Singular asymptotic expansions and delta waves for Burgers' equation

This paper is concerned with the characterization of the weak limits (delta waves) associated to the Cauchy problem for the Burgers' equation and the inviscid Burgers' equation with strongly singular initial data in the form of a regularization by smooth mollifiers of sums of derivatives of Dirac measures. By means of Laplace's method we give the precise asymptotic expansion of the solutionsu in powers of . Then we apply these asymptotics in order to classify completely all possible delta waves under a suitable nondegeneracy condition on some mollifiers regularizing the leading singular term of the initial data. We propose also certain stability results for the weak limits under suitable perturbations of the initial data.Partially supported by 60% of MURST, ItalyPartially supported by grant MM-410/94 with MES, Bulgaria and by 40% of MURST, Italy