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

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$u(t):=S_tu_0$ is actually an entropy solution to the Cauchy problem. When $p\le q\le \mathrm{\infty }$ 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.