Seminario di Equazioni Differenziali e Applicazioni: Generic regularity of solutions to a nonlinear wave equation

Lunedì 6 Luglio 2015, ore 14:00 - Aula 2BC45 - Alberto Bressan



Lunedì 6 Luglio 2015 alle ore 14:00 in Aula 2BC45, Alberto Bressan (Penn State University) terrà un seminario dal titolo "Generic regularity of solutions to a nonlinear wave equation".

Solutions to hyperbolic PDEs can have pathological behavior, such as a dense set of singularities. However, for "most" initial data solutions are much nicer. For example, a well known theorem by Schaeffer (1973) shows that for an open dense set of smooth initial data the solution to a scalar conservation laws develops finitely many shocks. A similar result, on generic structure of singularities, was recently proved for solutions to the second order variational wave equation $u_{tt} - c(u)(c(u)u_x)_x = 0$.
In this talk, I shall survey some basic definitions and results in differential geometry, used in the analysis of generic singularities.
To apply there techniques, we use a variable transformation, that reduces the nonlinear wave equation to a semilinear system. By studying generic solutions to this semilinear system, one obtains a generic regularity result for the original wave equation. A local Taylor expansion of solutions, followed by a change of coordinates, yields a detailed local description of all structurally stable singularities.

Rif. Int. M. Bardi, C. Marchi, F. Ancona

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