L²-type Lyapunov functions for hyperbolic scalar conservation laws

Abstract

We prove unexpected decay of the L²-distance from the solution u(t) of a hyperbolic scalar conservation law, to some convex, flow-invariant target sets.

Keywords:

• Conservation laws
• Lyapunov functions
• shock waves

AMS classification:

• 35L65
• 35B35

Acknowledgement

I am indebted to Marie-Françoise Roy, who guided me in the realm of Real Algebraic Geometry.

Notes

1 This is the reason why we choose a relative entropy, and not an arbitrary integrand $G(u(t),\pi u(t)).$

2 If we worked with the Lax-Friedrichs scheme, j would run over $\mathbb{Z}+\frac{k}{2},$ k being the index of the time step.

3 These assertions tell us that semialgebraic sets form an o-minimal structure.