On optimal initial value conditions for local strong solutions of the Navier–Stokes equations

Autor: Hermann Sohr, Werner Varnhorn, Reinhard Farwig
Rok vydání: 2009
Předmět:
Zdroj: ANNALI DELL'UNIVERSITA' DI FERRARA. 55:89-110
ISSN: 1827-1510
0430-3202
DOI: 10.1007/s11565-009-0066-4
Popis: Consider a smooth bounded domain $${\varOmega\subseteq{\mathbb R}^3}$$ , and the Navier–Stokes system in $${[0,\infty)\times\varOmega}$$ with initial value $${u_0\in L^2_\sigma(\varOmega)}$$ and external force f = div F, $${F\,{\in} \,L^2(0,\infty; L^2(\varOmega))\cap L^{s/2}(0,\infty; L^{q/2}(\varOmega))}$$ where $${2\,< \,s\,< \,\infty, 3\,< \,q\,< \,\infty, \frac{2}{s}+\frac{3}{q} \,{=} \,1}$$ , are so-called Serrin exponents. It is an important question what is the optimal (weakest possible) initial value condition in order to obtain a unique strong solution $${u\in L^s(0,T; L^q(\varOmega))}$$ in some initial interval [0, T), $${0 < T \leq \infty}$$ . Up to now several sufficient conditions on u 0 are known which need not be necessary. Our main result, see Theorem 1.1, shows that the condition $${\int_0^\infty||e^{-t A}u_0||_q^s {\rm d}t < \infty}$$ , A denotes the Stokes operator, is sufficient and necessary for the existence of such a strong solution u. In particular, if $${\int_0^\infty||e^{-t A}u_0||_q^s {\rm d}t = \infty}$$ , $${u_0\in L_\sigma^2(\varOmega)}$$ , then any weak solution u in the usual sense does not satisfy Serrin’s condition $${u\in L^s(0,T; L^q(\varOmega))}$$ for each 0
Databáze: OpenAIRE
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