On the existence of local strong solutions for the Navier-Stokes equations in completely general domains |
| |
Authors: | Reinhard Farwig Hermann Sohr |
| |
Affiliation: | a Fachbereich Mathematik, Technische Universität Darmstadt, D-64283 Darmstadt, Germany b Fakultät für Elektrotechnik, Informatik und Mathematik, Universität Paderborn, D-33098 Paderborn, Germany |
| |
Abstract: | There are only very few results on the existence of unique local in time strong solutions of the Navier-Stokes equations for completely general domains Ω⊆R3, although domains with edges and corners, bounded or unbounded, are very important in applications. The reason is that the Lq-theory for the Stokes operator A is available in general only in the Hilbert space setting, i.e., with q=2. Our main result for a general domain Ω is optimal in a certain sense: Consider an initial value and a zero external force. Then the condition is sufficient and necessary for the existence of a unique local strong solution u∈L8(0,T;L4(Ω)) in some interval [0,T), 0<T≤∞, with u(0)=u0, satisfying Serrin’s condition . Note that Fujita-Kato’s sufficient condition u0∈D(A1/4) is strictly stronger and therefore not optimal. |
| |
Keywords: | primary, 35Q30 secondary, 35B30, 76D05 |
本文献已被 ScienceDirect 等数据库收录! |
|