Conservation law with the flux function discontinuous in the space variable—II: Convex–concave type fluxes and generalized entropy solutions

We deal with a single conservation law with discontinuous convex–concave type fluxes which arise while considering sign changing flux coefficients. The main difficulty is that a weak solution may not exist as the Rankine–Hugoniot condition at the interface may not be satisfied for certain choice of the initial data. We develop the concept of generalized entropy solutions for such equations by replacing the Rankine–Hugoniot condition by a generalized Rankine–Hugoniot condition. The uniqueness of solutions is shown by proving that the generalized entropy solutions form a contractive semi-group in L¹$L^1$. Existence follows by showing that a Godunov type finite difference scheme converges to the generalized entropy solution. The scheme is based on solutions of the associated Riemann problem and is neither consistent nor conservative. The analysis developed here enables to treat the cases of fluxes having at most one extrema in the domain of definition completely. Numerical results reporting the performance of the scheme are presented.     

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.     

Renormalized entropy solutions to the Cauchy problem for first order quasilinear conservation laws in the class of periodic functions

We introduce the notion of a periodic in spatial variables renormalized entropy solution to the Cauchy problem for a first order quasilinear conservation law. We prove the existence and uniqueness theorems and comparison principle. We also clarify relations with generalized entropy solutions. Bibliography: 19 titles. Illustrations: 1 figure.     

Entropy Dissipation Methods for Degenerate ParabolicProblems and Generalized Sobolev Inequalities

We analyse the large-time asymptotics of quasilinear (possibly) degenerate parabolic systems in three cases: 1) scalar problems with confinement by a uniformly convex potential, 2) unconfined scalar equations and 3) unconfined systems. In particular we are interested in the rate of decay to equilibrium or self-similar solutions. The main analytical tool is based on the analysis of the entropy dissipation. In the scalar case this is done by proving decay of the entropy dissipation rate and bootstrapping back to show convergence of the relative entropy to zero. As by-product, this approach gives generalized Sobolev-inequalities, which interpolate between the Gross logarithmic Sobolev inequality and the classical Sobolev inequality. The time decay of the solutions of the degenerate systems is analyzed by means of a generalisation of the Nash inequality. Porous media, fast diffusion, p-Laplace and energy transport systems are included in the considered class of problems. A generalized Csiszár–Kullback inequality allows for an estimation of the decay to equilibrium in terms of the relative entropy.     

Entropy Dissipation Methods for Degenerate ParabolicProblems and Generalized Sobolev Inequalities

We analyse the large-time asymptotics of quasilinear (possibly) degenerate parabolic systems in three cases: 1) scalar problems with confinement by a uniformly convex potential, 2) unconfined scalar equations and 3) unconfined systems. In particular we are interested in the rate of decay to equilibrium or self-similar solutions. The main analytical tool is based on the analysis of the entropy dissipation. In the scalar case this is done by proving decay of the entropy dissipation rate and bootstrapping back to show convergence of the relative entropy to zero. As by-product, this approach gives generalized Sobolev-inequalities, which interpolate between the Gross logarithmic Sobolev inequality and the classical Sobolev inequality. The time decay of the solutions of the degenerate systems is analyzed by means of a generalisation of the Nash inequality. Porous media, fast diffusion, p-Laplace and energy transport systems are included in the considered class of problems. A generalized Csiszár–Kullback inequality allows for an estimation of the decay to equilibrium in terms of the relative entropy. (Received 11 October 2000; in revised form 13 March 2001)     

Solutions for a generalized fractional anomalous diffusion equation

In this paper, we investigate the solutions for a generalized fractional diffusion equation that extends some known diffusion equations by taking a spatial time-dependent diffusion coefficient and an external force into account, which subjects to the natural boundaries and the generic initial condition. We obtain explicit analytical expressions for the probability distribution and study the relation between our solutions and those obtained within the maximum entropy principle by using the Tsallis entropy.     

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.     

Entropy boundary and jump conditions in the theory of sedimentation with compression

We present an initial–boundary value problem of a quasilinear degenerate parabolic equation for the settling and consolidation of a flocculated suspension. The corresponding definition of generalized solutions is formulated. It is based on an entropy integral inequality in the sense of Kružkov. From this definition, jump and entropy conditions that have to be satisfied at discontinuities, and an entropy condition valid on one boundary of the computational domain are derived. The latter implies a set-valued reformulation of the original boundary condition. It is interpreted geometrically and characterized by the solution of an auxiliary hyperbolic Riemann problem. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons, Ltd.     

Existence of global L∞ solutions to a generalized n × n hyperbolic system of LeRoux type

In this article, we give the existence of global L^∞ bounded entropy solutions to the Cauchy problem of a generalized n × n hyperbolic system of LeRoux type. The main difficulty lies in establishing some compactness estimates of the viscosity solutions because the system has been generalized from 2 × 2 to n × n and more linearly degenerate characteristic fields emerged, and the emergence of singularity in the region {v₁=0} is another difficulty. We obtain the existence of the global weak solutions using the compensated compactness method coupled with the construction of entropy-entropy flux and BV estimates on viscous solutions.     

A Hamiltonian Regularization of the Burgers Equation

We consider a quasilinear equation that consists of the inviscid Burgers equation plus O(α²) nonlinear terms. As we show, these extra terms regularize the Burgers equation in the following sense: for smooth initial data, the α > 0 equation has classical solutions globally in time. Furthermore, in the zero-α limit, solutions of the regularized equation converge strongly to weak solutions of the Burgers equation. We present numerical evidence that the zero-α limit satisfies the Oleinik entropy inequality. For all α ≥ 0, the regularized equation possesses a nonlocal Poisson structure. We prove the Jacobi identity for this generalized Hamiltonian structure.     

On well-posedness classes of locally bounded generalized entropy solutions of the Cauchy problem for quasilinear first-order equations

We study scalar conservation laws with power-growth restriction on the flux vector. For such equations, we find correctness classes for the Cauchy problem among locally bounded generalized entropy solutions. These classes are determined by some exponents of admissible growth with respect to space variables. We give examples showing that increasing the growth exponent leads to failure of the well-posedness. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 5, pp. 175–188, 2006.     

On Wavewise Entropy Inequalities for High-Resolution Schemes. I: The Semidiscrete Case

We develop a new approach, the method of wavewise entropy inequalities for the numerical analysis of hyperbolic conservation laws. The method is based on a new extremum tracking theory and Volpert's theory of BV solutions. The method yields a sharp convergence criterion which is used to prove the convergence of generalized MUSCL schemes and a class of schemes using flux limiters previously discussed in 1984 by Sweby.

     

Delta shock wave for a 3 × 3 hyperbolic system of conservation laws

We study the one-dimensional Riemann problem for a hyperbolic system of three conservation laws of Temple class. The system is a simplification of a recently proposed system of five conservations laws by Bouchut and Boyaval that models viscoelastic fluids. An important issue is that the considered 3×3 system is such that every characteristic field is linearly degenerate. We study theRiemann problemfor this system and under suitable generalized Rankine-Hugoniot relation and entropy condition, both existence and uniqueness of particular delta shock type solutions are established.     

The Riemann problem with delta initial data for a class of coupled hyperbolic systems of conservation laws

In this paper, we solve the Riemann problem with the initial data containing Dirac delta functions for a class of coupled hyperbolic systems of conservation laws. Under suitably generalized Rankine–Hugoniot relation and entropy condition, the existence and uniqueness of solutions involving delta shock waves are proved. Further, four kinds of different structure for solutions are established uniquely.     

THE RIEMANN PROBLEM WITH DELTA INITIAL DATA FOR THE ONE-DIMENSIONAL CHAPLYGIN GAS EQUATIONS

In this article,we study the Riemann problem with delta initial data for the one-dimensional Chaplygin gas equations.Under the generalized Rankine-Hugoniot conditions and the entropy condition,we const...     

On a nonlinear elliptic-parabolic integro-differential equation with L-data

We generalize the concept of entropy solutions for parabolic equations with L¹-data and consider a class of nonlinear history-dependent degenerated elliptic-parabolic equations including problems with a fractional time derivative such as with Dirichlet boundary conditions and initial condition, where 0<γ?1. We show uniqueness of entropy solutions for general L¹-data by Kruzhkov's method of doubling variables. Moreover, existence in the nondegenerated case, i.e. b≡id, is shown by using the concept of generalized solutions of a corresponding abstract Volterra equation.     

The Riemann Problem with Delta Data for Zero-Pressure Gas Dynamics

In this paper, the Riemann problem with delta initial data for the one-dimensional system of conservation laws of mass, momentum and energy in zero-pressure gas dynamics is considered. Under the generalized Rankine-Hugoniot conditions and the entropy condition, we constructively obtained the global existence of generalized solutions which contains delta-shock. Moreover, the author obtains the stability of generalized solutions by making use of the perturbation of the initial data.     

Riemann problem with Delta initial data for the relativistic Chaplygin Euler equations

In this paper, we study the Riemann problem with the initial data containing the Dirac delta function for the relativistic Chaplygin Euler equations. Under the generalized Rankine-Hugoniot conditions and entropy condition, we constructively obtain the global existence of generalized solutions including delta shock waves that explicitly exhibit four kinds of different structures. Moreover, we obtain the stability of generalized solutions by making use of the perturbation of the initial data     

Convergence of difference schemes with high resolution for conservation laws

We are concerned with the convergence of Lax-Wendroff type schemes with high resolution to the entropy solutions for conservation laws. These schemes include the original Lax-Wendroff scheme proposed by Lax and Wendroff in 1960 and its two step versions-the Richtmyer scheme and the MacCormack scheme. For the convex scalar conservation laws with algebraic growth flux functions, we prove the convergence of these schemes to the weak solutions satisfying appropriate entropy inequalities. The proof is based on detailed estimates of the approximate solutions, compactness estimates of the corresponding entropy dissipation measures, and some compensated compactness frameworks. Then these techniques are generalized to study the convergence problem for the nonconvex scalar case and the hyperbolic systems of conservation laws.

     

The Riemann problem with delta initial data for the nonsymmetric Keyfitz-Kranzer system with Chaplygin pressure

In this paper, we study the Riemann problem with the initial data containing the Dirac delta function for the nonsymmetric Keyfitz-Kranzer system with Chaplygin pressure. Under the generalized Rankine-Hugoniot conditions and entropy condition, we constructively obtain the global existence of generalized solutions including delta shock waves that explicitly exhibit four kinds of different structures. Moreover, we obtain the stability of generalized solutions by making use of the perturbation of the initial data.