Archive for Rational Mechanics and Analysis

, Volume 229, Issue 3, pp 1091–1137 | Cite as

Lipschitz Metric for the Novikov Equation

  • Hong Cai
  • Geng Chen
  • Robin Ming Chen
  • Yannan Shen
Article
  • 57 Downloads

Abstract

We consider the Lipschitz continuous dependence of solutions for the Novikov equation with respect to the initial data. In particular, we construct a Finsler type optimal transport metric which renders the solution map Lipschitz continuous on bounded sets of \({H^1(\mathbb{R})\cap W^{1,4}(\mathbb{R})}\), although it is not Lipschitz continuous under the natural Sobolev metric from an energy law due to the finite time gradient blowup. By an application of Thom’s transversality theorem, we also prove that when the initial data is in an open dense subset of \({H^1(\mathbb{R})\cap W^{1,4}(\mathbb{R})}\), the solution is piecewise smooth. This generic regularity result helps us extend the Lipschitz continuous metric to the general weak solutions. Our method of constructing the metric can be used to treat other kinds of quasi-linear equations, provided a good knowledge about the energy concentration.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Hong Cai
    • 1
  • Geng Chen
    • 2
  • Robin Ming Chen
    • 3
  • Yannan Shen
    • 4
  1. 1.Department of MathematicsQingdao University of Science and TechnologyQingdaoPeople’s Republic of China
  2. 2.Department of MathematicsUniversity of KansasLawrenceUSA
  3. 3.Department of MathematicsUniversity of PittsburghPittsburghUSA
  4. 4.Department of MathematicsCalifornia State UniversityNorthridgeUSA

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