Smoothing solutions to initial-boundary problems for first-order hyperbolic systems

Abstract

We consider initial-boundary problems for general linear first-order strictly hyperbolic systems with local or nonlocal nonlinear boundary conditions. While boundary data are supposed to be smooth, initial conditions can contain distributions of any order of singularity. It is known that such problems have a unique continuous solution if the initial data are continuous. In the case of strongly singular initial data we prove the existence of a (unique) delta wave solution. In both cases, we say that a solution is smoothing if it eventually becomes k-times continuously differentiable for each k. Our main result is a criterion allowing us to determine whether or not the solution is smoothing. In particular, we prove a rather general smoothingness result in the case of classical boundary conditions.

Keywords:

• first-order hyperbolic systems
• initial-boundary problems
• delta wave solutions
• regularity of solutions

AMS Subject Classifications::

• 35B65
• 35L50
• 35B30
• 35D05

Acknowledgements

This work is supported by a Humboldt Research Fellowship. The author is grateful to Natal'ya Lyul'ko for helpful discussions.

Notes

Note

1. This sharply contrasts with the case of a Cauchy problem where solutions cannot be smoothing because the boundary part all the time ‘remembers’ the regularity of the initial data.