\input amstex \documentstyle{amsppt} \loadmsbm \magnification=\magstephalf \hcorrection{1cm} \vcorrection{-6mm} \nologo \TagsOnRight \NoBlackBoxes \headline={\ifnum\pageno=1 \hfill\else% {\tenrm\ifodd\pageno\rightheadline \else \leftheadline\fi}\fi} \def\rightheadline{EJDE--1999/05\hfil Regularity of solutions to the Navier-Stokes equation \hfil\folio} \def\leftheadline{\folio\hfil Dongho Chae \& Hi-Jun Choe \hfil EJDE--1999/05} \def\pretitle{\vbox{\eightrm\noindent\baselineskip 9pt % Electronic Journal of Differential Equations, Vol. {\eightbf 1999}(1999), No.~05, pp.~1--7.\hfil\break ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \hfill\break ftp ejde.math.swt.edu (login: ftp)\bigskip} } \topmatter \title Regularity of solutions to the Navier-Stokes equation \endtitle \thanks {\it 1991 Mathematics Subject Classifications:} 35Q35, 76C99.\hfil\break\indent {\it Key words and phrases:} Navier-Stokes equations, regularity conditions. \hfil\break\indent \copyright 1999 Southwest Texas State University and University of North Texas. \hfil\break\indent Submitted February 8, 1999. Published February 28, 1999. \hfil\break\indent The first author was partially supported by GARC-KOSEF, BSRI-MOE, KOSEF(K95070), \hfil\break\indent and SNU Research Fund.\hfil\break\indent The second author was supported by KOSEF and BSRI-prg. \endthanks \author Dongho Chae \& Hi-Jun Choe \endauthor \address Dongho Chae \hfill\break Department of Mathematics, Seoul National University \hfill\break Seoul 151-742, Korea \endaddress \email dhchae\@math.snu.ac.kr \endemail \address Hi-Jun Choe \hfill\break Department of Mathematics, KAIST \hfill\break Taejon, Korea \endaddress \email ch\@math.kaist.ac.kr \endemail \abstract Recently, Beir\~ao da Veiga [15] obtained regularity for the Navier-Stokes equation in $\Bbb R^3$ by imposing conditions on the vorticity rather than the velocity. In this article, we obtain regularity by imposing conditions on only two components of the vorticity vector. \endabstract \endtopmatter \def\diver{\operatorname{div}} \def\curl{\operatorname{curl}} \head 1. Introduction \endhead We are concerned with the initial value problem of the Navier-Stokes equation in $\Bbb R^3 \times (0,T)$, $$\gather {\partial v \over \partial t}+(v\cdot \nabla)v = -\nabla p+\nu \Delta\,,, \tag1 \\ \diver v=0\,, \tag2 \\ v(x,0)=v_0 (x)\,, \tag3 \endgather $$ where $ v(x,t) = (v_1(x,t) ,v_2(x,t),v_3(x,t))$, $x\in {\Bbb R^3 }$, and $t\in (0,T)$. For simplicity we assume that the external force is zero, but it is easy to extend our results to the nonzero-external-force case. Recall that a weak solution to the Navier-Stokes equation, which is called the Leray-Hopf weak solution, is defined as a vector field $ v\in L^{\infty} (0, T;L^2 (\Bbb R^3)) \cap L^2(0,T;H^1 (\Bbb R^3)) $ satisfying $\diver v=0 $ in the distributional sense, and $$ \int^T _0 \int_{\Bbb R^3 } [v\cdot \phi_t + (v\cdot \nabla)\phi \cdot v +v \cdot \Delta \phi] dx\,dt =0 $$ for all $\phi \in [C^{\infty} _0 (\Bbb R^3 \times (0,T))]^3$ with $\diver \phi=0$. For $v_0 \in L^2 (\Bbb R^3 )$ with $\diver v_0=0$, the existence of weak solutions was established by Leray[13] and Hopf in [11]. A weak solution of the Navier-Stokes equation that belongs to $L^\infty \Big(0,T;H^1 (\Bbb R^3 )\Big) \cap L^2 \Big(0,T;H^2 (\Bbb R^3 )\Big)$ is called a strong solution. It is well-known that for $ v_0 \in H^1 (\Bbb R^3 ) $ with $\diver v_0 =0$ there exists a local unique strong solution $v \in C([0,T);H^1 (\Bbb R^3 ))$, and that the maximal time of existence $T_*$ depends on the initial data ${\| v_0 \|}_{H^{1/2} (\Bbb R^3 )}$. Moreover, the strong solution belongs to the class $C\big((0,T_* );C^{\infty} (\Bbb R^3 )\big)$, i.e., the solution becomes regular in the space variable immediately after the initial moment. The global-in time continuation of the local strong solution is an outstanding open problem in mathematical fluid mechanics. Another notion of solution, useful for the study of the Navier-Stokes equation, is that of mild solution initiated by Fujita and Kato [9]. In this note we are concerned with obtaining a sufficient condition for the global continuation of strong solutions. In this direction there is a classical result due to Serrin[14], which states that if a weak solution belongs to $L^{\alpha, \gamma} _T$, then $v$ becomes the strong solution in $(0,T]$. Here $L^{\alpha,\gamma} _T = L^{\alpha ,\gamma } _{[0,T]} =L^\alpha (0,T;L^\gamma (\Bbb R^3 )$ with $ {2 \over\alpha} + {3\over\gamma } < 1$, and $\alpha<\infty$. Later, Fabes-Jones-Riviere [8] extended the above criterion to the case ${2\over\alpha} + {3\over\gamma} =1$. The problem of regularity and uniqueness for the marginal case $\alpha=\infty$, $\gamma =3$ in Serrin's condition has been extensively studied by many authors; see for example [3], [10], and [12]. Recently, Beir\~ao da Veiga [15] obtained a sufficient condition for regularity using the vorticity rather than the velocity. His result says that if the vorticity $\omega = \curl v $ of a weak solution $v$ belongs to the space $ L^{\alpha,\gamma}_T $ with $ {2\over\alpha}+{3\over\gamma} \leq 2$ with $1<\alpha < \infty $, then $v$ becomes the strong solution on $(0,T]$. Here we prove that it is sufficient to control only two components of the vorticity vector, or the gradients of the two components of the velocity vector. See Theorem~1 below. Given a velocity field $v$, the two-component vorticity field is denoted by $\tilde\omega =\omega_1 e_1 +\omega_2 e_2$, where $e_1 =(1,0,0)$, $e_2 =(0,1,0)$. \proclaim{Theorem 1} Let $v_0 \in L^2(\Bbb R^3 )$ with $\diver v_0 =0 $ and $\omega_0 =\curl v_0 \in L^2(\Bbb R^3 )$. If a Leray-Hopf weak solution $v$, satisfies $\tilde \omega \in L^{\alpha,\gamma} _T $ with ${2\over\alpha} +{3\over\gamma} \leq 2$, $1< \alpha < \infty $ and $\frac{3}{2} < \gamma <\infty$, or if $\|\tilde \omega \|_{L^{\infty, \frac{3}{2}} _T } $ is sufficiently small, then $v$ becomes the classical solution on $(0,T]$. \endproclaim \remark{Remark 1} As an immediate corollary of the above theorem, we find that if the classical solution of the 3-D Navier-Stokes equations blows up at time $T$, then $ {\|\tilde \omega \|}_{L^{\alpha, \gamma} _T }= \infty$, where $\tilde \omega $ is any two component vector of $\omega $, and $ (\alpha,\gamma)$ is a pair of real numbers satisfying $ { 2 \over \alpha} +{ 3\over\gamma}\leq 2$, $1 < \alpha < \infty $. In other words, at finite blow-up time at least two components of the vortices must simultaneously blow up. \endremark A related result is studied by Beale-Kato-Majda [1]. They show that for 3-D Euler equations, the blow up of full gradients of the velocity field is controlled by only three components of the vorticity field. \remark{Remark 2} As another immediate corollary we obtain the global regularity for the 2-D Navier-Stokes equations, since in this case $\tilde \omega (x,t)=0,$ for all $(x,t) \in { \Bbb R^3} \times (0,T)$. \endremark Our second theorem concerns the regularity criterion in terms of gradients of the two components of velocity. \proclaim{Theorem 2} Let $\tilde v = v_1 e_1 +v_2 e_2 $ be the first two components of a Leray-Hopf weak solution of the Navier-Stokes equation corresponding to $v_0 \in H^1 (\Bbb R^3 ) $ with $\diver v_0 =0$. Suppose that $ D\tilde v \in L^{ \alpha, \gamma } _T $ with $ {2\over \alpha} + {3\over \gamma} \leq 1 $, where $ 2 \leq \alpha \leq \infty$, and $3\leq \gamma \leq \infty$, then $v$ becomes a classical solution in $(0,T]$. \endproclaim \head 2. Proof of Main Theorems \endhead The key idea in booth proofs is a careful observation of the structure of the nonlinear terms of the vorticity equations for the Navier-Stokes system. The structure of the nonlinear term has been emphasized in the works by Constantin and Fefferman [4], [5], [6], and [7]. Below we use the notation $$ \|u\|_p =\bigg( \int_{\Bbb R^3} |u(x)|^p dx \bigg)^{1/p}, \quad 1\leq p< \infty\,. $$ We also use $C$ for various constants in the estimates below. \demo{Proof of Theorem 1} Taking the $\curl$ on (1), we obtain $$\omega _t +(v \cdot \nabla )\omega =(\omega \cdot \nabla)v + \nu \Delta \omega. \eqno(4) $$ Multiplying (4) by $\omega$ in $ L^2 (\Bbb R^3 )$, and integrating by parts, we obtain $$ {1\over 2} {d \over dt} \|\omega (t) \| ^2_2 +\nu \|\nabla \omega (t) \|^2_2 = \int _{\Bbb R^3} (\omega \cdot \nabla ) v \cdot \omega \,dx\,. \eqno(5)$$ Using the Biot-Savart law, $v$ is written in terms of $\omega$: $$ v(x,t) = -{1 \over 4\pi} \int_{\Bbb R^3} {{(x-y) \times \omega(y,t)} \over |{x-y|^3 }}\,dy\,. \eqno(6) $$ Substituting this into the right hand side of (5), we have $$\eqalign{\int_{\Bbb R^3 } (\omega\cdot \nabla )v\cdot \omega \,dx &={3\over {4\pi}} \iint {y \over |y|}\cdot \omega(x,t) \bigg\{ {y \over |y|^4 } \times \omega(x+y,t) \cdot \omega(x,t) \bigg\} \,dy\, dx \cr &\Big( \text{ We decompose $\omega = \tilde \omega + \omega' ,\,\tilde \omega = \omega_1 e_1 +\omega_2 e_2 , \, \omega' = \omega_3 e_3 $}\cr &\text{ for the vorticities in $\{ \cdot \}$} \Big)\cr &= {3 \over {4\pi}}\iint {y \over {|y|}}\cdot \omega(x,t)\bigg\{{y\over {|y|^4}}\times \tilde \omega (x+y,t)\cdot \omega' (x,t)\bigg\} \,dy\,dx \cr &+ {3\over {4\pi}} \iint {y \over |y|}\cdot \omega(x+y,t) \bigg\{ {y \over {|y|^4} } \times \tilde \omega(x+y,t) \cdot \tilde \omega (x,t) \bigg\} \,dy\,dx \cr &+ {3\over {4\pi}} \iint {y \over |y| }\cdot \omega(x,t) \bigg\{{y \over |y|^4 } \times \omega' (x+y,t) \cdot \tilde \omega(x,t) \bigg\} \,dy\,dx\,,} \eqno(7) $$ where all the integrations with respect to $y$ are in the sense of principal value. We have thus $$ \align \bigg | \int_{\Bbb R^3 } (\omega \cdot \nabla )v \cdot \omega dx \bigg| &\leq C\int_{\Bbb R^3 }| \omega (x,t) || P(\tilde \omega )| | \omega'(x,t)| dx\\ & +C\int_{\Bbb R^3 } | \omega (x,t) | | P(\tilde \omega )| | \tilde \omega(x,t)| dx\\ & +C\int_{\Bbb R^3 }| \omega (x,t) | | P(\omega')| | \tilde \omega(x,t) | dx\\ &\leq C\int_{\Bbb R^3 }| \omega |^2 | P(\tilde \omega ) | dx + C \int_{\Bbb R^3 } | \omega | | P(\omega') | | \tilde \omega | dx. \\ &=:I_1 +I_2, \endalign $$ where $P(\cdot)$ denotes the singular integral operator defined by the integrals with respect to $y$ in (7). We first consider the case $\frac{3}{2} < \gamma < \infty $. We have the following estimates $$ \eqalign{ I_1&\leq \|P(\tilde \omega)\|_{\gamma} \|\omega \|^2 _{2\gamma \over {\gamma -1}} \quad\text{(by H\"{o}lder's inequality)} \cr &\leq C \|\tilde \omega\|_{\gamma} \|\omega \|^{\frac{2\gamma-3}{\gamma}} _2 \|\nabla\omega\|^{\frac{3}{\gamma}} _2 \cr &\leq C \|\tilde \omega\|_{\gamma} ^{\frac{2\gamma}{2\gamma -3}}\|\omega\|^2 _2 +\frac{\nu}{4} \|\nabla\omega\|_2 ^2 \quad \text{(by Young's inequality)}, }\eqno(8) $$ where we used the Calderon-Zygmund and the Gagliardo-Nirenberg inequalities in the second inequality. For the second term of the right hand side of (8) we have by the H\"oder inequality and the Calderon-Zygmund inequality, $$\eqalign{ I_2 &\leq \|\tilde \omega \|_\gamma \|P( \omega')\|_{2\gamma \over {\gamma -1}}\|\omega \|_{2\gamma \over {\gamma -1}} \quad \text{(by H\"{o}lder's inequality)} \cr &\leq C\|\tilde \omega \|_\gamma \|\omega'\|_{2\gamma \over{\gamma -1}}\|\omega \|_{2\gamma \over {\gamma -1}} \quad \text{(by the Calderon-Zygmund inequality)} \cr &\leq C\|\tilde \omega \|_\gamma \|\omega \|^2_{2\gamma \over {\gamma -1}}\cr &\leq C \|\tilde \omega\|_{\gamma} ^{\frac{2\gamma}{2\gamma -3}}\|\omega\|^2 _2 +\frac{\nu}{4} \|\nabla\omega\|_2 ^2 \quad \text{(by the similar estimates to (8))} .} \eqno(9) $$ Thus, combining (8)-(9) with (5), we obtain $$ {d \over dt} \|\omega (t) \|^2_2 +\nu \|\nabla \omega (t) \|^2_2 \leq C \|\tilde \omega (t) \|_{\gamma} ^{\frac{2\gamma}{2\gamma -3}}\|\omega (t) \|^2 _2 . $$ Applying the standard Gronwall lemma, we have $$ \gather \|\omega (t)\|_2 ^2 +\int _0 ^t \|\nabla \omega (\tau )\|_2 ^2 \exp\left( C \int _{\tau} ^t\|\tilde \omega (s) \|_{\gamma} ^{\frac{2\gamma}{2\gamma -3}}ds\right) d\tau \\ \leq \|\omega_0 \|_2 ^2 \exp \left( C \int _{0} ^t \|\tilde \omega (s) \|_{\gamma} ^{\frac{2\gamma}{2\gamma -3}}ds\right). \endgather $$ Since $0< \frac{2\gamma}{2\gamma -3}\leq \alpha$, by the H\"{o}lder inequality we obtain $$ \align \sup_{0\leq t\leq T} \|\omega (t)\|_2 ^2 +\nu \int_0 ^T \|\nabla \omega (t) \|^2_2 dt &\leq \|\omega _0 \|_2 ^2 \exp \left( C \int _0 ^T \|\tilde \omega (t) \|_{\gamma} ^{\frac{2\gamma}{2\gamma -3}} dt \right) \cr &\leq \|\omega _0 \|_2 ^2 \exp \left( C \|\tilde \omega \|_{L^{\alpha , \gamma}_T} ^ {\frac{2\gamma}{2\gamma -3}} T^{\frac{2\gamma}{2\gamma -3} (2-\frac{2}{\alpha}-\frac{3}{\gamma})} \right). \endalign $$ Thus, in this case $\|\tilde \omega \|_{L^{\alpha , \gamma}_T} < \infty$ implies $$ \omega \in L^{\infty} \big(0,T : L^2(\Bbb R^3 )\big) \cap L^2 \big(0,T : H^1 (\Bbb R^3 )\big). $$ Using the regularity of the strong solution, we obtain the conclusion of the theorem for $1< \alpha < \infty$, $\frac{3}{2} < \gamma < \infty$. Next, we consider the case $\alpha=\infty, \gamma = 3/2$. In this case, instead of (8) and (9), we estimate as follows: $$ \{1\}\leq \|P(\tilde \omega)\|_{\frac{3}{2} } \|\omega \|^2 _{6} \leq C \|\tilde \omega\|_{\frac{3}{2} } \|\nabla\omega\|^{2} _2, \eqno(10) $$ and $$\eqalign{ I_2 &\leq \|\tilde \omega \|_{ \frac{3}{2} } \|P( \omega')\|_{6}\|\omega \|_{6} \cr &\leq C\|\tilde \omega \|_{\frac{3}{2} } \|\omega'\|_{6} \|\omega \|_{6} \cr &\leq C\|\tilde \omega \|_{ \frac{3}{2} } \|\omega \|^2_{6} \leq C \|\tilde \omega\|_{\frac{3}{2} } \|\nabla \omega\|^2 _2 .} \eqno(11) $$ Combining (10)-(11) with (5), integrating over $[0, T]$, we deduce $$ \sup_{0\leq t\leq T} \|\omega (t)\|_2 ^2 +\nu \int_0 ^T \|\nabla \omega (t) \|^2_2 dt \leq C \|\tilde \omega \|_{L^{\infty ,\frac{3}{2}}_T }\int _0 ^T \|\nabla \omega (t)\|^2 _2 dt\,. $$ Thus, if $C \|\tilde \omega \|_{L^{\infty ,3/2}_T} < \frac{\nu}{2}$, then we have again $$ \omega \in L^{\infty} \big(0,T : L^2(\Bbb R^3 )\big) \cap L^2 \big(0,T : H^1 (\Bbb R^3 )\big), $$ which implies the regularity of $v$ as previously. \qed \enddemo \demo{Proof of Theorem 2} We set $\tilde v = (v_1,v_2, 0)$. Then, taking the first two components of the vorticity equation (4), we obtain $$\tilde \omega_t + (v\cdot \nabla ) \tilde \omega = ( \omega \cdot \nabla ) \tilde v +\nu \Delta \tilde \omega \,. $$ Taking $L^2 (\Bbb R^3 )$ inner product (12) with $\tilde \omega$, we have, after integration by part, $$ {1\over 2} {d \over dt} \|\tilde \omega (t) \| ^2_2 +\nu \|\nabla \tilde\omega (t) \|^2_2 = \int _{\Bbb R^3} (\omega \cdot \nabla ) \tilde v \cdot \tilde \omega \,dx\,. \eqno(12) $$ We first consider the case $ 2\leq \alpha < \infty$, $3< \gamma \leq \infty$. Using the H\"{o}lder and the Gagliardo-Nirenberg inequalities, we estimate $$ \eqalign{ \Big| \int _{\Bbb R^3} (\omega \cdot \nabla ) \tilde v \cdot \tilde \omega \,dx\Big| &\leq\|\omega \|_2 \|\nabla \tilde v\|_{\gamma} \|\tilde \omega \|_{\frac{2\gamma}{\gamma-2}} \cr &\leq C \|\omega \|_2 \|\nabla \tilde v\|_{\gamma} \|\tilde \omega \|_2 ^{\frac{\gamma-3}{\gamma}} \|\nabla \tilde \omega \|_2 ^{\frac{3}{\gamma}} \cr &\leq C \|\omega \|_2 ^2 + C \|\nabla \tilde v\|_{\gamma} ^{\frac{2\gamma}{\gamma-3}} \|\tilde \omega \|_2 ^2 + \frac{\nu}{2} \|\nabla \tilde \omega \|_2 ^2,} \eqno(13) $$ where the case $\gamma= \infty$ corresponds to the obvious limit $\gamma\rightarrow \infty$ for the norms of the estimates. The estimates (13), combined with (12), yield $$ {d \over dt} \|\tilde \omega (t) \| ^2_2 +\nu \|\nabla \tilde\omega (t) \|^2_2 \leq C\|\omega (t)\|_2 ^2 +C \|\nabla \tilde v (t) \|_{\gamma} ^{\frac{2\gamma}{\gamma-3}} \|\tilde \omega (t) \|_2 ^2 . $$ Using the Gronwall lemma similarly to the proof of Theorem 1, we obtain $$ \multline \sup_{0\leq t\leq T} \|\tilde \omega (t)\|_2 ^2 +\nu \int_0 ^T \|\nabla \tilde \omega (t) \|^2_2 dt \cr \leq \left( \|\omega _0 \|_2 ^2 +\int_0 ^T \|\omega (t) \|^2 _2 dt \right) \exp \left( C \int _0 ^T \|\nabla \tilde v(t) \|_{\gamma} ^{\frac{2\gamma}{\gamma -3}}dt \right) \cr \leq \left( \|\omega _0 \|_2 ^2 +\int_0 ^T \|Dv (t) \|^2 _2 dt \right) \exp \left(C \|\nabla \tilde v\|_{L_T ^{\alpha, \gamma}} ^{\frac{2\gamma}{\gamma -3}}T^{ {\frac{2\gamma}{\gamma -3} } ( 1-\frac{2}{\alpha}-\frac{3}{\gamma} ) }\right), \endmultline $$ where we used the fact $\frac{2\gamma}{\gamma -3} \leq \alpha$ in the second inequality. Thus, if $\nabla \tilde v \in L^{ \alpha, \gamma } _T $, then we have by the Sobolev embedding, $H^1 (\Bbb R^3 )\hookrightarrow L^6 (\Bbb R^3 )$, $$ \tilde \omega \in L^{\infty} \big(0,T :L^2 (\Bbb R^3 )\big) \cap L^2 \big(0,T :L^6 (\Bbb R^3 )\big) . $$ Since $\alpha=2$ and $\gamma=6 $ satisfy ${2\over \alpha}+{3 \over \gamma} \leq 2 $, the conclusion of Theorem 2 for the case $ 2\leq \alpha < \infty$, $3< \gamma \leq \infty$ follows from Theorem 1. Next, we consider the case $\alpha =\infty$, $\gamma =3$. In this case we estimate $$ \align \Big| \int _{\Bbb R^3} (\omega \cdot \nabla ) \tilde v \cdot \tilde \omega \,dx \Big| &\leq \|\omega \|_2 \|\nabla \tilde v\|_{3} \|\tilde \omega \|_{6} \cr &\leq C \|\omega \|_2 \|\nabla \tilde v\|_{3} \|\nabla \tilde \omega \|_2 \cr &\leq C \|\omega \|_2 ^2 \|\nabla \tilde v\|_{3} ^2 + \frac{\nu}{2} \|\nabla \tilde \omega \|_2 ^2 \,. \endalign $$ This, together with (12), provide us with $$ \align \sup_{0\leq t\leq T} \|\tilde \omega (t)\|_2 ^2 +\nu \int_0 ^T \|\nabla \tilde \omega (t) \|^2_2 dt &\leq C \|\nabla \tilde v\|_{L^{\infty, 3} _T} ^2 \int _0 ^T \|Dv (t) \|^2 _2 dt \endalign $$ after integrating over $[0,T]$. This inequality, in turn, implies that if $\|\nabla \tilde v\|_{L^{\infty, 3} _T} < \infty$, then $$ \tilde \omega \in L^{\infty} \big(0,T :L^2 (\Bbb R^3 )\big) \cap L^2 \big(0,T :L^6 (\Bbb R^3 )\big)\,. $$ In a similar manner to the previous case, we conclude the present proof. \qed \enddemo \remark{Acknowledgement} The authors would like to thank Professor J. Serrin for his helpful comment on Remark 1. \endremark \Refs \ref \no 1 \by J. Beale, T. Kato and A. Majda \paper Remarks on the breakdown of smooth solutions for the 3-D Euler equations \jour Comm. Math. Phys. \vol 94 \yr 1984 \page 61-66 \endref % \ref \no 2 \by M. Cannone \paper Ondelettes, Paraproduits et Navier-Stokes \jour Diderot Editeur \yr 1995 \endref % \ref \no 3 \by M. Cannone and F. Planchon \paper On the non stationary Navier-Stokes equations with an external force \jour preprint \endref \ref \no 4 \by P. 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