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Abstract $$L^ p$$ estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains. (English) Zbl 0739.35067
The authors first generalize the result of G. Dore and A. Venni [Math. Z. 196, 189-201 (1987; Zbl 0615.47002)] on abstract Cauchy- problems $$u'+Au=f$$ to operators $$A$$, for which one has the estimate $\| A^{iy}\|\leq K\exp(\theta| y|)\quad\hbox { with } \theta<\pi/2 (*)$ for the pure imaginary powers of $$A$$, and show furthermore that the constant $$C$$ in the estimate $\int^ T_ 0(\| u'\|^ s+\| Au\|^ s)dt\leq C\int^ T_ 0\| f\|^ sdt$ is independent of $$T$$, which is important for ”global in time” results. The estimate (*) for the Stokes operator in $$L_{q,\sigma}$$ on exterior domains was proved by the authors in J. Fac. Sci., Univ. Tokyo, Sect. I A 36, 103-130 (1989; Zbl 0689.76012). Hence their abstract results imply then new $$L_ p-L_ q$$ estimates of integral type (global in time) for the solutions of the Navier-Stokes system $u'-\Delta u+(u\nabla u)+\nabla p=f,\quad\hbox { div} u=0\quad\hbox {and } u=0\hbox { on } \partial \Omega.$ If $$\|\centerdot\|_{q,s}$$ denotes the norm in $$L_ s(\mathbb{R}^ +,L_ q)$$, they get $\| u'\|_{q,s}+\| D^ 2u\|_{q,s}+\|\nabla p\|_{q,s}+\| p\|_{r,s}\leq C(\| f\|_{q,s}+\| f\|_{2,2}+\| u(0)\|_ Y)$ (for certain $$Y$$), if $$n+1={n\over q}+{2\over s}$$, $${n\over r}={n\over q}-1$$. For $$n=3$$ this implies the crucial result $$p\in L_{5/3}(\mathbb{R}_ +\times\Omega)$$ for proving regularity for large $$t$$.

##### MSC:
 35Q30 Navier-Stokes equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35B65 Smoothness and regularity of solutions to PDEs
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