We study the higher differentiability of stationary solutions up to the flat boundary for a class of systems of fluid mechanics modelling flows of incompressible fluids with shear and pressure dependent viscosity in 2D or 3D. We consider systems which are a kind of generalized Navier-Stokes system where stress tensor T has linear growth in symmetric velocity gradient and dependence of viscosity on the pressure is small with respect to an ellipticity constant. | JOURNAL OF SCIENCE OF HNUE 2011 Vol. 56 N . 1 pp. 3-10 ON THE W 2 2 -REGULARITY OF INCOMPRESSIBLE FLUIDS WITH SHEAR AND PRESSURE DEPENDENT VISCOSITY IN THE CASE OF FLAT BOUNDARY Nguyen Duc Huy Hanoi National University of Education E-mail ndhuyuk@ Abstract. We study the higher differentiability of stationary solutions up to the flat boundary for a class of systems of fluid mechanics modelling flows of incompressible fluids with shear and pressure dependent viscosity in 2D or 3D. We consider systems which are a kind of generalized Navier-Stokes system where stress tensor T has linear growth in symmetric velocity gradient and dependence of viscosity on the pressure is small with respect to an ellipticity constant. Keywords incompressible fluid elliptic system regularity up to the boundary 1. Introduction Let Ω Rd d 2 3 be a bounded domain with boundary Ω. We study a following problem For given f f1 fd Ω Rd and stress tensor T Dv p Rd d R Rd d we look for v v1 vd Ω Rd and p Ω R solving d X v vk div T p Dv p f in Ω k 1 xk div v 0 in Ω v 0 on Ω where Dv denotes the symmetric part of the velocity gradient v 1 1 v i v j Dv v T v with Dij v . 2 2 xj xi We assume throughout this section that T p Dv ν p D 2 Dv 3 Nguyen Duc Huy where a generalized viscosity ν is supposed to be continuously differentiable function of both variables. Moreover there exist positive constants λ0 λ1 and ν0 such that for arbitrary symmetric d d- matrices ξ D and any p R the following estimates hold T λ0 ξ 2 p D ξ ξ λ1 ξ 2 D ν p D D ν0 . p The existence of solutions to problem is proved in 2 under assumptions on growth conditions of T . For the regularity problems the smoothness of u and p is a more delicate problem. As we deal with a system of nonlinear elliptic PDEs we can not expect full regularity in space dimensions d 3. The local regularity of solutions for problem studied in 3 . In this paper we study the higher differentiability of these solutions up to the .
