Appl Math Optim 41:365–375 (2000)
2000 Springer-Verlag New York Inc.
Solvability of the Navier–Stokes System with L
Department of Mathematics, University of Zagreb,
Bijeniˇcka 30, 10000 Zagreb, Croatia
Communicated by R. Temam
Abstract. We prove the existence of the very weak solution of the Dirichlet prob-
lem for the Navier–Stokes system with L
boundary data. Under the small data
assumption we also prove the uniqueness. We use the penalization method to study
the linearized problem and then apply Banach’s ﬁxed point theorem for the non-
linear problem with small boundary data. We extend our result to the case with no
small data assumption by splitting the data on a large regular and small irregular
Key Words. Navier–Stokes equations, Very weak solution, Nonregular data.
AMS Classiﬁcation. 35Q30, 35D05.
Let ⊂ R
, n = 2, 3, be a bounded C
domain. Let g = (g
f = ( f
) be two given functions deﬁned on = ∂ and , respectively.
The Dirichlet problem for the stationary Navier–Stokes system in a dimensionless form
can be written as follows:
−u + R (u∇)u +∇p = f in , (1)
div u = 0in, u = g on , (2)
where R denotes the Reynolds number, u the velocity, p the pressure, f the density of
the body force, and g the prescribed velocity on the boundary of .
Theclassicalmethod of Leray(see,e.g.,)gives thesolutionfor f ∈ H
, g ∈
. That result was improved by Serre . He used Cattabriga’s regularity result