Stokes- and Navier-Stokes equations in weighted function spaces (Q2743181)

The work presents the author's approach to functional-analytic problems of solvability of the Stokes and Navier-Stokes equations. The considerations are performed in weighted spaces, fundamental are the spaces of the type NEWLINE\[NEWLINEL^q_\omega(\Omega)= \Biggl\{u\in L^1_{\text{loc}}(\Omega): \int_\Omega|u|^q_\omega\, d\Omega< \infty\Biggr\},\quad 1\leq q\leq \infty,\quad\Omega\subseteq \mathbb R^n.NEWLINE\]NEWLINE The book consists of 8 chapters and an appendix. In the first four chapters there are formulated the basic weight estimates for the Stokes resolvent and other basic characteristics of spaces. The resolvent characteristics are presented in Chapter 5, which is the heart of the work. In the 6th chapter the Helmholtz decomposition is proved for weighted \(L^q\) spaces. In the last two chapters problems of solvability of non-stationary equations are investigated. Estimates of nonlinear terms in the equations are given, and abstract existence theorems are formulated and proved. In the Appendix there is proved one part of the embedding theorem for Navier-Stokes equations.NEWLINENEWLINENEWLINEThe dissertation is concisely, successfully utilizing the newest results. It may be recommended to specialists in the field, but also to all interested in its possible utilization for advances in numerical analysis.