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Entropy Solutions to a Second Order Forward-Backward Parabolic Differential Equation

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Abstract

We prove that the first boundary value problem for a second order forward-backward parabolic differential equation in a bounded domain G T ⊂ ℝd+1, where d ≥ 2, has a unique entropy solution in the sense of F. Otto. Under some natural restrictions on the boundary values this solution is constructed as the limit with respect to a small parameter of a sequence of solutions to Dirichlet problems for an elliptic differential equation. We also show that the entropy solution is stable in the metric of L 1(G T ) with respect to perturbations of the boundary values in the metric of L 1(∂G T ).

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Authors and Affiliations

  1. Lavrent’ev Institute of Hydrodynamics, Novosibirsk, Russia

    I. V. Kuznetsov

Authors
  1. I. V. Kuznetsov
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Original Russian Text Copyright © 2005 Kuznetsov I. V.

The author was supported by the Russian Foundation for Basic Research (Grant 03-01-00829).

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Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 594–619, May–June, 2005.

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Kuznetsov, I.V. Entropy Solutions to a Second Order Forward-Backward Parabolic Differential Equation. Sib Math J 46, 467–488 (2005). https://doi.org/10.1007/s11202-005-0049-3

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  • DOI: https://doi.org/10.1007/s11202-005-0049-3

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