Lipschitz Metrics for Nonlinear Wave Equations
Lipschitz Metrics for Nonlinear Wave Equations
The talk is concerned with some classes of nonlinear wave equations: of first order, such as the Camassa-Holm equation, or of second order, asÌýthe variational wave equation $u_{tt}-c(u)(c(u)u_x)_x=0$.ÌýIn both cases, it is known that the equations determine a unique flow ofÌýconservative solutions within the natural ``energy" space $H^1(\mathbb{R})$. However, this flow is not continuous w.r.t.~the $H^1$ distance.ÌýLocal well-posedness is usually recovered only on spaces with higher regularity.ÌýOur goal is to construct a new metric, which renders thisÌýflow uniformly Lipschitz continuous on bounded subsets of $H^1$.ÌýFor this purpose, $H^1$ is given the structure ofÌýa Finsler manifold, where the norm of tangent vectors is definedÌýin terms of an optimal transportation problem. For paths ofÌýpiecewise smooth solutions, one can carefully estimate howÌýthe weighted length grows in time.ÌýÌýTo complete the construction, one needs an additional argumentÌýshowing that the family of piecewise smooth solutions is dense.ÌýThis generic regularity property can be proved usingÌýa variable transformation that reduces the equations to a semilinearÌýsystem, followed by an application of Thom's transversality theorem.