Appl Math Optim 38:1–19 (1998)
1998 Springer-Verlag New York Inc.
On the Two-Dimensional Navier–Stokes Equations
with the Free Boundary Condition
Department of Mathematics, and the Institute for Scientiﬁc Computing and Applied Mathematics,
Indiana University, Bloomington, IN 47405, USA
Communicated by R. Temam
Abstract. In this article we consider the two-dimensional Navier–Stokes equa-
tions with free boundary condition (open surface), and derive a number of different
results: a neworthogonalproperty for thenonlinear term, improved a prioriestimates
on the solution, an upper bound on the dimension of the attractor which agrees with
the conventional theory of turbulence; ﬁnally, for elongated rectangular domains,
an improved Lieb–Thirring (collective Sobolev) inequality leads to an upper bound
on the dimension of the attractor which might be optimal.
Key Words. Navier–Stokes equations, Trilinear form, Global attractors, Haus-
dorff and fractal dimensions, Grashof number, Elongated domains.
AMS Classiﬁcation. 34C35, 35Q30, 76D05.
In this article we study the two-dimensional Navier–Stokes equations with the classical
free boundary condition for which we establish three different results. Firstly, consider-
ing a general two-dimensional domain, we establish an orthogonality property for the
trilinear form similar to the one known in the periodic case. Based on this result, we
derive an upper bound of the dimension of the corresponding attractor which is given by
This work was partially supported by the National Science Foundation under Grant NSF-DMS-
9400615, by ONR under Grant N00149919J-1140, and by the Research Fund of Indiana University.
Current address: Center for Turbulence Research, Stanford University, Stanford, CA 94305, USA.