数理解析研究所講究録 1330 巻 2003 年 1-91

Remarks on the Perturbed Euler equations
Dongho Chae
Department of Mathematics
Seoul National University
Seoul 151-742, Korea
e-mail:dhchae@math.snu.ac.kr
Abstract

We consider two type of perturbations of the Euler equations forinviscid incompressible fluid flows in Rn, $n\geq 2$ . We present global-posedness result of these perturbed Euler system in the TriebelLizorkin spaces for intial vorticity which is smal in the critical TiebelLizorkin norms. Comparison type of theorems are obtained between
the Euler system and its perturbations.
$\mathrm{w}\mathrm{e}\mathrm{U}$1

Introduction and Main Results

We axe concerned with the perturbations of the following Euler equations forthe homogeneous incompressible fluid flows.
$\frac{\partial v}{\partial t}+(v\cdot\nabla)v=-\nabla p$

(E)

$\{$

,
$v(x,0)=v_{0}(x)$ ,$\mathrm{d}\mathrm{i}\mathrm{v}v=0$,$(x,t)\in \mathrm{R}^{n}\mathrm{x}(0, \infty)$$(x,t)\in \mathrm{R}^{n}\mathrm{x}$$(0, \infty)$$x\in \mathrm{R}^{l1}$where $v=(v_{1}, \cdots,v_{n})$ , $v_{j}=v_{j}(x, t)$ , $j=1$ , $\cdots,n$ , is the velocity of theflow, $p=p(x, t)$ is the scalar pressure, andis the given initial velocity,
satisfying. The local well-posedness of solution is established bymany authors in various function spaces[14, 15, 16, 7, 21, 22, 3, 4, 5]. The
question of finite(or infinite) time blow-up of such local regular solution of (E)is an outstanding open problem in the mathematical fluid mechanics. One$v_{0}$$\mathrm{d}\mathrm{i}\mathrm{v}v_{0}=0$

2
of the most significant achievements in this direction is the celebrated Beale
KataMajda(BKM) criterion for the blow-up of solutions [2], which states$\lim\sup_{t\nearrow T_{\star}}||v(t)||_{H^{m}}=\infty$

if and only if$\int_{0}^{T*}||\omega(s)||_{L^{\infty}}ds=\mathrm{o}\mathrm{o}$

,

where
is the vorticity of the flows. Bahouri and Dehman also
obtained similar blow-up criterion in the Holder space[l]. Recently the BKM
criterion has been refined by Kozono and Taniuch[17], replacing thenorm of vorticity by the $BMO$ norm, and by the author of this paper[3],replacing thenorm of vorticity bynorm and the Sobolev normby the Triebel-Lizorkin norm
respectively. We note here
, and
that
. We also mention that
there is ageometric type of blow-up criterion, using deep structure of thenonlinear term of the Euler equation[10].
In this paper we study the well-posednes/blow-up problems for perturbationsof the Euler equations, which are supposed to closer to the original Euler
system than the usual Navier-Stokes perturbation. In order to optimize theresults we use the TtLebel-Lizorkin spaces.
Our first perturbation of (E) is the following:$\omega$$=\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}v$$L^{\infty}$$L^{\infty}$$\dot{F}_{\infty,\infty}^{0}$$||u(t)||_{H^{m}}$$||u(t)||_{F_{\mathrm{p},q}^{e}}$$L^{\infty}arrow BMOrightarrow\dot{F}_{\infty,\infty}^{0}$$H^{m}(\mathrm{R}^{n})=F_{2,2}^{m}$$\frac{\partial u}{\partial t}+a(t)(u\cdot\nabla)u=-\nabla q$,$(x, t)\in \mathrm{R}^{n}\mathrm{x}(0, \infty)$$\mathrm{d}\mathrm{i}\mathrm{v}u=0$,$(x,t)\in \mathrm{R}^{n}\mathrm{x}(0, \infty)$$u(x,0)=u_{0}(x)$,$x\in \mathrm{R}^{n}$$(\mathrm{a}\mathrm{E})\{$

where $u(x,t)$ , $q=q(x,t)$ are similar to the above, andis agiven initial
vector field satisfying. $a(t)>0$ is agiven continuous real valued
function on. If we set $a(t)\equiv 1$ , then the system
reduces tothe well-known Euler equations for homogeneous incompressible fluid flows.
Below we will impose the condition that
. We observe that
if we choose e.g. $a(t)=1$ for $t\in[0, t_{0}]$ , and,forthen the systemcoincides with (E) during the time interval, anddistorts from (E) after that. For the system
we have the following small
data global existence result. For an introduction to the function spaces weuse below, we summarized basic facts about about the Triebel-Lizorkin and
the Besov spaces in the Appendix. The detailed proofs of the results below
are in [6].
$u_{0}$$\mathrm{d}\mathrm{i}\mathrm{v}u_{0}=0$$[0, \infty)$$(\mathrm{a}\mathrm{E})$$a(\cdot)\in L^{1}(0, \infty)$$a(t)= \frac{1}{1+(t-t_{0})^{2}}$$t\in(t_{0}, \infty)$$[0, t_{0}]$$(\mathrm{a}\mathrm{E})$$(\mathrm{a}\mathrm{E})$

Theorem 1.1 Let $s>n/p$ , with $(p, q)\in[1, \infty]^{2}$ , or $s=n$ with $p=1$ ,$q\in[0, \infty]$ .
Suppose $a(\cdot)\in L^{1}(0, \infty)$ . There exists an absolute constant$C_{0}>0$ such that
if initial vorticity
satisfies$\omega_{0}\in F_{p,q}^{\theta}$$|| \omega_{0}||_{p_{\infty,1}<}.(C_{0}\int_{0}^{\infty}a(t)dt)^{-1}$,3
of (aE) exists. Moreover,then a global unique solution u
the solution satisfies the estimate$\in \mathrm{C}([0, \infty);F_{\mathrm{p},q}^{s+1})$$\sup_{0\leq t<\infty}||\omega(t)||_{F_{\mathrm{p},q}^{\epsilon}}\leq||\omega_{0}||_{F_{\mathrm{p}.q}^{\epsilon}}\exp(\frac{C_{0}\int_{0}^{\infty}a(t)dt||\omega_{0}||_{\dot{B}_{\infty,1}^{0}}}{1-C_{0}\int_{0}^{\infty}a(t)dt||\omega_{0}||_{B_{\infty,1}^{\mathrm{O}}}|})$

.

(1.1)

is the usual ffactional order Sobolev
Remark 1.1 Since
space, Theorem 1.1 implies immediately the global well-posedness of
sufficiently small.
withfor initial datainnorm of vorticity.
We emphasize here that we need smallness only for(see Lemma 2.1 below), it would be
In view of the embedding
interesting to extend the above result to the case with smallness assumption.
on$W^{s,p}(\mathrm{R}^{n})=F_{p,2}^{s}$$(\mathrm{a}\mathrm{E})$$u_{0}\in W^{\epsilon,p}(\mathrm{R}^{n})$$W^{s\mathrm{p}}(\mathrm{R}^{n})$$||\omega_{0}||_{\dot{B}_{\infty.1}^{0}}$$\dot{F}_{\infty,1}^{0}$$\dot{F}_{\infty,1}^{0}arrow L^{\infty}$$||\omega_{0}||_{L^{\infty}}$

The following theorem states the equivalence of local existence of the Euler system with the global existence of the perturbed system with suitable
modification of initial data.

Theorem 1.2 The solution
blows up at $t=T_{*}<\infty$ in$v^{E}$$v_{0}^{E}$of the Eider system (E) with the initial data

$F_{p,q}^{s}$, namely$\lim\sup_{tarrow T_{*}}||v^{E}(t)||_{F_{\mathrm{p}_{1}q}^{l}}=\infty$

,

(1.2)

if and only if for solution of (E) associated with the initial data$u$$u_{0}(x)= \frac{T_{*}}{\int_{0}^{\infty}a(s)ds}v_{0}^{E}(x)$we have(1.3)$\int_{0}^{\infty}||\omega(t)||_{||_{B_{\infty.1}^{0}}}a(t)dt=\mathrm{o}\mathrm{o}$

for

$s$

$>n/p+1$ , $(p,q)$$\in[1, \infty]^{2}$, while(1.4)$\int_{0}^{\infty}||\omega(t)||_{\dot{B}_{\infty,1}^{0}}a(t)dt=\mathrm{o}\mathrm{o}$for $s=n+1$ , $p=1$ ,$q\in[1, \infty]$

respectively.

by
Remark 1.2 As in Remark 1.1 we can replace
the norm,
replacecan,we(1.2). Also, since, orin (1.3).
by$||v^{E}(t)||_{F_{\mathrm{p},q}^{\epsilon}}$$L^{\infty}arrow BMO\epsilonarrow||_{\dot{B}_{\infty,1}^{0}}$$||\omega(t)||_{BMO}$$||v^{E}(t)||w^{\epsilon,\mathrm{p}}(\mathrm{B}^{n})$in$||\omega(t)||_{bdn}$$||\omega(t)||_{L^{\infty}}$

Remark 1.3 By following exactly the same procedure as in [3] and [4] it is4easy to find that the following blow-up criterion holds for the system (aE):The solution $u(t)$ of the system (aE) blows up at t $=T_{*}<\infty$ in$\lim\sup_{tarrow T_{*}}||u(t)||_{F_{\mathrm{p},q}^{\epsilon}}=\infty$$F_{p,q}^{s}$,, namely(1.5)

if and only if(1.6)$\int_{0}^{T_{*}}||\omega(t)||_{||_{B_{\infty,1}^{0}}}a(t)dt=\infty$

for $s>n/p+1$ , $(p, q)\in[1, \infty]^{2}$ , while(1.7)$\int_{0}^{T_{*}}||\omega(t)||_{\dot{B}_{\infty.1}^{0}}a(t)dt=\infty$

for $s=n+1$ , $p=1$ , $q\in[1,\infty]$ respectively. Thus, the conditions (1.3)
and (1.4), in turn, are equivalent to the blow-up of solution $u(t)$ ofat
infinite time, namely.
(1.8)$(\mathrm{a}\mathrm{E})$$\lim\sup_{tarrow\infty}||u(t)||_{F_{\dot{\mathrm{p},}q}}=\infty$

Next, we consider the following ‘damping’ perturbation of the Euler equations:$(\mathrm{E})_{\epsilon}\{\begin{array}{l}\frac{\partial u}{\partial t}+(u\cdot\nabla)u=-\nabla q-\epsilon u,(x,t)\in \mathbb{R}^{n}\mathrm{x}(0,\infty)\mathrm{d}\mathrm{i}\mathrm{v}u=0,(x,t)\in \mathrm{R}^{n}\mathrm{x}(0,\infty)u(x,0)=u_{0}(x),x\in \mathrm{R}^{n}\end{array}$with, which could be considered as a‘milder’ perturbation of the Euler
system than the usual Navier-Stokes system. We will see below that thesystemcan be treated as aspecial case of. Applying Theorem 1.1
and 1.2, we establish the following two results regarding.
$\epsilon>0$$(\mathrm{E})_{\epsilon}$$(\mathrm{a}\mathrm{E})$$(\mathrm{E})_{\epsilon}$$(p, q)\in[1, \infty]^{2}$ , orCorollary 1.1 Let $s>n/p$ ,$s=n$ will $p=1$ ,$q\in[0, \infty]$ .
Therean absolute constant $C_{1}>0$ such that if initialvorticityand the ‘niscosit$y’$ satisfies$wid\iota$$e\dot{m}b$$\omega_{0}\in F_{\mathrm{p},q}^{s}$$\epsilon$$|| \omega_{0}||_{\dot{B}_{\infty,1}^{\mathrm{O}}}<\frac{\epsilon}{C_{1}}$then global unique solution
solution satisfies the estimate,$u\in C([0, \infty);F_{\mathrm{p},q}^{\epsilon+1})$

of$(E)_{\epsilon}$

exists. Moreover the$\sup_{0\leq t<\infty}||\omega(t)||_{F_{\mathrm{p},q}^{\epsilon}}\leq||\omega_{0}||_{F_{\dot{\mathrm{p},}q}}\exp(\frac{C_{1}||\omega_{0}||_{\dot{B}_{\infty.1}^{\mathrm{O}}}}{\epsilon-C_{1}||\omega_{0}||_{\dot{B}_{\infty,1}^{0}}})$

.

(1.9)

5
Similar remark to Remark 1.1, concerning the changes of the function spaces, also holds for Corollary
into to the more familiar spaces such as
1.1.
$W^{s,p}(\mathrm{R}^{n})$Corollary 1.2 The solution$T_{*}<\infty$ in, namely$v^{E}$of the Euler system (E) blows up at $t=$

$F_{p,q}^{s}$

Jim$\sup_{tarrow T_{k}}||v^{E}(t)||_{F_{\dot{\mathrm{p}},q}}=\infty$

if and only if for solution of$u$$(E_{\epsilon})$

with

,

(1.10)$\epsilon=\frac{\lambda}{T_{*}}we$have(1.11)$\int_{0\prime}^{\infty}||\omega(t)||_{||_{\dot{B}}0,\infty 1},dt=\infty$

for $s>n/p+1,$$(p, q)\in[1, \infty]^{2}$, while(1.12)$\int_{0}^{\infty}||\omega(t)||_{\dot{B}_{\infty,1}^{0}}dt=\infty$

for $s=n+1$ , $p=1$ ,$q\in[1, \infty]$

respectively.

with $m> \frac{n}{2}+1$ ,Remark 1.4 In terms of the usual Sobolev spaces,
Corollary 1.2 implies that if we have local solution $v^{E}\in C([0,T];H^{m}(\mathrm{R}^{n}))$ to
the problem (E) with initial data , then necessarily we have global solution$u\in \mathrm{C}([0, \infty);Hm\{Rn$ )) of.
, andwith the initial data
This resembles the comparison type of result between the Euler equations andthe Navier-Stokes equations obtained by Constantin(See Theorem 1.1[9]).
$H^{m}(\mathrm{R}^{n})$$v_{0}^{E}$$u_{0}=\lambda v_{0}^{E}$$(\mathrm{E})\mathrm{e}$$\epsilon=\frac{\lambda}{T}$As amodel problem of the perturbed Euler equation we also consider theConstantin-Lax-Majda equation[ll] first considered in [11]:(CLM)$\omega_{t}-H(\omega)\omega=0$$(x, t)\in \mathrm{R}\mathrm{x}\mathbb{R}^{+}$$\{$$\omega(x,0)=\omega_{0}(x)$$x\in \mathrm{R}$with $\omega=\omega(x, t)$ ascalar function, and $H(f)$ is the Hilbert transform ofdefined by$H(f)= \frac{1}{\pi}PV\int\frac{f(y)}{x-y}dy$.

$f$(1.13)For the problem (CLM), Constantin-Lax-Majda derived the following explicit solution[ll](see also Section 5.2.1 of [19])$\omega(x,t)=\frac{4v_{0}(x)}{(2-tH\omega_{0}(x))^{2}+t^{2}\omega_{0}^{2}(x)}$.

(1.14)

6
The perturbed equation we are concerned is$\sigma_{t}-H(\sigma)\sigma=-\epsilon\sigma$$(x, t)\in \mathbb{R}\cross \mathbb{R}^{+}$$(\mathrm{C}\mathrm{L}\mathrm{M})_{\xi}\{$$\sigma(x, 0)=\sigma_{0}(x)$$x\in \mathbb{R}$We have the following relation between the two solutions:$\sigma(x,t)$$\sigma_{0}(x)$$=e^{-\epsilon t}\omega(x,$$=\omega_{0}(x)$$\frac{1}{\epsilon}(1-e^{-\epsilon t}))$,(1.15).

Combining (1.15) with (1.14), we easily obtain the following explicit solutionof:$(CLM)_{\epsilon}$$\sigma(x, t)=\frac{4\epsilon^{2}\sigma_{0}(x)e^{-\epsilon t}}{(2\epsilon-H\sigma_{0}(x)(1-e^{-\epsilon t}))^{2}+(1-e^{-\epsilon t})^{2}\sigma_{0}^{2}(x)}$.

(1.16)

The formula leads us to the following proposition:

Proposition 1.1 In case $H\sigma_{0}(x)\leq 0$ for all
there is no blow up ofsolution. Otherwise, we consider the three cases. Let us put $S=\{x\in \mathrm{R}$ :$\sigma_{0}(x)=0$ , $H\sigma_{0}(x)>0\}$ .
$x\in \mathrm{R}$(i) If$\epsilon$$> \frac{1}{2}\sup_{x\in S}H\sigma_{0}(x)$, then there is no blow-up.

(ii) If there exists $x\in S$ such thatgiven by$\epsilon<\frac{1}{2}H\sigma_{0}(x)$, then solution blows up at$T_{*}$$T_{*}= \frac{1}{\epsilon}\ln(1-\frac{2\epsilon}{\sup_{x\in \mathrm{S}}H\sigma_{0}(x)})^{-1}$(1.17)(Hi) If the set $S_{1}= \{x\in \mathrm{R} :\sigma_{0}(x)=0, \epsilon =\frac{1}{2}H\sigma_{0}(x)\}$ is nonempty, and if

for all$x\in \mathrm{R}\backslash S_{1}$we have

blows up at $t=+\infty$ .

$\epsilon>\frac{1}{2}H\sigma_{0}(x)$, or $\sigma_{0}(x)>0$ , then the solution

Remark 4.1 We note that (i),(iii) above are the new phenomena of (CLM),

not occurred in (CLM).

2Appendix: Function spaces
We first set our notations, and recall definitions of the TViebel-Lizorkinspaces. We follow [20]. Let be the Schwartz class of rapidly decreasing
functions. Given $f\in S$ , its Fourier transform $F(f)=\hat{f}$ is defined by$S$$\hat{f}(\xi)=\frac{1}{(2\pi)^{n/2}}\int_{\mathrm{R}^{n}}e^{-\cdot x\cdot\xi}.f(x)dx$.

7
satisfying Supp:|$\frac{1}{2}\leq|\xi|\leq 2\}$ , andWe consider(In other words, $\varphi j(x)=$if $\frac{1}{2}<|\xi|<2$ . Setting, we can adjust the normalization constant in front of so that$\subset\{\xi\in \mathbb{R}^{n}$$\varphi\in S$$\hat{\varphi}_{j}=\hat{\varphi}(2^{-j}\xi)$$\hat{\varphi}(\xi)>0$$2^{jn}\varphi(2^{j}x).)$$\hat{\varphi}$$\forall\xi\in \mathbb{R}^{n}\backslash \{0\}$$\sum_{j\in \mathrm{Z}}\hat{\varphi}_{j}(\xi)=1$

Given$k\in \mathbb{Z}$.

, we define the function $S_{k}\in S$ by its Fourier transform$\hat{S}_{k}(\xi)=1-\sum_{j\geq k+1}\hat{\varphi}_{j}(\xi)$.

, , $q\in[0, \infty]$ . Given $f\in S’$ , we denote $Ajf=\varphi_{j}*f$ , and thenLet
.is defined by
the homogeneous Triebel-Lizorkin semi-norm$s\in \mathrm{R}$$p$$||f||_{F_{\mathrm{p},q}^{\epsilon}}$$||f||_{\dot{F}_{\mathrm{p}_{1}q}^{*}}= \{||(\sum_{j\in \mathrm{Z}}2^{jqs}|\Delta_{j}f(\cdot)|^{q})||_{L^{\mathrm{p}}}\mathrm{i}\mathrm{f}q\in[1, \infty)||\sup_{j\in \mathrm{Z}}(2^{js}|\Delta_{j}f(\cdot)|)||_{L^{\mathrm{p}}}\mathrm{i}\mathrm{f}q=\infty$

is aqausi-normed space with
The homogeneous Triebel-Lizorkin space. For $s>0$ , $(p, q)\in[1, \infty]^{2}$ we define thethe quasi-norm given by .
of $f\in S’$ as
inhomogeneous Triebel-Lizorkin space norm$\dot{F}_{\mathrm{p},q}^{\mathit{8}}$$||$$||_{\dot{F}_{\mathrm{p},q}^{\epsilon}}$$||f||_{F_{\mathrm{p},q}^{\epsilon}}$$||f||_{F_{\mathrm{p},q}^{\epsilon}}=||f||_{L^{\mathrm{p}}}+||f||_{\dot{F}_{\mathrm{p}.q}^{\epsilon}}$.

The inhomogeneous Triebel-Lizorkin space is aBanach space equipped withis. Similarly, the homogeneous Besov nom
the norm,,$||\cdot||_{\dot{B}_{\infty,1}^{0}}$$||f||_{||}.\cdot \mathrm{p},q\epsilon B$

defined by$||f||_{\dot{B}_{\mathrm{p},q}^{*}}= \{[_{-\infty}\sum_{\sup_{j}}^{\infty}2^{jqs}||\varphi_{j}*f||_{L^{\mathrm{p}}}^{q}]^{\frac{1}{q}}\mathrm{i}\mathrm{f}q\in[1, \infty)[2^{js}||\varphi_{j}*f||_{L^{\mathrm{p}}}]\mathrm{i}\mathrm{f}q=\infty$

is aquasi-normed space with the quasiThe homogeneous Besov space. For $s>0$ we define the inhomogeneous Besov space
norm given by.
of $f\in S’$ asnorm$\dot{B}_{p,q}^{s}$$||\cdot||_{\dot{B}^{\epsilon}}$$\mathrm{p}.q$$||f||_{B_{\mathrm{p},q}^{\epsilon}}$$||f||_{B_{\mathrm{p},q}^{\epsilon}}=||f||_{L^{\mathrm{p}}}+||f||_{\dot{B}_{\mathrm{p},q}^{s}}$Lemma 2.1 Let $s\in(0, n)$ , , $q\in[1, \infty]$ and $sp=n$ .
sequence of continuous embeddings hold.

Tfeen the following$p$$\dot{F}_{1,q}^{n}rightarrow\dot{B}_{p,1}^{s}rightarrow\dot{B}_{\infty,1}^{0}arrow\dot{F}_{\infty,1}^{0}rightarrow L^{\infty}$

.

(2.1)

8
The first imbedding of Lemma 2.1 is proved in [12], while the second one isproved in [4]. The others are obvious from the definitions of the corresponding
norms.
Acknowledgement$\mathrm{s}$This research is supported partially by the grant n0.2000-2-10200-002-5 ffom
the basic research program of the KOSEF.

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$4^{e}$