Source-solutions for the multi-dimensional Burgers equation

Abstract

We have shown in a recent collaboration that the Cauchy problem for the multi-dimensional Burgers equation is well-posed when the initial data u(0) is taken in the Lebesgue space L 1 (R n), and more generally in L p (R n). We investigate here the situation where u(0) is a bounded measure instead, focusing on the case n = 2. This is motivated by the description of the asymptotic behaviour of solutions with integrable data, as t $\rightarrow$ +$\infty$. MSC2010: 35F55, 35L65. Notations. We denote $\times$ p the norm in Lebesgue L p (R n). The space of bounded measure over R m is M (R m) and its norm is denoted $\times$ M. The Dirac mass at X $\in$ R n is $\delta$ X or $\delta$ x=X. If $\nu$ $\in$ M (R m) and $\mu$ $\in$ M (R q), then $\nu$ $\otimes$ $\mu$ is the measure over R m+q uniquely defined by $\nu$ $\otimes$ $\mu$, $\psi$ = $\nu$, f $\mu$, g whenever $\psi$(x, y) $\not\equiv$ f (x)g(y). The closed halves of the real line are denoted R + and R --. * U.M.P.A., UMR CNRS-ENSL \# 5669. 46 all{\'e}e d'Italie,