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

The paper recalls two of the regularity results for Burgers' equation, and discusses what happens in the case of genuinely nonlinear, strictly hyperbolic systems of conservation laws. The first regularity result which is considered is Oleinik-Ambroso-De Lellis SBV estimate: it provides bounds on the x-derivative of u when u is an entropy solution of the Cauchy problem for Burgers' equation with bounded initial data. Its extensions to the case of systems is then mentioned. The second regularity result of debate is Schaeffer's theorem: entropy solutions to Burgers' equation with smooth and generic, in a Baire category sense, initial data are piecewise smooth. The failure of the same regularity for general genuinely nonlinear systems is next described. The main focus of this paper is indeed including heuristically an original counterexample where a kind of stability of a shock pattern made by infinitely many shocks shows up, referring to [Caravenna-Spinolo] for the rigorous result.Comment: 10 pages, 1 figur

Schaeffer's regularity theorem for scalar conservation laws does not extend to systems

Several regularity results hold for the Cauchy problem involving one scalar conservation law having convex flux. Among these, Schaeffer's theorem guarantees that if the initial datum is smooth and is generic, in the Baire sense, the entropy admissible solution develops at most finitely many shocks, locally, and stays smooth out of them. We rule out with the present paper the possibility of extending Schaeffer's regularity result to the class of genuinely nonlinear, strictly hyperbolic systems of conservation laws. The analysis relies on careful interaction estimates and uses fine properties of the wave-front tracking approximation

A Proof of Monge Problem in R^n by Stability

The Monge problem in R^n, with a possibly asymmetric norm cost function and absolutely continuous first marginal, is generally underdetermined. An optimal transport plan is selected by a secondary variational problem, from a work on crystalline norms. In this way the mass still moves along lines. The paper provides a quantitative absolute continuity push forward estimate for the translation along these lines: the consequent area formula, for the disintegration of the Lebesgue measure w.r.t. the partition into these 1D-rays, shows that the conditional measures are absolutely continuous, and yields uniqueness of the optimal secondary transport plan non-decreasing along rays, recovering that it is induced by a map

On the extremality, uniqueness and optimality of transference plans

We consider the following standard problems appearing in optimal mass transportation theory: when a transference plan is extremal; when a transference plan is the unique transference plan concentrated on a set A,; when a transference plan is optimal

A Directional Lipschitz Extension Lemma, with Applications to Uniqueness and Lagrangianity for the Continuity Equation

We prove a Lipschitz extension lemma in which the extension procedure simultaneously preserves the Lipschitz continuity for two non-equivalent distances. The two distances under consideration are the Euclidean distance and, roughly speaking, the geodesic distance along integral curves of a (possibly multi-valued) flow of a continuous vector field. The Lipschitz constant for the geodesic distance of the extension can be estimated in terms of the Lipschitz constant for the geodesic distance of the original function. This Lipschitz extension lemma allows us to remove the high integrability assumption on the solution needed for the uniqueness within the DiPerna-Lions theory of continuity equations in the case of vector fields in the Sobolev space $W^{1,p}$, where $p$ is larger than the space dimension, under the assumption that the so-called "forward-backward integral curves" associated to the vector field are trivial for almost every starting point. More precisely, for such vector fields we prove uniqueness and Lagrangianity for weak solutions of the continuity equation that are just locally integrable

On the Structure of BV Entropy Solutions for Hyperbolic Systems of Balance Laws with General Flux Function

The paper describes the qualitative structure of BV entropy solutions of a general strictly hyperbolic system of balance laws with characteristic field either piecewise genuinely non- linear or linearly degenerate. In particular, we provide an accurate description of the local and global wave-front structure of a BV solution generated by a fractional step scheme combined with a wave-front tracking algorithm. This extends the corresponding results which were obtained in [Bianchini-Yu] for strictly hyperbolic system of conservation laws

H\"older regularity of continuous solutions to balance laws and applications in the Heisenberg group

We prove H\"older regularity of any continuous solution $u$ to a $1$-D scalar balance law $u_t + [f(u)]_x = g$, when the source term $g$ is bounded and the flux $f$ is nonlinear of order $\ell \in \mathbb{N}$ with $\ell \ge 2$. For example, $\ell = 3$ if $f(u) = u^3$. Moreover, we prove that at almost every point $(t,x)$, it holds $u(t,x+h) - u(t,x) = o(|h|^{\frac{1}{\ell}})$ as $h \to 0$. Due to Lipschitz regularity along characteristics, this implies that at almost every point $(t,x)$, it holds $u(t+k,x+h) - u(t,x) = o((|h|+|k|)^{\frac{1}{\ell}})$ as $|(h,k)|\to 0$. We apply the results to provide a new proof of the Rademacher theorem for intrinsic Lipschitz functions in the first Heisenberg group