High-order regularity and approximation for Hamilton-Jacobi equations

Bum Il Hong, Purdue University

Abstract

We prove new regularity results for solutions of first-order partial differential equations of Hamilton-Jacobi type posed as initial value problems on the real line. We show that certain spaces determined by quasinorms related to the solution's approximation properties in C($\IR$) by continuous, piecewise quadratic polynomial functions are invariant under the action of the differential equation. As a corollary, we show that if the initial data $u\sb{0}$ has $u\sbsp{0}{\prime}$ of bounded variation and if the flux is strictly convex and smooth enough, then whenever $u\sbsp{0}{\prime}$ is in the Besov space $B\sbsp{1/\alpha}{\alpha-1}(L\sp{1/\alpha})$ for $\alpha$ between 1 and 3, $u\sb{x}(\cdot,t)$ remains in the same space for all positive time t. As a result, we show that solutions of Hamilton-Jacobi equations have enough regularity to be approximated well in C($\IR$) by moving-grid finite element methods. The preceding results depend on a new stability theorem for Hamilton-Jacobi equations in any number of spatial dimensions.

Degree

Ph.D.

Advisors

Lucier, Purdue University.

Subject Area

Mathematics

COinS