Dirichlet Problem for the Stationary Navier-Stokes System on Lipschitz Domains

초록

We consider the stationary Navier-Stokes system on a bounded Lipschitz domain Omega in R-3 with connected boundary partial derivative Omega. The main concern is the solvability of the Dirichlet problem with external force (f) over right arrow and boundary data (g) over right arrow having minimal regularity, i.e., (f) over right arrow is an element of L-s+1/q-2(q)(Omega) and (g) over right arrow is an element of L-2(partial derivative Omega). Here L-s+1/q-2(q)(Omega) denotes the standard Sobolev space with the pair (s, q) being admissible for the unique solvability in L-s+1/q(q)(Omega) of the Stokes system. We show that if 1 + s >= 2/q in addition, then for any (f) over right arrow is an element of L-s+1/q-2(q)(Omega) and (g) over right arrow is an element of L-2(partial derivative Omega) satisfying the necessary compatibility condition, there exists at least one solution (u) over right arrow in L-s+1/q(q)(Omega) + L-1/2(2)(Omega) of the Dirichlet problem and this solution (u) over right arrow has a complete regularity property. The uniqueness of solutions is also shown under the smallness condition on the corresponding norms of the data.

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