**Title: **Quadratic Lyapunov potential for hyperbolic systems of conservation laws

**Abstract: **
After a brief introduction to the theory of conservation laws, I will present the result obtained by Stefano Bianchini and myself concerning the existence of a Lyapunov potential which bounds the total variation of the speed of the waves present in the solution of the Cauchy problem

\begin{equation}

\label{E_cauchy}

u_t + f(u)_x = 0, \quad u(0,x) = u_0(x),

\end{equation}

where $f: \R^n \to \R^n$ is strictly hyperbolic and $u: [0,\infty) \times \R \ni (t,x) \mapsto u(t,x) \in \R^n$.

Thanks to its properties, this potential can be used to prove useful results about the rate of convergence of the Glimm scheme, which is the approximation algorithm used to prove the existence of solutions to the Cauchy problem \eqref{E_cauchy} and about the structure of such solutions.

This seminar is part of the AJS series of seminars.