数理解析研究所講究録
1197 巻 2001 年 54-59

54

Global existence of smooth solutions
for two dimensional Navier-Stokes equations
with nondecaying initial velocity
北大院儀我美– (Yoshikazu Giga)
Department of Mathematics, Hokkaido University
道情報大松井伸也 (Shin’ya Matsui)

Hokkaido Information University
北大院沢田宙広 (Okihiro Sawada)
Department of Mathematics, Hokkaido University

1Introduction

We consider the nonstationary Navier-Stokes equations in the plane.

$(\mathrm{N}\mathrm{S})$where $u=u(X, t)=(u^{1}(X, t),$ $u^{2}(x, t))$ and $p=p(x, t)$ stand for the unknownvelocity vector field of the fluid and unknown scaler of its pressure, while$u_{0}=u_{0}(x)=(u_{0}^{1}(x), u(\mathrm{o}x)2)$ is a given initial velocity vector field.

Our goal is to prove the unique existance of-time smooth solution ofwhen initial velocity
belongs to merely bounded uniformly
continious, i.e., $u_{0}\in BUC=BUC(\mathbb{R}^{2})$ .
$\mathrm{g}\mathrm{l}\mathrm{o}\mathrm{b}\mathrm{a}1_{-}\mathrm{i}\mathrm{n}$$(\mathrm{N}\mathrm{S})$$u_{0}$

Theorem 1 Assume that $u_{0}\in BUC$ satisfies(in the sensein$u\inC([0,\infty);BUc)$distribution).
Thenthereexists
such
that
of
$\mathrm{d}\mathrm{i}\mathrm{v}u_{0}=0$$\mathbb{R}^{2}$$u(\mathrm{O})=u_{0}$55

and $(u(t), \nabla p(t))$ is a unique classical solution of $(NS)$ $fort>0$ , provided that$\nabla p(t)=\Sigma_{i,j=1}^{2}\nabla RiRju^{i}(t)u(jt)$ , whereis the Riesz$R_{j}=(-\triangle)^{-1/}2\partial/\partial_{X_{j}}$transform.
inWe may note that we do not impose any smallness assumptions onTheorem 1.
We consider that belongs to $BUC$ . In this case the initial velocity does
can be taken such as a constant
not decay at space infinity. For example
or a spatially periodic function.
$u_{0}$$u_{0}$$u_{0}$

There is a large literature on local solvability of Navier-Stokes equations
even in a various domain of. In particular Leray [Le] has already. The method of
obtained the time global smooth solutions if
his proof is based on the energy estimate. The kinematic energy is defined. This method does not apply directly to our situation because
bythe energy is infinite.
$\mathbb{R}^{n}(n\geq 2)$$u_{0}\in L^{2}(\mathbb{R}^{2})$$||u||_{L^{2}}^{2}/2$

On the other hand the time local solution in our situation is constructedby Cannon-Knightly [CK] in 1970, Cannone [Ca] in 1995, and Giga-InuiMatsui [GIM] in 1999. They show the time local solvability including higher
dimensional problems.
The relation
does not follow from the equations sincemay not decay at space infinity. Recently work of Jun .K ato [Ka] shows asufficient condition on to get this relation.
$\nabla p=\sum\nabla R_{i}R_{j}uu^{j}i$

$u$

$p$

Remark 1 (Jun Kato) Assume that the initial data $u_{0}\in BUC(\mathbb{R}^{n})$ sat. Let $(u,p)$ is classical solutions withisfying, where $BMO$ denotes the space of bounded
andmean oscillation functions. Then $(u(t), \nabla p(t))$ is a ique for $t>0$ , and therelation$u\in L^{\infty}((\mathrm{O}, T)\cross \mathbb{R}^{n})$$\mathrm{d}\mathrm{i}\mathrm{v}u_{0}=0$$p\in L_{loc}^{1}((\mathrm{o}, T);BMo(\mathbb{R}^{n}))$u.n$\nabla p(t)=\sum_{i,j=1}^{n}\nabla RiRju^{i}(t)u(jt)$holds.

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2Sketch of the proof of Theoreml
Let us briefly explain main ideas of proving Theorem 1. We use the integral
equation;(INT),$u(t)=e^{t\triangle}u_{0}- \int_{0}^{t}\nabla\cdot e^{(ts)\triangle_{\mathrm{p}}}-(u\otimes u)(s)d_{S}$where the matrix operator $\mathrm{P}=(P_{ij})_{i,j1,2}=’ P_{ij}=\delta_{ij}+R_{i}R_{j}$ . We denote
thatis Kronecker’s delta, andis formaly the orthogonal projector on
divergence-hee subspace.
is the solution operator of heat equation. We
call the solution of (INT) the mild solution.
$\mathrm{P}$$\delta_{ij}$$e^{t\Delta}$

In the literature [CK] and [GIM], the local solution $u\in C.([0, \tau_{0}];BUc)$ ,thisis estimates by,$T_{0}$$T_{0}\geq C/||u_{0}||_{\infty}^{2}$

where

$C$

is a numerical constant.

The main idea is to establish a priori bound forit, the local solution can be extended globally.

$||u(t)||_{\infty}$. Once we obtainTheorem 2 Assume that $u_{0}\in BUC$ , and assume that is the mild solution
in $[0, T]$ . Then there exists a positive constant $K$ independent of and ,such that$||u(t)||_{\infty}\leq K\exp(Ke^{Kt})$
for $t\in[0, T]$ .
$u$

$T$

It is easy to see that Theorem 2 implies Theorem 1.

We give an outline of the proof of Theorem 2. It consists of 3 steps.
(i) Maximum principle for vorticity equation.
(ii) Estimate of bilinear terms.

(iii) Logarithmic type Gronwall inequality.

(Step i) We take rotation to$(\mathrm{N}\mathrm{S})$to get$\omega_{t}-\triangle\omega+(u, \nabla)\omega=0$,$u$

57, which is a scalar function.
whereSince and are bounded, we can apply the maximum principle for thevorticity equation. We obtain the following inequality:$\omega(t)=\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}u(t)$$\omega$$u$$||\omega(t)||_{\infty}\leq||\omega_{0}||_{\infty}$for$t\in[0, T]$,where.
It is well-known result that there is a regularizing effect, so that for all$t_{0}\in(0, T),$ $\nabla u(t_{0})\in BUC$ . Thus we may assume that thisis an initialtime, so thatis finite.
$\omega_{0}=\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}u_{0}$$t_{0}$$||\omega_{0}||_{\infty}$

We note that in the case of the boundary exists the above estimate is
not expected since the vorticity is created near the boundary. In higher dimensional cases without boundary it is not expected to have similar vorticity
estimate because of vorticity stretching terms.

(Step ii) It is summarized in the following lemma:
Lemma 1 There exists a numerical positive constant$C$such that,$|| \nabla e^{t\triangle}(f\otimes f)||\infty\leq C\{(1+\frac{1}{\sqrt{t}}+\log R)||f||\infty||\mathrm{c}\mathrm{u}\mathrm{r}1f||\infty+\frac{1}{R}||f||_{\infty}^{2}\}$for all $t>0,$ $R>1$ , and$f\in C^{1}(\mathbb{R}^{2});\mathrm{d}\mathrm{i}_{\mathrm{V}}f=0$.

We refer to the similar estimate holds in higher dimensional cases.
The proof of this lemma is not difficult, but not short. We estimate theRiesz transform by using duality, but we skip the detail.

(Step iii) This step is summarized by the logarithmic type Gronwall
inequality:

Lemma 2 Let nonnegative function $a(t, s)$ be continious in $\{(t, s);0\leq s<$, and satisfies $a(t, \cdot)\in L^{1}(0, t)$ for $t\in(0, T]$ , with some $T>0$ .
Assume that there exists a positive constantand a constant $A\in(0,1)$
such that$t\leq T\}$$\epsilon_{0}$$\sup_{0\leq t\leq T}\int_{t-\epsilon}^{t}0a(t, s)ds\leq 1-A$.

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If positive constants$\alpha,$$\beta>0$ ,$C([0, T])$ satisfy thatand nonnegative continious function $f\in$,$f(t) \leq\alpha+\int_{0}^{t}a(t, S)f(s)ds+\beta\int_{0}^{t}\{1+\log(1+f(S))\}f(S)d_{S}$for $t\in[0, T]$ . Then the following inequality holds,$\cdot$$f(t) \leq-1+\frac{1}{e}[(1+\frac{\alpha}{A})e]^{\mathrm{e}}\mathrm{x}\mathrm{p}(^{\underline{\beta}+}At\Delta)$

for $t\in[0, T]$ , where the constant$\gamma$,is defined by$\gamma=\sup_{0\leq t\leq\tau}\{0\leq s\leq t-\sup_{0\epsilon}a(t, S)\}$.

The Gronwall inequality with the logarithmic terms is shown by Wolibner[Wo] in 1933 and Brezis-Gallouet [BG] in 1980. But in our case there is asingular term $a(t, s)$ , so ours is quite different from thiers.

Finally we estimate$L^{\infty}$-norm of the mild solution explicitly.

$||u(t)||_{\infty} \leq||e^{t\Delta}u_{0}||_{\infty}+\int_{0}^{t}||\nabla e^{(t}-s)\triangle \mathrm{P}(u\otimes u)(S)||_{\infty^{d}}s$

We use the estimate$||e^{t\triangle_{u_{0}}}||_{\infty}\leq||u_{0}||_{\infty}$.

, and by Lemma 1 to obtain,$||u(t)||_{\infty} \leq||u_{0}||\infty+\int_{0}^{t}c\{(1+\frac{1}{\sqrt{t-s}}+\log R)||u(s)||_{\infty}||\omega(s)||\infty+\frac{1}{R}||u(s)||^{2}\infty\}ds$

with some positive constant $C$ . We now take $R=1+||u(S)||_{\infty}$ ; this choiceof $R$ is similar to that of [BG]. By maximum principle for vorticity equation,
we have.

$||u(t)||_{\infty} \leq||u_{0}||_{\infty}+C(1+||\omega 0||_{\infty})\int_{0}^{t}\{1+\frac{1}{\sqrt{t-s}}+\log(1+||u(S)||\infty)\}||u(S)||\infty^{dS}$

We apply the logarithmic type Gronwall inequality (Lemma 2) to obtain
Theorem 2.

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References[BG] BREZIS, H. and GALLOUET, T. : Nonlinear Schr\"odinger evolutionequations, J. Nonlinear Anal. 4, (1980) 677-681.
[CK] CANNON, J. R. and KNIGHTLY, G. H. : A note on the Cauchy problem for the Navier-Stokes equations, SIAM J. Appl. Math. 18, (1984)

641-644.

[Ca] CANNONE, M. : Paraproduits et Navier-Stokes Diderot Editeur, Artset Sciences aris-Newam (1995).
$\acute{\mathrm{P}}$$\mathrm{Y}\mathrm{o}\mathrm{r}\mathrm{k}-\mathrm{A}\mathrm{m}\mathrm{S}\mathrm{t}\mathrm{e}\mathrm{r}_{\mathrm{v}}\mathrm{d}$[GIM] GIGA, Y. , INUI, K. and MATSUI, S. : On the Cauchy problem
for the Navier-Stokes equations with nondecaying initial data, Quaderni diMatematica, 4, (1999) 28-68.

[Ka] KATO, J. and: On the uniqueness of the nondecaying solution of theNavier-Stokes equations, (preprint)[Le] LERAY, J. : Sur le mouvement d’un liquide visquex emplissant l’espace,
Acta. Math. 63, (1934) 193-248.
[Wo] WOLIBNER, W. : Un theor\‘eme sur l’existence du mouvement plan d’unfluide parfait, homog\‘ene, incompreeible, pendant un temps infiniment long,
Math. Z. 37, (1933) 698-726.