18

数理解析研究所講究録
第2009巻 2016年 18-26

Initial value conditions for the Navier‐Stokes

equations in the weighted Serrin class
Reinhard Farwig (Technische Universität Darmstadt)

farwig@mathematik.tu‐darmstadt.de
Yoshikazu Giga (University of Tokyo)

labgiga@ms.u‐tokyo.ac.jp
Pen‐Yuan Hsu (University of Tokyo)

pyhsu@ms.u‐tokyo.ac.jp

Introduction

1

This manuscript contains a summary of [3] and new related references. For details
werefer the reader to [3] and [2]. In [3] we consider the initial value problem\partial_{t}u-\triangle u+u\cdot\nabla u+\nabla p=f,\mathrm{d}\mathrm{i}\mathrm{v}u=0 in (0, T)\times $\Omega$(1.1)

u|_{\partial $\Omega$}=0, u(0)=u_{0}in a bounded domain[0, T),

0<T\leq\infty

solutionas.

$\Omega$\subset \mathbb{R}^{3} with boundary \partial $\Omega$ of class C^{2,1} and a time interval

We define a new type of a strong solution, thefollows.

)

L_{ $\alpha$}^{s}(L^{q}) ‐strong

Definition 1.1. Let u_{0}\in L_{ $\sigma$}^{2}( $\Omega$) be an initial value and let f=\mathrm{d}\mathrm{i}\mathrm{v}F with F=(F_{ij})_{i,j=1}^{3}\in L^{2}(0, T;L^{2}( $\Omega$)) beanexternal force. A vector fieldu\in L^{\infty}(0, T;L_{ $\sigma$}^{2}( $\Omega$))\cap L^{2}(0, T;W_{0}^{1,2}( $\Omega$))
is called a weak solution

(1.1)

(1.2)

(in the sense of Leray‐Hopf) of the Navier‐Stokes systemwith data u_{0\mathrm{z} f , if the relation-\{u, w_{t}\}_{ $\Omega$,T}+\{\nabla u, \nabla w\}_{ $\Omega$,T}- \{uu, \nabla w\rangle_{ $\Omega$,T}=\langle u_{0}, w(0)\}_{ $\Omega$}
holds for each test function

w\in C_{0}^{\infty}([0, T);C_{0, $\sigma$}^{\infty}( $\Omega$)),—\{F, \nabla w\}_{ $\Omega$,T}

and if the energy inequality\displaystyle\frac{1}{2}\Vertu(t)\Vert_{2}^{2}+\int_{0}^{t}\Vert\nablau\Vert_{2}^{2}\mathrm{d}$\tau$\leq\frac{1}{2}\Vertu_{0}\Vert_{2}^{2} \displaystyle\int_{0}^{t}(F,\nablau)\mathrm{d}$\tau$
—

(1.3)

(1.4)

19

is satisfied for

0\leq t<T.

of (1.1) is called an L_{ $\alpha$}^{s}(L^{q}) ‐strong solution with exponents
2<s<\infty, 3<q<\infty and weight $\tau$^{ $\alpha$} in time, 0< $\alpha$<\displaystyle \frac{1}{2} where \displaystyle \frac{2}{s}+\frac{3}{q}=1-2 $\alpha$such that additionally the weighted Serrin condition
A weak solution u,u\in L_{ $\alpha$}^{s}(0, T;L^{q}( $\Omega$))
is

,

i e
.

.

\displaystyle\int_{0}^{T}($\tau$^{$\alpha$}\Vertu($\tau$)\Vert_{q})^{s}\mathrm{d}$\tau$<\infty

satisfied. If in (1.5) $\alpha$=0 and \displaystyle \frac{2}{s}+\frac{3}{q}=1

,

(1.5)

then u is called a strong solution(L^{s}(L^{q}) ‐strong solution).

(1.1) is well‐known since thepioneering work of [7, 9]. The existence of a strong solution u of (1.1) could beshown up to now at least in a sufficiently small interval [0, T ), 0<T\leq\infty and
The existence of at least one weak solution u of,under additional smoothness conditions on the initial data u_{0} and the external forcef.

The first sufficient condition on the initial data for a bounded domain seems

to be due to[8], yielding a solution class of so‐called local strong solutions. Since

then many results on suffcient initial value conditions for the existence of local
strong solutions have been developed. Recent results in [4, 5] yield sufficient andnecessary conditions, also written in terms of (solenoidal) Besov spaces\mathb {B}_{q_{)}s ^{-1+\frac{3}{q} ( $\Omega$) where \displaystyle \frac{2}{s}+\frac{3}{q}=1 In this work,
.

we

are

\mathb {B}_{q}^{-\frac{2}{s^{s} ( $\Omega$)=interested in a weighted Serrin

condition with respect to time and L_{ $\alpha$}^{s}(L^{q}) ‐strong solutions. Our result in [3] yields
a sufficientcondition on initial data and external force to guarantee the existence of

local L_{ $\alpha$}^{s}(L^{q}) ‐strong solutions. The motivation for this approach is an extension of
the results in [4, 5] wheree^{- $\tau$ A}u_{0}\not\in L^{s}(0, T;L^{q}( $\Omega$))with somecondition\mathb {B}_{q,s(q,0)}^{-1+\frac{3}{q} ($\Omega$)

O.

$\alpha$>\displaystyle \frac{2}{s}+\frac{3}{q}=1 to the,caseu_{0}\not\in \mathb {B}_{q,s}^{-1+\frac{3}{q} ( $\Omega$) i.e.,,\displaystyle\int_{0}^{T}($\tau$^{$\alpha$}\Verte^{-$\tau$A}u_{0}\Vert_{q})^{s}\mathrm{d}$\tau$<\infty, \displaystyle \frac{2}{s}+\frac{3}{q}=1-2 $\alpha$butMore precisely, for the case $\alpha$=0(classical Serrin class), thee^{- $\tau$ A}u_{0}\in L^{s(q,0)}(0, T;L^{q}( $\Omega$)) with \displaystyle \frac{2}{s(q,0)}+\frac{3}{q}=1 is equivalent to u_{0}\in,whereas for$\alpha$with

0< $\alpha$<\displaystyle \frac{1}{2} (weighted Serrin class) the conditione^{- $\tau$ A}u_{0}\in L_{ $\alpha$}^{s(q, $\alpha$)}(0, T;L^{q}( $\Omega$)) with \displaystyle \frac{2}{s(q, $\alpha$)}+\frac{3}{q}=1-2 $\alpha$ is equivalent to u_{0}\in \mathb {B}_{q,s(q_{)} $\alpha$)}^{-1+\frac{3}{q} ( $\Omega$)\mathb {B}_{q,s(q,0)}^{-1+\frac{3}{q} ($\Omega$)\subset\mathb {B}_{q,s(q,$\alpha$)}^{-1+\frac{3}{\mathrm{q} ($\Omega$).

Since s(q, $\alpha$)>s(q, 0) by embedding theorems we know
Therefore, the spaces to yield strong solutions are larger than the classical Serrin
class discussed in the literature, and the theory of [4, 5] is extended to the scale of,

Besov spaces\mathb {B}_{q,s(q,$\alpha$)}^{-1+\frac{3}\mathrm{q} ($\Omega$) filling the

gap between\mathb {B}_{q,s(q,0)}^{-1+\frac{3}{q} ($\Omega$) and \mathb {B}_{q,\infty}^{-1+\frac{3}{q} ($\Omega$).

.

We state our main result in [3] in a more precise way as follows.

([3, Theorem 1.2]) Let $\Omega$\subseteq \mathbb{R}^{3} be a bounded domain with boundary

Theorem 1.2.

\partial $\Omega$ of class C^{2,1},

and let 0<T\leq\infty, 2<s<\infty, 3<q<\infty,

0< $\alpha$<\displaystyle \frac{1}{2} with \displaystyle \frac{2}{s}+\frac{3}{q}=20

1-2 $\alpha$ be given. Consider the Navier‐Stokes equation with initial value u_{0}\in L_{ $\sigma$}^{2}( $\Omega$)

and an external force f=\mathrm{d}\mathrm{i}\mathrm{v}F where

F\in L^{2}(0, T;L^{2}( $\Omega$))\cap L_{2 $\alpha$}^{s/2}(0, T;L^{q/2}( $\Omega$))

Then there exists a constant $\epsilon$_{*}=$\epsilon$_{*}(q, s, $\alpha$, $\Omega$)>0 with the following property:\Vert e^{- $\tau$ A}u_{0}\Vert_{L_{ $\alpha$}^{s}(0,T;L^{\mathrm{q} )}+\Vert F\Vert_{L_{2 $\alpha$}^{s/2}(L^{q/2})}\leq$\epsilon$_{*}
then the Navier‐Stokes system
u_{0},

f on the interval [0, T).

Theorem 1.3.

.

If

(1.6)

,

(1.1) has a unique L_{ $\alpha$}^{s}(L^{q}) ‐strong solution with data([3, Theorem 1.3]) Let $\Omega$ be as in Theorem 1.2, let 2<s<\infty,

3<q<\infty, 0< $\alpha$<\displaystyle \frac{1}{2} with \displaystyle \frac{2}{s}+\frac{3}{q}=1-2 $\alpha$ be given, and let u_{0}\in L_{ $\sigma$}^{2}( $\Omega$) and an
external force f=\mathrm{d}\mathrm{i}\mathrm{v}F where(1)F\in L^{2}(0, \infty;L^{2}( $\Omega$))\cap L_{2 $\alpha$}^{s/2}(0, \infty;L^{q/2}( $\Omega$)).

The condition\displaystyle\int_{0}^{\infty}($\tau$^{$\alpha$}\Verte^{-$\tau$A}u_{0}\Vert_{q})^{s}\mathrm{d}$\tau$<\infty(1.7)

is sufficient and necessary for the existence of a unique L_{ $\alpha$}^{s}(L^{q}) ‐strong solution u\in

L_{ $\alpha$}^{s}(0, T;L^{q}) of the Navier‐Stokes system (1.1), with data u_{0}, f in some interval[0, T) 0<T\leq\infty.
(2) Let u be a weak solution of the system (1.1) in [0, \infty ) \times $\Omega$ with data u_{0)}f,,

and let\displaystyle\int_{0}^{\infty}($\tau$^{$\alpha$}\Verte^{-$\tau$A}u_{0}\Vert_{q})^{s}\mathrm{d}$\tau$=\infty

.

(1.8)

weighted Serrins condition u\in L_{ $\alpha$}^{s}(0, T;L^{q}( $\Omega$)) does not hold for each0<T\leq\infty Moreover, the system (1.1) does not have a L_{ $\alpha$}^{s}(L^{q}) ‐strong solution withdata u_{0\mathrm{z} f and weighted Serrin exponents s q) $\alpha$ in any interval [0, T), 0<T\leq\infty.
Then the.

,Besides, we also prove a restricted Serrins uniqueness theorem in [3]. A weak‐strong uniqueness theorem in the sense of the classical Serrin Uniqueness Theorem
seems to be out of reach for
L_{ $\alpha$}^{s}(L^{q}) ‐strong solutions within the full class of weaksolutions satisfying the energy inequality. The reason is based on the algebraic iden‐
tities and sharp use of norms and Hölder estimates in the proof of Serrins Theorem,cf.

[10, Ch. V, Sect. 1.5]. However, we prove uniqueness within the subclass ofwell‐chosen weak solutions describing weak solutions constructed by concrete ap‐

proximation procedures. We refer to Assumptions 1.5, 1.8 and Remarks 1.6, 1.7 for
precise definitions.
Theorem 1.4.

([3, Theorem 1.4]) Let $\Omega$\subset \mathbb{R}^{3} be a bounded domain with boundaryof class C^{2,1} and let 2<s<\infty, 3<q<\infty,
Moreover, suppose that0< $\alpha$<\displaystyle \frac{1}{2} with \displaystyle \frac{2}{s}+\frac{3}{q}=1-2 $\alpha$ be given.

u_{0}\in L_{ $\sigma$}^{2}( $\Omega$)\cap \mathb {B}_{q,s}^{-1+\frac{3}{q} andF\in L^{2}(0, \infty;L^{2}( $\Omega$))\cap L_{2 $\alpha$}^{s/2}(0, \infty;L^{q/2}( $\Omega$))

are

an

external force f=\mathrm{d}\mathrm{i}\mathrm{v}F where

given. Then the unique L_{ $\alpha$}^{s}(L^{q}) ‐strong

solution u\in L_{ $\alpha$}^{s}(0, T;L^{q}( $\Omega$)) is unique on a time interval [0, T ), T>0 , in the classof all well‐chosen weak solutions.

21

Assumption 1.5. Let $\Omega$\subset \mathbb{R}^{3} be a bounded domain with boundary of class C^{2,1}(1) Given u_{0}\in L_{ $\sigma$}^{2}( $\Omega$) and an external force f=\mathrm{d}\mathrm{i}\mathrm{v}F where F\in L^{2}(0, \infty;L^{2}( $\Omega$))
we assume

the existence of approximating sequences (u_{0n})\subset L_{ $\sigma$}^{2}( $\Omega$)inu_{0n}\rightarrow u_{0}and (F_{n})\subset L^{2}(0, \infty;L^{2}( $\Omega$))of u_{0} such that

L_{ $\sigma$}^{2}( $\Omega$)of F such that

F_{n}\rightarrow FinL^{2}(0, \infty;L^{2}( $\Omega$))asn\rightarrow\infty.

(2) Let (J_{n}) denote a family of bounded operators in \mathcal{L}(L_{ $\sigma$}^{q}( $\Omega$), D(A_{q}^{1/2}) such
1<q<\infty there exists a constant C_{q}>0 such thatthat for each\displaystyle\VertJ_{n}\Vert_{\mathcal{L}(L_{$\sigma$}^{q}) +\Vert\frac{1}{n}A_{q}^{1/2}J_{n}\Vert_{\mathcal{L}(L_{$\sigma$}^{q}) \leqC_{q}and

J_{n}u\rightarrow u in L_{ $\sigma$}^{q}( $\Omega$) as n\rightarrow\infty.

(3) For each n\in \mathbb{N} let u_{n} denote the weak solution of the approximate Navier‐Stokes system\partial_{t}u_{n}-\triangle u_{n}+(J_{n}u_{n})\cdot\nabla u_{n}+\nabla p_{n}=\mathrm{d}\mathrm{i}\mathrm{v}F_{n}, \mathrm{d}\mathrm{i}\mathrm{v}u_{n}=0u_{n}|_{\partial $\Omega$}=0, u_{n}(0)=u_{n0}in(0, T)\times $\Omega$(1.9)

Remark 1.6. A typical example of operators (J_{n}) in Assumption 1.5 is given by thefamily of Yosida operators J_{n}=(I+\displaystyle \frac{1}{n}A_{q}^{1/2})^{-1}

It is well known that this family of

operators is uniformly bounded on L_{ $\sigma$}^{q}( $\Omega$) as well as on D(A_{q}^{1/2}) for each 1<q<\infty.

Moreover, J_{n}u\rightarrow u in L_{ $\sigma$}^{q}( $\Omega$) as n\rightarrow\infty
have the same properties.
We know from [10, Ch.

V, Thm.

.

By analogy, the operators J_{n}=e^{-A_{q}^{1/2}/n}2.5.11 (with a minor modification\backslash in the caseof J_{n}=e^{-A_{q}^{1/2}/n}) that there exists a unique weak solution u_{n}\in \mathcal{L}H_{T} :=L^{\infty}(L^{2})\cap
L^{2}(H_{0}^{1}) of (1. 9) satisfying the uniform estimate\Vert u_{n}\Vert_{L^{\infty}(L^{2})}+\Vert u_{n}\Vert_{L^{2}(H^{1})}\leq C(\Vert u_{0n}\Vert_{2}+\Vert F_{n}\Vert_{L^{2}(L^{2})})\leq C(\Vert u_{0}\Vert_{2}+\Vert F\Vert_{L^{2}(L^{2})}+1)
for all suficiently large n\in \mathbb{N}(u_{n_{k}}) of (u_{n}) such thatu_{n_{k}}\rightarrow v inL^{2}(H_{0}^{1}),.

Therefore, there exists v\in \mathcal{L}H_{T} and a subsequence

u_{n_{k} \rightarrow*v in L^{\infty}(L^{2}),u_{n}k\rightarrow v inL^{2}(L^{2}).

From the last convergence we also conclude that u_{n_{k}}(t_{0})\rightarrow v(t_{0}) in L^{2}( $\Omega$) for a.a.

t_{0}\in(0, T) Actually, v\in \mathcal{L}H_{T} is a weak solution of (1.1).
.

(1) Since we do not know whether weak solutions of (1.1) are unique,v may depend on the subsequence(u_{n_{k}}) chosen above. In this case, we say that
Remark 1.7.

v

is a well‐chosen weak solution of (1.1).

(1.10)

22

Note that a well‐chosen weak solution v is always related to a concrete approximation
procedure as in Assumption 1.5 and the choice of an adequate (weakly-*) convergent
subsequence of a sequence of approximate solutions (un).

(2) The question whether solutions constructed by the Galerkin method fall into
the scope of a modified Assumption 1.5 and yield uniqueness in the senseof The‐1.4 has not been settled. A similar question concerning the property to be asuitable weak solution, cf. H. Beirão da Veiga [1, p.3211, has been answered in theoremaffirmative, see J.‐L. Guermond [6].

Assumption 1.8. Under the assumptions of Assumption 1.5 additionally let 2<s<\infty, 3<q<\infty, 0< $\alpha$<\displaystyle \frac{1}{2} with \displaystyle \frac{2}{s}+\frac{3}{q}=1-2 $\alpha$ be given. Suppose that evenu_{0},

u_{0n}\in \mathb {B}_{q_{)}s}^{-1+\frac{3}{q} and F, F_{n}\in L_{2 $\alpha$}^{s/2}(0, \infty;L^{q/2}( $\Omega$)) such that also
u_{0n}\rightarrow u_{0}in\mathb {B}_{q,s}^{-1+\frac{3}{q} , F_{n}\rightarrow FinL_{2 $\alpha$}^{s/2}(0, \infty;L^{q/2}( $\Omega$))asn\rightarrow\infty.

(1.1) we also assume that the

From now on by a well‐chosen weak solution of

approximation satisfies Assumption 1.8 as well as Assumption 1.5.
Remark 1.9. In [2], the assumptions on well‐chosen weak solutions had been weaken
to improve or extend the restricted Serrins uniqueness theorem of [3].

In the next section, for readers convenience, we summary the proof for the main

theorems in [3].

2Proof of Theorems 1.2 and 1.3

Now we are in the position to state the proof of the main theorem in [3].

Proof of Theorem 1.2. Let u be a weak solution of (1.1) with initial value u_{0}\in L_{ $\sigma$}^{2}F\in L^{2}(L^{2})\cap L_{2 $\alpha$}^{s/2}(L^{q/2}) Furthermore, let E_{f,u_{0}}denote the solution of the Stokes problem
and external force f=\mathrm{d}\mathrm{i}\mathrm{v}F where.

\partial_{t}v-\triangle v+\nabla p=f, \mathrm{d}\mathrm{i}\mathrm{v}v=0

v|_{\partial $\Omega$}=0, v(0)=u_{0},i.e.

E_{f,u_{0} (t)=e^{-tA}u_{0}+\displaystyle \int_{0}^{t}A^{1/2}e^{-(t- $\tau$)A}A^{-1/2}\mathrm{P}\mathrm{d}\mathrm{i}\mathrm{v}F( $\tau$)\mathrm{d} $\tau$=:E_{0,u_{0}}(t)+E_{f,0}(t)AssumeE_{0,u_{0}}\in L_{ $\alpha$}^{s}(L^{q})

,

i.e.

.

\displaystyle \int_{0}^{t}\Vert$\tau$^{ $\alpha$}e^{- $\tau$ A}u_{0}\Vert_{q}^{s}\mathrm{d} $\tau$<\infty

.

Since

u_{0}\in L_{ $\sigma$}^{2} and F\inL^{2}(L^{2}) we know that E_{f,u_{0}}\in C^{0}([0, T];L^{2})\cap L^{2}(H^{1}) satisfying the energy equality.
,

,

23

Moreover, by using the estimates (3.1) and (3.2) (see Appendix) with 2 $\beta$+\displaystyle \frac{3}{q}=\frac{3}{q/2}with q>3 i.e.
,$\beta$=\displaystyle \frac{3}{2q}<\frac{1}{2},オ\displaystyle \Vert E_{f,0}(t)\Vert_{q}\leq C\int_{0} \VertA^{\frac{1}{2}+$\beta$}e^{-(t-$\tau$)A}(A^{-\frac{1}{2} P\mathrm{d}\mathrm{i}\mathrm{v})F($\tau$)\Vert_{\frac{q}{2} \mathrm{d}$\tau$\displaystyle\leqc\int_{0}^{t}(-$\tau$)^{-$\beta$-\frac{1}{2}\VertF($\tau$)\Vert_{\frac{\mathrm{q}{2}\mathrm{d}$\tau$.

By applying the weighted Hardy‐Littlewood‐Sobolev inequality (see Lemma 3.1 inAppendix) with the exponents s_{2}=s, $\alpha$_{2}= $\alpha$, s_{1}=s/2, $\alpha$_{1}=2 $\alpha$, $\lambda$= $\beta$+\displaystyle \frac{1}{2}\in(0 1 ),-\displaystyle \frac{2}{s}<2 $\alpha$<1-\frac{2}{s}and−\displaystyle \frac{1}{s}< $\alpha$<1-\frac{1}{s}

,

we

,

have

(2.1)

\Vert E_{f_{)}^{0} \Vert_{L_{ $\alpha$}^{s}(L^{q})}\leq c\Vert F\Vert_{L_{2 $\alpha$}^{s/2}(L^{q/2})}

provided \displaystyle \frac{2}{s}+(\frac{3}{2q}+\frac{1}{2}+2 $\alpha$- $\alpha$)=1+\frac{1}{s} (which is equivalent to \displaystyle \frac{2}{s}+\frac{3}{q}=1-2 $\alpha$ ). We
then set ũ=u—Ef, u_{0} which solves the (Navier‐)Stokes system\partialtũ— \triangle ũ + u \cdot\nabla u +\nabla p =0 , divũ =0\~{u}|\partial $\Omega$=0, \~{u}(0)=0.
So we can write at least formally\displaystyle \~{u}(t)=-\int_{0}^{t}e^{-(t- $\tau$)A}P\mathrm{d}\mathrm{i}\mathrm{v}(u\otimes u)( $\tau$)\mathrm{d} $\tau$=-\displaystyle \int_{0}^{t}A^{1/2}e^{-(t- $\tau$)A}(A^{-1/2}P\mathrm{d}\mathrm{i}\mathrm{v})(u\otimes u)( $\tau$)\mathrm{d} $\tau$.
With$\beta$=\displaystyle \frac{3}{2q}

as

(2.2)

above we getオ|\displaystyle \~{u}(t)\Vert_{q}\leq C\int_{0} \VertA^{\frac{1}{2}+$\beta$}e^{-(t-$\tau$)A}\Vert\VertA^{-\frac{1}{2} P\mathrm{d}\mathrm{i}\mathrm{v}\Vert\Vert(u\otimesu)\Vert_{\frac{\mathrm{q} {2} \mathrm{d}$\tau$\displaystyle\leqc\int_{0}^{t}( -$\tau$)^{-\frac{1}{2}-$\beta$}\Vertu\Vert_{q}^{2}\mathrm{d}$\tau$(2.3)

Then the Hardy‐Littlewood‐SoUolev inequality as above implies that

\Vert ũ( t ) \Vert_{L_{ $\alpha$}^{s}(L^{q})}\leq c\Vert(\Vert u\Vert_{q}^{2})\Vert_{L_{2 $\alpha$}^{s/2} =c\Vert u\Vert_{L_{ $\alpha$}^{s}(L^{\mathrm{q} )}^{2}

.

(24)

Since u=\~{u}+E_{f,u_{0}} we have

\Vert\~{u}\Vert_{L_{ $\alpha$}^{8} (0,T;L^{\mathrm{q} )\leq C (||\~{u} | L;(0,T;L^{q}) +\Vert F\Vert_{L_{2 $\alpha$}^{s/2} (0,T;L^{q/2})+\Vert e^{- $\tau$ A}u_{0}\Vert_{L_{ $\alpha$}^{s} (0,T;L^{q}) ^{2}

(25)

24

As in [5, p. 99] there exists by Banachs Fixed Point Theorem an $\epsilon$_{*}=$\epsilon$_{*}(q, s, $\alpha$, $\Omega$)>
0 such that we get the existence of a unique fixed point \~{u}\in L_{ $\alpha$}^{s}(0, T;L^{q}) solving\partial_{t}\~{u}-\triangle\~{u}+(\~{u}+E_{f,u_{0}})\cdot\nabla(\~{u}+E_{f,u_{0}})+\nabla p=0,divũ =0\~{u}|\partial $\Omega$=0, \~{u}(0)=0
provided (1.6) is satisfied, i.e.

\tilde{u}+E_{f,u_{0}}\in L_{ $\alpha$}^{s}(0, T;L^{q})\Vert e^{- $\tau$ A}u_{0}\Vert_{L_{ $\alpha$}^{s}(0,T;L^{q})}+\Vert F\Vert_{L_{2 $\alpha$}^{s/2}(L^{q/2})}\leq$\epsilon$_{*} Hence.

.

Now we need to prove that this constructed mild solution u is indeed

solution under the following conditions, cf.
some

u=

a

weak

the assumptions in Theorem 1.2 andfacts already proved:u, \~{u}\in L_{ $\alpha$}^{s}(L^{q}) , u_{0}\in L_{ $\sigma$}^{2}, e^{- $\tau$ A}u_{0}\in L_{ $\alpha$}^{s}(L^{q}) , F\in L^{2}(L^{2})\cap L_{2 $\alpha$}^{s/2}(L^{q/2}).

To this aim we need the following lemmata which had been proved in [3].

([3, Lemma 3.1]) The mild solution u constructed in the above proce‐

Lemma 2.1.

dure satisfies \nabla u\in L^{2}(0, T;L^{2}( $\Omega$)).

Lemma 2.2. ([3, Lemma 3.2]) Under the assumptions of Lemma 2.1 we have thatu\in L^{s_{2}}(0, T;L^{q_{2}}( $\Omega$)) for all \displaystyle \frac{2}{s_{2} +\frac{3}{q_{2} =\frac{3}{2}, 2\leq s_{2}\leq\infty, 2\leq q_{2}\leq 6 Moreover,\Vert\~{u}(t)\Vert_{2}\rightarrow 0 and u(t)\rightarrow u_{0} in L^{2}( $\Omega$) as t\rightarrow 0+.
.

([3, Lemma 3.3]) Under the assumptions of Lemma 2.1,

Lemma 2.3.

u\in L_{ $\alpha$/(2+8 $\alpha$)}^{4}(0, T;L^{4}( $\Omega$)).

By Lemma 2.3 we may use that u\in L_{ $\alpha$/(2+8 $\alpha$)}^{4}(L^{4}) Hence u\in L^{4}( $\epsilon$, T;L^{4}) for
all 0< $\epsilon$<T So, by [10, IV. Thm. 2.3.1, Lemma 2.4.2] and for a.a. $\epsilon$\in(0, T) u
is the unique weak solution in L^{4}( $\epsilon$, T;L^{4}) on ( $\epsilon$, T) of the linear Stokes problem.

.

,\partial_{t}u-\triangle u+\nabla p=\mathrm{d}\mathrm{i}\mathrm{v}\tilde{F}, \mathrm{d}\mathrm{i}\mathrm{v}u=0
u|_{\partial $\Omega$}=0, u|_{t= $\epsilon$}=u( $\epsilon$)with external force \mathrm{d}\mathrm{i}\mathrm{v}\tilde{F},L^{4}( $\Omega$)\subset L^{2}( $\Omega$).

\tilde{F}=F-u\otimes u\in L^{2}( $\epsilon$, T;L^{2}) and initial value u( $\epsilon$)\in

Therefore, u satisfies the energy equality on ( $\epsilon$, T)

,

i.e.

\displaystyle\frac{1}{2}\Vertu(t)\Vert_{2}^{2}+\int_{$\epsilon$}^{t}\Vert\nablau\Vert_{2}^{2}\mathrm{d}$\tau$=\frac{1}{2}\Vertu($\epsilon$)\Vert_{2}^{2}-\int_{$\epsilon$}^{t}(F,\nablau)\mathrm{d}$\tau$t\in( $\epsilon$, T) and a.a. $\epsilon$\in(0, T) Moreover, u\in C^{0}([ $\epsilon$, T);L^{2}) and henceu\in C^{0}((0, T);L^{2}) see [10, IV 2.1‐2.3]. Furthermore, since by Lemma 2.2 u\inL^{\infty}((0, T);L^{2}) it also satisfies the energy equality on [0, T ). Hence u is a weak
for all.

,,solution; this completes the proof of Theorem 1.2.

\square25

Appendix3

For the readers convenience, we explain some well‐known properties of the Stokes

operator. Let $\Omega$ be as in Theorem 1.2, let [0, T ), 0<T\leq\infty be a time interval,

P_{q}:L^{q}( $\Omega$)\rightarrow L_{ $\sigma$}^{q}( $\Omega$) denotes the Helmholtz projection,and the Stokes operator A_{q}=-P_{\mathrm{q}}\triangle : D(A_{q})\rightarrow L_{ $\sigma$}^{q}( $\Omega$) is defined with domain
D(A_{q})=W^{2,q}( $\Omega$)\cap W_{0}^{1,q}( $\Omega$)\cap L_{ $\sigma$}^{q}( $\Omega$) and range R(A_{q})=L_{ $\sigma$}^{q}( $\Omega$) Since P_{q}v=P_{ $\gamma$}v
for v\in L^{q}( $\Omega$)\cap L^{ $\gamma$}( $\Omega$) and A_{q}v=A_{ $\gamma$}v for v\in D(A_{q})\cap D(A_{ $\gamma$}) 1< $\gamma$<\infty wesometimes write A_{q}=A to simplify the notation if there is no misunderstanding.
and let

1<q<\infty

.

Then

.

,

In particular, if q=2 ,wealways write P=P_{2} and A=A_{2}

A_{q}^{ $\alpha$} D(A_{q}^{ $\alpha$})\rightar ow L_{ $\sigma$}^{q}( $\Omega$) -1\leq $\alpha$\leq 1
D(A_{q})\subseteq D(A_{q}^{ $\alpha$})\subseteq L_{ $\sigma$}^{q}( $\Omega$) R(A_{q}^{ $\alpha$})=L_{ $\sigma$}^{q}( $\Omega$) if 0\leq $\alpha$\leq 1(A_{q}^{- $\alpha$}) and (A_{q})'=A_{q'} where \displaystyle \frac{1}{q}+\frac{1}{q}=1.
:

,

,

.

,

Furthermore, letdenote the fractional powers of A_{q},.

We note that.

It holds(A_{q}^{ $\alpha$})^{-1}=Now we recall the embedding estimate\Vert v\Vert_{q}\leq c\Vert A_{ $\gamma$}^{ $\alpha$}v\Vert_{ $\gamma$}, v\in D(A_{ $\gamma$}^{ $\alpha$}),1< $\gamma$\leq q,

2$\alpha$+\displaystyle\frac{3}{q}=\frac{3}{$\gam a$},0\leq $\alpha$\leq 1

,

(3.1)

and the estimate\Vert A_{q}^{ $\alpha$}e^{-tA_{q} v\Vert_{q}\leq ct^{- $\alpha$}e^{- $\delta$ t}\Vert v\Vert_{q}, v\in L_{ $\sigma$}^{q}( $\Omega$) , 0\leq $\alpha$\leq 1, t>0
with constants c=c( $\Omega$, q)>0,

,

(3.2)

$\delta$= $\delta$( $\Omega$, q)>0.

Then we recall a weighted version of the Hardy‐Littlewood‐Sobolev inequality.
For $\alpha$\in \mathbb{R} and s\geq 1 we consider the weighted L^{8} ‐space

L_{ $\alpha$}^{s}(\displaystyle \mathb {R})=\{u:\Vert u\Vert_{L_{ $\alpha$}^{s} =(\int_{\mathb {R} (| $\tau$|^{ $\alpha$}|u( $\tau$)| ^{s}\mathrm{d} $\tau$)^{1/s}<\infty\}.
Lemma 3.1. Let 0< $\lambda$<1, 1<s_{1}\leq s_{2}<\infty,

and\displaystyle \frac{1}{s_{1} +( $\lambda$+$\alpha$_{1}-$\alpha$_{2})=1+\frac{1}{s_{2} , $\alpha$_{2}\leq$\alpha$_{1}.

-\displaystyle \frac{1}{s_{1} <$\alpha$_{1}<1-\frac{1}{s_{1} , -\displaystyle \frac{1}{s_{2} <$\alpha$_{2}<1-\frac{1}{s_{2}Then the integral operator

I_{ $\lambda$}f(t)=\displaystyle \int_{\mathb {R} (t- $\tau$)^{- $\lambda$}f( $\tau$)\mathrm{d} $\tau$is bounded as operator I_{ $\lambda$} :L_{$\alpha$^{1}1}^{s}(\mathbb{R})\rightar ow L_{(x_{2}^{2} ^{s}(\mathbb{R}).

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