Multi-dimensional Burgers equation with unbounded initial data: well-posedness and dispersive estimates

Abstract

The Cauchy problem for a scalar conservation laws admits a unique entropy solution when the data $u_0$ is a bounded measurable function (Kruzhkov). The semi-group $(S_t{)}_{t\ge 0}$ is contracting in the $L^1$-distance. For the multi-dimensional Burgers equation, we show that $(S_t{)}_{t\ge 0}$ extends uniquely as a continuous semi-group over $L^p({\mathbb{R}}^n)$ whenever $1\le p<\infty$, and $u(t):=S_t{u}_0$ is actually an entropy solution to the Cauchy problem. When $p\le q\le \infty$ and $t>0$, $S_t$ actually maps $L^p({\mathbb{R}}^n)$ into $L^q({\mathbb{R}}^n)$. These results are based upon new dispersive estimates. The ingredients are on the one hand Compensated Integrability, and on the other hand a De Giorgi-type iteration.

This article supersedes arXiv:1807.10474. arXiv admin note: text overlap with arXiv:1807.10474

References in corpus (3)

• Multi-dimensional scalar conservation laws with unbounded integrable initial data
• Estimating the number and the strength of collisions in molecular dynamics
• A dispersive estimate for the multidimensional Burgers equation

Cited by in corpus (4)

• Nonlocal nonlinear diffusion equations. Smoothing effects, Green functions, and functional inequalities
• Source-solutions for the multi-dimensional Burgers equation
• Large-time behaviour for anisotropic stable nonlocal diffusion problems with convection
• Lower Semi-Continuity for $\mathcal{A}$-Quasiconvex Functionals under Convex Restrictions