A study of constrained Navier-Stokes equations and related problems

Abstract: 

The talk will be divided in two parts: In the first part I will talk about deterministic and stochastic Navier-Stokes equations with a constraint on the L 2 energy of the solution. I will speak about the existence and the uniqueness of local strong solutions and the existence of a global solutions for the constrained 2D Navier-Stokes equations. So far we have been able to show the latter only on the torus and on the whole Euclidean space.

In the second part of the talk I will focus on the tamed 3D Navier-Stokes equations which was introduced by Röckner and Zhang [2, 3]. They proved the existence of a unique strong solution to a stochastic tamed 3D Navier-Stokes equation in the whole space and for the periodic boundary conditions case, using a result from Stroock and Varadhan [4]. We have found an alternative self contained approach based on a different approximation scheme and Skorohod-Jakubowski Theorem [1]. We also proved the existence of invariant measures in the whole space for damped equations.

References

[1] Z. Brzeźniak and E. Motyl, Existence of a martingale solution of the stochastic Navier-Stokes equations in unbounded 2D and 3D-domains, J. Differential Equations, 254(4), 1627-1685 (2013).

[2] M. Röckner and X. Zhang, Tamed 3D Navier-Stokes Equation: Existence, Uniqueness and Regularity, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 12(4), 525-549 (2009).

[3] M. Röckner and X. Zhang, Stochastic Tamed 3D Navier-Stokes Equation: Existence, Uniqueness and Ergodicity, Probab. Theory Relat. Fields, 145, 211-267 (2009).

[4] D.W. Stroock and S.R.S. Varadhan, Multidimensional Diffusion Proceses, Springer, Berlin (1979).