Solvability of Hyperbolic IBVPs Through Filtering

Abstract

In a recent paper in collaboration with S. Benzoni, F. Rousset and K. Zumbrun, we uncovered the amazing fact that besides the classes of Hadamard unstable (no estimate at all) and of strongly stable hyperbolic IBVPs (estimates without loss of derivatives), there is a third “open” class, in the sense that it is stable under small disturbances of the coefficients (we apologize for the use of the word “stable” with two different meanings) in the PDEs and in the boundary conditions. We called the latter class (WR), because it has a Weak stability property (estimates with loss of one derivative) and because it is characterized by a “Real” characteristic set for the Lopatinsliĭ determinant.

We show here that with an appropriate filtering, systems in the class WR can be recast as strongly stable ones. It is remarkable that the same filtering is applied to both the solution and the data. Thus there is a hope to apply this linear theory to fixed point methods in non-linear problems, like the stability of two-dimensional vortex sheets when the jump in the velocity is larger than 2√2 times the sound speed.