Source-solutions for the multi-dimensional Burgers equation

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 $ u$ $in$ M (R m) and $mu$ $in$ M (R q), then $ u$ $otimes$ $mu$ is the measure over R m+q uniquely defined by $ u$ $otimes$ $mu$, $psi$ = $ u$, f $mu$, g whenever $psi$(x, y) $ otequiv$ 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 dItalie,