数理解析研究所講究録
第 1905 巻 2014 年 164-179

164

Long time existence for the $3D$ incompressible
Euler equations with high-speed rotation
Ryo Takada
Mathematical Institute, Tohoku University
Sendai 980-8578, JAPANE-mail: ryo@m.tohoku.ac.jp

1 Introduction and Main Result
This note is a brief survey of our paper [14], which is ajoint work with Professors
Youngwoo Koh and Sanghyuk Lee.
We consider the initial value problem for the $3D$ rotating incompressible Euler equations:$\{\begin{array}{ll}\partial_{t}u+\Omega e_{3}\cross u+(u\cdot\nabla)u+\nabla p=0 t>0, x\in \mathbb{R}^{3},divu=0 t>0, x\in \mathbb{R}^{3},u(O, x)=u_{0}(x) x\in \mathbb{R}^{3},\end{array}$$(E_{\Omega})$

where $u=(u_{1}(t, x), u_{2}(t, x), u_{3}(t, x))$ and $p=p(t, x)$ denote the unknown velocity field
and the unknown scalar pressure, respectively, while $u_{0}=(u_{0,1}(x), u_{0,2}(x), u_{0,3}(x))$ denotes the given initial velocity field satisfying $divu_{0}=0$ . The constantrepresents
the speed of rotation of the fluids around the vertical unit vector $e_{3}=(0,0,1)$ , which iscalled the Coriolis parameter.
In this note, we prove the long time existence of unique classical solutions to the Euler
equations with the Coriolis force provided the speed of rotation is suﬃciently high. Moreprecisely, we shall show that for given initial velocity
and for given time $0<T<\infty$
there exists a positive constantdepending on
and the norm ofsuch that if
thenadmits a unique classical solution on the given time interval $[0, T].$Let
be the Helmholtz projection onto the divergence-freefields,vectorwhere
denotes the Riesz transform in . Applying the projection to, we obtain the following system for the velocity
both sides of the first equation offields:$\Omega\in \mathbb{R}$$u_{0}$$\Omega_{0}$$|\Omega|\geq\Omega_{0}$$\mathbb{P}$$s,$ $T$$u_{0}$$(E_{\Omega})$$:=(\delta_{jk}+R_{j}R_{k})_{1\leq j,k\leq 3}$$\mathbb{R}^{3}$$R_{j}$$\mathbb{P}$$(E_{\Omega})$$\{\begin{array}{ll}\partial_{t}u+\Omega \mathbb{P}(e_{3}\cross u)+\mathbb{P}(u\cdot\nabla)u=0 t>0, x\in \mathbb{R}^{3},divu=0 t>0, x\in \mathbb{R}^{3},u(O, x)=u_{0}(x) x\in \mathbb{R}^{3}.\end{array}$$(E_{\Omega}’)$165
We now summarize the local existence results on the original Euler equations for $\Omega=0.$
Kato [10] proved that for given integer
with $m>5/2$ and for given initial velocity
with $divu_{0}=0$ , there exists a positive time $T=T(m, \Vert u_{0}\Vert_{H^{m}})$ such thatthe equation (E\’o) admits a unique classical solution in the class $C([O, T];H^{m}(\mathbb{R}^{3}))\cap$$C^{1}([0, T];H^{m-1}(\mathbb{R}^{3}))$ . Kato and Ponce [11] extended this result to the fractional ordered
Sobolev spaces
for $s,>3/p+1,$ $1<p<\infty$ . Chae [2] and Chen, Miao andZhang [7] obtained a local well-posedness for (E\’o) in the Triebel-Lizorkin spaces
for $s>3/p+1,$ $1<p,$ $q<\infty$ . Chae [3] also obtained the local well-posedness in theBesov spaces
for $s>3/p+1,$ $1<p<\infty,$ $1\leq q\leq\infty$ and for $s=3/p+1,$$1<p<\infty,$ $q=1$ . Pak and Park [17] extended these results to the Besov space.
, Dutrifoy [9] showed the asymptotics of solutions
For large Coriolis parameterto vortex patches or Yudovich solutions as the Rossby number goes to zero for someparticular initial data, and gave a lower bound on the lifespanof the solution toas. Similar results are obtained for the quasigeostrophic systems by
Dutrifoy [8] and Charve [4].
Now, let us state our main result.
$m\in \mathbb{Z}$$u_{0}\in H^{m}(\mathbb{R}^{3})$$\backslash$$u$$W^{s,p}(\mathbb{R}^{3})$$F_{p,q}^{s}(\mathbb{R}^{3})$$B_{p,q}^{s}(\mathbb{R}^{3})$$B_{\infty,1}^{1}(\mathbb{R}^{3})$$|\Omega|$$(E_{\Omega}’)$$T_{\Omega}$$T_{\Omega}\sim>\log\log|\Omega|$

Theorem 1.1. Let
satisfy $s>5/2$ . Then, for $0<T<\infty$ andsatisfying $divu_{0}=0$, there exists a positive constantpossesses a unique classical solution satisfying
then$s\dot{\in}\mathbb{R}$$u_{0}\in H^{s+1}(\mathbb{R}^{3})$$\Omega_{0}=\Omega_{0}(s, T, \Vert u_{0}\Vert_{H^{s+1}})$$if|\Omega|\geq\Omega_{0}$$(E_{\Omega}’)$such that$u$$u\in C([O, T];H^{s+1}(\mathbb{R}^{3}))\cap C^{1}([0, T];H^{s}(\mathbb{R}^{3}))$.

In particular, for $2<q<\infty$ there exist a positive absolute constantconstant $C_{1}=C_{1}(s, q)$ such that the parametercan be taken so that

$C_{0}$

and a positive$\Omega_{0}$$\Omega_{0}\geq C_{0}[1+\Vert u_{0}\Vert_{H^{s+1}}T\exp\{C_{1}T^{1-\frac{1}{q}}\Vert u_{0}\Vert_{H^{s+1}}\}]^{q}$(1.1)

Remark 1.2. From the characterization (1.1) it follows that for suﬃciently high speed ofrotation , the maximal existence timeof the solution tohas alower bound$T_{\Omega}\geq 1$$|\Omega|$$T_{\Omega} \geq\frac{C_{1}’}{\Vert u_{0}\Vert_{H^{s+1}}}\log(\frac{|\Omega|}{C_{0}})$$(E_{\Omega}’)$(1.2)

with some positive constants $C_{0}’=C_{0}’(q)$ and $C_{1}’=C_{1}’(s, q)$ . On the other hand, in [S,9]it was shown thatastends to infinity. Although Theorem 1.1 does
not cover the data such as vortex patches or Yudovich solutions which are treated in [8,9],
the lower bound (1.2) can be regarded as an improvement for the lifespan of the solutionto. This is due to the use of a single exponential estimate for the blow-up criterion(see Section 3).
$\tau_{\Omega}>\sim\log\log|\Omega|$$|\Omega|$$(E_{\Omega}’)$This note is organized as follows. In Section 2, we recall the definitions of function
spaces, and the commutator estimates in these spaces. In Section 3, we state the local
existence results and the blow-up criterion of Beale-Kato-Majda type for the local solutions. In Section 4, we shall give the Strichartz estimates for the propagator generated by
the Coriolis force. In Section 5, we present the proof of Theorem 1.1.

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Throughout this paper, we denote by the constants which may diﬀer from line toline. In particular, $C=C(\cdot, \ldots, \cdot)$ will denote the constant which depends only on thequantities appearing in parentheses. For $A,$ $B\geq 0,$ $A<B\sim$ means that there exists somepositive constant such that $A\leq CB$ . Also, $A>\sim B$ is defined in the same way as$A<B\sim.$ $A\sim B$ means that $A<B\sim$ and $A>B\sim.$
$C$

$C$

2 Preliminaries
be the space of tempered disLet
be the Schwartz class, and let
Littlewood-Paley
decomposition. Let
tributions. We first recall the definition of the
supp
for allsatisfying
be a function in$\{\xi\in \mathbb{R}^{3}|1/2\leq|\xi|\leq 2\}$ and$\mathscr{S}’(\mathbb{R}^{3})$$\mathscr{S}(\mathbb{R}^{3})$$\xi\in \mathbb{R}^{3},$$0\leq\hat{\varphi_{0}}(\xi)\leq 1$$\mathscr{S}(\mathbb{R}^{3})$$\varphi_{0}$$\sum_{j\in \mathbb{Z}}\hat{\varphi_{j}}(\xi)=1$

for all$\hat{\varphi_{0}}\subset$$\xi\in \mathbb{R}^{3}\backslash \{0\},$be the. Let. We setwhere$:=\varphi_{j}*f$ for. Then, we reLittlewood-Paley operator defined bycall the definitions of the inhomogeneous and the homogeneous Besov spaces.
$:=2^{3j}\varphi_{0}(2^{j}x)$$\varphi_{j}(x)$$\hat{\chi}(\xi)$$:=1- \sum_{j\geq 1}\hat{\varphi_{j}}(\xi)$$\{\triangle_{j}\}_{j\in \mathbb{Z}}$$f\in \mathscr{S}’(\mathbb{R}^{3})$$\triangle_{j}f$

and $1\leq p,$ $q\leq\infty$ , the inhomogeneous Besov space
Definition 2.1. (i) For
such thatis defined to be the set of all tempered distributions$s\in \mathbb{R}$$f\in \mathscr{S}’(\mathbb{R}^{3})$$B_{p,q}^{s}(\mathbb{R}^{3})$$\Vert f\Vert_{B_{p,q}^{s}}:=\Vert\chi*f\Vert_{L^{p}}+\Vert\{2^{sj}\Vert\triangle_{j}f\Vert_{L^{p}}\}_{j=1}^{\infty}\Vert_{l^{q}}<\infty.$

and $1\leq p,$ $q\leq\infty$ , the homogeneous Besov spacesuch thatbe the set of all tempered distributions(ii) For$s\in \mathbb{R}$$\dot{B}_{p,q}^{s}(\mathbb{R}^{3})$is defined to$f\in \mathscr{S}’(\mathbb{R}^{3})$$\Vert f\Vert_{\dot{B}_{p,q}^{s}}:=\Vert\{2^{sj}\Vert\triangle_{j}f\Vert_{Lr}\}_{j\in \mathbb{Z}}\Vert_{\ell^{q}}<\infty.$

Next, let$\langle f,$$H^{s}(\mathbb{R}^{3})$denotes the Sobolev space of order$s\in \mathbb{R}$with the inner product$g \rangle_{H^{s}}:=\int_{\mathbb{R}^{3}}(1-\Delta)^{\frac{s}{2}}f(x)\overline{(1-\triangle)^{\frac{s}{2}}g(x)}dx=\frac{1}{(2\pi)^{3}}\int_{\mathbb{R}^{3}}(1+|\xi|^{2})^{s}\hat{f}(\xi)_{\hat{9}}\overline{(\xi)}d\xi$

and the norm$\Vert f\Vert_{H^{s}}$$:=\sqrt{\langle f,f\rangle_{H^{s}}}$. For $s>0$ , it is known that the norm equivalence$\Vert f\Vert_{H^{s}}\sim\Vert f\Vert_{L^{2}}+||f\Vert_{\dot{H}^{\epsilon}}$(2.1)

holds, where$\Vert f\Vert_{\dot{H}^{s}}:=\Vert f\Vert_{\dot{B}_{2,2}^{s}}=(\sum_{j\in \mathbb{Z}}2^{2sj}\Vert\triangle_{j}f\Vert_{L^{2}}^{2})^{\frac{1}{2}}$We end this section by recalling several commutator estimates in the Sobolev spaces.

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Lemma 2.2 $(Kato-$Ponce $[11 For s\geq 0, there$ exists
such that$a$positive constant $C=C(s)$$\Vert(1-\triangle)^{\frac{s}{2}}(fg)-f(1-\triangle)^{\frac{s}{2}}g\Vert_{L^{2}}\leq C(\Vert\nabla f\Vert_{L^{\infty}}\Vert g\Vert_{H^{s-1}}+\Vert g\Vert_{L^{\infty}}\Vert f\Vert_{H^{s}})$for all$f\in\dot{W}^{1,\infty}(\mathbb{R}^{3})\cap H^{s}(\mathbb{R}^{3})$and$g\in L^{\infty}(\mathbb{R}^{3})\cap H^{s-1}(\mathbb{R}^{3})$.

Lemma 2.3 (Chen Miao Zhang [7] and [19]). For $s>0$, there exists a positive constant$C=C(s)$ such that$( \sum_{j\in \mathbb{Z}}2^{2sj}\Vert[u\cdot\nabla, \triangle_{j}]f\Vert_{L^{2}}^{2})^{\frac{1}{2}}\leq C(\Vert\nabla u\Vert_{L^{\infty}}\Vert f\Vert_{\dot{H}^{s}}+\Vert\nabla f\Vert_{L^{\infty}}\Vert u\Vert_{\dot{H}^{s}})$for all$(u, f)\in\dot{W}^{1,\infty}(\mathbb{R}^{3})\cap\dot{H}^{s}(\mathbb{R}^{3})^{3+1}$

with $divu=$ O.

3 Blow-up Criterion
In this section, we shall prove the blow up criterion of the Beale-Kato Majda type[1]. We first state the uniform local existence results of the unique classical solutions to$(E_{\Omega}’)$.

Theorem 3.1. Let
satisfy $s>5/2$ . Then, for $u_{0}\in H^{s}(\mathbb{R}^{3})satis\beta ingdivu_{0}=0,$
there exists a positive time $T=T(s, \Vert u_{0}\Vert_{H^{s}})$ depending only on andsuch thatpossesses a unique classical solution for all
satisfying$s\in \mathbb{R}$$\Vert u_{0}\Vert_{H^{s}}$$s$$(E_{\Omega}’)$$u$$\Omega\in \mathbb{R}$$u\in C([O, T];H^{s}(\mathbb{R}^{3}))\cap C^{1}([0, T];H^{s-1}(\mathbb{R}^{3}))$.

As in the case of the original Euler equations $\Omega=0$ , Theorem 3.1 follows from theclassical energy method and the Kato-Ponce commutator estimates (Lemma 2.2). Indeed,
thanks to the skew-symmetry of the Coriolis force:$\int_{\mathbb{R}^{3}}\Omega e_{3}\cross u(t, x)\cdot u(t, x)dx=0,$

we can obtain the uniformsee [14].

$H^{s}$

energy estimates with respect to$\Omega\in \mathbb{R}$. For the details,

Proposition 3.2. Let $s>5/2$ , and letsatisp $divu_{0}=0$ . Suppose that
$C([O,T);H^{s}(\mathbb{R}^{3}))\cap
C^{1}([0,T);H^{s-1}(\mathbb{R}^{3}))$ .
a solution ofin the classIf$u_{0}\in H^{8}(\mathbb{R}^{3})$

$u$

is

$(E_{\Omega}’)$$\int_{0}^{T}\Vert\nabla u(t)\Vert_{L^{\infty}}dt<\infty,$

then

$u$

can be continued to the solution insome $T’>T.$$C([O, T H^{s}(\mathbb{R}^{3})$) $\cap C^{1}([0, T H^{s-1}(\mathbb{R}^{3})$ ) $for$

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An immediate consequence of the above proposition is the following.
Corollary 3.3. Let $s>5/2$, and letsatisﬃ $divu_{0}=0$ . Suppose that is a
$C([O,T);H^{s}(\mathbb{R}^{3}))\cap
C^{1}([0,T);H^{s-1}(\mathbb{R}^{3}))$ . Suppose that $T$in the classsolution ofis maximal, that is, cannot be continued to $C([O, T H^{s}(\mathbb{R}^{3})$ ) $\cap C^{1}([0, T H^{s-1}(\mathbb{R}^{3})$ )for any $T’>T$. Then,$u_{0}\in H^{s}(\mathbb{R}^{3})$$u$$(E_{\Omega}’)$$u$$\int_{0}^{T}\Vert\nabla u(t)\Vert_{L^{\infty}}dt=\infty.$

Proof of Proposition 3.2. By Theorem 3.1, we see that if$u_{0}\in H^{s}(\mathbb{R}^{3})$with $s>5/2$ , thein the class $C([O, T);H^{s}(\mathbb{R}^{3}))\cap$time interval $[0, T$ ) of the existence of solutions to. Hence by the standard argument
depends only on andsolutions,
it suﬃces to establish an a priori estimate for inwhich continues the local
in terms ofandwith , we have
Taking the -inner product of$(E_{\Omega}’)$$C^{1}([0, T);H^{s-1}(\mathbb{R}^{3}))$$\Vert u_{0}\Vert_{H^{s}}$$\mathcal{S}$$u$$H^{s}(\mathbb{R}^{3})$$s,$$\Vert u_{0}\Vert_{H^{s}}$$L^{2}$$\int_{0}^{T}\Vert\nabla u(t)\Vert_{L}\infty dt.$$(E_{\Omega}’)$$u$$\frac{1}{2}\frac{d}{dt}\Vert u(t)\Vert_{L^{2}}^{2}=0$

by the skew-symmetry of$e_{3}\cross u$

and the divergence-free condition. Hence we see that
(3.1)$\Vert u(t)\Vert_{L^{2}}=||u_{0}\Vert_{L^{2}}$-estimate for . Applying the Littlewood-Paleyfor all $t\in[0, T$ ). We next derive the, we haveto both sides ofoperator$\dot{H}^{s}$$u$$(E_{\Omega}’)$$\triangle_{j}$$\partial_{t}\triangle_{j}u+\mathbb{P}\Omega e_{3}\cross\triangle_{j}u+\mathbb{P}\triangle_{j}(u\cdot\nabla)u=0$, and using
Taking the -inner product of (3.2) withwhich follows from the divergence-free condition, we have$L^{2}$$\triangle_{j}u$$\frac{1}{2}\frac{d}{dt}\Vert\triangle_{j}u(t)\Vert_{L^{2}}^{2}=-\langle\triangle_{j}(u(t)\cdot\nabla)u(t)$

,

.

(3.2)

$\langle(u\cdot\nabla)\triangle_{j}u,$$\triangle_{j}u\rangle_{L^{2}}=0,$$\triangle_{j}u(t)\rangle_{L^{2}}=\langle[u(t)\cdot\nabla, \triangle_{j}]u(t)$,$\triangle_{j}u(t)\rangle_{L^{2}}.$

From this and the Schwartz inequality, we see that$\frac{1}{2}\frac{d}{dt}\Vert\triangle_{j}u(t)\Vert_{L^{2}}^{2}\leq\Vert[u(t)\cdot\nabla, \triangle_{j}]u(t)\Vert_{L^{2}}1\triangle_{j}u(t)\Vert_{L^{2}}.$This gives$\frac{d}{dt}\Vert\triangle_{j}u(t)\Vert_{L^{2}}\leq\Vert[u(t)\cdot\nabla, \Delta_{j}]u(t)\Vert_{L^{2}}.$Integrating both sides on $[0, t]$ , we have$\Vert\Delta_{j}u(t)\Vert_{L^{2}}\leq\Vert\triangle_{j}u_{0}\Vert_{L^{2}}+\int_{0}^{t}\Vert[u(\tau)\cdot\nabla, \Delta_{j}]u(\tau)\Vert_{L^{2}}d\tau$

.

(3.3)

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Multiplying both sides of (3.3) by
Minkowski inequality that$2^{sj}$and then taking the$\ell^{2}(\mathbb{Z})$-norm, we have from the$\Vert u(t)\Vert_{\dot{H}^{s}}\leq\Vert u_{0}\Vert_{\dot{H}^{s}}+\int_{0}^{t}(\sum_{j\in \mathbb{Z}}2^{2sj}\Vert[u(\tau)\cdot\nabla, \triangle_{j}]u(\tau)\Vert_{L^{2}}^{2})^{\frac{1}{2}}d\tau.$

By Lemma 2.3, it follows that$\Vert u(t)\Vert_{\dot{H}^{s}}\leq\Vert u_{0}\Vert_{\dot{H}^{s}}+C\int_{0}^{t}\Vert\nabla u(\tau)\Vert_{L^{\infty}}\Vert u(\tau)\Vert_{H^{s}}d\tau$with some positive constant $C=C(s)$ . Combining (3.1) and the above inequality, by(2.1) we see that there exists a positive constant $C=C(s)$ such that$\Vert u(t)\Vert_{H^{s}}\leq C\Vert u_{0}\Vert_{H^{s}}+C\int_{0}^{t}\Vert\nablau(\tau)\Vert_{L^{\infty}}\Vert u(\tau)\Vert_{H^{s}}d\tau.$Hence from the Gronwall inequality we obtain(3.4)$\Vert u(t)\Vert_{H^{s}}\leq C\Vert u_{0}\Vert_{H^{s}}\exp\{C\int_{0}^{t}\Vert\nabla u(\tau)\Vert_{L}\infty d\tau\}$

for all $t\in[0, T$ ). This completes the proof of Proposition 3.2.

$\square$

4 Strichartz Estimates
In this section, we shall give the sharp Strichartz estimates for the propagator gen. As it is already observed in [5, 6, 9], the systemerated by the Coriolis force
exhibits a dispersion phenomenon which is due to the presence of the Coriolis force.
$\Omega e_{3}\cross u$$(E_{\Omega}’)$This is closely related to the Strichartz estimates for the operator
Fourier integral$e^{\pm i\Omega t_{D|}^{D}}\#$

defined by the$e^{\pm i\Omega t_{\overline{|}D|}^{D_{\Delta}}}f(x):= \int_{\mathbb{R}^{3}}e^{ix\cdot\xi\pm i\Omega t\frac{\xi}{1}\xi_{1}}\hat{f}(\xi)d\xi, (t, x)\in(0, \infty)\cross \mathbb{R}^{3}.$Here denotes the Fourier transform of . We are interested in the space-time Strichartz
estimate(4.1),$\hat{f}$$f$$\Vert e^{\pm it\frac{D}{1}4}D|f\Vert_{L_{t}^{q}L_{x}^{r}}<\sim\Vert f\Vert_{\dot{H}^{s}}$, it is easy to see thatforis replaced by. By the scaling
whereis establishedthisestimateand should satisfy $s=3/2-3/r$ . Oncethe exponents
shows that the eﬀect of the Coriolis parameter
the simple change of variables
is homogeneous of degree , by the. Also, since the function
is given by$\Omega t$$t\in \mathbb{R}$$q,$ $r$$f\mapsto f(\lambda\cdot)$$\lambda>0$$s$$t\mapsto\Omega t$$C|\Omega|^{-\frac{1}{q}}$$\Omega$$\xi_{3}/|\xi|$

$0$

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Littlewood-Paley decomposition and the scaling argument, the matter is reduced to thefrequency localized case. Now we consider the operator$\mathscr{G}_{\pm}(t)f(x):=\int_{\mathbb{R}^{3}}\ovalbox{\tt\small REJECT}, (t,x)\in \mathbb{R}^{1+3},$

wheresatisfies supp
andon $\{2^{-1}\leq|\xi|\leq 2\}.$The operator
was previously studied by Dutrifoy [9, Corollary 2], who obtained thespace-time estimates$\hat{\Phi}\subset\{2^{-2}\leq|\xi|\leq 2^{2}\}$$\Phi\in \mathscr{S}(\mathbb{R}^{3})$$\hat{\Phi}=1$$\mathscr{G}_{\pm}$(4.2)$\Vert \mathscr{G}_{\pm}(t)f\Vert_{L_{t}^{q}L_{x}^{r}\sim}<\Vert f\Vert_{L^{2}}$for the exponents and which satisfy $4<q\leq\infty,$ $2\leq r\leq\infty$ and the admissible
relation $1/q+1/(2r)<1/4$ . We extend the estimate to the optimal range except an
endpoint case. More precisely, we shall prove the following.
$q$

$r$

Theorem 4.1. Let $2\leq q,$ $r\leq\infty$ with $(q, r)\neq(2, \infty)$ . Then, the space-time estimate(4.2) holds if and only if$\frac{1}{q}+\frac{1}{r}\leq\frac{1}{2}$

.

(4.3)

As it was shown by Montgomery-Smith [16], it seems likely that one can show the
failure of the endpoint case $(q, r)=(2, \infty)$ .
By (4.2), the Littlewood-Paley theory and the embeddingfor$2\leq r<\infty$ , we can prove the Strichartz estimates (4.1) for the original operator as acorollary of Theorem 4.1. Then by the scaling
we have the following whichshows the dispersive eﬀect of the Coriolis forces.
$\dot{B}_{r,2}^{0}(\mathbb{R}^{3})\mapsto L^{r}(\mathbb{R}^{3})$$t\mapsto\Omega t$

Corollary 4.2. Let $2\leq q\leq\infty$ and $2\leq r<\infty$ , and letspace-time estimate$\Omega\in \mathbb{R}\backslash \{0\}$. Then, the$\Vert e^{\pm i\Omega t\frac{D}{1}A}D|f\Vert_{L_{t}^{q}L_{x}^{r}}<|\Omega|^{-\frac{1}{q}}\Vert f\Vert_{\dot{H}}$holds if and satisfy (4.3).
$r$

$q$

Let us set$\Psi(\xi):=\frac{\xi_{3}}{|\xi|}, \xi\neq 0.$

As it is well-known, the boundedness of the estimate (4.2) is closely related to the curvature of the surface$\Sigma_{0}:=\{(\xi, \rho)\in \mathbb{R}^{3}\cross \mathbb{R}|\rho=\Psi(\xi) , \frac{1}{4}\leq|\xi|\leq 4\}.$A direct computation shows that the Hessian matrix $H\Psi=$
to$( \frac{\partial^{2}\Psi}{\partial\xi_{j}\partial\xi_{k}})_{1\leq j,k\leq 3}$

of$H \Psi(\xi)=\frac{1}{|\xi|^{5}}(\xi_{1}(3\xi_{3}^{2}-|\xi|^{2})\xi_{3}(3\xi_{1}^{2}-|\xi|^{2})3\xi_{1}\xi_{2}\xi_{3} \xi_{2}(3\xi_{3}^{2}-|\xi|^{2})\xi_{3}(3\xi_{2}^{2}-|\xi|^{2})3\xi_{1}\xi_{2}\xi_{3} -3\xi_{3}(\xi_{1}^{2}+\xi_{2}^{2})\xi_{2}(3\xi_{3}^{2}-|\xi|^{2})\xi_{1}(3\xi_{3}^{2}-|\xi|^{2}))$$\Psi$,

is equal

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and$\det H\Psi(\xi)=\frac{(\xi\xi_{2}^{2})\xi_{3}}{9}$

.

(4.4)

This shows that the surface
has 3 non-vanishing principal curvatures (non-vanishing$(\xi_{1},\xi_{2})=0$ orcurvature)
GaussianunlessO. Hence, taking a suﬃciently small$c>0$ , we decompose
as follows:$\Sigma_{0}$$\xi_{3}=$$\mathscr{G}_{\pm}(t)$$\mathscr{G}_{\pm}(t)f=\mathscr{G}_{\pm}^{1}(t)f+\mathscr{G}_{\pm}^{2}(t)f+\mathscr{G}_{\pm}^{3}(t)f,$

where$\mathscr{G}_{\pm}^{1}(t)f(x):=\int_{\mathbb{R}^{3}}e^{ix\cdot\xi\pm it^{\xi}}d_{\xi}(1-\psi(|\xi_{h}|/c))(1-\psi(|\xi_{3}|/c))\hat{\Phi}(\xi)\hat{f}(\xi)d\xi,$$\mathscr{G}_{\pm}^{2}(t)f(x):=\int_{\mathbb{R}^{3}}e^{ix\cdot\xi\pm it_{\xi 1}^{\xi}}\#\psi(|\xi_{h}|/c)\hat{\Phi}(\xi)\hat{f}(\xi)d\xi,$$\mathscr{G}_{\pm}^{3}(t)f(x):=\int_{\mathbb{R}^{3}}e^{ix\cdot\xi\pm it_{|\xi|}^{\xi}}\psi(|\xi_{3}|/c)\hat{\Phi}(\xi)\hat{f}(\xi)d\xi\Delta.$) such that $\psi=1$
is a compactly supported smooth function on the half line
. Note that
, 1 , and we denote
on , 1/2 and supp
supp
supp
if is small enough, to say $c<2^{-3}.$
For the proof of Theorem 4.1, let us recall the following abstract Strichartz estimatesby Keel and Tao [12].
Here,$[0, \infty$$\psi$$[0$$\psi\subset[0$$]$$\psi(|\xi_{h}|/c)\psi(|\xi_{3}|/c)\cap$$\xi=(\xi_{h}, \xi_{3})\in \mathbb{R}^{2}\cross \mathbb{R}$$]$$\hat{\Phi}(\xi)=\emptyset$

Theorem 4.3 (Keel-Tao [12]). Let
all

$c$

$\{U(t)\}_{t\in \mathbb{R}}$be a family ofoperators. Suppose thatfor$t,$ $s\in \mathbb{R}$$\Vert U(s)(U(t))^{*}f\Vert_{L^{\infty\sim}}<(1+|t-s|)^{-\sigma}\Vert f\Vert_{L^{1}}$$\Vert U(s)(U(t))^{*}f\Vert_{L^{2}\sim}<\Vert f\Vert_{L^{2}}$

.

,

(4.5)
(4.6)

Then the estimate$\Vert U(t)f\Vert_{L_{t}^{q}L_{x}^{r}\sim}<\Vert f\Vert_{L^{2}}$holds for all $2\leq q,$$r\leq\infty$with $(q, r, \sigma)\neq(2, \infty, 1)$ satisfying$\frac{1}{q}+\frac{\sigma}{r}\leq\frac{\sigma}{2}.$Therefore, for the proof of the suﬃciency part of Theorem 4.1, it suﬃces to show
the dispersive estimates (4.5) since the second property (4.6) follows from the Plancherel

theorem.

Proposition 4.4 ([13]). The dispersive estimates,$\Vert \mathscr{G}_{\pm}^{1}(t)f\Vert_{L\infty\sim}<(1+|t|)^{-\frac{3}{2}}\Vert f\Vert_{L^{1}}$$\Vert \mathscr{G}_{\pm}^{2}(t)f\Vert_{L^{\infty\sim}}<(1+|t|)^{-1}\Vert f\Vert_{L^{1}}$$\Vert \mathscr{G}_{\pm}^{3}(t)f\Vert_{L}\infty\sim<(1+|t|)^{-1}\Vert f\Vert_{L^{1}}$hold for all$t\in \mathbb{R}$and$f\in L^{1}(\mathbb{R}^{3})$

.

,

(4.7)
(4.8)

(4.9)

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For the readers’ convenience, we present the proof of Proposition 4.4. To this end, werecall the following lemma due to Littman [15].

Lemma 4.5 (Littman [15], Stein [18, page 361 Let
be a surface measure on asmooth surface in , and let. Suppose that for all $x\in S$, at least of
the principal curvatures are not zero. Then, it holds$d\mu$$S$$\mathbb{R}^{n}$$k$$\phi\in C_{0}^{\infty}(\mathbb{R}^{n})$$|\overline{\phi d\mu}(\eta)|\leq A|\eta|^{-\frac{k}{2}}.$For the surface , the number of non-vanishing principal curvatures is equal to thenumber of the nonzero eigenvalues, that is, the rank of the symmetric matrix$\Sigma_{0}$$H\Psi.$

Proofof roposition 4 . The desired estimates (4.7), (4.8) and (4.9) follow from the es$.4$$P^{H}$

timates$| \int_{\mathbb{R}^{3}}e^{ix\cdot\xi\pm it^{\xi}}\iota 3(1-\psi(|\xi_{h}|/c))(1-\psi(|\xi_{3}|/c))\hat{\Phi}(\xi)d\xi|\leq C(1+|t|)^{-\frac{3}{2}}$$| \int_{\mathbb{R}^{3}}e^{ix\cdot\xi\pm it_{\xi 1}^{\xi}}\not\in\psi(|\xi_{h}|/c)\hat{\Phi}(\xi)d\xi|\leq C(1+|t|)^{-1}$,,(4.10)(4.11)

and$| \int_{\mathbb{R}^{3}}e^{ix\cdot\xi\pm it^{\xi}}d_{\epsilon}\psi(|\xi_{3}|/c)\hat{\Phi}(\xi)d\xi|\leq C(1+|t|)^{-1}$(4.12)

Since $\det H\Psi\neq 0$ on the support of $(1-\psi(|\xi_{h}|/c))(1-\psi(|\xi_{3}|/c))\hat{\Phi}(\xi)$ by (4.4), thefirst estimates (4.10) follows from the standard stationary phase method or Lemma 4.5.
In order to $show\wedge(4.11)$ and (4.12), we shall use Lemma 4.5. We show that on the support ofandthehas two eigenvalues with their absolutefact,values near to 1. In
the determinant ofis zero if $\xi_{1}=\xi_{2}=0$ or $\xi_{3}=0$ . Hence
it is suﬃcient to consider the cases $\xi_{1}=\xi_{2}=0$ and $\xi_{3}=0$ while $|\xi|\sim 1$ (more precisely, $1/4\leq|\xi|\leq 4$ ), and show that there are two non-zero eigenvalues. Then choosing
suﬃciently small $c>0$ , we obtain the desired estimates by the continuity.
In the case $\xi_{1}=\xi_{2}=0$ , we see that$\psi(|\xi_{h}|/c)\Phi(\xi)$$\psi(|\xi_{3}|/c)\Phi(\xi)$$H\Psi$$H\Psi$$H \Psi(0,0, \xi_{3})=\frac{1}{|\xi_{3}|^{5}}(\begin{array}{lll}-\xi_{3}^{3} 0 00 -\xi_{3}^{3} 00 0 0\end{array})$

and is small,. Thus, two non-zero eigenvalues
are of absolute value near to 1.
In the case $\xi_{3}=0$ , we have thatSince$|\xi|\sim 1$$c$$|\xi_{3}|\sim 1$$-\xi_{3}^{3}|\xi_{3}|^{-5}$$H \Psi(\xi_{1}, \xi_{2},0)=\frac{1}{|\xi_{h}|^{5}}(00 -\xi_{2}|\xi_{h}|^{2}00 -\xi_{2}|\xi_{h}|^{2}-\xi_{1}|\xi_{h}|^{2}0)$$-\xi_{3}^{3}|\xi_{3}|^{-5}$

and

173
A calculation gives$\det(\lambda I-H\Psi(\xi_{1}, \xi_{2},0))=\lambda(\lambda-\frac{1}{|\xi_{h}|^{2}})(\lambda+\frac{1}{|\xi_{h}|^{2}})$Since. Therefore, two non-zero eigenvalues
and is small, we see thatare of absolute value near to 1. This completes the proof of Proposition 4.4.
$|\xi|\sim 1$$|\xi_{h}|\sim 1$$c$$\square$

Proofof the necessity part of Theorem 4.1. We shall show the necessity of the condition(4.3). For $0<\delta\ll 1$ and $N\gg 1$ , let us consider$\mathcal{R}:=\{\xi\in \mathbb{R}^{3}|\xi_{1}, \xi_{2}\in[1/2, 1], \xi_{3}\in[\delta, 2\delta]\},$$A :=\{(t, x)\in \mathbb{R}^{1+3}||x_{1}|, |x_{2}|\leq N^{-1}, |x_{3}|\leq N^{-1}\delta^{-1}, |t|\leq N^{-1}\delta^{-1}\}$, which denotes the characteristic function on
and define by setting
ing suﬃciently small , we see that
and$\hat{f_{\delta}}=\chi_{\mathcal{R}}$$f_{\delta}$$\delta$$\mathcal{R}$. Choos-$\mathcal{R}\subset\{\xi\in \mathbb{R}^{3}|1/2\leq|\xi|\leq 2\}$$|x \cdot\xi\pm t\frac{\xi_{3}}{|\xi|}|\sim^{\frac{1}{N}}<$if$\xi\in \mathcal{R}$and $(t, x)\in A$ . Hence we have$| \mathscr{G}_{\pm}(t)f_{\delta}(x)|\geq|\int_{\mathcal{R}}\cos(x\cdot\xi\pm t\frac{\xi_{3}}{|\xi|})d\xi|\sim>|\mathcal{R}|\sim\delta$

for $(t, x)\in A$ with suﬃciently large $N$ . Therefore (4.2) implies that$\delta^{1-\frac{1}{q}-\frac{1}{r}}<\sim\Vert \mathscr{G}_{\pm}(t)f_{\delta}\Vert_{L_{t}^{q}L_{x}^{r}(A)}\leq\Vert \mathscr{G}_{\pm}(t)f_{\delta}\Vert_{L_{t}^{q}L_{x}^{r}\sim}<\Vert f_{\delta}\Vert_{L^{2}}\sim\delta^{\frac{1}{2}}.$The condition (4.3) follows by letting$\deltaarrow 0.$$\square$

5 Proof of Theorem 1.1
We shall prove that the local solution to
constructed by Theorem 3.1 can beextended to any time interval $[0, T]$ provided the speed of rotation is high enough. To thisend we adapt the ideas in [4, 9].
Let
in the classsatisfy $divu_{0}=0$ , and let be the solution to, where $0<T_{\Omega}<\infty$ denotes the maximal
time of the existence. We define the projection operatorby$u$$u_{0}\in H^{s+1}(\mathbb{R}^{3})$$(E_{\Omega}’)$$(E_{\Omega}’)$$u$$C([O, T_{\Omega});H^{s+1}(\mathbb{R}^{3}))\cap C^{1}([0,T_{\Omega});H^{s}(\mathbb{R}^{3})))$$P_{\pm}$$P_{\pm}v:= \frac{1}{2}(\mathbb{P}v\pm i\frac{D}{|D|}\cross v)$

for$v\in L^{2}(\mathbb{R}^{3})$. Note that

$P_{\pm}\mathbb{P}=P\pm\cdot$

Furthermore, for the divergence-free vector field$v,$$v=P_{+}v+P_{-}v,$$\mathbb{P}e_{3}\cross v=-i\frac{D_{3}}{|D|}(P_{+}v-P_{-}v)$,$P_{\pm}P_{\pm}v=P_{\pm}v,$$P_{\pm}P_{\mp}v=0.$

(5.1)

174
Hence applying$P_{\pm}$

to both sides of$(E_{\Omega}’)$, we have$\{\begin{array}{l}\partial_{t}P_{\pm}u\mp i\Omega\frac{D}{1}A{}_{D|}P_{\pm}u+P_{\pm}(u\cdot\nabla)u=0,P_{\pm}u(O, x)=P_{\pm}u_{0}(x) .\end{array}$

By the Plancherel theorem and the Lebesgue dominated convergence theorem, we see thatonwithis the infinitesimal generator of the semigroup. Therefore by the Duhamel principle wethe domain of the generator
obtain$e^{\pm i\Omega t_{D}^{D}}e$$C_{0}$$A_{\pm}:=\pm i\Omega_{\overline{|}D|}^{D_{\Delta}}$$L^{2}(\mathbb{R}^{3})$$D(A_{\pm})=L^{2}(\mathbb{R}^{3})$$P_{\pm}u(t)=e^{\pm i\Omega t} T^{D}\#_{P_{\pm}u_{0}}-\int_{0}^{t}e^{\pm i\Omega(t-\tau)^{D}}\Theta_{P_{\pm}(u(\tau)\cdot\nabla)u(\tau)d\tau}$

.

(5.2)

Since $s>5/2$ , we can take $\alpha=\alpha(s)\in(0,1)$ so that $s\geq 5/2+\alpha$ . In what follows
we show the-estimate for the solution . By (5.1) it suﬃces to show the estimates, separately. By Theorem 4.1 and scaling, for $2<q<\infty$ there exists aforpositive constant $C=C(q)$ such that$u$

$B_{\infty,\infty}^{1+\alpha}$$P_{+}u,$$P_{-}u$(5.3)$\Vert f\leq C|\Omega|^{-\frac{1}{q}}(2^{j})^{\frac{3}{2}}\Vert\triangle_{i}f\Vert_{L^{2}}$for alland. For the high frequency part of the first term in the righthand side of (5.2), by the Minkowski inequality and (5.3), we see that$j\in \mathbb{Z}$$\Omega\in \mathbb{R}\backslash \{0\}$$\Vert\sup_{j\geq 1}2^{(1+\alpha)j}\Vert\triangle_{j}e^{\pm i\Omega t_{1\#_{P_{\pm}u_{0\Vert_{L^{\infty}}}}}^{D}}\Vert_{L_{t}^{q}(0,\infty)}$$\leq\Vert(\sum_{j\geq 1}2^{2(1+\alpha)j}\Vert\Delta_{j}e^{\pm i\Omega t}T^{D}ff_{P_{\pm}u_{0\Vert_{L\infty}^{2})^{\frac{1}{2}}}}\Vert_{L_{t}^{q}(0,\infty)}$$\leq(\sum_{j\geq 1}2^{2(1+\alpha)j}\Vert\triangle_{j}e^{\pm i\Omega t_{T}^{D}}fi_{P_{\pm}u_{0\Vert_{L^{q}(0,\infty;L^{\infty})}^{2})^{\frac{1}{2}}}}$$\leq C|\Omega|^{-\frac{1}{q}}(\sum_{j\geq 1}2^{2(\frac{5}{2}+\alpha)j}\Vert\triangle_{j}P_{\pm}u_{0}\Vert_{L^{2}}^{2})^{\frac{1}{2}}$Hence, by the Plancherel theorem, we have$\Vert\sup_{j\geq 1}2^{(1+\alpha)j}\Vert\triangle_{j}e^{\pm i\Omega t\frac{D}{1}A}D|P_{\pm}u_{0}\Vert_{L^{\infty}}\Vert_{L_{t}^{q}(0,\infty)}\leq C|\Omega|^{-\frac{1}{q}}\Vert u_{0}\Vert_{\dot{H}^{5}}2^{+\alpha}$

We now handle the low frequency part. Since
and(5.3) that$H^{s+1}(\mathbb{R}^{3})$$u_{0}\in H^{s+1}(\mathbb{R}^{3})$$e^{\pm i\Omega t_{D|}^{D}} \neq P_{\pm}u_{0}=\sum_{j\in \mathbb{Z}^{\triangle_{j^{e^{\pm i\Omega t}T^{D}}}\triangleleft_{D}}}P_{\pm}u_{0}$in, we have$L^{\infty}(\mathbb{R}^{3})$

.

(5.4)

$e^{\pm i\Omega t^{D}}k_{P_{\pm}u_{0}}\in$. Hence it follows from$\Vert\chi*e^{\pm i\Omega t}T^{D}\# P_{\pm}u_{0}\Vert_{Lq(0,\infty;L^{\infty})}\leq C\Vert\sum_{j=-\infty}^{2}\Vert\Delta_{j}e^{\pm i\Omega t_{\uparrow\#}^{D}}P_{\pm}u_{0}\Vert_{L^{\infty}}\Vert_{Lq(0,\infty)}$175$\leq c\sum_{j=-\infty}^{2}\Vert\triangle_{j}e^{\pm i\Omega t_{\overline{|}D|P_{\pm}u_{0\Vert_{Lq(0,\infty;L^{\infty})}}}^{D_{A}}}$$\leq C|\Omega|^{-\frac{1}{q}}\sum_{j=-\infty}^{2}2^{\frac{3}{2}j}\Vert\triangle_{j}u_{0}\Vert_{L^{2}}$$\leq C|\Omega|^{-\frac{1}{q}}(\sum_{j=-\infty}^{2}2^{3j})^{\frac{1}{2}}(\sum_{j=-\infty}^{2}\Vert\triangle_{j}u_{0}\Vert_{L^{2}}^{2})^{\frac{1}{2}}$Hence we have

.

(5.5)

$\Vert e^{\pm i\Omega t_{D}^{D}}rightarrow P_{\pm}u_{0\Vert_{Lq(0,\infty;B_{\infty,\infty}^{1+\alpha})}}\leq C|\Omega|^{-\frac{1}{q}}\Vert u_{0}\Vert_{H^{5}}z+\alpha$(5.6)$\Vert*\#\infty\leq C|\Omega|^{-\frac{1}{q}}\Vert u_{0}\Vert_{L^{2}}$Combining (5.4) and (5.5), by (2.1) we obtainfor some constant $C=C(q, \alpha)>$ O. Next we consider the nonlinear term in (5.2). It
follows from the Minkowski inequality and (5.3) that$\Vert\triangle_{j}\int_{0}^{t}e^{\pm i\Omega(t-\tau)_{D}^{D}}rightarrow P_{\pm}(u(\tau)\cdot\nabla)u(\tau)d\tau\Vert_{L^{q}(0,T;L^{\infty})}$$\leq\int_{0}^{T}\triangle_{j}\#_{P_{\pm}(u(\tau)\cdot\nabla)u(\tau)\Vert_{L_{t}^{q}(\tau,T;L^{\infty})}d\tau}$(5.7)$\leq C|\Omega|^{-\frac{1}{q}}(2^{j})^{\frac{3}{2}}\int_{0}^{T}\Vert\Delta_{j}(u(\tau)\cdot\nabla)u(\tau)\Vert_{L^{2}}d\tau$for all $0<T<T_{\Omega}$ . Hence for the high frequency part, by the Minkowski inequality andthe above inequality we see that$\Vert\sup_{j\geq 1}2^{(1+\alpha)j}\Vert\triangle_{j}\int_{0}^{t}e^{\pm i\Omega(t-\tau)}$di$P_{\pm}(u(\tau)\cdot\nabla)u(\tau)d\tau\Vert_{L^{\infty}}\Vert_{L^{q}(0,T)}$$\leq\Vert(\sum_{j\geq 1}2^{2(1+\alpha)j}\Vert\triangle_{j}\int_{0}^{t}e^{\pm i\Omega(t-\tau)^{D}3_{P_{\pm}(u(\tau)\cdot\nabla)u(\tau)d_{\mathcal{T}\Vert_{L^{\infty}}^{2})^{\frac{1}{2}}\Vert_{L^{q}(0,T)}}}}\Gamma D$$\leq(\sum_{j\geq 1}2^{2(1+\alpha)j}\Vert\triangle_{j}\int_{0}^{t}e^{\pm i\Omega(t-\tau)_{D}^{D}}e_{P_{\pm}(u(\tau)\cdot\nabla)u(\tau)d\tau\Vert_{Lq(0,T;L^{\infty})}^{2})^{\frac{1}{2}}}$$\leq C|\Omega|^{-\frac{1}{q}}\{\sum_{j\geq 1}2^{2(\frac{5}{2}+\alpha)j}(\int_{0}^{T}\Vert\triangle_{j}(u(\tau)\cdot\nabla)u(\tau)\Vert_{L^{2}}d\tau)^{2}\}^{\frac{1}{2}}$$\leq C|\Omega|^{-\frac{1}{q}}\int_{0}^{T}(\sum_{j\geq 1}2^{2(\frac{5}{2}+\alpha)j}\Vert\triangle_{j}(u(\tau)\cdot\nabla)u(\tau)\Vert_{L^{2}}^{2})^{\frac{1}{2}}d\tau.$

176
Therefore, we obtain$\Vert\sup_{j\geq 1}2^{(1+\alpha)j}\Vert\triangle_{j}\int_{0^{e^{\pm i\Omega(t-\tau)}}}^{t}\mathfrak{s}^{D}4_{P_{\pm}(u(\tau)\cdot\nabla)\dot{u}(\tau)d\tau\Vert_{L^{\infty}}}\Vert_{Lq(0,T)}$(5.8)$\leq C|\Omega|^{-\frac{1}{q}}\int_{0}^{T}5.$

For the low frequency part, we repeat the line of argument which was used to show (5.5).
In fact, by the Minkowski inequality and (5.7) we see that$\Vert\chi*\int_{0}^{t}e^{\pm i\Omega(t-\tau)_{D}^{D}}*P_{\pm}(u(\tau)\cdot\nabla)u(\tau)d\tau\Vert_{Lq(0,T;L}\infty)$$\leq C\sum_{j=-\infty}^{2}\Vert\triangle_{j}\int_{0^{e^{\pm i\Omega(t-\tau)}}}^{t}\mathfrak{s}^{D}fl_{P_{\pm}(u(\tau)\cdot\nabla)u(\tau)d\tau\Vert_{L^{q}(0,T,\cdot L^{\infty})}}$$\leq C|\Omega|^{-\frac{1}{q}}\sum_{j=-\infty}^{2}(2^{j})^{\frac{3}{2}}\int_{0}^{T}\Vert(u(\tau)\cdot\nabla)u(\tau)\Vert_{L^{2}}d\tau.$

Then, by the Schwarz inequality as before, we have$\Vert\chi*\int^{t}0^{e^{\pm i\Omega(t-\tau)_{D}^{D}}}e_{P_{\pm}(u(\tau)\cdot\nabla)u(\tau)d\tau\Vert_{Lq(0,\tau;L^{\infty})}}$$\leq C|\Omega|^{-\frac{1}{q}}\int_{0}^{T}(\sum_{j=-\infty}^{2}\Vert\triangle_{j}(u(\tau)\cdot\nabla)u(\tau)\Vert_{L^{2}}^{2})^{\frac{1}{2}}d\tau$$\leq C|\Omega|^{-\frac{1}{q}}\int_{0}^{T}\Vert(u(\tau)\cdot\nabla)u(\tau)\Vert_{L^{2}}d\tau.$

From this and (5.8), we see that$\Vert\int_{0}^{t}e^{\pm i\Omega(t-\tau)}$Pi$P_{\pm}(u(\tau)\cdot\nabla)u(\tau)d\tau\Vert_{L^{q}(0,T;B_{\infty,\infty}^{1+\alpha})}$$\leq C|\Omega|^{-\frac{1}{q}}\int_{0}^{T}5+\alpha$$\leq C|\Omega|^{-\frac{1}{q}}\int_{0}^{T}\Vert(u(\tau)\cdot\nabla)u(\tau)\Vert_{H}\S+\alpha d\tau$for all $0<T<T_{\Omega}$ with some constant $C=C(q, \alpha)>$ O. Therefore, by combining
this and (5.6) and by recalling (5.2) and (5.1), we see that for $2<q<\infty$ there exists a
positive constant $C=C(q, \alpha)$ such that$\Vert u\Vert_{L^{q}(0,T;B_{\infty,\infty}^{1+\alpha})}\leq C|\Omega|^{-\frac{1}{q}}(\Vert u_{0}\Vert_{H^{5+\alpha}}2+\int_{0}^{T}\Vert(u(\tau)\cdot\nabla)u(\tau)\Vert_{H^{5}}z+\alpha d\tau)$

(5.9)

177
for all $0<T<T_{\Omega}.$Now, let us define$V(t) := \int_{0}^{t}\Vert\nabla u(\tau)\Vert_{L^{\infty}}d\tau, 0<t<T_{\Omega}.$

Since $s\geq 5/2+\alpha$ , by applying the continuous embeddings
H\"older inequality, (5.9) and (3.4), successively, we see that

$B_{\infty,\infty}^{1+\alpha}(\mathbb{R}^{3})\mapsto C^{1}(\mathbb{R}^{3})$, the$V(t) \leq C\int_{0}^{t}\Vert u(\tau)\Vert_{B_{\infty,\infty}^{1+\alpha}}d\tau\leq Ct^{1-\frac{1}{q}}\Vert u\Vert_{Lq(0,t;B_{\infty_{)}\infty}^{1+\alpha})}$$\leq Ct^{1-\frac{1}{q}}|\Omega|^{-\frac{1}{q}}(\Vert u_{0}\Vert_{H^{s}}+\int_{0}^{t}\Vert(u(\tau)\cdot\nabla)u(\tau)\Vert_{H^{s}}d\tau)$$\leq Ct^{1-\frac{1}{q}}|\Omega|^{-\frac{1}{q}}(\Vert u_{0}\Vert_{H^{s+1}}+\int_{0}^{t}\Vert u(\tau)\Vert_{H^{s+1}}^{2}d\tau)$$\leq Ct^{1-\frac{1}{q}}|\Omega|^{-\frac{1}{q}}(\Vert u_{0}\Vert_{H^{s+1}}+\Vert u_{0}\Vert_{H^{s+1}}^{2}\int_{0}^{t}e^{CV(\tau)}d\tau)$Hence, there exist positive constants $C_{1}=C_{1}(q, s)$ and $C_{2}=C_{2}(s)$ such that
$V(t)\leq C_{1}t^{1-\frac{1}{q}}|\Omega|^{-\frac{1}{q}}\Vert u_{0}\Vert_{H^{s+1}}(1+\Vert u_{0}\Vert_{H^{s+1}}te^{C_{2}V(t)})$(5.10)

for all $0<t<T_{\Omega}$ . Now, for given $0<T<\infty$ , let us define$X_{T,\Omega}:=\{t\in[0, T]\cap[O, T_{\Omega})|V(t)\leq C_{1}T^{1-\frac{1}{q}}\Vert u_{0}\Vert_{H^{s+1}}\},$We shall prove that
Then we can take so that
$V(t)$ is uniformly continuous on$\tilde{T}_{\Omega}=\min\{T, T_{\Omega}\}$$\tilde{T}$by contradiction. Assume that
. Since, and it holds that$\tilde{T}_{\Omega}<\tilde{T}<\min\{T, T_{\Omega}\}$$[0, \tilde{T}]$$V(\tilde{T}_{\Omega})\leq C_{1}T^{1-\frac{1}{q}}\Vert u_{0}\Vert_{H^{s+1}}$

Take a suﬃciently large$\Omega\in \mathbb{R}\backslash \{O\}$$\tilde{T}_{\Omega}:=\sup X_{T,\Omega}.$$\tilde{T}_{\Omega}<\min\{T, T_{\Omega}\}.$$u\in C([0,\tilde{T}];H^{s+1}(\mathbb{R}^{3}))$.

$\tilde{T}_{\Omega}<T$(5.11)

so that$|\Omega|^{\frac{1}{q}}\geq 2(1+\Vert u_{0}\Vert_{H^{s+1}}Te^{C_{1}C_{2}T^{1-\frac{1}{q}}\Vert u_{0}\Vert_{H^{s+1}}})$

Then, since,, by (5.10), (5.11) and (5.12) we have

$V(\tilde{T}_{\Omega})\leq C_{1}(\tilde{T}_{\Omega})^{1-\frac{1}{q}}|\Omega|^{-\frac{1}{q}}\Vert u_{0}\Vert_{H^{s+1}}(1+\Vert u_{0}\Vert_{H^{s+1}}\tilde{T}_{\Omega}e^{C_{2}V(\tilde{T}_{\Omega})})$$\leq C_{1}T^{1-\frac{1}{q}}\Vert u_{0}\Vert_{H^{s+1}}|\Omega|^{-\frac{1}{q}}(1+\Vert u_{0}\Vert_{H^{s+1}}Te^{C_{1}C_{2}T^{1-\frac{1}{q}}\Vert uo\Vert_{H^{s+1}}})$$\leq\frac{1}{2}C_{1}T^{1-\frac{1}{q}}\Vert u_{0}\Vert_{H^{s+1}}.$(5.12)

178, whichandHence there exists such thatprovided the Coriolis
contradicts the definition of . Thus, we haveparameters satisfy (5.12).
, we have. Therefore, it holds that
If
$V(S)\leq C_{1}T^{1-\frac{1}{q}}\Vert u_{0}\Vert_{H^{s+1}}$$\tilde{T}_{\Omega}<S<\tilde{T}$$S$$\tilde{T}_{\Omega}$$\tilde{T}_{\Omega}=\min\{T, T_{\Omega}\}$$\Omega$$T_{\Omega}= \tilde{T}_{\Omega}=\sup X_{T,\Omega}$$T_{\Omega}<T$$V(t)= \backslash \int_{0}^{t}\Vert\nabla u(\tau)\Vert_{L}\infty d\tau\leq C_{1}T^{1-\frac{1}{q}}\Vert u_{0}\Vert_{H^{s+1}}<\infty$

for all $0\leq t<T_{\Omega}$ . By Corollary 3.3, this contradicts the maximality of . Hence we
if is as large as in (5.12). This completes the proof of Theorem 1.1.
obtain that
$T_{\Omega}$$T_{\Omega}\geq T$$\Omega$Acknowledgements This work was supported by JSPS Grant-in-Aid for Research Activity Start-up Grant Number 25887005.

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