Laura Caravenna

Intrinsic Lipschitz graphs in Heisenberg groups and continuous solutions of a balance equation

Abstract We provide a characterization of intrinsic Lipschitz graphs in the sub-Riemannian Heisenberg groups in terms of their distributional gradients. Moreover, we prove the equivalence of different notions of continuous weak solutions to the equation ∂ ϕ ∂ z + ∂ ∂ t [ ϕ 2 / 2 ] = w , where w is a bounded measurable function.

NEW INTERACTION ESTIMATES FOR THE BAITI-JENSSEN SYSTEM

We establish new interaction estimates for a system introduced by Baiti and Jenssen. These estimates are pivotal to the analysis of the wave front-tracking approximation. In a companion paper we use them to construct a counter-example which shows that Schaeffers Regularity Theorem for scalar conservation laws does not extend to systems. The counter-example we construct shows, furthermore, that a wave-pattern containing infinitely many shocks can be robust with respect to perturbations of the initial data. The proof of the interaction estimates is based on the explicit computation of the wave fan curves and on a perturbation argument.

A note on regularity and failure of regularity for systems of conservation laws via Lagrangian formulation

The paper reviews recent progresses on regularity results which have been studied since Oleĭnik and Schaeffer. It also outlines a limit introducing heuristically an original counterexample obtained with L. Spinolo.

SBV Regularity Results for Solutions to 1D Conservation Laws

A well-posedness theory has been established for entropy solutions to strictly hyperbolic systems of conservation laws, in one space variable, with small total variation. We give in this note an introduction to SBV-regularity results: when the characteristic fields are genuinely nonlinear, the derivative of an entropy solution consists only of the absolutely continuous part and of the jump part, while a fractal behavior (the Cantor part) is ruled out. We first review the scalar uniformly convex case, related to the Hopf-Lax formula. We then turn to the case of systems: one has a decay estimate for both positive and negative waves, obtained considering the interaction-cancellation measures and balance measures for the jump part of the waves. When the Cantor part of the time restriction of the entropy solution does not vanish, either the Glimm functional has a downward jump, or there is a cancellation of waves or this wave balance measure is positive, and this can occur at most at countably many times. We then remove the assumption of genuine nonlinearity. The Cantor part is in general present. There are however interesting nonlinear functions of the entropy solution which still enjoy this regularity.

AN ENTROPY BASED GLIMM-TYPE FUNCTIONAL

We consider the Cauchy problem for a scalar conservation law in one space dimension We introduce, in this simple setting, a new Glimm-type interaction potential: the time marginal of the entropy dissipation measure of a uniformly convex entropy. We show that the Glimm estimates hold for this functional.