Premium
Malliavin calculus for the stochastic 2D Navier—Stokes equation
Author(s) -
Mattingly Jonathan C.,
Pardoux Étienne
Publication year - 2006
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20136
Subject(s) - mathematics , malliavin calculus , forcing (mathematics) , lebesgue measure , mathematical analysis , projection (relational algebra) , navier–stokes equations , boundary (topology) , viscosity , compressibility , periodic boundary conditions , measure (data warehouse) , lebesgue integration , boundary value problem , partial differential equation , stochastic partial differential equation , physics , algorithm , quantum mechanics , database , computer science , engineering , aerospace engineering
We consider the incompressible, two‐dimensional Navier‐Stokes equation with periodic boundary conditions under the effect of an additive, white‐in‐time, stochastic forcing. Under mild restrictions on the geometry of the scales forced, we show that any finite‐dimensional projection of the solution possesses a smooth, strictly positive density with respect to Lebesgue measure. In particular, our conditions are viscosity independent. We are mainly interested in forcing that excites a very small number of modes. All of the results rely on proving the nondegeneracy of the infinite‐dimensional Malliavin matrix. © 2006 Wiley Periodicals Inc.