Lipschitz Metrics for Nonlinear Wave Equations

Lipschitz Metrics for Nonlinear Wave Equations

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Alberto Bressan , Penn State
Fine Hall 110

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