Source-solutions for the multi-dimensional Burgers equation
Résumé
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 → +∞. MSC2010: 35F55, 35L65. Notations. We denote · 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 · M. The Dirac mass at X ∈ R n is δ X or δ x=X. If ν ∈ M (R m) and µ ∈ M (R q), then ν ⊗ µ is the measure over R m+q uniquely defined by ν ⊗ µ, ψ = ν, f µ, g whenever ψ(x, y) ≡ 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 allée d'Italie,
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