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Ergodicity of the 2D Navier-Stokes equations with degenerate multiplicative noise

Ergodicity of the 2D Navier-Stokes equations with degenerate multiplicative noise Consider the two-dimensional, incompressible Navier-Stokes equations on torus T 2 = [−π, π]2 driven by a degenerate multiplicative noise in the vorticity formulation (abbreviated as SNS): dw t = νΔw t dt + B(Kw t ,w t )dt + Q(w t )dW t . We prove that the solution to SNS is continuous differentiable in initial value. We use the Malliavin calculus to prove that the semigroup {P t }t≥0 generated by the SNS is asymptotically strong Feller. Moreover, we use the coupling method to prove that the solution to SNS has a weak form of irreducibility. Under almost the same Hypotheses as that given by Odasso, Prob. Theory Related Fields, 140: 41–82 (2005) with a different method, we get an exponential ergodicity under a stronger norm. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

Ergodicity of the 2D Navier-Stokes equations with degenerate multiplicative noise

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References (25)

Publisher
Springer Journals
Copyright
Copyright © 2018 by Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/s10255-018-0728-z
Publisher site
See Article on Publisher Site

Abstract

Consider the two-dimensional, incompressible Navier-Stokes equations on torus T 2 = [−π, π]2 driven by a degenerate multiplicative noise in the vorticity formulation (abbreviated as SNS): dw t = νΔw t dt + B(Kw t ,w t )dt + Q(w t )dW t . We prove that the solution to SNS is continuous differentiable in initial value. We use the Malliavin calculus to prove that the semigroup {P t }t≥0 generated by the SNS is asymptotically strong Feller. Moreover, we use the coupling method to prove that the solution to SNS has a weak form of irreducibility. Under almost the same Hypotheses as that given by Odasso, Prob. Theory Related Fields, 140: 41–82 (2005) with a different method, we get an exponential ergodicity under a stronger norm.

Journal

Acta Mathematicae Applicatae Sinica – Springer Journals

Published: Mar 8, 2018

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