The Stokes and Navier-Stokes equations in an aperture domain
The author deals with the nonstationary Navier-Stokes initial boundary value problem (IBVP) \[ \begin{gathered} \partial_t u-\Delta u+(u,\nabla) u+\nabla\pi= 0\quad\text{in }\Omega\times (0,\infty),\\ \nabla\cdot u= 0\quad\text{in }\Omega\times (0,\infty),\\ u(x,t)= 0\quad\text{on }\partial\Omega\times (0,\infty),\\ u(x,0)= u_0(x)\quad\text{in }\Omega,\end{gathered}\tag{1} \] where \(u\) is the unknown field, \(u= (u_1,\dots, u_n)\in (W^{2,p}(\Omega))^n\) and the unknown scalar pressure term \(\nabla\pi\in (L^p(\Omega))^n\), \(1< p<\infty\) and \(\Omega\) is an aperture domain. The author proves the global existence of a unique solution to the (1) with the vanishing flux condition and some decay properties as \(t\to\infty\), when the initial velocity is sufficiently small in \(L^n\) space. Moreover, he can prove the time-local existence of an unique solution to the (1) with the non-trivial flux condition.
scientific article; zbMATH DE number 5190375