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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

##### MSC:
 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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