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.     

Vanishing Viscosity Limit for Incompressible Viscoelasticity in Two Dimensions

This paper studies the inviscid limit of the two-dimensional incompressible viscoelasticity, which is a system coupling a Navier-Stokes equation with a transport equation for the deformation tensor. The existence of global smooth solutions near the equilibrium with a fixed positive viscosity was known since the work of [35]. The inviscid case was solved recently by the second author [28]. While the latter was solely based on the techniques from the studies of hyperbolic equations, and hence the two-dimensional problem is in general more challenging than that in higher dimensions, the former was relied crucially upon a dissipative mechanism. Indeed, after a symmetrization and a linearization around the equilibrium, the system of the incompressible viscoelasticity reduces to an incompressible system of damped wave equations for both the fluid velocity and the deformation tensor. These two approaches are not compatible. In this paper, we prove global existence of solutions, uniformly in both time t ∈ [0, +∞) and viscosity μ ≥ 0 . This allows us to justify in particular the vanishing viscosity limit for all time. In order to overcome difficulties coming from the incompatibility between the purely hyperbolic limiting system and the systems with additional parabolic viscous perturbations, we introduce in this paper a rather robust method that may apply to a wide class of physical systems of similar nature. Roughly speaking, the method works in the two-dimensional case whenever the hyperbolic system satisfies intrinsically a “strong null condition.” For dimensions not less than three, the usual null condition is sufficient for this method to work. © 2019 Wiley Periodicals, Inc.     

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.     

Remarks on Vanishing Viscosity Limits for the 3D Navier-Stokes Equations with a Slip Boundary Condition

The authors study vanishing viscosity limits of solutions to the 3-dimensional incompressible Navier-Stokes system in general smooth domains with curved boundaries for a class of slip boundary conditions. In contrast to the case of flat boundaries, where the uniform convergence in super-norm can be obtained, the asymptotic behavior of viscous solutions for small viscosity depends on the curvature of the boundary in general. It is shown, in particular, that the viscous solution converges to that of the ideal Euler equations in C([0, T];H1(Ω)) provided that the initial vorticity vanishes on the boundary of the domain.     

Remarks on Ramanujam-Kawamata-Viehweg Vanishing Theorem

In this article we prove a general result on a nef vector bundle E on a projective manifold X of dimension n depending on the vector space H^n,n(X,E): It is also shown that H^n,n(X,E) = 0 for an indecomposable nef rank 2 vector bundles E on some specific type of n dimensional projective manifold X. The same vanishing shown to hold for indecomposable nef and big rank 2 vector bundles on any variety with trivial canonical bundle.     

Delta Shocks and Vacuum States in Vanishing Pressure Limits of Solutions to the Relativistic Euler Equations

The Riemann problems for the Euler system of conservation laws of energy and momentum in special relativity as pressure vanishes are considered. The Riemann solutions for the pressureless relativistic Euler equations are obtained constructively. There are two kinds of solutions, the one involves delta shock wave and the other involves vacuum. The authors prove that these two kinds of solutions are the limits of the solutions as pressure vanishes in the Euler system of conservation laws of energy and momentum in special relativity.     

Remarks on shear flows of the Euler equations

The gradient estimates for renormalization solutions to the Euler equations are derived. The proof is based on the boundedness of the solutions to the linear transport equation, component-wisely. A shear flow is a unique globally-in-time strong solution in certain class due to the argument of renormalization solutions. Shear flows give lower bounds for gradient estimates as well as the analyticity rate. The threshold between locally well-posedness and ill-posedness of the Euler equations is clarified in terms of functions spaces of initial data.     

Remarks on the derivation of the hydrostatic Euler equations

The motion of an inviscid incompressible fluid between two horizontal plates is studied in the limit when the plates are infinitesimally close. The convergence of the solutions of the Euler equations to those of their formal ‘hydrostatic’ limit can be established in the case when the initial velocity field satisfies a local Rayleigh conditions. This result, originally obtained by Grenier through weighted energy estimates based on Arnold's stability analysis of the Euler equations, is proven here by a more straightforward method even closer to Arnold's method.     

Stability of Solutions of Parabolic PDEs with Random Drift and Viscosity Limit

Let u _α be the solution of the It? stochastic parabolic Cauchy problem , where ξ is a space—time noise. We prove that u _α depends continuously on α , when the coefficients in L _α converge to those in L ₀ . This result is used to study the diffusion limit for the Cauchy problem in the Stratonovich sense: when the coefficients of L _α tend to 0 the corresponding solutions u _α converge to the solution u ₀ of the degenerate Cauchy problem . These results are based on a criterion for the existence of strong limits in the space of Hida distributions (S) ^* . As a by-product it is proved that weak solutions of the above Cauchy problem are in fact strong solutions. Accepted 22 May 1998     

Remarks on the nonlinear instability of incompressible Euler equations

In this paper, we consider the nonlinear instability of incompressible Euler equations. If a steady density is non-monotonic, then the smooth steady state is a nonlinear instability. First, we use variational method to find a dominant eigenvalue which is important in the construction of approximate solutions, then by energy technique and analytic method, we obtain the dynamical instability result.     

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.     

Global Weak Solutions to One-Dimensional Compressible Navier-Stokes Equations with Density-Dependent Viscosity Coefficients

We prove the global existence of weak solutions of the one-dimensional compressible Navier-stokes equations with density-dependent viscosity. In particular, we assume that the initial density belongs to L^1 and L^∞, module constant states at x = -∞ and x = ＋∞, which may be different. The initial vacuum is permitted in this paper and the results may apply to the one-dimensional Saint-Venant model for shallow water.     

Hydrodynamic Limit and Viscosity Solutions for a Two-Dimensional Growth Process in the Anisotropic KPZ Class

We study a two-dimensional stochastic interface growth model that is believed to belong to the so-called anisotropic KPZ (AKPZ) universality class [4,5]. It can be seen either as a two-dimensional interacting particle process with drift that generalizes the one-dimensional Hammersley process [1,24], or as an irreversible dynamics of lozenge tilings of the plane [5,29]. Our main result is a hydrodynamic limit: the interface height profile converges, after a hyperbolic scaling of space and time, to the solution of a nonlinear first-order PDE of Hamilton-Jacobi type with nonconvex Hamiltonian (nonconvexity of the Hamiltonian is a distinguishing feature of the AKPZ class). We prove the result in two situations: (1) for smooth initial profiles and times smaller than the time T_shock when singularities (shocks) appear or (2) for all times, including t > T_shock, if the initial profile is convex. In the latter case, the height profile converges to the viscosity solution of the PDE. As an important ingredient, we introduce a Harris-type graphical construction for the process. © 2018 Wiley Periodicals, Inc.     

A New Proof for the Equivalence of Weak and Viscosity Solutions for the p-Laplace Equation

In this paper, we give a new proof for the fact that the distributional weak solutions and the viscosity solutions of the p-Laplace equation ?div(|Du| ^p?2 Du) = 0 coincide. Our proof is more direct and transparent than the original proof of Juutinen et al. [8 Juutinen , P. , Lindqvist , P. , Manfredi , J.J. ( 2001 ). On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation . SIAM J. Math. Anal. 33 : 699 – 717 .[Crossref], [Web of Science ®] , [Google Scholar]], which relied on the full uniqueness machinery of the theory of viscosity solutions. We establish a similar result also for the solutions of the non-homogeneous version of the p-Laplace equation.     

关于导函数极限定理的注记

本文旨在给出导函数极限的几个结果,依据此结果可以判断函数在一点的不可导性;同时,指出文献[2]中的一个不妥之处.     

消失的弱~Morrey 型空间上某些算子的刻画

In this paper we mainly give some characterizations for the boundedness of the weight Hardy operator, maximal operator, potential operator and singular integral operator on the vanishing generalized weak Morrey spaces V W L_Π~(p,φ)(?) with bounded set ?.