Contractive metrics for nonsmooth evolutions

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Given an evolution equation, a standard way to prove the well posedness of the Cauchy problem is to establish a Gronwall type estimate, bounding the distance between any two trajectories. There are important cases, however, where such estimates cannot hold, in the usual distance determined by the Euclidean norm or by a Banach space norm. In alternative, one can construct different distance functions, related to a Riemannian structure or to an optimal transportation problem. This paper reviews various cases where this approach can be implemented, in connection with discontinuous ODEs on ℝn, nonlinear wave equations, and systems of conservation laws. For all the evolution equations considered here, a metric can be constructed such that the distance between any two solutions satisfies a Gronwall type estimate. This yields the uniqueness of solutions, and estimates on their continuous dependence on the initial data.

Original languageEnglish (US)
Title of host publicationNonlinear Partial Differential Equations
Subtitle of host publicationThe Abel Symposium 2010
Pages13-35
Number of pages23
DOIs
Publication statusPublished - Dec 1 2012
EventAbel Symposium 2010: Nonlinear Partial Differential Equations - Oslo, Norway
Duration: Sep 28 2010Oct 2 2010

Publication series

NameNonlinear Partial Differential Equations: The Abel Symposium 2010

Other

OtherAbel Symposium 2010: Nonlinear Partial Differential Equations
CountryNorway
CityOslo
Period9/28/1010/2/10

    Fingerprint

All Science Journal Classification (ASJC) codes

  • Geometry and Topology

Cite this

Bressan, A. (2012). Contractive metrics for nonsmooth evolutions. In Nonlinear Partial Differential Equations: The Abel Symposium 2010 (pp. 13-35). (Nonlinear Partial Differential Equations: The Abel Symposium 2010). https://doi.org/10.1007/978-3-642-25361-4_2