Abstract

Consider the classical initial, boundary-value problem for the 2D Euler equations, which describes the motion of an ideal, incompressible, fluid in a impermeable vessel. In the early eighties we introduced and studied a Banach space, denoted, which enjoys the following property: if the curl of the initial velocity belongs to, and the curl of the external forces is integrable in time with values in the above space, then all derivatives appearing in the differential equations and in the boundary conditions are continuous in space-time, up to the boundary (we call these solutions classical solutions). At that time this conclusion was know if is replaced by a Hölder space. In the proof of the above result we appealed to a regularity result for solutions to the Poisson equation, vanishing on the boundary and with external forces in. Actually, at that time, we have proved this regularity result for solutions to more general second-order linear elliptic boundaryvalue problems. However the proof remained unpublished. Recently, we have published an adaptation of the proof to solutions of the Stokes system. We recall these results in Section 1.1 below. On the other hand, attempts to prove the above regularity results for data in functional spaces properly containing, have also been done. Bellow we prove some partial results in this direction. This possibly unfinished picture leads to interesting open problems.

The Euler and Stokes equations with data in.

In these notes we want to give an overview on some results, both old and new. Some are old, but remained unpublished for a long time. The starting point will be Beirão da Veiga (1981, 1982, 1984).

We start by introducing some notation. Ω is an open, bounded, connected set in Rn, n ≥ 2, locally situated on one side of its boundary Γ. We assume that Γ is of class, for some positive λ. By we denote the Banach space of all real, continuous functions in with the norm

In the sequel we use the notation

and appeal to the canonical spaces and, with the norms

respectively. Further, for each λ ∈ (0, 1], we define the semi-norm

and the Hölder space, with the norm

In particular, is the space of Lipschitz continuous functions in.