Oscillation Properties of Scalar Conservation Laws

Total Oscillation Dimishing Property for Scalar Conservation Laws

Abstract. We prove a BV estimate for scalar conservation laws that generalizes the classical Total Variation Diminishing property. In fact, for any smooth function Φ, we have that Φ(u) is TVD with the correct definition of the initial seminorm. We call this property Total Oscillation Diminishing. The reason being that it is in contradiction with the oscillations observed recently on some numeri...

Total oscillation diminishing property for scalar conservation laws

We prove a BV estimate for scalar conservation laws that generalizes the classical Total Variation Diminishing property. In fact, for any Lipschitz continuous monotone Φ : R→ R, we have that |Φ(u)|TV (R) is nonincreasing in time. We call this property Total Oscillation Diminishing because it is in contradiction with the oscillations observed recently on some numerical computations based on TVD ...

Variational properties of the kinetic solutions of scalar conservation laws

We consider the variational kinetic formulation of the Cauchy problem for a scalar conservation law due to Brenier[] and Panov[]. The solutions in this formulation are represented by a kinetic density function Y that solves a certain differential inclusion ∂tY ∈ A(Y ), in a suitable Hilbert space. In this paper we establish a sufficient “nondegeneracy” condition under which the operator A is th...

Viscous Conservation Laws, Part I: Scalar Laws

Viscous conservation laws are the basic models for the dissipative phenomena. We aim at a systematic presentation of the basic ideas for the quantitative study of the nonlinear waves for viscous conservation laws. The present paper concentrates on the scalar laws; an upcoming Part II will deal with the systems. The basic ideas for scalar viscous conservation laws originated from two sources: th...

L2 formulation of multidimensional scalar conservation laws