Space-time stationary solutions for the Burgers equation

Bakhtin, Yuri¹ ; Cator, Eric² ; Khanin, Konstantin³

¹ School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, Georgia 30332-0160
² Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands
³ Department of Mathematics, University of Toronto, 40 St George Street, Toronto, Ontario, M5S 2E4, Canada

Journal of the American Mathematical Society, Tome 27 (2014) no. 1, pp. 193-238

Résumé

We construct space-time stationary solutions of the $1$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 space-time we prove that there is a unique global solution with any prescribed average velocity. These global solutions serve as one-point random attractors for the infinite-dimensional 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 action-minimizing curves. We prove that for an arbitrary value of the average velocity, with probability 1 there exists a unique field of action-minimizing curves initiated at all of the Poisson points. Moreover, action-minimizing curves corresponding to different starting points merge with each other in finite time.

Détail

DOI : 10.1090/S0894-0347-2013-00773-0

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