Remark on the \(L_q-L_\infty\) estimate of the Stokes semigroup in a 2-dimensional exterior domain
We show the \(L_q-L_\infty\) type estimate of the Stokes semigroup in a 2-dimensional exterior domain \(\Omega\). In our previous study, we obtained the \(L_q\)-\(L_r\) estimates of the Stokes semigroup in a 2-dimensional exterior domain, but it does not include the case that \(r=\infty\). If we go back to the representation formula of solutions to the resolvent equation, by combining several known results concerning the estimates of Stokes resolvent in \(\mathbb{R}^n\) and the asymptotic behavior of Stokes resolvent in the exterior domain near the boundary we obtain the estimate \[ \|e^{-t\mathbb{A}} {\mathbf f} \|_{L_\infty} \leq Ct^{-1/q} \|{\mathbf f} \|_{L_q} \quad \forall t>0\text{ and }{\mathbf f}\in \mathbb{J}_q(\Omega), \] where \(\mathbb{A}\) is the Stokes operator, and \(\mathbb{J}_q (\Omega)\) denotes the closure in \(L_q(\Omega) \times L_q(\Omega)\) of all solenoidal vector fields with compact support. This result can be applied to the Navier-Stokes equation to obtain the \(L_\infty\) estimate of the solution in a 2-dimensional exterior domain.
scientific article; zbMATH DE number 1356077