LisMath Seminar 

01/04/2016, 16:00 — 17:00 — Room 6.2.33, Faculty of Sciences of the Universidade de Lisboa
Guoping Liu, Universidade de Lisboa

A stochastic variational approach to viscous Burgers equations

In 1966, V.I. Arnold [1] showed that the Lagrangian flows corresponding to the Euler equations can be characterised as geodesics in the group of diffeomorphisms of the underlying manifold when considered with the $L^2$ metric. One can consider different metrics in this infinite-dimensional group (or in other groups) and derive many other conservative systems as the corresponding geodesics.

For the dissipative systems, replacing geodesics by irregular (stochastic) paths, it is possible to consider stochastic variational principles whose critical points are the Lagrangian trajectories of the systems. The velocity corresponds in these contexts to the drift of the stochastic paths. In [2, 3, 4], stochastic variational principle for Navier-Stokes equations, Camassa-Holm equations and Leray-alpha equations have been described.

In the first part of the seminar, we will give a stochastic variational principle for one-dimensional viscous Burgers equations considering a $L^q$ metric. After that, we will prove the existence of the critical diffusion. Finally, we will give an alternative probabilistic derivation of Burgers equations via a stochastic forward-backward differential systems. In [5,6], the Navier-Stokes equation was studied using this relation.

Bibliography

1. V. I. Arnold, Sur la géometrie differentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits.  Ann. Inst. Fourier., 16, 316-361 (1966)
2. F. Cipriano, A. B. Cruzeiro, Navier-Stokes equation and diffusions on the group of homeomorphisms of the torus. Comm. Math. Phys., 275, no.1, 255-269 (2007).
3. M. Arnaudon, X. Chen, A. B. Cruzeiro, Stochastic Euler-Poincaré reduction. J. Math. Physics., 55, 081507 (2014).
4. A. B. Cruzeiro, G. P. Liu, A stochastic variational approach to the viscous Camassa-Holm and related equations. arxiv:1508.04064.
5. A. B. Cruzeiro, E. Shamarova, Navier-Stokes equations and forward-backward SDEs on the group of diffeomorphisms of the torus. Stoch. Proc. and their Applic., 119, 4034-4060(2009)
6. A. B. Cruzeiro, Z. M. Qian, Backward Stochastic Differential Equations Associated with the Vorticity Equations. J. Funct. Anal., 267, no.3, 660--677(2014)

See also

Burgers equations.pdf