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Solutions in \(L_ r\) of the Navier-Stokes initial value problem. (English) Zbl 0587.35078
The authors prove the existence of a unique strong solution in \(L_ r\) space \((1<r<\infty)\) of the initial value problem of the Navier-Stokes equations \(u_ t+(u,\nabla)u-\Delta u=f-\nabla p,\) div u\(=0\) in \(D\times (0,T)\), \(u=0\) on \(S\times (0,T)\), \(u(0,x)=a(x)\) in D, where D is a bounded domain in \(R^ n\) (n\(\geq 2)\) with smooth boundary S. Let \(X_ r\) be the closure in \((L_ r(D))^ n\) of \(\{u\in (C_ 0^{\infty}:\) div u\(=0\}\). Then the appropriate Stokes operator \(-A_ r\) \((1<r<\infty)\) generates a bounded holomorphic semigroup of class \(C_ 0\) in \(X_ r\) [see the first author, Math. Z. 178, 297-329 (1981; Zbl 0473.35064)], and the domain of fractional power \(D(A_ r^{\alpha})\) \((0<\alpha <1)\) is the complex interpolation space \([X_ r,D(A_ r)]_{\alpha}\) [see the first author, Proc. Jap. Acad., Ser. A 57, 85-89 (1981; Zbl 0471.35069) and Arch. Ration. Mech. Anal. 89, 251-265 (1985)]. On the base of these facts, they obtain the following remarkable results:
(i) Fix \(\gamma\) and choose \(\delta\geq 0\) such that n/2r-\(\leq \gamma <1\) (n\(\geq 2)\), \(-\gamma <\delta <1-| \gamma |\). Assume \(a\in D(A_ r^{\gamma})\) and \(\| A_ r^{-\delta}P_ rf(t)\|\) is continuous on (0,T) and \(\| A_ r^{-\delta}P_ rf(t)\| =o(t^{\gamma +\delta -1})\) as \(t\to 0\), where \(P_ r\) is the projection of \(L_ r(D))^ n\) on \(X_ r\). Then there exists a local solution of the integral equation \[ (*)\quad u(t)=\exp (-tA_ r)a+\int^{t}_{0}\exp (-(t-s)A_ r)\{-P_ r(u,\nabla)u+P_ rf(s)\} ds \] such that (a) \(u\in C([0,T_*];D(A_ r^{\gamma}))\), \(u(0)=a\), (b) \(u\in C((0,T_*];D(A_ r^{\alpha}))\) for some \(T_*>0\), (c) \(\| A_ r^{\alpha}u(t)\| =o(t^{\gamma -\alpha})\) as \(t\to 0\) for all \(\alpha\), \(\gamma <\alpha <1-\delta.\)
(ii) Any solution of (*) satisfying (a) and (b’) \(u\in C((0,T_*];D(A_ r^{\beta}))\), (c’) \(\| A_ r^{\beta}u(t)\| =o(t^{\gamma - \beta})\) for some \(\beta\), \(| \gamma | <\beta\) is unique.
(iii) If \(P_ rf: (0,T]\to X_ r\) is Hölder continuous on each [\(\epsilon\),T] \((0<\epsilon <T)\), the solution u(t) of (*) given in (i) satisfies the differential equation in \(X_ r:\) \[ u_ t+A_ ru=-P_ r(u,\nabla)u+P_ rf\quad on\quad (0,T_*]\quad and\quad u(t)\in D(A_ r)\quad for\quad t\in (0,T_*]. \] (iv) Let \(a\in D(A_ r^{\gamma})\) and \(P_ rf\in C((0,\infty);X_ r).\) Then the solution u(t) given by (i) exists on (0,\(\infty)\) provided the data a and \(P_ rf\) are small in some sense.
The authors also consider the regularity of solutions and show that if the external force f is smooth, their solutions given by (i) are smooth up to the boundary.
Reviewer: R.Iino

35Q30 Navier-Stokes equations
47D03 Groups and semigroups of linear operators
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B65 Smoothness and regularity of solutions to PDEs
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