From Boltzmann to Euler: Hilbert’s 6th problem revisited
Marshall Slemrod, University of Wisconsin – Madison
This talk addresses the hydrodynamic limit of the Boltzmann equation, namely the compressible Euler equations of gas dynamics. An exact summation of the Chapman–Enskog expansion originally given by Gorban and Karlin is the key to the analysis. An appraisal of the role of viscosity and capillarity in the limiting process is then given where the analogy is drawn to the limit of the Korteweg–de Vries–Burgers equations as a small parameter tends to zero.
What is between conservative and dissipative solutions for the Camassa–Holm equation?
Helge Holden, Norwegian University of Science and Technology
Abstract: The Camassa–Holm (CH) equation reads $u_t-u_{txx}+kappa u_x+3uu_x-2u_xu_{xx}-uu_{xxx}=0$ where $kappa$ is a real parameter.
We are interested in the Cauchy problem on the line with initial data in $H^1$. There is a well-known and well-studied dichotomy between two distinct classes of solutions of the CH equation. The two classes appear exactly at wave breaking where the spatial derivative of the solution becomes unbounded while its $H^1$ norm remains finite. We here introduce a novel solution concept gauged by a continuous parameter
$alpha$ in such a way that $alpha=0$ corresponds to conservative solutions and $alpha=1$ gives the dissipative solutions. This allows for a detailed study of the difference between the two classes of solutions and their behavior at wave breaking. We also extend the analysis to a two-component Camassa–Holm system.
This is joint work with Katrin Grunert (NTNU) and Xavier Raynaud (SINTEF).