Regular solutions of the stationary Navier-Stokes equations on \(\mathbb{R}^ 5\)
The author investigates the existence of regular solutions to the stationary Navier-Stokes equations in 5 dimensions; this case is closely connected to the 3-dimensional nonstationary Navier-Stokes system and hence of particular interest. It is shown that for smooth exterior forces there exists a classical solution provided the domain is \(\mathbb{R}^5\) and the exterior force has either compact support or else periodic boundary values in the unit cube. The proof uses a local estimate for \({1\over 2}|u|^2+p\) due to \textit{J. Frehse} and \textit{M. Růžička} [Ann. Sci. Norm. Sup. Pisa 21, 63-94 (1994; Zbl 0823.35141); these authors treat the periodic case, too (cf. Math. Ann. 302, 699-717 (1995; reviewed above))]. The bulk of the paper consists of a priori bounds for smooth solutions, first in Sobolev norms, which are then improved to estimates for potentials of \({1\over 2}|u|^2+p\) and \([u(x)\cdot(x-x_0)]^2\). These hold uniformly for \((u,p)\) in a bounded domain of the underlying function space which finally leads to an existence proof that is based on a degree argument. In the nonperiodic case the solution decays at infinity like the fundamental solution to the Stokes equations.
regular solutions to the stationary Navier-Stokes equations
scientific article; zbMATH DE number 800586