lectronic Journal of Differential quations, Vol. 2010(2010), No. 15, pp. 1 7. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu RGULARITY FOD NAVIR-STOKS QUATIONS IN TRMS OF TWO COMPONNTS OF TH VORTICITY SADK GALA Abstract. We establish regularit conditions for the D Navier-Stokes equation via two components of the vorticit vector. It is known that if a Lera- Hopf weak solution u satisfies ω L 2/(2 r) (0, T ; L /r ( )) with 0 < r < 1, where ω form the two components of the vorticit, ω = curl u, then u becomes the classical solution on (0, T ] (see [5]). We prove the regularit of Lera-Hopf weak solution u under each of the following two (weaker) conditions: ω L 2/(2 r) (0, T ; Ṁ 2,/r ( )) for 0 < r < 1, ũ L 2/(2 r) (0, T ; Ṁ 2,/r ( )) for 0 r < 1, where Ṁ2,/r( ) is the Morre-Campanato space. Since L /r ( ) is a proper subspace of Ṁ 2,/r ( ), our regularit criterion improves the results in Chae- Choe [5]. 1. Introduction We consider the Navier-Stokes equations, in, t u + (u )u u + p = 0, (x, t) (0, T ), div u = 0, (x, t) (0, T ), u(x, 0) = u 0 (x), x, (1.1) where u = u(x, t) is the velocit field, p = p(x, t) is the scalar pressure and u 0 (x) with div u 0 = 0 in the sense of distribution is the initial velocit field. For simplicit, we assume that the external force has a scalar potential and is included in the pressure gradient. In their well known articles, Lera [15] and Hopf [10] independentl constructed a weak solution u of (1.1) for arbitrar u 0 L 2 ( ) with div u 0 = 0. The solution is called the Lera-Hopf weak solution. Regularit of such Lera-Hopf weak solutions is one of the most significant open problems in mathematical fluid mechanics. Here we mean b the weak solution a function u L (0, T ; L 2 ( )) L 2 (0, T ; Ḣ1 ( )) satisfing (1.1) in sense of distributions. 2000 Mathematics Subject Classification. 5Q5, 76C99. Ke words and phrases. Navier-Stokes equations; regularit conditions; Morre-Campaanto spaces. c 2010 Texas State Universit - San Marcos. Submitted Ma 20, 2010. Published October 28, 2010. 1
2 S. GALA JD-2010/15 A weak solution of the Navier-Stokes equation that belongs to L (0, T ; H 1 ( )) L 2 (0, T ; H 2 ( )) is called a strong solution. Introducing the class L α (0, T ; L q ( )), it is shown that if a Lera-Hopf weak solution u belongs to L α ((0, T ); L q ( )) with the exponents α and q satisfing 2 α + q 1, 2 α <, < q, then u(x, t) C ( (0, T )) [20, 18, 19, 8, 21, 22, 9], while the limit case α =, q = was covered much later scauriaza, Seregin and Sverak in [7]. See also [12] for recent improvements of the criteria, using the negative order Triebel-Lizorkin spaces. On the other hand, Beirão da Veiga [1] obtained a sufficient condition for regularit using the vorticit rather than velocit. His result sas that if the vorticit ω =curl u of a weak solution u belongs to the space L α (0, T ; L q ( )) with 2 α + q 2 and 1 α <, then u becomes the strong solution on (0, T ]. Later, Chae-Choe [5] obtained an improved regularit criterion of [1] imposing condition on onl two components of the vorticit, namel if ω = (ω 1, ω 2, 0) L α (0, T ; L q ( )) with 2 α + q 2, 1 α <, (1.2) then the weak solution becomes smooth. The purpose of this article is to prove the result of [5] in the other cases, proving that if ω L 2/(2 r) (0, T ; Ṁ 2,/r ( )) with 0 < r < 1, then the weak solution becomes smooth. Here M 2,/r ( ) is the Morre-Campanato space, which is strictl bigger than L /r ( ) (see the next section for the related embedding relations). We remark that in the limiting case r = 0, Kozono-Yatsu [1] previousl weakened the condition ω L 1 (0, T ; L ( )) of [5] into ω L 1 (0, T ; Ḃ0, ( )), where Ḃ0, is the Besov space. 2. Preliminaries and the main theorems Now, we recall the definition and some properties of the space that we are going to use. These spaces pla an important role in studing the regularit of solutions to partial differential equations (see e.g. [11, 2]). Definition 2.1. For 1 < p q +, the Morre-Campanato space Ṁ p,q ( ) is defined as Ṁ p,q ( ) = { f L p loc (R ) : f Ṁp,q = sup It is eas to check that sup x R>0 /q /p f()1 B(x,R) () L p (d) < } f(λ ) Ṁp,q = 1 λ /q f Ṁ p,q, λ > 0, Ṁ p, ( ) = L ( ) for 1 p. Additionall, for 2 p /r and 0 r < /2 we have the following embedding relations: L /r ( ) L /r, ( ) Ṁp,/r( ), where L p, denotes the weak L p space. The second relation L /r, ( ) Ṁp, r (R )
JD-2010/15 RGULARITY FOD NAVIR-STOKS QUATIONS is shown as follows. f Ṁp,/r ( sup r 1 p ( = sup pr 1 f() p d ( = sup R>0 f() p d) 1/p (f L /r, ( )) ) 1/p R {x : f() p > R} pr/) 1/p = sup R {x R p : f() > R} r/ R>0 = f L /r,. Remark 2.2. For the case q = /2 in (1.2), we can show that there exists an absolute constant δ such that if the weak solution u of (1.1) on (0, T ) with energ inequalit satisfies sup ω(t) L /2, δ, 0<t<T then u is actuall regular (see [1, p. 60], []). As another tpe of criterion, Neustupa-Novotn-Penel [17] considered suitable weak solution u = (u 1, u 2, u ) introduced b Caffarelli-Kohn-Nirenberg [4] and showed regularit of u under the hpothesis that u L α (0, T ; L q ( )) with 2 α + q = 1 2 (see [6]). The corresponding result for u to the above case was obtained b Berselli [2] who proved regularit under the assumption that sup 0<t<T ũ(t) L, is sufficientl small, where ũ = (u 1, u 2, 0). We need the following lemma which is essentiall due to Lemarié -Rieusset [14]. Lemma 2.. For 0 r < /2, the space Żr( ) is defined as the space of f(x) L 2 loc (R ) such that f Żr = sup fg L 2 <. g Ḃr 1 Then f Ṁ2,/r( ) if and onl if f Żr( ) with equivalence of norms. Since L /r ( ) Ṁ2,/r( ), the above regularit criterion is an improvement on Chae-Choe s result and hence our regularit criterion covers the recent result given b Chae-Choe [5]. Our result on (1.1) reads as follows. Theorem 2.4. Let u 0 L 2 ( ) with.u 0 = 0 and ω 0 =curl u 0 L 2 ( ). If the first two components of the vorticit ω = ω 1 e 1 + ω 2 e 2 of the Lera-Hopf weak solution u, satisfies ω L 2/(2 r) (0, T, Ṁ 2,/r ( )) with 0 < r < 1, then u becomes the classical solution on (0, T ]. Remark 2.5. As an immediate consequence of the above theorem, we find that if the classical solution of the Navier-Stokes equations blow-up at time T, then ω L 2/(2 r) (0,T,Ṁ2,/r( )) =, where ω is an two component vector of ω. Our second theorem concerns the regularit criterion in terms of gradients of the components of velocit.
4 S. GALA JD-2010/15 Theorem 2.6. Let ũ = u 1 e 1 + u 2 e 2 be the first two components of a Lera-Hopf weak solution of the Navier-Stokes equation corresponding to u 0 H 1 ( ) with div u 0 = 0. Suppose that ũ L 2/(2 r) (0, T, Ṁ 2,/r ( )) with 0 r < 1, then u becomes the classical solution on (0, T ].. Proof of Theorem 2.4 Now we are in a position to prove our main result. Proof. Taking the curl on (1.1), we obtain t ω ω + (u )ω (ω )u = 0. (.1) Multipling (.1) b ω in L 2 ( ) and integrating b parts, we obtain Here we have used the identit 2 dt ω(t, ) 2 L + ω(t, 2 ) 2 L2 = ω u, ω. (.2) u ω, ω = 0. Using the Biot-Savart law, u is written in terms of ω: u(x, t) = 1 (x ) ω(x, t) 4π R x d. Substituting this into the right hand side of (.2), we obtain ω u, ω = 4π ω(x, t){ 4 ω(x +, t) ω(x, t)} d dx. We decompose ω for the vorticities in { } as follows ω = ω + ω, ω = ω 1 e 1 + ω 2 e 2, ω = ω e. Since ω = (0, 0, ω ), there holds 4π ω(x, t){ 4 ω (x +, t) ω (x, t) } d dx = 0 for all 0 < t < T. Then, it follows ω u, ω = ω(x, t){ 4π R 4 ω(x +, t) ω (x, t)} d dx + ω(x, t){ ω(x +, t) ω(x, t)} d dx 4π R 4 + ω(x, t){ 4π 4 ω (x +, t) ω(x, t)} d dx, (.) where all the integrations with respect to are in the sense of principal value. Using the following interpolation inequalit [16]: w Ḃr C w 1 r L w r 2 L2, r (0, 1)
JD-2010/15 RGULARITY FOD NAVIR-STOKS QUATIONS 5 it is eas to see that b Lemma 2. ω u, ω C ω(x, t) P ( ω) ω (x, t) dx R + C ω(x, t) P ( ω) ω(x, t) dx R + C ω(x, t) P (ω ) ω(x, t) dx R C ω 2 P ( ω) dx + C ω P (ω ) ω dx C ω L 2 ω P ( ω) L 2 + C ω P (ω ) L 2 ω L 2 C ω L 2 ω Ḃr P ( ω) Ṁ2,/r + C ω L 2 P (ω ) Ḃr ω Ṁ2,/r C ω Ṁ2,/r ω L 2 ω Ḃr + C ω L 2 ω Ṁ2,/r ω Ḃr C ω Ṁ2,/r ω 2 r L 2 ω r L 2 where P (.) denotes the singular integral operator defined b the integrals with respect to in (.). B Young s inequalit, we find Substituting (.4) in (.2), we have ω u, ω C ω 2/(2 r) ω 2 Ṁ 2,/r L + 1 2 2 ω 2 L2. (.4) d dt ω(, t) 2 L + ω(, 2 t) 2 L C ω 2 2 2 r ω 2 Ṁ 2,/r L2. (.5) B Gronwall s inequalit we have that This implies ω(, t) L 2 ( ω(0, ) L 2 exp C T ω L ([0, T ); L 2 ( )) L 2 ([0, T ); H 1 ( )) 0 ) ω(, τ) 2/(2 r) dτ. (.6) Ṁ 2, r provided that ω satisfies the condition ω L 2 2 r (0, T ; Ṁ2,/r ( )). This proves Theorem 2.4. 4. Proof of Theorem 2.6 Now we are in a position to prove our second result. Proof. We set ũ = (u 1, u 2, 0). Then, taking the first two components of the vorticit equation (.1), we obtain t ω ω + (u ) ω (ω )ũ = 0. Multipling (.1) b ω in L 2 ( ) and integrating b parts, we obtain 2 dt ω(t, ) 2 L + ω(t, 2 ) 2 L2 = ω ũ, ω. (4.1)
6 S. GALA JD-2010/15 We first consider the case 0 < r < 1. Using the Hölder inequalit and Lemma 2., we estimate ω ũ, ω = ( ω + ω ) ũ, ω ω L 2 ω ũ L 2 where we used C ũ Ṁ2,/r ω L 2 ω Ḃr C ũ Ṁ2,/r ω L 2 ω 1 r L 2 ω r L 2 = C( ũ 2/(2 r) ω 2 2 r Ṁ 2,/r L2) 2 ω r L 2 C ũ 2/(2 r) Ṁ 2,/r ω 2 L 2 + C 2 ω 2 L 2, ω ũ, ω = 0. stimates (4.2) combined with (4.1), ield 2 dt ω(t, ) 2 L + ω(t, 2 ) 2 L 2 B Gronwall s inequalit we have ω(t, ) L 2 ω(0, ) L 2 exp ( T C 0 C ũ 2/(2 r) Ṁ 2,/r ω 2 L 2. ) ũ(, τ) 2/(2 r) dτ. M 2,/r Next we consider the case r = 0. In this case we estimate (4.2) ω ũ, ω ω L 2 ω ũ L 2 ω 2 L 2 ũ L. (4.) This estimate combined with (4.1), ield 2 dt ω(t, ) 2 L + ω(t, 2 ) 2 L C ũ 2 L ω 2 L 2. B Gronwall s inequalit we have ( T ) ω(t, ) L 2 ω(0, ) L 2 exp C ũ L dτ. 0 This completes the proof of Theorem 2.6. Acknowledgements. The author would like to express his gratitude to Professor Dongho Chae for man helpful discussions, constant encouragement and valuable suggestions; also to the anonmous referee for his or her useful comments and suggestions. References [1] H. Beirão da Veiga; A new regularit class for the Navier-Stokes equations in R n, Chinese Ann. Math. Ser. B, 16, (1995), 407-412. [2] L. C. Berselli; A note on regularit of weak solutions of the Navier-Stokes equations in R n, Japan. J. Math. 28 (2002), 51 60. [] L. C. Berselli; On a regularit criterion for the solutions to the D Navier-Stokes equations, Differential Integral quations, 15 (2002), 1129 117. [4] L. Caffarelli, R. Kohn and L. Nirenberg; Partial regularit of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math. 5 (1982), 771-81. [5] D. Chae, D. and H. -J. Choe; Regularit of solutions to the Navier-Stokes equation, lectron. J. Differential quations (1999), No. 05, (electronic) 1-7. [6] C. Cao and. S. Titi; Global regularit criterion for the D Navier-Stokes equations involving one entr of the velocit gradient tensor, Archive of Rational Mechanics & Analsis, (to appear)
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