We consider here a model of fluid-structure evolution problem which, in particular, has been largely studied from the numerical point of view. We prove the existence of a strong solution to this problem. Mathematics Subject Classification (2000). 35Q30, 74F10, 76D05.

Navier-Stokes equations with shear dependent viscosity under the classical nonslip boundary condition were introduced and studied in the 1960s by O. A. Ladyzhenskaya and, in the case of gradient dependent viscosity, by J.-L. Lions. A particular case is the well-known Smagorinsky turbulence model. This is nowadays a central subject of investigation. On the other hand, boundary conditions of slip type seems to be more realistic in some situations, in particular in numerical applications. They are a main research subject. The existence of weak solutions u to the above problems, with slip-(or nonslip-) type boundary conditions, is well-known in many cases. However, regularity up to the boundary still presents many open questions. In what follows we present some regularity results, in the stationary case, for weak solutions to this kind of problem; see Theorem 3.1 and Theorem 3.2. The evolution problem is studied in a forthcoming paper [6]. A cornerstone in our proof is the classical Nirenberg translation method; see [38].

We consider the initial-boundary value problem for the 3D Navier-Stokes equations. The physical domain is a bounded open set with a smooth boundary on which we assume a condition of free-boundary type. We show that if a suitable hypothesis on the vorticity direction is assumed, then weak solutions are regular. The main tool we use in the proof is an explicit representation of the velocity in terms of the vorticity, by means of Green's matrices.

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In this article we prove some sharp regularity results for the stationary and the evolution Navier-Stokes equations with shear dependent viscosity, see (1.1), under the no-slip boundary condition (1.4). We are interested in regularity results for the second order derivatives of the velocity and for the first order derivatives of the pressure up to the boundary, in dimension n ≥ 3. In reference [4] we consider the stationary problem in the half space R n + under slip and no-slip boundary conditions. Here, by working in a simpler context, we concentrate on the basic ideas of proofs. We consider a cubic domain and impose our boundary condition (1.4) only on two opposite faces. On the other faces we assume periodicity, as a device to avoid unessential technical difficulties. This choice is made so that we work in a bounded domain Ω and, at the same time, with a flat boundary. In the last section we provide the extension of the results from the stationary to the evolution problem. (2000). Primary 35J25, 35Q30; secondary 76D03, 76D05. Mathematics Subject Classification