A variational principle for nonlinear transport equations

Nassif Ghoussoub

doi:10.3934/cpaa.2005.4.735

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2005, Volume 4, Issue 4: 735-742. Doi: 10.3934/cpaa.2005.4.735

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A variational principle for nonlinear transport equations

Nassif Ghoussoub^1,

1.
Department of Mathematics, The University of British Columbia, Vancouver BC Canada V6T 1Z2

Received: December 31, 2004

Revised: May 31, 2005

Published: November 2005

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Abstract

We verify -after appropriate modifications- an old conjecture of Brezis-Ekeland [4] concerning the feasibility of a global and variational approach to the problems of existence and uniqueness of solutions of non-linear transport equations, which do not normally fit in an Euler-Lagrange framework. Our method is based on a concept of "anti-self duality" that seems to be inherent in many problems, including gradient flows of convex energy functionals treated in [10] and other parabolic evolution equations ([7]).

Keywords:

Transport equation,
convex duality.

Mathematics Subject Classification: 35F30, 35A15.

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