics in Two Dimensions

We prove that for sufficiently small initial displacements in some weighted Sobolev space, the Cauchy problem of the systems of incompressible isotropic Hookean elastodynamics in two space dimensions admits a uniqueness global classical solution. © 2016 Wiley Periodicals, Inc.     

A High-Order Kernel-Free Boundary Integral Method for Incompressible Flow Equations in Two Space Dimensions

This paper presents a fourth-order kernel-free boundary integral method for the time-dependent, incompressible Stokes and Navier-Stokes equations defined on irregular bounded domains. By the stream function-vorticity formulation, the incompressible flow equations are interpreted as vorticity evolution equations. Time discretization methods for the evolution equations lead to a modified Helmholtz equation for the vorticity, or alternatively, a modified biharmonic equation for the stream function with two clamped boundary conditions. The resulting fourth-order elliptic boundary value problem is solved by a fourth-order kernel-free boundary integral method, with which integrals in the reformulated boundary integral equation are evaluated by solving corresponding equivalent interface problems, regardless of the exact expression of the involved Green's function. To solve the unsteady Stokes equations, a four-stage composite backward differential formula of the same order accuracy is employed for time integration. For the Navier-Stokes equations, a three-stage third-order semi-implicit Runge-Kutta method is utilized to guarantee the global numerical solution has at least third-order convergence rate. Numerical results for the unsteady Stokes equations and the Navier-Stokes equations are presented to validate efficiency and accuracy of the proposed method.     

Density-Dependent Incompressible Fluids with Non-Newtonian Viscosity

We study the system of PDEs describing unsteady flows of incompressible fluids with variable density and non-constant viscosity. Indeed, one considers a stress tensor being a nonlinear function of the symmetric velocity gradient, verifying the properties of p-coercivity and (p–1)-growth, for a given parameter p > 1. The existence of Dirichlet weak solutions was obtained in [2], in the cases p 12/5 if d = 3 or p 2 if d = 2, d being the dimension of the domain. In this paper, with help of some new estimates (which lead to point-wise convergence of the velocity gradient), we obtain the existence of space-periodic weak solutions for all p 2. In addition, we obtain regularity properties of weak solutions whenever p 20/9 (if d = 3) or p 2 (if d = 2). Further, some extensions of these results to more general stress tensors or to Dirichlet boundary conditions (with a Newtonian tensor large enough) are obtained.     

Burgers方程初边值问题的粘性消失极限

在本文中,我们研究粘性Burgers方程在物理边界存在时的粘性消失极限,并证明相应的无粘Burgers方程初边值问题(IBVP)满足熵条件的弱解的唯一性和对初边值的连续依赖性.     

二维不可压缩流的谱方法

本文以二维涡度方程为模型,介绍了谱方法和拟谱方法以及它们与差分方法和有限元法相结合的混合解法．这些方法可推广应用于其它一些类似的非线性问题．本文还给出了这些方法的某些数值例子和误差估计结果     

A Probability Approach to the Method of Vanishing Viscosity

We clarify conditions under which solutions to the Cauchy problem for a general (fully nondiagonal) system of linear and nonlinear parabolic equations admit probability representations. Such representations are also used for constructing and studying solutions to the Cauchy problem for nonlinear hyperbolic systems. Bibliography: 26 titles.     

On Regularity of Incompressible Fluid with Shear Dependent Viscosity

The authors consider a non-Newtonian fluid governed by equations with p-structure in a cubic domain.A fluid is said to be shear thinning(or pseudo-plastic) if 1 < p < 2,and shear thickening(or dilatant) if p > 2.The case p > 2 is considered in this paper.To improve the regularity results obtained by Crispo,it is shown that the secondorder derivatives of the velocity and the first-order derivative of the pressure belong to suitable spaces,by appealing to anisotropic Sobolev embeddings.     

粘性系数依赖于密度的Korteweg型不可压流体的强解问题

研究一类Korteweg型不可压流体模型的强解问题.针对粘性系数依赖于密度的情形,当初始值满足兼容性条件(9)对,证明了强解的局部存在性和唯一性.我们在这指出,本文允许初始真空存在.     

On the Vanishing Electron-Mass Limit in Plasma Hydrodynamics in Unbounded Media

We consider the zero-electron-mass limit for the Navier?CStokes?CPoisson system in unbounded spatial domains. Assuming smallness of the viscosity coefficient and ill-prepared initial data, we show that the asymptotic limit is represented by the incompressible Navier?CStokes system, with a Brinkman damping, in the case when viscosity is proportional to the electron-mass, and by the incompressible Euler system provided the viscosity is dominated by the electron mass. The proof is based on the RAGE theorem and dispersive estimates for acoustic waves, and on the concept of suitable weak solutions for the compressible Navier?CStokes system.     

On the Limit as the Density Ratio Tends to Zero for Two Perfect Incompressible Fluids Separated by a Surface of Discontinuity

We study the asymptotic limit as the density ratio ρ^?/ρ⁺ → 0, where ρ⁺ and ρ^? are the densities of two perfect incompressible 2-D/3-D fluids, separated by a surface of discontinuity along which the pressure jump is proportional to the mean curvature of the moving surface. Mathematically, the fluid motion is governed by the two-phase incompressible Euler equations with vortex sheet data. By rescaling, we assume the density ρ⁺ of the inner fluid is fixed, while the density ρ^? of the outer fluid is set to ε. We prove that solutions of the free-boundary Euler equations in vacuum are obtained in the limit as ε → 0.     

Probabilistic Interpretation of the Vanishing Viscosity Method for Systems of Conservation and Balance Laws

Mathematical Notes - Refinements of some relations between mixed moduli of smoothness of fractional orders in the metrics $$L_p$$ and $$L_\infty$$ are obtained.     

Global Well-Posedness of the 2D Incompressible Micropolar Fluid Flows with Partial Viscosity and Angular Viscosity

This paper is concerned with the two-dimensional equations of incompressible micropolar fluid flows with mixed partial viscosity and angular viscosity. The global existence and uniqueness of smooth solution to the Cauchy problem is established.     

Global Existence of Classical Solutions for Some Oldroyd-B Model via the Incompressible Limit

In this paper, we prove local and global existence of classical solutions for a system of equations concerning an incompressible viscoelastic fluid of Oldroyd-B type via the incompressible limit when the initial data are sufficiently small.     

Generalized Solutions of Nonlinear Parabolic Systems and the Vanishing Viscosity Method

We show in this paper that stochastic processes associated with nonlinear parabolic equations and systems allow one to construct a probabilistic representation of a generalized solution to the Cauchy problem. We also show that in some cases the derived representation can be used to construct a solution to the Cauchy problem for a hyperbolic system via the vanishing viscosity method. Bibliography: 12 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 311, 2004, pp. 7–39.     

Incompressible Limit of the Compressible Q-tensor System of Liquid Crystals

We study the connection between the compressible Navier-Stokes equations coupled by the Qtensor equation for liquid crystals with the incompressible system in the periodic case, when the Mach number is low. To be more specific, the convergence of the weak solutions of the compressible nematic liquid crystal model to the incompressible one is proved as the Mach number approaches zero, and we also obtain the similar results in the stochastic setting when the equations are driven by a stochastic fo...     

二维Landau-Lifshitz静态方程组的渐近性质

在本文中，我们讨论了ｕ∧（－Δｕ＋λ（ｕ，ｎ）ｎ）＝０，｜ｕ｜＝１，ｘ∈Ｂ１和ｕ＝（ｘ１，ｘ２，０）ｘ∈Ｂ１的轴对称解ｕλ的渐近行为，其中Ｂ１是Ｒ２中单位圆，ｕ＝（ｕ１，ｕ２，ｕ３）．我们证明了ｕλｕ∞在Ｈ２Ｂ１＼Ｂε，Ｒ３中．     

一个极限问题的两种解法

针对第13届美国普特南数学竞赛的一道数列极限题,将原本不具有单调性质的数列转化为单调数列,并利用单调有界原理及数学归纳法,得出其极限,此方法可推广用以解决同类的一般性数列求极限.此外,此题也可采用不动点原理,简化计算过程.