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 \(\nu\) \(\in\) 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{é}e d'Italie,