DEMONSTRATIO MATHEMATICA Vol. XLVIII No 1 015 Adam Kuica A REGULARITY CRITERION FOR POSITIVE PART OF RADIAL COMPONENT IN THE CASE OF AXIALLY SYMMETRIC NAVIER-STOKES EQUATIONS Communicated y K. Chełmiński Astact. We examine the conditional egulaity of the solutions of Navie Stokes euations in the entie thee-dimensional sace unde the assumtion that the data ae axially symmetic. We show that if ositive at of the adial comonent of velocity satisfies a weighted Sein tye condition and in addition angula comonent satisfies some condition then the solution is egula. 1. Intoduction We will conside the Navie Stokes euations in entie thee-dimensional sace Bu Bt ` u u ν u ` 0 in 0 T ˆ R3 (1) div u 0 in 0 T ˆ R 3 u0 x u 0 x in R 3 whee u : 0 T ˆ R 3 Ñ R 3 is the velocity field : 0 T ˆ R 3 Ñ R is the essue 0 ă T 8 ν is the viscosity coefficient u 0 is the initial velocity and the focing tem is fo the sake of simlicity consideed to e zeo. U to now it is not known whethe euations (1) have gloal in time smooth solutions. In this ae we analyze the secial class of solutions which ae axially symmetic i.e. ae in the fom ut x u t ze ` u t ze ` u z t ze z whee a x 1 ` x e x 1 x 0 e x x 1 0 and e z 0 0 1 hence cylindical coodinates u u u z do not deend on the angle. In this 010 Mathematics Suject Classification: 35Q30 76D05. Key wods and hases: Navie Stokes euations axial symmety egulaity citeia weighted saces. DOI: 10.1515/dema-015-0006 c Coyight y Faculty of Mathematics and Infomation Science Wasaw Univesity of Technology Download Date 1/6/18 7:49 PM
A egulaity citeion fo ositive at of adial comonent... 63 case euations (1) have a simle fom. Howeve the issue of existence of smooth axially symmetic solutions is still oen and thee ae only atial esults. The fist of them deal with flows without swil (i.e. u 0) and thee is oved that solutions emain smooth if the data wee smooth (see [5] [11] and [6] fo anothe oof). In the case of u 0 thee ae many conditional esults. Let us mention some of them: thee is no low-u solutions if in addition one of the following conditions is satisfied () u P L 0 T ; L s Ω with ` 3 1 fo s ą 3 ([8]) s (3) u P L 0 T ; L s Ω with ` 3 ă 1 fo s ą 4 ([9]) s (4) u P L 0 T ; L s Ω with ` 3 s fo s ą 3 ([1]) (5) u P L 8 0 T ; L 8 Ω ([] ) (6) d P L 0 T ; L s Ω with ` 3 s ` d 1 (7) d u P L 0 T ; L s Ω with ` 3 s ` d ă 1 fo s ą 3 ą 1 and d P 1 1 ([4]) fo s ą 4 ą and d P (8) d u P L 8 0 T ; L 8 Ω with d ă 5 6 ([4]) 0 s 4 ([4]) s whee maxt u 0u. We will denote Ω δ1 tx P R 3 : ă δ 1 u and u` u` is ositive at of adial comonent. Ou main esult is following Theoem 1. Let u e a weak solution to olem 1 satisfying the enegy ineuality with u 0 P W R 3 u 0 P L 8 R 3 and 1 ` u 0 P L 1 R 3. Let u 0 e axisymmetic. If in addition u` a ositive at of adial comonent of velocity satisfies (9) d u` P L w 0 T ; L s Ω δ1 fo some s P 3 8 w P 1 8 and d P 1 1 such that w ` 3 s ` d 1 and fo some ositive δ 1 and (10) 1 δ 0 u P L 8 0 T ˆ R 3 fo some ositive δ 0 then u whee is the coesonding essue is axially symmetic stong solution to olem 1 which is uniue in the class of all weak solutions satisfying the enegy ineuality. Download Date 1/6/18 7:49 PM
64 A. Kuica Remak 1. If u 0 P L 8 R 3 then weak solutions of (1) satisfies u P L 8 0 T ; R 3. Hence the assumtion (10) is aitay close to this oeties of weak solutions. Remak. It is woth to mention that the nom }u} L ws } d u} d L w R`;L s R 3 is scaling invaiant if and only if w ` 3 s `d 1. Indeed fo such exonents we get }u} L ws }u d λ } L ws whee u d λ t x λuλt λ x. Theefoe conditions () (4) (5) (6) and (9) involve scaling invaiant noms. In ode to ove Theoem 1 we wite the euations (1) in cylindical coodinates (11) u t ` u u ` u z u z 1 1 u ` ν `u ` u zz u 0 (1) u t ` u u ` u z u z ` 1 1 u u ν `u ` u zz u 0 1 `uz (13) u zt ` u u z ` u z u zz ` z ν ` u zzz 0 (14) u ` u ` u zz 0 whee the last one is a continuity euation. If we denote ω cul u then in cylindical coodinates we have ω u z ω u z u z ω z u ` u. Theefoe the euations fo ω in cylindical coodinates ae following 1 `ω (15) ω t`u ω `u z ω z u ω u z ω z ν `ω zz ω 0 (16) ω t`u ω `u z ω z u ω ` u 1 ν ω `ω `ω zz ω 0 1 `ωz (17) ω zt`u ω z`u z ω zz u z ω u zz ω z ν `ω zzz 0. We will ove Theoem 1 y contadiction. Theefoe suose that 0ăt ăt is the time of the fist low-u of solution i.e. the smalle ositive nume such that su tp0t } ut } L R 3 8. Then fo 0 ă t ă t the euations (11) (17) ae satisfied in 0 t ˆ R 3 in stong sense. We will show that if u` and u satisfy assumtions of Theoem 1 then } ut } L R 3 emains unifomly ounded fo t P 0 t and we will get contadiction. We shall otain a unifom estimate fo t P 0 t. Fo this uose we multily euation (1) y u u m u and integate ove R 3. Then µ Download Date 1/6/18 7:49 PM
A egulaity citeion fo ositive at of adial comonent... 65 integating y ats and continuity euation (14) yield (18) 1 d u 4 1ν dt µ ` u µ u ` ν1 µ Next we multily euation (16) y get (19) 1 1 µ ` 1 ` µ ω ω α α d ω 4ν 1 dt α ` ω α ω ` ν1 α 1 α ` 1 α 1 α u` Summing u aove eualities we otain (0) u u` u µ 1 ` µ u. µ and then in a simila way we ω α ` ω α u ω zω α α. u 1 d u dt µ ` 1 d ω 4 1ν dt α ` u µ 4ν 1 ` ω α ` ν1 µ ` 1 ` µ u` u ` 1 α ω µ α u ω zω α ` 1 ` µ α u µ ` 1 α u` ω I1 α ` 1 ` µi ` 1 αi 3. u µ 1 ` ν1 α ω α 1 We will show that fo some exonents α and µ the ight hand side can e estimated.. Estimate of I 1 Poosition 1. Assume that γ P 0 3 P 4 γ 4 γ µ P 1 1 and a P 0 1. Then fo ε 1 ε ε 3 ą 0 the following estimate Download Date 1/6/18 7:49 PM
66 A. Kuica (1) u u z u 1 ω I 1 ε 1 ` ε µ µ µ ` ε 1 ω 3 α ` C α holds whee () α µ γ 1 1 ` µ 1 a and C Cγ a ε 1 ε ε 3. Poof. Using Young ineuality we get u ω u I 1 u z ω α α 1 u z u µ µ u u z u 4 ω ε 1 ` C1{ε1 µ µ µ 1`α u s u z ε 1 ` C1{ε1 µ µ u γ µ 1`α α 1 1 ω α u 4 γ ˆ ω 1a ω 11 a 4 1 4 1 `α µ`γ a α 1a` a α 11 a. It is well known that (18) leads to the estimate }u } L 8 }u 0 } L 8 thus alying this estimate and next Young ineuality with exonents 1a 11 a we otain I 1 ε 1 u µ u z µ ` ε ` ε 3 ω α 1 ` C ω α u 4 γs 4 1 `α µ`γ as whee C deends on a ε 1 ε ε 3 and }u 0 } L 8. By simle calculations we get 4 γs and 4 1 ` α µ ` γ as µ `. 3. Estimate of I We egin with the following Remak 3. Assume that P 1 8 α and ε 0 satisfy ` ε 0 ă α ă ε 0. Then thee exists a constant C C α ε 0 such that u 1 ω (3) 1`α ε 0 C 1 α ε 0. Poof. Fom Lemma. [3] we have } u } c}ω } fo all P 1 8. Thus we only have to veify that α` ε0 is A weight (see comment Download Date 1/6/18 7:49 PM
A egulaity citeion fo ositive at of adial comonent... 67 efoe Lemma.6 [3]). In view of Examle 1..5 [10] it holds if i.e. ` ε 0 ă α ă ε 0. ă α ` ε 0 ă 1 Poosition. Assume that ϖ } 1 δ 0 u } L 8 ă 8 fo some δ 0 P 0 1 6. Then fo all γ P 0 3 P 4 γ a P 1 4 γδ 0 4 1 4 4 γ δ 0 X 0 1 (4) µ P δ 0 1 δ 0 ` γ X 1 1 4 γ and fo ε 4 ε 5 P 0 1 the following estimate holds u 1 (5) I u ε4 µ µ ` ε 5 whee and 4 γ α µ γ C Cε 4 ε 5 a δ 0 γ ϖ. ω α 1 ` C 1 1 ` µ 1 a ω α Poof. We denote κ 1 1 a. Then the assumtion concening u and Young ineuality with exonents 1 yield I u µ u µ` 1 ε 4 u µ 1 ` C γ ϖ ε 4 1 δ 0 u 1`α` κ δ 0 u 1 1`α κ δ 0. We define ε 0 y euality κ δ 0 4 γ δ 0 1 1 as ε 0. Then the assumtion (4) leads to ` ε 0 ă α ă ε 0 hence we can aly Remak 3 and we get u 1 ω I ε 4 µ ` C γ ϖ ε 1 4 a µ α ε 0. Using the assumtion on a we deduce that 1 ε 0 satisfies P 0 1 hence we can aly Young ineuality with exonents 1 1 1 and then we get ω 1 α ε 0 ω 1 ω 1 ω α` ω ε5 α α ` Cε 5. α Download Date 1/6/18 7:49 PM
68 A. Kuica Poositions 1 and immediately give Coollay 1. Assume that ϖ } 1 δ 0 u } L 8 ă 8 fo some δ 0 P 0 1 6. Then fo all γ P 0 3 P 4 γ a P 1 4 γδ 0 4 1 4 4 γ δ 0 X 0 1 (6) µ P δ 0 1 δ 0 ` γ X 1 1 4 γ and fo ε 1 ε ε 3 P 0 1 the following estimate holds u 1 ω (7) I 1 ` I ε 1 µ ` ε 1 α ` ε 3 u ω µ ` C α whee 4 γ α µ γ 1 1 ` µ 1 a and C Cε 1 ε ε 3 a δ 0 γ ϖ. The aove estimate involve many exonents and it is not clea at once whethe we can get the estimate with useful ange of exonents. Theefoe we fomulate Coollay. Assume that ϖ } 1 δ 0 u } L 8 ă 8 fo some δ 0 P 0 1 6. Then fo ε P 0 1 7 such that (8) 41 ε1 εε δ 0 and fo all ε 1 ε ε 3 P 0 1 the following estimate holds u 1 ω (9) I 1 ` I ε 1 µ ` ε 1 α ` ε 3 u µ ` C ω α whee 1 ε 1 ε µ 1 ε 1`ε and α 1 ε1 ` εε and C Cε 1 ε ε 3 δ 0 ϖ ε. In aticula fo such exonents we have u ω (30) c α 1`α α and (31) 8 ` 3 1 ` α 1 1 ` 7ε ă 1. Poof. We have to veify the assumtions of Coollay 1. Theefoe we ut γ 1 ε. Then γ P 0 3 and P 4 γ and we set a 1 1 ε ε. Then condition (8) imlies a P 1 4 γδ 0 4 1 4 4 γ δ 0. Finally δ 0 is ositive hence µ 1 ε 1`ε satisfies (6) thus assumtions of Coollay 1 ae satisfied and we get (7) with 4 γ 1 ε and α µ γ 1 1`µ 1 a 1 ε1 ` εε. Download Date 1/6/18 7:49 PM
A egulaity citeion fo ositive at of adial comonent... 69 In ode to get (30) we have to veify that α is A weight. It is euivalent to ă α ă and holds tue ecause ε is small enough. Finally y diect calculations we otain ineuality (31). 4. Estimate of I 3 Poosition 3. Assume that s P 3 8 w P 1 8 and d P 1 1 ae such that w ` 3 s ` d 1. If P 1 8 α P 1 1 and δ 1 ą 0 then fo all ε 4 ε 5 P 0 1 the following estimate ω 1 (3) I 3 ε 4 α ` ε 5 ω ω α ` Cft ` gts α holds whee and gt j w u` 10 3 ft Ω d u` s s dx 1 C Cε 4 ε 5 δ 1 s w. Remak 4. Recall that Ω δ1 denotes tx P R 3 : ă δ 1 u. It is known that if u is weak solutions of (1) then with the hel of Soolev emedding theoem we deduce that u P L 10 3 0 T ˆ Ω hence function gt is integale on 0 T. Howeve u to now thee is no oof of integaility of ft on 0 T and its integaility is ou main assumtion in oving smoothness of axially symmetic solutions. Fo weak solutions we have u P L 8 0 T ; L Ω 1 and u P L 0 T ; L Ω 1 i.e. the exonents satisfy too weak conditions: 8 ` 3 3 and ` 3 1 3. Poof. Let η η e smooth cut off function such that η 1 fo ă δ 1 { and η 0 fo ą δ 1. Then we can wite I 3 ηu` 1 ω ` ηu` α ω α I31 ` I 3. We fist estimate I 31. We set a w ` 3 s w s ą 3 and alying Young ineuality with exonents a ` 3. Then a ą 1 and a a 1 we otain I 31 ηu` ω a 1 ω α α ηu` a a 1 a a ω a α ω 1 ε 1 α ` cε 1 a ηu` a a 1 a a 1 ω. α To estimate the last integal on the ight hand side we use twice Hölde ineuality with exonents and 3 Download Date 1/6/18 7:49 PM
70 A. Kuica ηu` a a 1 a a 1 ηu` ηu` ω α ηu` a a 1 a a 1 a a 1 a a 1 j ω j 1 ω 3 3 ε ` cε α a a 1 a a 1 α j ω α ηu` 3 j 3 j ω α ω α 3 a a 1 a a 1 j ω α 3 whee we also alied Young ineuality with exonents 3 a a we have a 1 s a 1 ds and 3 w s thus (33) I 31 ε 1 ω α 1 ` ε ` cε 1 ε w sft ` ε ` cε 1 ε ω 3 α j 1 3 ω α j 1 j j 3 ω α j. j 3. By definition In ode to estimate I 3 we ut a 4 and 5 and oceeding analogously we get j 1 ω 1 ω 3 3 (34) I 3 ε 1 α α 1 ηu` 10 3 5 3 j j ω α. Clealy ş 1 ηu` 10 3 5 3 {δ 1 5{3 ş u` 10 3. Finally alying Soolev emedding theoem in estimates (33) and (34) we get (3). Coollay 3. Assume that s P 3 8 w P 1 8 and d P 1 1 ae such that w ` 3 s ` d 1 and ϖ }1 δ 0 u } L 8 ă 8 fo some δ 0 P 0 1 6 and δ 1 is ositive. Then fo all ε P 0 1 14 such that (35) 41 ε1 εε δ 0 the following estimate d (36) u dt µ ` d ω 4 1ν dt α ` u µ 4ν 1 ` ω α ` ν1 µ ` ν1 α u µ 1 ω 1 α C1 ` ft ` gts ω α Download Date 1/6/18 7:49 PM
A egulaity citeion fo ositive at of adial comonent... 71 holds whee 1 ε 1 ε µ 1 ε 1`ε and α 1 ε1 ` εε gt ş u` 10 3 ft ş Ω 1 d u` s dxs w s and C Cν ε δ 0 δ 1 ϖ s w. In aticula if ft is integale on 0 t then (37) su ω α C 1 u µ 0 ` ω α 0 tp0t whee C 1 C 1 C }f} L 1 0T }u 0 } L R 3. Poof. The assumtions of Coollay and Poosition 3 ae satisfied hence we get (36). If ft is integale on 0 t then y Gonwall lemma we otain (37). Poof of Theoem 1. Unde the assumtions of Theoem 1 we get estimate (37) whee the ight hand side is finite y the assumtion concening u 0. Fo such and α ineuality (30) holds thus we deduce that u 1`α j su tp0t is finite whee exonents and α satisfy (31). Theefoe we can aly Theoem 1(i) ([4]) and we deduce that u is egula on 0 t i.e. thee is no low-u at time t t which gives the contadiction. Remak 5. The condition (9) can e weakened a it. Namely it is enough to assume that ş (38) t ÞÑ ft Ω 1 d u` s dx w s 1 ` ln` ` u µ ` ω is integale on 0 T α whee d w s ae as in Theoem 1. Indeed fom (36) we have ˆ d u dt µ ` ω ˆ u α C1 ` ft ` gts µ ` ω α hence aguing similaly as in [7] we can wite ˆ d dt ln u 1 ` ln` µ ` ω j 1 ` ft ` gts α C 1 ` ln` ` u µ ` ω. α Acknowledgements. This wok has een suoted y the Euoean Union in the famewok of Euoean Social Fund though the Wasaw Univesity of Technology Develoment Pogamme ealized y Cente fo Advanced Studies. Refeences [1] D. Chae J. Lee On the egulaity of the axisymmetic solutions of the Navie Stokes euations Math. Z. 39(4) (00) 645 671. Download Date 1/6/18 7:49 PM
7 A. Kuica [] C. C. Chen R. M. Stain T. P. Tsai H. T. Yau Lowe ounds on the low-u ate of the axisymmetic Navie Stokes euations. II Comm. Patial Diffeential Euations 34(1 3) (009) 03 3. [3] O. Keml M. Pokoný A egulaity citeion fo the angula velocity comonent in axisymmetic Navie Stokes euations Electon. J. Diffeential Euations 007 No. 08 10. [4] A. Kuica M. Pokoný W. Zajączkowski Remaks on egulaity citeia fo axially symmetic weak solutions to the Navie Stokes euations Math. Methods Al. Sci. 35(3) (01) 360 371. [5] O. A. Ladyzhenskaya Uniue gloal solvaility of the thee-dimensional Cauchy olem fo the Navie Stokes euations in the esence of axial symmety Za. Nauchn. Sem. Leningad. Otdel. Mat. Inst. Steklov. (LOMI) 7 (1968) 155 177. [6] S. Leonadi J. Málek J. Nečas M. Pokoný On axially symmetic flows in R 3 Z. Anal. Anwendungen 18 (1999) 639 649. [7] S. Montgomey-Smith Conditions imlying egulaity of the thee dimensional Navie Stokes euation Al. Math. 50(5) (005) 451 464. [8] J. Neustua M. Pokoný An inteio egulaity citeion fo an axially symmetic suitale weak solution to the Navie Stokes euations J. Math. Fluid Mech. (4) (000) 381 399. [9] M. Pokoný A egulaity citeion fo the angula velocity comonent in the case of axisymmetic Navie Stokes euations Ellitic and Paaolic Polems (Rolduc/Gaeta 001) 33 4 Wold Sci. Pul. Rive Edge NJ 00. [10] B. O. Tuesson Nonlinea Potential Theoy and Weighted Soolev Saces Lectue Notes in Mathematics 1736. Singe-Velag Belin 000. [11] M. R. Uchovskii B. I. Yudovich Axially symmetic flows of an ideal and viscous fluid in the whole sace (in Russian also J. Al. Math. Mech. 3 (1968) 5 61) Pikl. Mat. Mekh. 3 (1968) 59 69. A. Kuica FACULTY OF MATHEMATICS AND INFORMATION SCIENCE WARSAW UNIVERSITY OF TECHNOLOGY Pl. Politechniki 1 00-661 WARSAW E-mail: a.kuica@mini.w.edu.l Received Mach 10 014. Download Date 1/6/18 7:49 PM