Appl Math Optim 48:195–210 (2003)
2003 Springer-Verlag New York Inc.
Existence and Regularity of the Pressure for the
Stochastic Navier–Stokes Equations
Jos´e A. Langa,
and Jacques Simon
Department of Differential Equations and Numerical Analysis,
University of Sevilla,
Tarﬁa s/n, E-41012 Sevilla, Spain
Laboratoire de Math´ematiques Appliqu´ees, CNRS
Universit´e Blaise Pascal,
63177 Aubi`ere cedex, France
Communicated by A. Bensoussan
Abstract. We prove, on one hand, that for a convenient body force with values in
the distribution space (H
, where D is the geometric domain of the ﬂuid,
there exist a velocity u and a pressure p solution of the stochastic Navier–Stokes
equation in dimension 2, 3 or 4.
On the other hand, we prove that, for a body force with values in the dual space
of the divergence free subspace V of (H
, in general it is not possible to
solve the stochastic Navier–Stokes equations. More precisely, although such body
forces have been considered, there is no topological space in which Navier–Stokes
equations could be meaningful for them.
Key Words. Stochastic, Navier–Stokes equations, Pressure.
AMS Classiﬁcation. 60H15, 60H30, 35R15, 35Q30.
The ﬁrst two authors were partially supported by MCYT (Feder), Proyecto BFM2002-03068.