Hermano Frid

On the theory of divergence-measure fields and its applications

Divergence-measure fields are extended vector fields, including vector fields inLp and vector-valued Radon measures, whose divergences are Radon measures. Such fields arise naturally in the study of entropy solutions of nonlinear conservation laws and other areas. In this paper, a theory of divergence-measure fields is presented and analyzed, in which normal traces, a generalized Gauss-Green theorem, and product rules, among others, are established. Some applications of this theory to several nonlinear problems in conservation laws and related areas are discussed. In particular, with the aid of this theory, we prove the stability of Riemann solutions, which may contain rarefaction waves, contact discontinuities, and/or vacuum states, in the class of entropy solutions of the Euler equations for gas dynamics.

Uniqueness and asymptotic stability of Riemann solutions for the compressible Euler equations

We prove the uniqueness of Riemann solutions in the class of entropy solutions in L∞ ∩ BVloc for the 3 × 3 system of compressible Euler equations, under usual assumptions on the equation of state for the pressure which imply strict hyperbolicity of the system and genuine nonlinearity of the first and third characteristic families. In particular, if the Riemann solutions consist of at most rarefaction waves and contact discontinuities, we show the global L2-stability of the Riemann solutions even in the class of entropy solutions in L∞ with arbitrarily large oscillation for the 3 × 3 system. We apply our framework established earlier to show that the uniqueness of Riemann solutions implies their inviscid asymptotic stability under L1 perturbation of the Riemann initial data, as long as the corresponding solutions are in L∞ and have local bounded total variation satisfying a natural condition on its growth with time. No specific reference to any particular method for constructing the entropy solutions is made. Our uniqueness result for Riemann solutions can easily be extended to entropy solutions U(x, t), piecewise Lipschitz in x, for any t > 0.

Vanishing shear viscosity in the equations of compressible fluids for the flows with the cylinder symmetry

We analyze the question of the limit process when the shear viscosity goes to zero for global solutions to the Navier--Stokes equations for compressible heat conductive fluids for the flows which are invariant over cylindrical sheets.

On a free boundary problem for a strongly degenerate quasi-linear parabolic equation with an application to a model of pressure filtration

We consider a free boundary problem of a quasi-linear strongly degenerate parabolic equation arising from a model of pressure filtration of flocculated suspensions. We provide definitions of generalized solutions of the free boundary problem in the framework of L2 divergence-measure fields. The formulation of boundary conditions is based on a Gauss--Green theorem for divergence-measure fields on bounded domains with Lipschitz deformable boundaries and avoids referring to traces of the solution. This allows one to consider generalized solutions from a larger class than BV. Thus it is not necessary to derive the usual uniform estimates of spatial and time derivatives of the solutions of the correspondingregularized problem, as required by the BV approach. We first prove the existence and uniqueness of the solution of the regularized parabolic free boundary problem and then apply the vanishing viscosity method to prove the existence of a generalized solution to the degenerate free boundary problem.

Periodic and almost periodic solutions of conservation laws: Global existence and decay

In this paper we survey recent results on the decay of periodic and almost periodic solutions of conservation laws. We also recall some recent results on the global existence of periodic solutions of conservation laws systems which lie inBVloc and are constructed through Glimm scheme. The latter motivates a discussion on a possible strategy for solving the open problem of the global existence of periodic solutions of the Euler equations for nonisentropic gas dynamics. We base our decay analysis on a general result about space-time functions which are almost periodic in the space variable, established here for the first time. This result is an abstract version of Theorem 2.1 in [31], which in turn is an extention of the combined result given by Theorems 3.1–3.2 in [9].

A quasi-linear parabolic system for three-phase capillary flow in porous media

A parabolic system is proposed to describe three-phase capillary flows in porous media. The assumptions imposed on the phase interaction mean that the capillarity matrix is triangular. The unique solvability of an associated nondegenerate parabolic system is proved for the Cauchy problem in a class of x-periodic solutions.

Asymptotic stability of Riemann waves for conservation laws

Abstract.We are concerned with the asymptotic behavior of entropy solutions of conservation laws. A new notion about the asymptotic stability of Riemann solutions is introduced, and corresponding analytical frameworks are developed. The correlation between the asymptotic problem and many important topics in conservation laws and nonlinear analysis is recognized and analyzed, such as zero dissipation limits, uniqueness of entropy solutions, entropy analysis, and divergence-measure fields in

Neumann Problems for Quasi-linear Parabolic Systems Modeling Polydisperse Suspensions

L^\infty

Initial Boundary Value Problems for a Quasi-linear Parabolic System in Three-Phase Capillary Flow in Porous Media

. Then this theory is applied to understanding the asymptotic behavior of entropy solutions for many important systems of conservation laws.

Homogenization of Nonlinear PDEs in the Fourier–Stieltjes Algebras

We discuss the well-posedness of a class of Neumann problems for n x n quasi-linear parabolic systems arising from models of sedimentation of polydisperse suspensions in engineering applications. This class of initial-boundary value problems includes the standard (zero-flux) Neumann condition in the limit as a positive perturbation parameter theta goes to 0. We call, in general, the problem associated with theta \ge 0 the theta-flux Neumann problem. The Neumann boundary conditions, although natural and usually convenient for integration by parts, are nonlinear and couple the different components of the system. An important aspect of our analysis is a time stepping procedure that considers linear boundary conditions for each time step in order to circumvent the difficulties arising from the nonlinear coupling in the original boundary conditions. We prove the well-posedness of the theta-flux Neumann problems for theta > 0 and obtain a solution of the standard (zero-flux) Neumann problem as the limit for the...