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Stochastic Burgers’ equation. (English) Zbl 0824.35112
The authors prove that the stochastic Burger’s equation forced by a cylindrical Wiener process with Dirichlet boundary conditions and the initial condition has a unique global solution. Also the existence of an invariant measure for the corresponding transition semigroup is established.
MSC:
35Q53KdV-like (Korteweg-de Vries) equations
35R60PDEs with randomness, stochastic PDE
References:
[1]D. H. CHAMBERS, R. J. ADRIAN, P. MOIN, D. S. STEWART, H. J. SUNG, Karhunen-Loéve expansion of Burgers’ model of turbulence,Phys. Fluids 31, (9), 2573-2582 (1988) · doi:10.1063/1.866535
[2]H. CHOI, R. TEMAM, P. MOIN, J. KIM, Feedback control for unsteady flow and its application to Burgers equation, Center for Turbulence Research, Stanford University, CTR Manuscript 131. To appear on J. Fluid Mechanics (1992)
[3]H. CRAUEL, F. FLANDOLI, Attractors for Random Dynamical Systems, Preprint n. 148 Scuola Normale Superiore di Pisa (1992)
[4]DAH-TENG JENG Forced Model Equation for Turbulence,The Physics of Fluids 12, 10, 2006-2010 (1969) · Zbl 0187.51604 · doi:10.1063/1.1692305
[5]F. FLANDOLI, Dissipativity and invariant measures for stochastic Navier-Stokes equations Preprint n. 24 Scuola Normale Superiore di Pisa (1993)
[6]I. HOSOKAWA, K. YAMAMOTO, Turbolence in the randomly forced one dimensional Burgers flow,J. Stat. Phys. 13, 245 (1975) · doi:10.1007/BF01012841
[7]M. KARDAR, M. PARISI, J. C. ZHANG Dynamical scaling of growing interfaces,Phys. Rev. Lett. 56, 889 (1986) · Zbl 1101.82329 · doi:10.1103/PhysRevLett.56.889
[8]G. DA PRATO, J. ZABCZYK, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, Cambridge University Press (1992)
[9]F. ROTHE (1984), Global Solutions of Reaction-Diffusion Systems,Lecture Notes in Mathematics,1072, Springer-Verlag (1984)