Phase transition of a heat equation with Robin’s boundary conditions and exclusion process
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- by Tertuliano Franco, Patrícia Gonçalves and Adriana Neumann PDF
- Trans. Amer. Math. Soc. 367 (2015), 6131-6158 Request permission
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.References
- Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
- Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
- Tertuliano Franco, Patrícia Gonçalves, and Adriana Neumann, Hydrodynamical behavior of symmetric exclusion with slow bonds, Ann. Inst. Henri Poincaré Probab. Stat. 49 (2013), no. 2, 402–427 (English, with English and French summaries). MR 3088375, DOI 10.1214/11-AIHP445
- Tertuliano Franco and Claudio Landim, Hydrodynamic limit of gradient exclusion processes with conductances, Arch. Ration. Mech. Anal. 195 (2010), no. 2, 409–439. MR 2592282, DOI 10.1007/s00205-008-0206-5
- Giovanni Leoni, A first course in Sobolev spaces, Graduate Studies in Mathematics, vol. 105, American Mathematical Society, Providence, RI, 2009. MR 2527916, DOI 10.1090/gsm/105
- Thomas M. Liggett, Interacting particle systems, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 276, Springer-Verlag, New York, 1985. MR 776231, DOI 10.1007/978-1-4613-8542-4
- Claude Kipnis and Claudio Landim, Scaling limits of interacting particle systems, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 320, Springer-Verlag, Berlin, 1999. MR 1707314, DOI 10.1007/978-3-662-03752-2
- Roger Temam, Navier-Stokes equations. Theory and numerical analysis, Studies in Mathematics and its Applications, Vol. 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. MR 0609732
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
- 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
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3356932