### Trace theorems for three-dimensional, time-dependent solenoidal vector fields and their applications

A. Fursikov, M. Gunzburger, L. Hou
2001 Transactions of the American Mathematical Society
We study trace theorems for three-dimensional, time-dependent solenoidal vector fields. The interior function spaces we consider are natural for solving unsteady boundary value problems for the Navier-Stokes system and other systems of partial differential equations. We describe the space of restrictions of such vector fields to the boundary of the space-time cylinder and construct extension operators from this space of restrictions defined on the boundary into the interior. Only for two
more » ... Only for two exceptional, but useful, values of the spatial smoothness index, the spaces for which we construct extension operators is narrower than the spaces in which we seek restrictions. The trace spaces are characterized by vector fields having different smoothnesses in directions tangential and normal to the boundary; this is a consequence of the solenoidal nature of the fields. These results are fundamental in the study of inhomogeneous boundary value problems for systems involving solenoidal vector fields. In particular, we use the trace theorems in a study of inhomogeneous boundary value problems for the Navier-Stokes system of viscous incompressible flows. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1080 A. FURSIKOV, M. GUNZBURGER, AND L. HOU for the following boundary value problem: NS ( v)(t, x)w(t, x) Here, NS ( v) is the derivative of the Navier-Stokes operator evaluated at an optimizer v, Q T = (0, T ) × Ω is a time-space cylinder on which (1.1) is posed, where Ω ⊂ R 2 is the spatial domain with boundary ∂Ω, and Σ T = (0, T ) × ∂Ω is the lateral surface of the cylinder Q T . The solvability of the problem (1.1) should be proved for each f ∈ F, g ∈ G, and v 0 ∈ W 0 , where F, G, and W 0 are appropriate function spaces. The correct choice of W, F, and G is very important because it is closely connected to the correct mathematical formulation of the original drag reduction problem. In the two-dimensional case, the space W is generated naturally (see  ) by the drag functional and is the "energy space" are the usual Sobolev spaces with smoothness index k and H k (Ω) = [H k (Ω)] d , with d denoting the space dimension. Evidently, F = {f ∈ L 2 (0, T ; H −1 (Ω)) : div f = 0} and A. FURSIKOV, M. GUNZBURGER, AND L. HOU Some notations are in order. Throughout, Ω is a bounded domain in R d , ∂Ω is the boundary of Ω, and ∂Ω is a compact and closed C ∞ manifold consisting of J connected components {Γ j }, ∂Ω = J j=1