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Transactions of the American Mathematical Society

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Phase transition of a heat equation with Robin's boundary conditions and exclusion process


Authors: Tertuliano Franco, Patrícia Gonçalves and Adriana Neumann
Journal: Trans. Amer. Math. Soc. 367 (2015), 6131-6158
MSC (2010): Primary 60K35, 26A24, 35K55
DOI: https://doi.org/10.1090/S0002-9947-2014-06260-0
Published electronically: December 3, 2014
MathSciNet review: 3356932
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Abstract: For a heat equation with Robin's boundary conditions which depends on a parameter $ \alpha >0$, we prove that its unique weak solution $ \rho ^\alpha $ converges, when $ \alpha $ goes to zero or to infinity, to the unique weak solution of the heat equation with Neumann's boundary conditions or the heat equation with periodic boundary conditions, respectively. To this end, we use uniform bounds on a Sobolev norm of $ \rho ^\alpha $ obtained from the hydrodynamic limit of the symmetric slowed exclusion process, plus a careful analysis of boundary terms.


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Additional Information

Tertuliano Franco
Affiliation: Instituto de Matemática, Universidade Federal de Bahia, Campus de Ondina, Av. Adhemar de Barros, S/N. CEP 40170-110, Salvador, Brazil
Address at time of publication: UFBA, Instituto de Matemática, Campus de Ondina, Av. Adhemar de Barros, S/N. CEP 40170-110, Salvador, Brazil
Email: tertu@impa.br, tertu@ufba.br

Patrícia Gonçalves
Affiliation: Departamento de Matemática, PUC-RIO, Rua Marquês de São Vicente, no. 225, 22453-900, Rio de Janeiro, Brazil – and – CMAT, Centro de Matemática da Universidade do Minho, Campus de Gualtar, 4710-057 Braga, Portugal
Email: patg@math.uminho.pt, patricia@mat.puc-rio.br

Adriana Neumann
Affiliation: Instituto de Matemática, Universidade Federal do Rio Grande do Sol, Campus do Vale, Av. Bento Gonçalves, 9500, CEP 91509-900, Porto Alegre, Brazil
Email: aneumann@mat.ufrgs.br

DOI: https://doi.org/10.1090/S0002-9947-2014-06260-0
Keywords: Phase transition, heat equation, Robin boundary conditions, hydrodynamic limit, slowed exclusion
Received by editor(s): October 13, 2012
Received by editor(s) in revised form: February 12, 2013, and June 4, 2013
Published electronically: December 3, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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