The Navier-Stokes equations in the weak-\(L^n\) space with time-dependent external force
The boundary value problem and the initial-boundary value problem for the nonstationary Navier-Stokes system are investigated: \[ \begin{aligned} &\frac {\partial v}{\partial t}-\Delta v+ v\cdot\nabla v+\nabla p=f(x,t), \quad \nabla\cdot v=0 \quad \text{in \;} \{x\in\Omega,\;t\in \mathbb{R}\}\\ &v(x,t)=0 \quad \text{on \;}\{x\in\partial \Omega,\;t\in\mathbb{R}\}\tag{1}\\ &v(x,t)\to 0\;\text{as \;} |x|\to\infty.\end{aligned} \] \[ \begin{aligned} &\frac {\partial v}{\partial t}-\Delta v+ v\cdot\nabla v+\nabla p=f(x,t),\quad \nabla\cdot v=0 \quad \text{in \;} \{x\in\Omega,\;t\in (0,\infty)\}\\&v(x,t)=0 \quad \text{on \;}\{x\in\partial \Omega,\;t\in (0,\infty)\}\tag{2}\\ &v(x,0)=a(x), \;\;v(x,t)\to 0\;\text{as \;} |x|\to\infty.\end{aligned} \] Here \(\Omega=\mathbb{R}^n\) or \(\Omega=\{x\in \mathbb{R}^n,\;x_n>0\}\), or \(\Omega\) is an exterior domain in \(\mathbb{R}^n\) with smooth boundary \(\partial \Omega,\;n\geqslant 3\). The mild solutions of the problems (1) and (2) are studied in the paper. This solution is the function, bounded in an appropriate function space, which solves the integral equation associated with the Navier-Stokes equations in the sense of distributions. The author gives conditions on the function \(f\) sufficient for the unique existence of a small solution in the Lorentz-type space \(L_{\sigma}^{n,\infty}\times \mathbb{R}\). In particular, sufficient conditions for the unique existence and the stability of small time-periodic solutions are obtained. The regularity of the mild solutions is studied too.
scientific article; zbMATH DE number 1515368