Lipschitz Metrics for a Class of Nonlinear Wave Equations

2015/06/21 by Alberto Bressan, Geng Chen, Bressan, Alberto +1 · 2 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Advanced Differential Geometry Research #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1506.06310

Abstract

The nonlinear wave equation u_tt-c(u)(c(u)u_x)_x=0 determines a flow of conservative solutions taking values in the space H¹(ℝ). However, this flow is not continuous w.r.t. the natural H¹ distance. Aim of this paper is to construct a new metric which renders the flow uniformly Lipschitz continuous on bounded subsets of H¹(ℝ). For this purpose, H¹ 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. By the generic regularity result proved in [7], these piecewise regular paths are dense and can be used to construct a geodesic distance with the desired Lipschitz property.

Cited by

• Generic regularity and Lipschitz metric for a two-component Novikov system
• Poiseuille flow of hyperbolic Ericksen-Leslie system in dimension two

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