Invariant Manifolds for Stochastic Partial Differential Equations

In order to investigate stochastic partial differential equations from a dynamical systems point of view, we need to establish a theory for invariant manifolds for stochastic partial differential equations. As in deterministic systems, we expect that invariant manifolds, especially stable and unstable manifolds, to be essential for describing and understanding dynamical behavior of nonlinear random systems. Some new development in understanding dynamics of stochastic partial differential equations include:

Uniqueness of invariant measures and ergodicity of two dimensional stochastic Navier-Stokes equations [8,9,10] and geophysical flow models [6];

Random attractors and inertial manifolds for a class of stochastic partial differential equations [5];

Qualitative theory for the numerics of non-autonomous and random dynamical systems [4].

Three of the team members of the present proposal, Duan, Lu and Schmalfuss [7] have proved the existence of a global unstable invariant manifold for a class of stochastic differential equation. The idea is based on a random graph transform, and a random and nonautonomous generalization of the Banach fixed point theorem. But our method only works for establishing a random unstable manifold of a deterministic stationary state. Infinite dimensional random dynamical systems are often generated by parabolic stochastic partial differential equations and its solution operator is not defined for the negative time. Therefore our method in [7] generally does not work for showing the existence of random stable invariant manifolds. In a paper by Bates, Lu and Zeng [2] a method has been introduced to establish invariant manifolds for deterministic systems. Their method can be used to show the existence of stable manifolds for deterministic parabolic partial differential equations. It is also important to consider stationary states (of stochastic partial differential equations) given by true stationary processes.

Caraballo, Langa and Robinson [3] have studied a pitchfork bifurcation for a stochastic reaction-diffusion equation. Invariant manifolds connecting the stable and unstable states of this random dynamical system are used to detect the bifurcation.

Our first objective during our Research in Teams stay at Banff is to establish a unified theory for stable, unstable and center manifolds for stochastic parabolic partial differential equations, by exploring the ideas in [7] and [2].

Our second objective is to investigate the perturbation and bifurcation of stable, unstable and center manifolds, and the connection between random attractors, stationary states and invariant manifolds for stochastic parabolic partial differential equations, by combining the approaches in [3] and [2].

Two of the team members, Duan and Schmalfuss, were Research in Pairs Fellows at the Oberwolfach Mathematical Research Institute, Germany, during July 2000. They had very productive collaboration there. We believe the Banff centre would provide great environment for our proposed research, whih the potential of making a fundamental contribution to the dynamical study of stochastic partial differential equations.

[1] L. Arnold. Random Dynamical Systems. Springer-Verlag. New York and Berlin. 1998.

[2] P. Bates, K. Lu and C. Zeng. Existence and Persistence of Invariant Manifolds for Semiflows in Banach Space. Memoirs of the AMS, vol 135, 1998.

[3] T. Caraballo, J. Langa and J. Robinson. A stochastic pitchfork bifurcation in a reaction-diffusion equation. Proceedings of the Royal Society, 2001, to appear.

[4] D. Cheban, P.E. Kloeden, and B. Schmalfuss. Variable time step discretization of a random cocycle attractor. J. Dyn. Diff. Eqns.,13:185--213, 2001.

[5] I. Chueshov and M. Scheutzow. Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations, J. Dyn. Diff. Eqns., 13:355--380, 2001. [6] J. Duan and B. Goldys. Ergodicity of Stochastically Forced Large Scale Geophysical Flows. International J. Math. Math. Sci., 2001. In press.

[7] J. Duan, K. Lu and B. Schmalfuss. Unstable manifolds for non-autonomous and stochastic partial differential equations. preprint. 2001.

[8] W. E, J. C. Mattingly and Y. Sinai. Gibbsian dynamics and ergodicity for the stochastically forced 2D Navier-Stokes Equation, Ann. Math, 2001. In press.

[9] F. Flandoli and B. Maslowski. Ergodicity of the 2D Navier-Stokes equation under randon perturbations. Commun. Math. Phys. 172:119--141, 1995.

[10] S. Kuksin and A. Shirikyan. Stochastic dissipative PDE's and Gibbs measures. Commun. Math. Phys., 213:291--330, 2000.

Confirmed Participants

Final Report (in PDF format