On a Compatibility Condition for the Navier-Stokes Solutions in Maximal \(L^p\) -regularity Class

We consider a compatibility condition on the initial data a and the external force f for the initial-boundary value problem of the Navier-Stokes equations with no-slip condition on \(\partial \Omega \) in a bounded domain \(\Omega \subset \mathbb R^n\) . Our class of solutions is based on that of maximal \(L^s\) -regularity as \(W^{1,s}(0, T; \mathcal {D}(A_r^\varepsilon )) \cap L^s(0, T; \mathcal {D}(A^{1+\varepsilon }_r))\) , where \(A_r\) denotes the Stokes operator in \(L^r_\sigma (\Omega )\) . We show that if the solution belongs to such a class for \(\varepsilon > 1/s + 1/2r\) , then a and f necessarily satisfy \(\begin{aligned} A_ra + P_r(a\cdot \nabla a) = P_rf(0) \quad \text {on }\partial \Omega , \end{aligned}\) where \(P_r\) denotes the \(L^r\) -Helmholtz projection in \(\Omega \) . Simultaneously, we construct the solution in such a class with \(0\le \varepsilon < 1/2r\) for a and f without such a compatibility condition as above provided \(2/s + n/r =3 + 2\varepsilon \) .