Decay rates and uniqueness for nonlinear conservation laws

Meeting Details

For more information about this meeting, contact Rachel Weaver, Toan T. Nguyen, Alberto Bressan.

Speaker: Alberto Bressan, Penn State University

Abstract Link: http://sites.psu.edu/nguyen/files/2024/08/abstracts-Bressan.pdf

Abstract: For a solution to the heat equation, the decay rate of the norm of the Laplacian $\|\Delta u(t) \|_{L^2}$ is related to the Sobolev regularity of the initial data. For a scalar conservation law with strictly convex flux, Oleinik's one-sided Lipschitz estimates imply that the total variation of a solution decays like $t^{-1}$. As proved by Glimm and Lax (1970), the same decay rate holds for solutions to a genuinely nonlinear $2\times 2$ systems of conservation law, with initial data having small $L^\infty$ norm. This talk will focus on solutions having a faster decay rate: $\mbox{Tot.Var.} \{ u(t)\} =O(1)\cdot t^{\alpha-1}$, for some $\alpha>0$. In the scalar case, conditions on the initial data are found, which imply such a decay rate. This leads to a new class of metric interpolation spaces. For a $2\times 2$ hyperbolic system, the eventual goal of this research is to identify domains of initial data, possibly with unbounded total variation, where solutions are unique and yield a H\"older continuous semigroup w.r.t. the $L^1$ distance.

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Room Reservation Information

Room Number: 114 McAllister

Date: 09/04/2024

Time: 2:30pm - 3:30pm