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Least action principle and the incompressible Euler equations with variable density


Authors: Milton C. Lopes Filho, Helena J. Nussenzveig Lopes and Juliana C. Precioso
Journal: Trans. Amer. Math. Soc. 363 (2011), 2641-2661
MSC (2010): Primary 35Q31; Secondary 49J45
DOI: https://doi.org/10.1090/S0002-9947-2010-05206-7
Published electronically: December 22, 2010
MathSciNet review: 2763730
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Abstract: In this article we study a variational formulation, proposed by V. I. Arnold and by Y. Brenier, for the incompressible Euler equations with variable density. We consider the problem of minimizing an action functional, the integral in time of the kinetic energy, among fluid motions considered as trajectories in the group of volume-preserving diffeomorphisms beginning at the identity and ending at some fixed diffeomorphism at a given time. We show that a relaxed version of this variational problem always has a solution, and we derive an Euler-Lagrange system for the relaxed minimization problem which we call the relaxed Euler equations. Finally, we prove consistency between the relaxed Euler equations and the classical Euler system, showing that weak solutions of the relaxed Euler equations with the appropriate geometric structure give rise to classical Euler solutions and that classical solutions of the Euler system induce weak solutions of the relaxed Euler equations. The first consistency result is new even in the constant density case. The remainder of our analysis is an extension of the work of Y. Brenier (1999) to the variable density case.


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Additional Information

Milton C. Lopes Filho
Affiliation: Departamento de Matemática, IMECC, Rua Sérgio Buarque de Holanda, 651, Cidade Universitária Zeferino Vaz, Universidade Estadual de Campinas - UNICAMP, Campinas, SP 13083-859, Brazil
Email: mlopes@ime.unicamp.br

Helena J. Nussenzveig Lopes
Affiliation: Departamento de Matemática , IMECC, Rua Sérgio Buarque de Holanda, 651, Cidade Universitária Zeferino Vaz, Universidade Estadual de Campinas - UNICAMP, Campinas SP 13083-859, Brazil
Email: hlopes@ime.unicamp.br

Juliana C. Precioso
Affiliation: Departamento de Matemática, Universidade Estadual Paulista - UNESP, 15054-000, S. J. do Rio Preto, SP, Brazil
Email: precioso@ibilce.unesp.br

DOI: https://doi.org/10.1090/S0002-9947-2010-05206-7
Received by editor(s): January 27, 2009
Received by editor(s) in revised form: August 24, 2009, and September 2, 2009
Published electronically: December 22, 2010
Additional Notes: The first author’s research was partially supported by CNPq Grant #303301/2007-4
The second author’s research was partially supported by CNPq Grant #302214/2004-6.
The third author’s research was part of the Ph.D. thesis of J. C. Precioso, partially supported by FAPESP Grant #2001/06984-5.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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