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Approximation of entropy solutions to degenerate nonlinear parabolic equations

  • Eduardo Abreu
  • Mathilde ColombeauEmail author
  • Evgeny Yu Panov
Article

Abstract

We approximate the unique entropy solutions to general multidimensional degenerate parabolic equations with BV continuous flux and continuous nondecreasing diffusion function (including scalar conservation laws with BV continuous flux) in the periodic case. The approximation procedure reduces, by means of specific formulas, a system of PDEs to a family of systems of the same number of ODEs in the Banach space \(L^\infty \), whose solutions constitute a weak asymptotic solution of the original system of PDEs. We establish well posedness, monotonicity and \(L^1\)-stability. We prove that the sequence of approximate solutions is strongly \(L^1\)-precompact and that it converges to an entropy solution of the original equation in the sense of Carrillo. This result contributes to justify the use of this original method for the Cauchy problem to standard multidimensional systems of fluid dynamics for which a uniqueness result is lacking.

Keywords

Partial differential equations Degenerate parabolic equations Entropy solutions Approximate solutions Stability 

Mathematics Subject Classification

35K55 35L65 65M12 

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Universidade Estadual de CampinasCampinasBrazil
  2. 2.Universidade de São PauloSão PauloBrazil
  3. 3.Universidade Estadual de CampinasCampinasBrazil
  4. 4.Novgorod State UniversityVeliky NovgorodRussian Federation
  5. 5.St. Petersburg State UniversitySaint PetersburgRussian Federation

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