Global stability of solutions with discontinuous initial data containing vacuum states for the isentropic Euler equations

The global stability of Lipschitz continuous solutions with discontinuous initial data is established in a broad class of entropy solutions in L^￥L^\infty containing vacuum states. In particular, the uniqueness of Lipschitz solutions with discontinuous initial data is obtained in the broad class of entropy solutions in L^￥L^\infty .     

Global stability of solutions with discontinuous initial data containing vacuum states for the isentropic Euler equations

The global stability of Lipschitz continuous solutions with discontinuous initial data is established in a broad class of entropy solutions in containing vacuum states. In particular, the uniqueness of Lipschitz solutions with discontinuous initial data is obtained in the broad class of entropy solutions in .     

Global entropy solutions to the relativistic Euler equations for a class of large initial data

We are concerned with global entropy solutions to the relativistic Euler equations for a class of large initial data which involve the interaction of shock waves and rarefaction waves. We first carefully analyze the global behavior of the shock curves, the rarefaction wave curves, and their corresponding inverse curves in the phase plane. Based on these analyses, we use the Glimm scheme to construct global entropy solutions to the relativistic Euler equations for the class of large discontinuous initial data.     

Global entropy solutions to the relativistic Euler equations for a class of large initial data

We are concerned with global entropy solutions to the relativistic Euler equations for a class of large initial data which involve the interaction of shock waves and rarefaction waves. We first carefully analyze the global behavior of the shock curves, the rarefaction wave curves, and their corresponding inverse curves in the phase plane. Based on these analyses, we use the Glimm scheme to construct global entropy solutions to the relativistic Euler equations for the class of large discontinuous initial data.Received: May 23, 2004     

基于向量Liapunov函数不连续系统的稳定性研究

基于局部Lipschitz连续且正则(Clarke意义下)的向量Liapunov函数,讨论不连续自治系统的稳定性(Filippov解意义下)．通过定义一类新的向量Liapunov函数的“集值导数”,给出了关于不连续系统的广义比较原理．基于Lipschitz连续且正则的向量Liapunov函数,进一步的给出不连续自治系统的Liapunov稳定性定理．     

Entropy solutions to a strongly degenerate anisotropic convection-diffusion equation with application to utility theory

We study the deterministic counterpart of a backward-forward stochastic differential utility, which has recently been characterized as the solution to the Cauchy problem related to a PDE of degenerate parabolic type with a conservative first order term. We first establish a local existence result for strong solutions and a continuation principle, and we produce a counterexample showing that, in general, strong solutions fail to be globally smooth. Afterward, we deal with discontinuous entropy solutions, and obtain the global well posedness of the Cauchy problem in this class. Eventually, we select a sufficient condition of geometric type which guarantees the continuity of entropy solutions for special initial data. As a byproduct, we establish the existence of an utility process which is a solution to a backward-forward stochastic differential equation, for a given class of final utilities, which is relevant for financial applications.     

On global discontinuous solutions of Hamilton–Jacobi equationsSur des solutions globales discontinues des équations d'Hamilton–Jacobi

The uniqueness of classical semicontinuous viscosity solutions of the Cauchy problem for Hamilton–Jacobi equations is established for globally Lipschitz continuous and convex Hamiltonian H=H(Du), provided the discontinuous initial value function ?(x) is continuous outside a set Γ of measure zero and satisfies

(1)$\text{?(x)??}{}_7\text{(x):=}\text{lim}\text{inf}\text{y→x,}\text{y∈}\text{R}{}^{\mathrm{d}}\text{?Γ}\text{?(y).}$

We prove that the discontinuous solutions with almost everywhere continuous initial data satisfying ($\text{1}$) become Lipschitz continuous after finite time for locally strictly convex Hamiltonians. The L¹-accessibility of initial data and a comparison principle for discontinuous solutions are shown for a general Hamiltonian. The equivalence of semicontinuous viscosity solutions, bi-lateral solutions, L-solutions, minimax solutions, and L^∞-solutions is clarified. To cite this article: G.-Q. Chen, B. Su, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 113–118     

Continuous dependence on the initial and flux functions for solutions of conservation laws

We prove the continuous dependence on the initial and flux functions for the entropy solutions to the Cauchy problem for conservation laws. Accordingly, we can show that the continuous dependence on the flux function for the entropy solutions depends only on the sup norm, not on the Lipschitz norm.     

A Convergent Lagrangian Discretization for a Nonlinear Fourth-Order Equation

A fully discrete Lagrangian scheme for numerical solution of the nonlinear fourth-order DLSS equation in one space dimension is analyzed. The discretization is based on the equation’s gradient flow structure in the \(L^2\)-Wasserstein metric. By construction, the discrete solutions are strictly positive and mass conserving. A further key property is that they dissipate both the Fisher information and the logarithmic entropy. Our main result is a proof of convergence of fully discrete to weak solutions in the limit of vanishing mesh size. Convergence is obtained for arbitrary nonnegative, possibly discontinuous initial data with finite entropy, without any CFL-type condition. The key estimates in the proof are derived from the dissipations of the two Lyapunov functionals. Numerical experiments illustrate the practicability of the scheme.     

Existence and multiplicity results for hemivariational inequalities with two parameters

A multiplicity result of Ricceri, based on a minimax inequality, is applied to locally Lipschitz functionals and the existence of one or two non-trivial solutions for a class of two-parameter hemivariational inequalities is proved. An application is given to elliptic problems with discontinuous nonlinearities on unbounded domains.     

Existence of Entropy Solutions to Two-Dimensional Steady Exothermically Reacting Euler Equations

We are concerned with the global existence of entropy solutions of the two-dimensional steady Euler equations for an ideal gas, which undergoes a one-step exothermic chemical reaction under the Arrhenius-type kinetics. The reaction rate function ?(T) is assumed to have a positive lower bound. We first consider the Cauchy problem (the initial value problem), that is, seek a supersonic downstream reacting flow when the incoming flow is supersonic, and establish the global existence of entropy solutions when the total variation of the initial data is sufficiently small. Then we analyze the problem of steady supersonic, exothermically reacting Euler flow past a Lipschitz wedge, generating an additional detonation wave attached to the wedge vertex, which can be then formulated as an initial-boundary value problem. We establish the global existence of entropy solutions containing the additional detonation wave (weak or strong, determined by the wedge angle at the wedge vertex) when the total variation of both the slope of the wedge boundary and the incoming flow is suitably small. The downstream asymptotic behavior of the global solutions is also obtained.     

CONVERGENCE OF THE LAX-FRIEDRICHS SCHEME AND STABILITY FOR CONSERVATION LAWS WITH A DISCONTINUOUS SPACE-TIME DEPENDENT FLUX

The authors give the first convergence proof for the Lax-Friedrichs finite differencescheme for non-convex genuinely nonlinear scalar conservation laws of the formu_t f(k(x, t), u)_x = 0,where the coefficient k(x, t) is allowed to be discontinuous along curves in the (x, t)plane. In contrast to most of the existing literature on problems with discontinuouscoefficients, here the convergence proof is not based on the singular mapping approach,but rather on the div-curl lemma (but not the Young measure) and a Lax type en-tropy estimate that is robust with respect to the regularity of k(x, t). Following [14],the authors propose a definition of entropy solution that extends the classical Kruzkovdefinition to the situation where k(x, t) is piecewise Lipschitz continuous in the (x, t)plane, and prove the stability (uniqueness) of such entropy solutions, provided that theflux function satisfies a so-called crossng condition, and that strong traces of the solu-tion exist along the curves where k(x, t) is disco     

ANTICIPATIVE STOCHASTIC INTEGRALS EQUATIONS DRIVEN BY SEMIMARTINGALES

In this paper, we use the Riemann sum approach to construct the anticipative stochastic integrals and consider the Cauchy problem (non-adapted initial value) for stochastic integral equations driven by discontinuous semimartingales. For general equations with Lipschitz coefficients, we prove the existence of the solutions. Apropos of semilinear equations, we find that under some conditions uniqueness of solutions will also hold.     

A STUDY ON GRADIENT BLOW UP FOR VISCOSITY SOLUTIONS OF FULLY NONLINEAR, UNIFORMLY ELLIPTIC EQUATIONS

We investigate sharp conditions for boundary and interior gradient estimates of continuous viscosity solutions to fully nonlinear,uniformly elliptic equations under Dirichlet boundary conditions. When ...     

Global entropy solutions to a class of quasi-linear wave equations with large time-oscillating sources

This research explores the Cauchy problem for a class of quasi-linear wave equations with time dependent sources. It can be transformed into the Cauchy problem of hyperbolic integro-differential systems of nonlinear balance laws. We introduce the generalized Glimm scheme in new version and study its stability which is proved by Glimm-type interaction estimates in a dissipativity assumption. The generalized solutions to the perturbed Riemann problems, the building blocks of generalized Glimm scheme, are constructed by Riemann problem method modeled on the source free equations. The global existence for the Lipschitz continuous solutions and weak solutions to the systems is established by the consistency of scheme and the weak convergence of source. Finally, the weak solutions are also the entropy solutions which satisfy the entropy inequality.     

On the piecewise smoothness of entropy solutions to scalar conservation laws

The behavior and structure of entropy solutions of scalar convex conservation laws are studied. It is well known that such entropy solutions consist of at most countable number of C¹-smooth regions. We obtain new upper. bounds on the higher order derivatives of the entropy solution in any one of its C¹-smoothness regions. These bounds enable us to measure the high order piecewise smoothness of the entropy solution. To this end we introduce an appropriate new Cⁿ-semi norm - localized to the smooth part of the entropy solution, and we show that the entropy solution is stable with respect to this norm. We also address the question regarding the number of C¹-smoothness pieces; we show that if the initial speed has a finite number of decreasing inflection points then it bounds the number of future shock discontinuities. Loosely speaking this says that in the case of such generic initial data the entropy solution consists of a finite number of smooth pieces, each of which is as smooth as the data permits. It is this type of piecewise smoothness which is assumed - sometime implicitly - in many finite-dimensional computations for such discontinuous problems.     

Dynamical behavior of delayed Hopfield neural networks with discontinuous activations

In this paper, a general class of neural networks with arbitrary constant delays is studied, whose neuron activations are discontinuous and may be unbounded or nonmonotonic. Based on the Leray–Schauder alternative principle and generalized Lyapunov approach, conditions are given under which there is a unique equilibrium of the neural network, which is globally asymptotically stable. Moreover, the existence and global asymptotic stability of periodic solutions are derived, where the neuron inputs are periodic. The obtained results extend previous works not only on delayed neural networks with Lipschitz continuous neuron activations, but also on delayed neural networks with discontinuous neuron activations.     

The genuinely nonlinear Graetz—Nusselt ultraparabolic equation

We study a second-order quasilinear ultraparabolic equation whose matrix of the coefficients of the second derivatives is nonnegative, depends on the time and spatial variables, and can change rank in the case when it is diagonal and the coefficients of the first derivatives can be discontinuous. We prove that if the equation is a priori known to enjoy the maximum principle and satisfies the additional “genuine nonlinearity” condition then the Cauchy problem with arbitrary bounded initial data has at least one entropy solution and every uniformly bounded set of entropy solutions is relatively compact in L _loc ¹ . The proofs are based on introduction and systematic study of the kinetic formulation of the equation in question and application of the modification of the Tartar H-measures proposed by E. Yu. Panov.     

Uniqueness of Solutions to Hyperbolic Balance Laws in Several Space Dimensions

We introduce the concept of modular family of entropy vectors for general r x r systems of balance laws. We then define the notion of entropy solution to the Cauchy problem compatible with the modular family, assuming that the system admits such a family. We show that this concept reduces to the usual one, introduced by S.N. Kruzkov, in the scalar case and when we restrict ourself to Lipschitz continuous solutions. We also show how the compatibility condition appears in the cases of symmetric systems, 2 x 2 psystems and equations of hyperelasticity and electromagnetism, the last two considered earlier by C.M. Dafermos. We demonstrate that generalized Oleinik's condition implies our compatibility condition in the case of symmetric systems. We prove the uniqueness and stability relatively to the initial data of the entropy solutions compatible with the modular family. This theorem has as corollary uniqueness results due to O.A. Oleinik, S.N. Kruzkov, A.E. Hurd, R.J. DiPerna and C. Bardos. We give also two uniqueness theorems to solutions of equations of hyperelasticity and electromagnetism.     

Uniform invariance principle of discontinuous systems with parameter variation

An extension of the invariance principle for a class of discontinuous righthand sides systems with parameter variation in the Filippov sense is proposed. This extension allows the derivative of an auxiliary function V, also called a Lyapunov-like function, along the solutions of the discontinuous system to be positive on some sets. The uniform estimates of attractors and basin of attractions with respect to parameters are also obtained. To this end, we use locally Lipschitz continuous and regular Lyapunov functions, as well as Filippov theory. The obtained results settled in the general context of differential inclusions, and through a uniform version of the LaSalle invariance principle. An illustrative example shows the potential of the theoretical results in providing information on the asymptotic behavior of discontinuous systems.