Existence of regular solutions to the stationary Navier-Stokes equations
With this paper the authors continue their interesting investigations on the regularity of solutions to the stationary Navier-Stokes equations [cf. Ann. Sc. Norm. Sup. Pisa 21, 63-95 (1994; Zbl 0823.35141) and Acta Appl. Math. 37, 53-66 (1994; Zbl 0814.35094)]. Starting from potential estimates for \({1\over 2}u^2+p\) and \((u(x)\cdot(x-x_0))^2\) from their previous work, where \((u,p)\) is a weak solution with periodic boundary conditions in \(N\) dimensions, \(5\leq N<10\), they first show that the potential of \((u(x)\cdot(x-x_0))^2\), if restricted to a suitably small ball \(B(x_0;r)\), becomes uniformly small with respect to \(x_0\) for all solutions \(u\) that are in a bounded set in \(W^{1,2}\). While the methods to show this inequality exploit the structure of the nonlinearity \((u\cdot\nabla)u\) in a delicate way the estimates of the Hölder seminorms of \(u\) and \(p\) are proven with arguments from the regularity theory for nonlinear elliptic systems. Finally, existence of regular solutions that belong to \(W^{2,q}\times W^{1,q} \forall q<\infty\) is shown by a degree argument, which can be applied because of the bounds on the Hölder norms. [For \(N=5\) the existence of regular solutions was also obtained by \textit{M. Struwe} in ibid. 719-741 (1995; reviewed below)].
regularity of solutions to the stationary Navier-Stokes equations
scientific article; zbMATH DE number 800585