On quantitative compactness estimates for hyperbolic conservation laws and Hamilton-Jacobi equations

Inspired by a question posed by Lax, in recent years it has received  

an increasing attention the study of quantitative compactness  

estimates for the solution operator $S_t$, $t>0$ that associates to  

every given initial data $u_0$ the corresponding solution $S_t u_0$ of  

a conservation law or of a first order Hamilton-Jacobi equation.



Estimates of this type play a central roles in various areas of  

information theory and statistics as well as of ergodic and learning  

theory. In the present setting, this concept could provide a measure  

of the order of ``resolution'' of a numerical method for the  

corresponding equation.



In this talk we shall first review the results obtained in  

collaboration with O. Glass and K.T. Nguyen, concerning the  

compactness estimates for solutions to conservation laws. Next, we  

shall turn to the  analysis of the Hamilton-Jacobi equation pursued in  

collaboration with P. Cannarsa and K.T.~Nguyen.