Appl Math Optim 46:31–53 (2002)
2002 Springer-Verlag New York Inc.
Stochastic 2-D Navier–Stokes Equation
and Sivaguru S. Sritharan
Department of Mathematics, Wayne State University,
Detroit, MI 48202, USA
US Navy, SPAWAR SSD – Code D73H,
San Diego, CA 92152-5001, USA
Abstract. In this paper we prove the existence and uniqueness of strong solutions
for the stochastic Navier–Stokes equation in bounded and unbounded domains.
These solutions are stochastic analogs of the classical Lions–Prodi solutions to the
deterministic Navier–Stokes equation. Local monotonicity of the nonlinearity is
exploited to obtain the solutions in a given probability space and this signiﬁcantly
improves the earlier techniques for obtaining strong solutions, which depended on
pathwise solutions to the Navier–Stokes martingale problem where the probability
space is also obtained as a part of the solution.
Key Words. Stochastic Navier–Stokes equation, Maximal monotone operator,
Markov–Feller semigroup, Stochastic differential equations.
AMS Classiﬁcation. 35Q30, 76D05, 60H15.
The mathematical theory of the Navier–Stokes equation is of fundamental importance
to a deep understanding, prediction and control of turbulence in nature and in techno-
logical applications such as combustion dynamics and manufacturing processes. The
incompressible Navier–Stokes equation is a well accepted model for atmospheric and
ocean dynamics. The stochastic Navier–Stokes equation has a long history (e.g., see
 and  for two of the earlier studies) as a model to understand external random
The research by S. S. Sritharan was supported by the ONR Probability and Statistics Program.