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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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