Spacetime stationary solutions for the Burgers equation
Authors:
Yuri Bakhtin, Eric Cator and Konstantin Khanin
Journal:
J. Amer. Math. Soc. 27 (2014), 193238
MSC (2010):
Primary 37L40; Secondary 37L55, 35R60, 37H99, 60K35, 60G55
Published electronically:
May 14, 2013
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Abstract: We construct spacetime stationary solutions of the D Burgers equation with random forcing in the absence of periodicity or any other compactness assumptions. More precisely, for the forcing given by a homogeneous Poisson point field in spacetime we prove that there is a unique global solution with any prescribed average velocity. These global solutions serve as onepoint random attractors for the infinitedimensional dynamical system associated with solutions to the Cauchy problem. The probability distribution of the global solutions defines a stationary distribution for the corresponding Markov process. We describe a broad class of initial Cauchy data for which the distribution of the Markov process converges to the above stationary distribution. Our construction of the global solutions is based on a study of the field of actionminimizing curves. We prove that for an arbitrary value of the average velocity, with probability 1 there exists a unique field of actionminimizing curves initiated at all of the Poisson points. Moreover, actionminimizing curves corresponding to different starting points merge with each other in finite time.
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Proceedings of the International Congress of Mathematicians, Vol.\ 1, 2
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 [AD95]
 D. Aldous and P. Diaconis, Hammersley's interacting particle process and longest increasing subsequences, Probab. Theory Related Fields 103 (1995), no. 2, 199213. MR 1355056 (96k:60017), http://dx.doi.org/10.1007/BF01204214
 [Arn98]
 Ludwig Arnold, Random dynamical systems, Springer Monographs in Mathematics, SpringerVerlag, Berlin, 1998. MR 1723992 (2000m:37087)
 [Bak07]
 Yuri Bakhtin, Burgers equation with random boundary conditions, Proc. Amer. Math. Soc. 135 (2007), no. 7, 22572262 (electronic). MR 2299503 (2008b:35223), http://dx.doi.org/10.1090/S0002993907087369
 [Bak12]
 Yuri Bakhtin, Burgers equation with Poisson random forcing, Accepted at Ann. of Probab., available at http://arxiv.org/abs/1109.5668 (2012).
 [CP12]
 Eric Cator and Leandro P. R. Pimentel, Busemann functions and equilibrium measures in last passage percolation models, Probab. Theory Related Fields 154 (2012), no. 12, 89125. MR 2981418, http://dx.doi.org/10.1007/s0044001103636
 [CP11]
 Eric Cator and Leandro P. R. Pimentel, A shape theorem and semiinfinite geodesics for the Hammersley model with random weights, ALEA Lat. Am. J. Probab. Math. Stat. 8 (2011), 163175. MR 2783936 (2012f:60331)
 [CGGK93]
 J. Theodore Cox, Alberto Gandolfi, Philip S. Griffin, and Harry Kesten, Greedy lattice animals. I. Upper bounds, Ann. Appl. Probab. 3 (1993), no. 4, 11511169. MR 1241039 (94m:60202)
 [DVJ03]
 D. J. Daley and D. VereJones, An introduction to the theory of point processes. Vol. I, 2nd ed., Probability and its Applications (New York), SpringerVerlag, New York, 2003. Elementary theory and methods. MR 1950431 (2004c:60001)
 [E99]
 Weinan E, AubryMather theory and periodic solutions of the forced Burgers equation, Comm. Pure Appl. Math. 52 (1999), no. 7, 811828. MR 1682812 (2000b:37068), http://dx.doi.org/10.1002/(SICI)10970312(199907)52:7811::AIDCPA23.0.CO;2D
 [EKMS00]
 Weinan E, K. Khanin, A. Mazel, and Ya. Sinai, Invariant measures for Burgers equation with stochastic forcing, Ann. of Math. (2) 151 (2000), no. 3, 877960. MR 1779561 (2002e:37134), http://dx.doi.org/10.2307/121126
 [Fat12]
 Albert Fathi, The weak KAM theorem in lagrangian dynamics, Cambridge Studies in Advanced Mathematics, Cambridge University Press, in press, 2012.
 [GK94]
 Alberto Gandolfi and Harry Kesten, Greedy lattice animals. II. Linear growth, Ann. Appl. Probab. 4 (1994), no. 1, 76107. MR 1258174 (95e:60104)
 [GIKP05]
 Diogo Gomes, Renato Iturriaga, Konstantin Khanin, and Pablo Padilla, Viscosity limit of stationary distributions for the random forced Burgers equation, Mosc. Math. J. 5 (2005), no. 3, 613631, 743 (English, with English and Russian summaries). MR 2241814 (2007f:35310)
 [HK03]
 Viet Ha Hoang and Konstantin Khanin, Random Burgers equation and Lagrangian systems in noncompact domains, Nonlinearity 16 (2003), no. 3, 819842. MR 1975784 (2004f:35151), http://dx.doi.org/10.1088/09517715/16/3/303
 [HN97]
 C. Douglas Howard and Charles M. Newman, Euclidean models of firstpassage percolation, Probab. Theory Related Fields 108 (1997), no. 2, 153170. MR 1452554 (98g:60182), http://dx.doi.org/10.1007/s004400050105
 [HN99]
 C. Douglas Howard and Charles M. Newman, From greedy lattice animals to Euclidean firstpassage percolation, Perplexing problems in probability, Progr. Probab., vol. 44, Birkhäuser Boston, Boston, MA, 1999, pp. 107119. MR 1703127 (2001j:60186)
 [HN01]
 C. Douglas Howard and Charles M. Newman, Geodesics and spanning trees for Euclidean firstpassage percolation, Ann. Probab. 29 (2001), no. 2, 577623. MR 1849171 (2002f:60189), http://dx.doi.org/10.1214/aop/1008956685
 [IK03]
 R. Iturriaga and K. Khanin, Burgers turbulence and random Lagrangian systems, Comm. Math. Phys. 232 (2003), no. 3, 377428. MR 1952472 (2004b:76079)
 [Joh00]
 Kurt Johansson, Transversal fluctuations for increasing subsequences on the plane, Probab. Theory Related Fields 116 (2000), no. 4, 445456. MR 1757595 (2001e:60210), http://dx.doi.org/10.1007/s004400050258
 [Kes93]
 Harry Kesten, On the speed of convergence in firstpassage percolation, Ann. Appl. Probab. 3 (1993), no. 2, 296338. MR 1221154 (94m:60205)
 [Kre85]
 Ulrich Krengel, Ergodic theorems, de Gruyter Studies in Mathematics, vol. 6, Walter de Gruyter & Co., Berlin, 1985. With a supplement by Antoine Brunel. MR 797411 (87i:28001)
 [Lio82]
 PierreLouis Lions, Generalized solutions of HamiltonJacobi equations, Research Notes in Mathematics, vol. 69, Pitman (Advanced Publishing Program), Boston, Mass., 1982. MR 667669 (84a:49038)
 [New95]
 Charles M. Newman, A surface view of firstpassage percolation, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 10171023. MR 1404001 (97h:60127)
 [Vil09]
 Cédric Villani, Optimal transport, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338, SpringerVerlag, Berlin, 2009. Old and new. MR 2459454 (2010f:49001)
 [Wüt02]
 Mario V. Wüthrich, Asymptotic behaviour of semiinfinite geodesics for maximal increasing subsequences in the plane, In and out of equilibrium (Mambucaba, 2000) Progr. Probab., vol. 51, Birkhäuser Boston, Boston, MA, 2002, pp. 205226. MR 1901954 (2003e:60227)
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Additional Information
Yuri Bakhtin
Affiliation:
School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, Georgia 303320160
Email:
bakhtin@math.gatech.edu
Eric Cator
Affiliation:
Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands
Email:
e.cator@math.ru.nl
Konstantin Khanin
Affiliation:
Department of Mathematics, University of Toronto, 40 St George Street, Toronto, Ontario, M5S 2E4, Canada
Email:
khanin@math.toronto.edu
DOI:
http://dx.doi.org/10.1090/S089403472013007730
PII:
S 08940347(2013)007730
Received by editor(s):
May 30, 2012
Received by editor(s) in revised form:
March 21, 2013, March 24, 2013, and March 25, 2013
Published electronically:
May 14, 2013
Additional Notes:
The first author was supported by NSF CAREER Award DMS0742424 and grant 040.11.264 from the Netherlands Organisation for Scientific Research (NWO). He is grateful for the hospitality of the Fields Institute in Toronto, Delft Technical University, and CRM in Barcelona where parts of this work have been written.
The second author is grateful for the hospitality of the Fields Institute in Toronto.
The third author was supported by NSERC Discovery Grant RGPIN 328565
Article copyright:
© Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
