Abstract
We study the relativistic heat equation in one space dimension. We prove a local regularity result when the initial datum is locally Lipschitz in its support. We propose a numerical scheme that captures the known features of the solutions and allows for analysing further properties of their qualitative behavior.
Highlights
In this work, we explore both analytically and numerically the implications of a new strategy to study flux-dominated nonlinear diffusions in one dimension
Many other models of nonlinear degenerate parabolic equations with flux saturation as the gradient becomes unbounded have been proposed by Rosenau and his coworkers [19, 37], and Bertsch and Dal Passo [12, 26]
The general class of flux limited diffusion equations and the properties of the relativistic heat equation have been studied in a series of papers [5, 4, 6, 21]
Summary
We explore both analytically and numerically the implications of a new strategy to study flux-dominated nonlinear diffusions in one dimension. The local regularity of entropy solutions to (1.3) will be done by a change variables, writing (1.3) in terms of its inverse distribution function This change of variables has its origin in using mass transport techniques to study diffusion equations [18, 13]. Transport distances between probability measures in one dimension are much easier to compute since they can be written in terms of distribution functions and their generalized inverses (pseudo-inverse), the so-called Hoeffding-Frechet Lemma [39, Section 2.2] This result led to the following change of variables based on the distribution function F associated to the probability measure u, defined as x. We emphasize that the new parts of this result with respect to the literature discussed above refer to the regularity stated on points (ii) and (iii) This result implies that sharp corners on the support of the initial data are immediately smoothed out by the evolution of the RHE. We include in Appendix A some basic material to describe the notion of entropy solutions for (1.3) for the sake of completeness
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