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Pointwise decay for solutions of the 2D Neumann exterior problem for the wave equation

Paolo Secchi — 2004

Bollettino dell'Unione Matematica Italiana

We consider the exterior problem in the plane for the wave equation with a Neumann boundary condition. We are interested to the asymptotic behavior for large times for the solution, and in particular to the dependence on the norms of the initial data in the estimate for the pointwise decay rate. In the paper we prove such an estimate, by a combination of the estimate of the local energy decay and decay estimates for the free space solution.

Nonlinear compressible vortex sheets in two space dimensions

Jean-François CoulombelPaolo Secchi — 2008

Annales scientifiques de l'École Normale Supérieure

We consider supersonic compressible vortex sheets for the isentropic Euler equations of gas dynamics in two space dimensions. The problem is a free boundary nonlinear hyperbolic problem with two main difficulties: the free boundary is characteristic, and the so-called Lopatinskii condition holds only in a weak sense, which yields losses of derivatives. Nevertheless, we prove the local existence of such piecewise smooth solutions to the Euler equations. Since the a priori estimates for the linearized...

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