Finite difference schemes for the symmetric Keyfitz-Kranzer system

We are concerned with the convergence of numerical schemes for the initial value problem associated to the Keyfitz-Kranzer system of equations. This system is a toy model for several important models such as in elasticity theory, magnetohydrodynamics, and enhanced oil recovery. In this paper we prove the convergence of three difference schemes. Two of these schemes is shown to converge to the unique entropy solution. Finally, the convergence is illustrated by several examples.Comment: 31 page

Relative entropy and the stability of shocks and contact discontinuities for systems of conservation laws with non BV perturbations

We develop a theory based on relative entropy to show the uniqueness and L^2 stability (up to a translation) of extremal entropic Rankine-Hugoniot discontinuities for systems of conservation laws (typically 1-shocks, n-shocks, 1-contact discontinuities and n-contact discontinuities of large amplitude) among bounded entropic weak solutions having an additional trace property. The existence of a convex entropy is needed. No BV estimate is needed on the weak solutions considered. The theory holds without smallness condition. The assumptions are quite general. For instance, strict hyperbolicity is not needed globally. For fluid mechanics, the theory handles solutions with vacuum.Comment: 29 page

Weak-strong uniqueness of dissipative measure-valued solutions for polyconvex elastodynamics

For the equations of elastodynamics with polyconvex stored energy, and some related simpler systems, we define a notion of dissipative measure-valued solution and show that such a solution agrees with a classical solution with the same initial data when such a classical solution exists. As an application of the method we give a short proof of strong convergence in the continuum limit of a lattice approximation of one dimensional elastodynamics in the presence of a classical solution. Also, for a system of conservation laws endowed with a positive and convex entropy, we show that dissipative measure-valued solutions attain their initial data in a strong sense after time averaging

On Almost Periodic Viscosity Solutions to Hamilton-Jacobi Equations

We establish that a viscosity solution to a multidimensional Hamilton-Jacobi equation with Bohr almost periodic initial data remains to be spatially almost periodic and the additive subgroup generated by its spectrum does not increase in time. In the case of one space variable and a non-degenerate hamiltonian we prove the decay property of almost periodic viscosity solutions when time t -> +infinity. For convex hamiltonian we also provide another proof of this property using the Hopf-Lax-Oleinik formula. For periodic solutions the more general result is proved on unconditional asymptotic convergence of a viscosity solution to a traveling wave

Decay of periodic entropy solutions to degenerate nonlinear parabolic equations

Under a precise nonlinearity-diffusivity condition we establish the decay of space-periodic entropy solutions of a multidimensional degenerate nonlinear parabolic equation. © 2020 Elsevier Inc

On decay of entropy solutions to multidimensional conservation laws

Under a precise genuine nonlinearity assumption we establish the decay of entropy solutions of a multidimensional scalar conservation law with merely continuous flux. © 2020 Society for Industrial and Applied Mathematics Publications. All rights reserved

To the theory of entropy sub-solutions of degenerate nonlinear parabolic equations

We prove existence of the largest entropy sub-solution and the smallest entropy super-solution to the Cauchy problem for a nonlinear degenerate parabolic equation with only continuous flux and diffusion functions. Applying this result, we establish the uniqueness of entropy solution with periodic initial data. The more general comparison principle is also proved in the case when at least one of the initial functions is periodic. © 2020 John Wiley & Sons, Ltd

On decay of entropy solutions to multidimensional conservation laws in the case of perturbed periodic initial data

Under a precise genuine nonlinearity assumption we establish the decay of entropy solutions of a multidimensional scalar conservation law with merely continuous flux and with initial data being a sum of periodic function and a function vanishing at infinity (in the sense of measure). © 2022 World Scientific Publishing Co. Pte Ltd. All rights reserved

On Decay of Entropy Solutions to Nonlinear Degenerate Parabolic Equation with Almost Periodic Initial Data

Abstract: We study the Cauchy problem for nonlinear degenerate parabolic equations with almost periodic initial data. Existence and uniqueness (in the Besicovitch space) of entropy solutions are established. It is demonstrated that the entropy solution remains to be spatially almost periodic and that its spectrum (more precisely, the additive group generated by the spectrum) does not increase in the time variable. Under a precise nonlinearity-diffusivity condition on the input data we establish the long time decay property in the Besicovitch norm. For the proof we use reduction to the periodic case and ergodic methods. © 2021, Pleiades Publishing, Ltd

ON THE DECAY OF VISCOSITY SOLUTIONS TO HAMILTON–JACOBI EQUATIONS WITH ALMOST PERIODIC INITIAL DATA

The Cauchy problem is treated for a multidimensional Hamilton–Jacobi equation with a merely continuous nonstrictly convex Hamiltonian and a Bohr almost periodic initial function. Under the condition that the Hamiltonian is not degenerate in resonant directions (laying in the additive group generated by the spectrum of the initial function), the uniform decay of the viscosity solution to the constant equal to the infimum of the initial function is established. © 2021. All rights reserved