On the regularity to the solutions of the Navier Stokes equations via one velocity component

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1 On the regularity to the olution of the Navier Stoke equation via one velocity component Milan Pokorný and Yong Zhou. Mathematical Intitute of Charle Univerity, Sokolovká 83, Praha 8, Czech Republic Zhejiang Normal Univerity, Abtract We conider the regularity criteria for the incompreible Navier Stoke equation connected with one velocity component. Baed on the method from [4] we prove that the weak olution i regular, provided u 3 L t (, T; L (, t , > 3 or provided u 3 L t (, T; L (, t if (3 9, 3] or t if (3, ]. A a corollary, we alo improve the regularity criteria p expreed by the regularity of or u 3. Introduction We conider the incompreible Navier Stoke equation in the full three-dimenional pace, i.e. } u + u u ν u + p = f. (. t in (,T, div u = u(,x = u (x in, where u : (,T i the velocity field, p : (,T R i the preure, f : (,T i the given external force, ν > i the vicoity. In what follow, we conider ν = and f. The value of the vicoity doe not play any role in our further conideration. We could alo eaily formulate uitable regularity aumption on f o that the main reult remain true. However, it would partially complicate the calculation, thu we kip it. The exitence of a weak olution to (.. (provided u and f atify certain regularity aumption i well known ince the famou paper by Leray []. It regularity and Mathematic Subject Claification (. 35Q3 Keyword. Incompreible Navier Stoke equation, regularity of olution, regularity criteria

2 uniquene remain till open. However, many criteria enuring the moothne of the olution are known. The claical Prodi Serrin condition (ee [7], [8] and for = 3 [7] ay that if the weak olution u additionally belong to L t (,T;L (, + 3 =, t [3, ], then the olution i a regular a the data allow and unique in the cla of all weak olution atifying the energy inequality. Similar reult on the level of the velocity gradient, i.e. u L t (,T;L (, + 3 =, t [3, ], i due to Beirão da Veiga (ee []. Note that the cae = 3 i a conequence of the Sobolev embedding theorem and [7]. Later on, criteria jut for one velocity component appeared. The firt reult in thi direction i due to Neutupa, Novotný and Penel [], where the author howed that if u 3 L t (,T;L (, + 3 =, (6, ], then the olution i mooth. Similar t reult, for the gradient of one velocity component, i independently due to Zhou [9] and Pokorný [5]. Further criteria, including everal component of the velocity gradient, preure or other quantitie can be found e.g. in [4], [6], [5], [3], [6], [], [3].... Recently, two intereting improvement appeared. In [9], Kukavica and Ziane proved that if u 3 L t (,T;L (, + 3 = 5, t 8 (4, ] (the author claim the 5 reult for 4, but their technique doe not work for the cae 5 L (,T;L ( or if u 3 L t (,T;L (, + 3 =, t 6 [54, 8 ], the weak olution i regular. Next, 3 5 in [4], Cao and Titi ued different method, intead of technical etimate they applied the multiplicative embedding theorem and howed the moothne under the aumption u 3 L t (,T;L ( (actually, they work for the pace periodic boundary condition, but the proof for the Cauchy problem i exactly the ame, + 3 < +, > 7. Note that t 3 3 thi reult i tronger than the reult in [9], even though the criterion doe not correpond the natural caling of the Navier Stoke equation. Although the reult of Cao and Titi i the tronget one among all the criteria for one velocity variable, it eem that the author did not ue all the poibilitie of their method. In thi hort note we want to extend their reult in everal apect. Firt, we how that for the regularity of the weak olution it i enough to aume u 3 L t (,T;L (, t , > 3 (or the norm in L (,T;L 3 ( i ufficiently mall, which i an improvement of the reult by Cao and Titi. Next, uing a very imilar method, we will how that it i enough to have 9 u 3 L t (,T;L (, t ( 3, 9, 3] 3 + 3, (3, ], 4 (or again the norm in L (,T;L 3 9( ufficiently mall. Note that for < 3 it preciely correpond to the reult for u 3, uing the Sobolev embedding theorem. The reult for u 3 will be further applied to improve the regularity criteria for p from [4] and for u 3 from [4]. (There, u 3 L ((,T wa required. Note finally that for 3+, , i.e. we are not o far away from + 3 = which enure the regularity if t u L t (,T;L ( (but alo if only either 3 u belong to thi pace ee [] or only u x and u 3, ee [4].

3 In the whole paper, we will ue the tandard notation for Lebegue pace L p ( endowed with the norm p and for Sobolev pace W k,p ( endowed with the norm k,p. We do not ditinguih between the pace X and their vector analogue X N, however, all vector- and tenor-valued function are printed boldfaced. Main reult The aim of thi paper i to how the following t Theorem Let u be a weak olution to the Navier Stoke equation correponding to u W, div (R3 which atifie the energy inequality. Let additionally u 3 L t (,T;L (, + 3 t 3 4 +, > 3. Then u i a mooth a the data allow, thu in our cae u C ((,T and u i unique in the cla of all weak olution atifying the energy inequality. t Theorem Let u be a weak olution to the Navier Stoke equation correponding to u W, div (R3 which atifie the energy inequality. Let additionally u 3 L t (,T;L (, + 3 t 9 + ( 3, 9, 3], 3 + 3, (3, ]. 4 Then u i a mooth a the data allow, thu in our cae u C ((,T and u i unique in the cla of all weak olution atifying the energy inequality. In the proof we will follow the idea from [4]. Firt, we know that there exit T > uch that on (,T there i a trong olution to the Navier Stoke equation, i.e. u L (,T ;W, ( L (,T ;W, ( with u L (,T ;L ( ; actually a t f, u C ((,T ]. Let T < T be the firt time of the blow up, i.e. necearily lim up t T u(t = +. We will how that for any τ < T we have u(τ C < with C independent of T. Thi contradict to the definition of T and thu T = T. 3 Proof of Theorem t Denote. (3. J (t = up h u(τ + τ (,t with. (3. h u = ( u x, u x 3 h u(τ dτ,

4 and.3 (3.3 V (t = ( u 3 (τ 8 3 u(τ dτ, 3, ], (i.e. V = u 3(τ 8 3 u(τ dτ. Then, teting (.. (recall ν = and f = by u = u i= lead to x i d.4 (3.4 dt hu(t + h u(t = (u u udx. Then.5 (3.5 We have (u u udx R 3 = u i u j dx + i,j= R i = J + J + J 3. J = = i,j= u 3 u 3 i= u 3 u i u i dx R 3 x j x j u i u i u 3 x i u 3 dx + u i dx + u 3 x j x j R ( 3 u u + u u dx, x x x x j= u 3 u j dx u 3 u u dx + R 3 x x ( u u + u u dx x x x x u 3 u u dx R 3 x x ee e.g. [9]. It expree the well-known fact that for u W, div (R W, (R, R (u u udx =. Further and thu Further J = i= k= u i u 3 u 3 dx = R k x i x k i= k= u 3 u i u 3 dx R i x k x k J + J C u 3 h u h u dx C u 3 h u h u C u 3 h u +3 h u 3 4 hu + u 3 h u. J 3 C u 3 u h u dx C u 3 h u u 3 u 3 6. Exactly a in [4] we apply the multiplicative Gagliardo Nirenberg inequality u 6 C u 3 u 3 u 3 x x 4

5 which lead to J 3 C u 3 h u + u 3 u C u 3 ( u u + 4 hu. Thu, integrating ( over time interval (,t, t < T and uing etimate above yield +C hu(t + u 3 (τ h u(τ dτ h u( + h u(τ dτ + C u 3 (τ h u(τ dτ ( u(τ u(τ dτ, i.e., taking the upremum over time interval in the firt term and uing Hölder inequality.6 (3.6 J (t K + C u 3 (τ ( +C u 3 (τ h u(τ dτ ( R3 h u(τ dτ u(τ dτ, where K = K ( u,. Up to now, the proof jut copied the proof from [4], with poibly different notation and lightly different argument. Next we ue, a in [4], a tet function u. However, we etimate the convective term more carefully. We have d dt u(t + u(t = (u u u. Now We have (u u u = R 3 = u 3 u j dx + j= R 3 = K + K + K 3. K = K = K 3 = j= k= i= i= j= u 3 dx + R k x k j= u i dx + j= k= R k x i x k u 3 dx = R 3 j= u i x i u j dx + j= k= u j u 3 dx R 3 j= x 3 u 3 dx, R 3 x k x k u i dx, j= k= R i x k x k ( u + u uj dx. x x i= Thu u(t + C t u(τ dτ C h u u dxdτ + u h u(τ u(τ u(τ 3 6 dτ + u. 5

6 Uing again the multiplicative embedding theorem yield u(t + u(τ dτ K + C h u L (,t;l ( h u L (,t;l ( u L (,t;l ( ( K + CJ (t u(τ 4 dτ. ( u(τ 4 dτ Now, we can ue the etimate of J(t from ( and get (the ret of the proof follow again the approach from [4] ( u(τ 4 dτ J (t ( + u 3 (τ u(τ dτ [ C K + ( u(τ dτ u 3 (τ ]( u(τ dτ u(τ 4 dτ Now, let β. Uing Hölder and Young inequality we get ( ( A CK u(τ 4 dτ + C u 3 (τ β u(τ β dτ ( β ( ( u(τ β dτ u(τ 4 dτ + C u 3 (τ β u(τ dτ ( β ( u(τ β dτ u(τ 4 dτ + ( CK + C u 3 (τ β 4 u(τ 3β dτ ( +C u 3 (τ β 4( u(τ β 3 dτ + u(τ dτ, a + 4( < for > 3. Thu u(t + u(τ dτ K + C ( V (t 4 4( 3β + V, β(3 = A. β which allow to ue the Gronwall lemma provided β = 4(. Note that the other condition, β 4 β(3, i le retrictive. A β = 8, we get that u(t 3 3 i bounded independently of T, provided u 3 L 8 3 (,T;L (, >. Finally, if u 3 3 L (,T;L << 3 (, we can tranfer the correponding term to the left-hand ide. The proof of Theorem i finihed. 4 Proof of Theorem The proof i imilar to the proof of Theorem. t We replace ( by ( u 3 (τ u(τ dτ,.7 (4. V (t = 9, 3], u 3 (τ u(τ dτ, (3, ] 6

7 with the tandard convention V (t = u 3(τ 4 3 u(τ dτ. A before, firt we multiply (.. by u and integrate over Ω. We get a before d dt hu(t + h u(t = (u u udx. But (u u udx = u 3 u i u i u 3 u u u 3 u u dx dx + dx i,j= R 3 x j x j R 3 x x R 3 x x u i u 3 u 3 dx + u 3 u j dx. R k x i x k R 3 i= k= The firt four term can be etimated by j= u 3 h u dx u 3 h u while the lat term j= u 3 u j dx = ( j,k= C u 3 h u + 4 hu, u 3 dx + R k x k The econd term can be etimated a above, and the firt term j,k= u 3 dx R k x k u 3 u j dx. R 3 x k x k C u 3 h u u dx = B. Now we etimate eparately B for 3 and > 3. We have a [ 3, 3]: B C u 3 h u 6 u C u 3 h u + 3 C u 3 h u u 3 u u 3 where we ued the multiplicative embedding theorem. Thu It yield h u(t + B 4 hu + C u 3 6 +C t h u(τ dτ K + C u 3 (τ ( u 6( 5 6 3, u u 3 (τ u 3 6 ( (3 5 6 u(τ u(τ dτ. 7 u(τ dτ

8 for 3, we get due to the Hölder inequality after taking the upremum over time on the left-hand ide A 6( 5 6.7a (4. J (t K + C ( C u 3 (τ u(τ dτ u 3 (τ 3( u(τ dτ ( u(τ 5 6 dτ. b (3, ]: We etimate the convective term differently B u 3 h u u dx C u 3 u h u C u 3 u h u 3 h u hu + C u 3 u. Thu J (t K + C u 3 (τ u(τ dτ. Next we ue a tet function u. We get d dt u(t + u(t = (u u udx R 3 = u i u j dx + u 3 u j dx i= j= R i j= R 3 u i = dx + i= j= k= R k x i x k i= j= k= u 3 dx + j= k= R k x k j= k= C h u u dx + C u 3 u dx = D + D. We have a before D u + C u 3 u, while uing the multiplicative embedding theorem u i dx R i x k x k u 3 dx R 3 x k x k D h u u 4 h u u u 3 6 C h u u h u u, i.e. u(t + +C t u(τ dτ K + C u 3 (τ h u(τ u(τ h u(τ u(τ dτ. u(τ dτ 8

9 The lat term can be etimated h u(τ u(τ h u(τ u(τ dτ ( h u L (,t;l ( h u L (,t;l ( u t L (,t;l ( u(τ 4 dτ ( CJ (t u(τ 4 dτ. We may therefore employ etimate of J (t and get eparately a 3: u(t + u(τ dτ K + C u 3 (τ u(τ dτ [ ( +C K + u 3 (τ t u(τ dτ + u 3 (τ 3( u(τ 5 6 dτ ( 3 ]( u 5 6 dτ u(τ 4 dτ. Again, for β we have Thu u(t + u(τ dτ CK + u(τ dτ ( +C u 3 (τ β ( u(τ β dτ + C u 3 (τ β u(τ dτ ( +C u 3 (τ β ( u(τ 9 β dτ. A ( u(t + u(τ dτ CK + CV β + CV 3 4β + CV ( for (3, 3], we have for = β 9 9 u(t + u(τ dτ K + CV (t, ( 9 β. where K = K( u,,β. Thu the Gronwall inequality implie that u(t C < for any t < T. A in Theorem, t if = 3, we need u 9 3 L (,T;L 3 <<. 9 ( Similarly, b > 3: u(t + u(τ dτ K + C u 3 (τ u(τ dτ ( ( +C K + u 3 (τ t u(τ dτ u(τ 4 dτ ( CK + C u 3 (τ u(τ β dτ ( +C u 3 (τ 4 u(τ 3β dτ + u(τ dτ β

10 Thu, if β = 4 3, 4 = , we have u(t + u(τ dτ K + CV, with K = K( u,,β. The Gronwall lemma finihe the proof of Theorem t. 5 Two additional criteria Note that we proved that if for a q > 3 we have u 3 L (,T;L q ( <, then the olution to the Navier Stoke equation i regular. Thi fact enable u to prove the following two corollarie: c Corollary 5. Let u be a weak olution to the Navier Stoke equation correponding to u W, div (R3 which atifie the energy inequality. Let additionally Then the olution i regular. u 3 L t (,T;L (, t + 3 < 4 ( 5 5, 4, ]. r Remark 5. In [4] i ha been proved that the regularity i enured if u 3 L ((,T. Thi reult i thu quite a big improvement of thi reult. Proof. We proceed a before, i.e. we work on the time interval where the olution i mooth. We tet the equation for u 3 by u 3 q u 3, q >. Then 3 d q dt u 3 q q + C(q u 3 q dx = (q p u 3 u 3 q dx. R 3 We need to etimate the right-hand ide. We have p u 3 u 3 q dx u 3 u 3 q q p R 3 x q C u 3 u q 3 q q u q 3 C u 3 u 3 q 6q q 3q 6+3q ( q q q u u 6 C u 3 q q u 6 + u 3 where 3q 6+3q 3q 6 q 5 4 q q 6q q, i.e. 3q. Thu, paing with q + q 6 q 3 6q q 4 5. The corollary i proved. r Remark 5. Indeed, in the proof of Corollary 5., c intead of the etimate of u 3 in L q (, we could conider the etimate in L 3q (, or in between. It would lead to the ame reult. We could alo ue part of the information from the firt energy etimate to increae the range of. However, thi would lead to wore (i.e. more regular cale for u 3, thu we omit it. Next reult concern the criterion on p. We get lightly better reult than in [4], due to the fact that we improved the criterion on u 3. We have +, q, +,

11 c Corollary 5. Let u be a weak olution to the Navier Stoke equation correponding to u W, div (R3 which atifie the energy inequality. Let additionally Then the olution i regular. p L t (,T;L (, t + 3 < 9 ( 3, 3, ]. 3 Proof. We proceed a above, only in the term on the right-hand ide we do not integrate by part. Thu we have d q dt u 3 q q + C(q u 3 q p dx = u 3 q u 3 dx. R 3 Now (note that q (q 3q 3q implie q q+ p u 3 q u 3 dx p u 3 q R 3 p u 3 q+q q u 3 3(q 3q ε u 3 q 3q + C(ε p Thu q d dt u 3 q q C p q q+3q (q q q+3q u 3 q q+q q+3q q. u 3 q q+q q+3q q. The Gronwall inequality implie the reult, provided p L t (,T;L ( with + 3 = t + 3, 3q q. Paing with q + we get the reult. q q+ 3 Acknowledgment. The work of the firt author i a part of the reearch project MSM 6839 financed by MSMT and partly upported by the grant of the Czech Science Foundation No. /8/35 and by the project LC65 (Jindřich Neča Center for Mathematical Modeling. The econd author.... Reference BV [] H. Beirão da Veiga, A new regularity cla for the Navier Stoke equation in R n, Chin. Ann. Math. Ser. B 6 ( BVBe [] H. Beirão da Veiga, L. Berelli, On the regularizing effect of the vorticity direction in incompreible vicou flow, Differential Integral Equation 5 (, No. 3, BeGa [3] L. Berelli, G.P. Galdi, Regularity criteria involving the preure for the weak olution to the Navier-Stoke equation, Proc. Am. Math. Soc. 3 (, No., CaTi [4] C. Cao, E.S. Titi, Regularity criteria for the three dimenional Navier Stoke equation, Indiana Univ. Math. J. 57 (8 No. 6, ChCh [5] D. Chae, H.J. Choe, Regularity of olution to the Navier Stoke equation, Electron. J. Differ. Equ. 5 (999 7.

12 CoFe [6] P. Contantin, C. Fefferman, Direction of vorticity and the problem of global regularity for the Navier-Stoke equation, Indiana Univ. Math. J. 4 (993, No. 3, ESeSv [7] L. Ecauriaza, G. Seregin, V. Šverák, Backward uniquene for parabolic equation, Arch. Ration. Mech. Anal. 69 (3 No., Ga [8] G.P. Galdi, An introduction to the Navier-Stoke initial-boundary value problem, Galdi, Giovanni P. (ed. et al., Fundamental direction in mathematical fluid mechanic, Bael: Birkhuer, -7,. KuZi [9] I. Kukavica, M. Ziane, Navier-Stoke equation with regularity in one direction, J. Math. Phy. 48 (7 No. 6, 653, pp. KuZi [] I. Kukavica, M. Ziane, One component regularity for the Navier-Stoke equation, Nonlinearity 9 (6 No., Le [] J. Leray, Sur le mouvement d un liquide viqueux empliant l epace, Acta Math. 63 (934, NeNoPe [] J. Neutupa, A. Novotný, P. Penel, An interior regularity of a weak olution to the Navier-Stoke equation in dependence on one component of velocity, Topic in mathematical fluid mechanic, Quad. Mat. ( 63 83; ee alo A Remark to Interior Regularity of a Suitable Weak Solution to the Navier Stoke Equation, CIM preprint No. 5 (999. NePe [3] J. Neutupa, P. Penel, Aniotropic and geometric criteria for interior regularity of weak olution to the 3D Navier Stoke Equation, in: Mathematical Fluid Mechanic (Recent Reult and Open Problem, J. Neutupa, P. Penel (ed., Advance in Mathematical Fluid Mechanic, Birkhäuer, PePo [4] P. Penel; M. Pokorný, Some new regularity criteria for the Navier Stoke equation containing the gradient of velocity, Appl. Math. 49 (4 No. 5, ; ee alo preprint of Charle Univerity MATH-MU-9/. Po [5] M. Pokorný, On the reult of He concerning the moothne of olution to the Navier- Stoke equation, Electron. J. Differ. Equ. (3 8. Po [6] M. Pokorný, A hort note on regularity criteria for the Navier Stoke equation containing the velocity gradient, Banach Center Publication, Vol. 7: Regularity and other apect of the Navier-Stoke Equation ( Pr [7] G. Prodi, Un teorema di unicità per el equazioni di Navier Stoke, Ann. Mat. Pura Appl. IV 48 ( Se [8] J. Serrin, The initial value problem for the Navier Stoke equation, in: Nonlinear Problem, Langer R.E. (ed., Univerity of Wiconin Pre, 963. Zh [9] Y. Zhou, A new regularity reult for the Navier-Stoke equation in term of the gradient of one velocity component, Method Appl. Anal. 9 ( No. 4,

A New Regularity Criterion for the 3D Navier-Stokes Equations via Two Entries of the Velocity Gradient

Acta Appl Math (014) 19:175 181 DOI 10.1007/s10440-013-9834-3 A New Regularity Criterion for the 3D Navier-Stokes Euations via Two Entries of the Velocity Gradient Tensor Zujin Zhang Dingxing Zhong Lin