COMPONENT REDUCTION FOR REGULARITY CRITERIA OF THE THREE-DIMENSIONAL MAGNETOHYDRODYNAMICS SYSTEMS

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1 Electronic Journal of Differential Equations, Vol. 4 4, No. 98,. 8. ISSN: UR: htt://ejde.math.txstate.edu or htt://ejde.math.unt.edu ft ejde.math.txstate.edu COMPONENT REDUCTION FOR REGUARITY CRITERIA OF THE THREE-DIMENSIONA MAGNETOHYDRODYNAMICS SYSTEMS KAZUO YAMAZAKI Abstract. We study the regularity of the three-dimensional magnetohydrodynamics system, and obtain criteria in terms of one velocity field comonent and two magnetic field comonents. In contrast to the revious results such as [], we have eliminated the condition on the third comonent of the magnetic field comletely while reserving the same uer bound on the integrability condition.. Introduction and statement of results We study the three-dimensional magnetohydrodynamics MHD system u + u u b b + π = ν u, t b + u b b u = η b, t u = b =, u, bx, = u, b x, t R + {},. where u : R R + R reresents the velocity field, b : R R + R the magnetic field, π : R R + R the ressure field, ν, η > the viscosity and diffusivity constants resectively. Hereafter let us assume ν = η = and write t = t and = i and the comonents of u and b by x i u = u, u, u, b = b, b, b, b h := b, b,. Due to the works of [], we know that. ossesses at least one global - weak solution air for any initial data air u, b. However, whether the local solution air remains smooth for all time remains oen as in the case of the Navier-Stokes equations NSE, the system. at b. To show that the weak solution air is actually strong, there has been a large amount of research conducted by many mathematicians to obtain a sufficient condition on u, b so that imosing such conditions lead to the H -norm bound on u, b. We discuss some of them in articular. Mathematics Subject Classification. 5B65, 5Q5, 5Q86. Key words and hrases. Magnetohydrodynamics system; Navier-Stokes equations; regularity criteria. c 4 Texas State University - San Marcos. Submitted December 9,. Published Aril, 4.

2 K. YAMAZAKI EJDE-4/98 Following the ioneering work by Serrin [], Bei rao da Veiga [] obtained regularity criteria on u. Similar results followed for the MHD system; in articular, Zhou [4] showed that it suffices to bound only u droing the conditions on b comletely. For examle, the following regularity criteria was obtained by He and Xin []. u r dτ <, +, <. r In an accomanying aer [9], the author reduced this criteria to any two comonents of u. For the regularity criteria in terms of other quantities for the NSE such as vorticity, π, we refer readers to [,, 7, 4,, 5]. Results related on the reduction of comonents aeared for examle in Kukavica and Ziane [6]: u r dτ <, + r 5 8, 4 5, see also [7,, ]. A few of the most recent results are the following: u r dτ <, + r + <, 7 <, for the NSE see Cao and Titi [4] also [5, 4,,, 6] followed by many in the case of the MHD system e.g. [6,, 8, 5]. In articular, Jia and Zhou [] showed that if u r + b r dτ <, + r 4 +, <,. then the solution air u, b remains smooth cf. [7] for the case of the NSE. More variations of. were also obtained in [, 9]; however, in any case, if condition is given only on u and no other comonent of u, then without a new idea, it seems we need to imose some integrability condition on every comonent of b. This is due to the difficulty in decomosing the four non-linear terms in the h u + h b -estimates so that every term has either have u or b h, where h =,, see the Aendix for details. Now we resent our results. Theorem.. Suose u, b solves. in time interval [, T ] and satisfies u 8/ + b h r dτ <, b h = b, b,,. where / < < and 8 r satisfy + r 4 +, < 5, r = 8, 5 <..4 Then there is no singularity u to time T. The boarder-line case of = / may be obtained as well, via a slight modification of the roof for Theorem.. Theorem.. Suose u, b solves. in time interval [, T ] and u 8/ dτ + τ [,T ] Then there is no singularity u to time T. b h τ / <, b h = b, b,.

3 EJDE-4/98 COMPONENT REDUCTION Theorem.. Suose u, b solves. in time interval [, T ] and satisfies u 8/ + u 8/ + b r dτ <, where / < <, 8 r satisfy.4. Then there is no singularity u to time T. Theorem.4. Suose u, b solves. in time interval [, T ] and u 8/ + u 8/ dτ + Then there is no singularity u to time T. τ [,T ] b τ / <. Remark.5. We may relace the role of u with any other comonent of u as long as the two comonents of b will be the different two comonents. We emhasize that in articular in Theorem., we have eliminated the condition on b comletely while reserving the integrability condition on b h and =, r = 8/ also satisfies.. Thus, it is clear that.4 is an imrovement of the secial case of.. We were also able to obtain results in case when for u in.; however, the integrability conditions became worse; thus, for simlicity, we chose not to resent those results. We also remark that in contrast to results from [], Theorem. is not a smallness result. We also wish to emhasize that reviously when the regularity criteria for the three-dimensional MHD system was obtained in terms of three terms, they have always been all from the velocity vector field; e.g. u, u, u from [6] and [], any three artial derivatives of u, u, u from [5, 8, 5]. 4 The new idea in our roof is to make use of the structure of the magnetic vector field equation and estimate b and obtain its bound in terms of b h and u. Our roof was insired by the others including [7], in articular [8, 9] concerning the [6, Theorems.-.4] and [8, Proositions.-.]. Modification of Proositions.-. are ossible indicating that in the future to obtain a regularity criteria of the MHD system in terms of one comonent of the velocity vector field, which has been done for the NSE but not for the MHD system, it suffices to discover a decomosition of the four non-linear terms that searate u and b, not necessarily just u. 5 After this manuscrit was comleted, the author discovered in [] a new decomosition of the four non-linear terms of 4. which led to a regularity criteria of. in terms of u and j where j is the third comonent of the current density j := b. In the Preliminary section, we set notation. Thereafter, we rove two crucial roositions and then rove Theorems..4.. Preliminary et us denote a constant that deends on a, b by ca, b and A B when there exists a constant c of no significance such that A cb. We shall also denote f = R fxdx, h =,,, h = +, Xt := ut + bt, Y t := h ut + hbt, Zt := ut + bt,

4 4 K. YAMAZAKI EJDE-4/98 M := N := u 8/ dτ, u, 8/ dτ, M := b / + b h t /, t [,T ] N := b, / + b t /, t [,T ] where e.g. b, is a two dimensional vector of two entries b and b. We have the following secial case of Troisi inequality cf. [6, ] f 6 c f / f / f /.. Finally, we obtain the basic energy inequality by taking -inner roducts of. with u, b resectively, integrating by arts and using the incomressibility of u and b to deduce after integrating in time ut + bt +.. t [,T ]. Two roositions Proosition.. Suose u, b is the solution to. in time interval [, T ]. Then for any,, the following inequality holds: for any distinct choices of j, j, j {,, } b j t b j R T e uj dλ t [,T ]. + e R T τ uj dλ u j b j,j dτ, where b j,j is a two dimensional vector of two entries b j and b j. Remark.. We remark that we cannot obtain an analogous bound for u due to the π-term in the first equation of.. Moreover, we were able to obtain various modifications of this inequality; however, we emhasize that. in articular imlies that if we have a sufficient bound on u j, then we may bound the -norm of b j by the same -norm of b j,j. Proof of Proosition.. From the second equation of., we have the equation that governs the growth of b j in time t b j + u b j b u j = b j.. We multily by b j b j and integrate in sace to obtain t b j b j b j b j = u b j b j b j + b u j b j b j. By the incomressibility condition, we see that the first term on the right hand side after integrating by arts equals zero. We comute the diffusive term after integrating by arts as follows: b j b j b j = kkb j b j b j = k= k= k b j b j.

5 EJDE-4/98 COMPONENT REDUCTION 5 Therefore, we obtain by integrating by arts and using the incomressibility condition of b, k t b j + b b j j = = k= k= k b k u j b j b j + b k u j k b j b j k= / b k u j b j / b j k b j b k u j b j + k= k= k= k= k b j b j by Young s inequality. Absorbing the diffusive term, we have t b j + k b j b j b k u j b j. Therefore, Hölder s inequalities and then dividing by b j lead to t b j u j b j u j b j,j. This leads to. comleting the roof of Proosition.. The next roosition may be obtained by an identical rocedure. Proosition.. Suose u, b is the solution air to. in time interval [, T ]. Then for any,, the following inequality holds: for any distinct choices of j, j, j {,, } b j,j t b j,j R T e uj,j dλ t [,T ] + e R T τ uj,j dλ u j,j b j dτ.. 4. Proof of Theorem. h u + h b -estimate. We now fix and r that satisfy.4, take -inner roducts of the first equation of. with h u and the second with h b to estimate ty t + h u + hb = u u h u b b h u + u b h b b u h b 4. := I + I + I + I 4, where Y t := h ut + h bt. The following decomosition was obtained in []; we rovide details in the Aendix for convenience of readers: I u u h u, I + I + I 4 b b h u + b u h b. 4.

6 6 K. YAMAZAKI EJDE-4/98 Now by Hölder s and Young s inequalities we immediately obtain I u u h u 4 hu + c u u. 4. Next, by Hölder s inequalities and interolation inequalities we estimate I + I + I 4 b b h u + b u h b b b b / 6 By. and Young s inequalities we have h u + b u I + I + I 4 b b h b / b / h u + b u h u / u / h b u / 6 h b. 4 hu + hb + c b b b + b u u. 4.4 Thus, with 4. and 4.4 alied to 4., absorbing the dissiative and diffusive terms, integrating in time we obtain Y τ + τ [,t] + t t h u + hb dτ u u + b X τz τdτ. 4.5 u + b -estimate. For both equations in., we take the -inner roducts with u and b resectively and integrate by arts to obtain txt + Zt = u u u b u + u b b b u b := 4 II i. i= For II and II 4 we have the estimate II + II 4 b b u + b u b c b b + b u + 8 Zt, 4.6 by Hölder s and Young s inequalities. Now we use a Gagliardo-Nirenberg inequality f f f / 4.7

7 EJDE-4/98 COMPONENT REDUCTION 7 and Young s inequalities to obtain II + II 4 c b b b 6 + b u u Zt c b X + 4 Zt. 4.8 For II, we integrate by arts twice to deduce II = k u i i b j k b j + u i ikb j k b j = = i,j,k= i,j,k= i,j,k= k u i i b j k b j iu i k b j k u i b j ikb j u b b so that similarly as before, Hölder s and Young s inequalities, 4.7 and another Young s inequality lead to II b u b c b u c b Xt + 4 Zt. u b 4.9 Finally, on II, we write II = u i i u u + u u h u + u u i= and then integrate by arts on each to obtain II = k u i i u k u + u i iku k u + i= k= k u u k u + u ku k u + u u k= = k u i i u k u iu i k u + k u u k u i= k= k= u k u u + u u h u u. Now Hölder s, interolation inequalities, and. lead to II h u u 4 h u u / u / 6 h u u / h u u /. 4.

8 8 K. YAMAZAKI EJDE-4/98 We aly the bounds of into 4.6 to obtain, after absorbing the dissiative and diffusive terms, t Xt + Zt b Xt + hu u / h u u /. 4. Integrating in time, we obtain Xt + t X + c + t t b b t + h uτ τ [,t] + c t h u u / h u u / dτ / t /4 t h u dτ by Hölder s inequality. By 4.5 and. we obtain where Xt + + t t + + c + b t 4 III i, i= u u + b X τz τdτ t t t /4, III = c b, III = c t t /4, III = c u u dτ t III 4 = c b X τz t /4, τdτ /4 /4 and c does not deend on t. By. with j =, j =, j =, we have b t R T dλ cec uλ + t [,T ] Using the elementary inequality we obtain bτ b c h τ c b h u λ b hλ dλ. 4. a + b a + b, a, b, 4. + b τ + ec R T u dλ + u b h dλ. 4.4

9 EJDE-4/98 COMPONENT REDUCTION 9 Thus, by., III c t b h τ + e c R T uλ dλ + u λ b hλ dλ. 4.5 The estimate on III is immediate by Young s inequality t /4 t III = c c Next, by Young s and Hölder s inequalities and., III c t u τ 8/ uτ dτ + 8 t. 4.7 Finally, by successive alications of Hölder s and Young s inequalities, t t + III 4 b 4 where using 4.4, we may obtain and therefore, t bτ 4 t c b + 8 t 8 c b bτ bτ = 8 8 c b h + t c b h + 4 t t, + ec R T u dλ u b h dλ 4 8 Xdτ + ec R T u dλ 4.8 u b h dλ due to.. We aly 4.9 into 4.8 and along with alied to 4., obtain after absorbing dissiative and diffusive terms Xt t t t b h Xdτ + R T dλ ec u + u b h dλ t u 8/ u dτ + b h + e c R T u dλ + 8 Xdτ u b h dλ 4

10 K. YAMAZAKI EJDE-4/98 + t + b h + e c R T u dλ t t u 8/ u dτ 8 4 Xdτ b h + u 8/ Xdτ + e ct /4 R T u 8/ dλ /4 + + t + t 8 u b h dλ b h + u 8/ Xdτ + /4 ect R T u 8/ dλ /4 4 u b h dλ 4 u 8/ dλ /4 b h 8 dλ /4 4 8 b h + u 8/ T Xdτ + c, M + b h 8 dλ. By Gronwall s inequality, the roof of Theorem. is comlete if 8 b h τ + u τ 8/ + b hτ 8 dτ <. For /, 5, we use Hölder s inequality to obtain b h τ 8 dτ T b h τ 8 dτ < by.4 whereas if 5,, Hölder s inequality again by.4 imlies b h τ 8 dτ T b h τ 8 dτ <. 5. Proof of Theorem. h u + h b -estimate. For fixed T >, firstly, from. with j =, j =, j =, by Hölder s inequalities we have b t / t [,T ] b / e 7 T /4 R T /4 uλ 8/ dτ R /4 e T /4 T dτ uλ 8/ b h t T /4 /4 u / τ 8/ dτ t [,T ] M e 7 T /4 M / e T /4 M /4 M T /4 M /4.

11 EJDE-4/98 COMPONENT REDUCTION Thus, using 4., we comute bt 8/5 b / h t + b / t / t [,T ] t [,T ] t [,T ] 8/5 M + M e 7 T /4 M / e T /4 M /4 M T /4 M /4. 5. Next, we choose t [, T ] to be secified subsequently and as before, we may obtain by which only required > and integrating in time over [, t ] as in 4.5 t Y t + t [,t ] + t h u + hb dτ u u + b 5 / X /4 τz /4 τdτ. u + b -estimate. By , all of which only required >, alied to 4.6 we have t Xt + Zt b Xt + / h u u / h u u / as in 4.. Integrating in time and by Hölder s inequality as before in 4., we have Xt + t [,t ] where X + c t t b + / t /4 t t [,t ] t + 4 IIII i, i= bt / t IIII = c t [,t ] u dτ /4 t / h uτ h u dτ t u u + b 5 X /4 τz /4 τdτ / t [,t ] bt /, IIII = c t /4, t t /4, IIII = c u u dτ t t /4, IIII 4 = c b 5 X /4 τz /4 τdτ / /4 5.

12 K. YAMAZAKI EJDE-4/98 for c indeendent of time t. Now due to., we can choose t [, T ] so that c 8/5 M + M e 7 T /4 M / e T /4 M /4 5/ M T /4 M /4 t / Then, by 5., IIII c 8/5 M + M e 7 T /4 M /4 e 7 T /4 M /4 M T /4 M / The estimate of IIII is same as before in 4.6 and the estimate of IIII is also same as 4.7. Finally, IIII 4 c t c t [,t ] /4 t b / t /4 t bt 5 / c 8/5 M + M e 7 T /4 M /4 8 t t, + 4 e 7 T /4 M /4 M T /4 M /4 5/ 5.5 by Hölder s inequality, 5. and 5.. Using 5.4 and 5.5 in 5., absorbing the dissiative and diffusive terms, we have t [,t ] Xt + t + By Gronwall s inequality, we have the bound t [,t ] Xt + t t. u 8/ u dτ. We restart on time interval [t, t ] and after finite number of reetitions obtain the same bound on [, T ]. This comletes the roof of Theorem.. 6. Proof of Theorem. This roof is similar to that of Theorem.. We sketch it for comleteness. h u + h b - estimate. As before, from which only required > leading to 4.5, we have Y τ + τ [,t] t h u + hb dτ

13 EJDE-4/98 COMPONENT REDUCTION + t u u + b X τz τdτ. u + b -estimate. As in the roof of Theorem., from 4.6, and using h u + h b -estimate leading to 4., we have Xt + t c + By. with j =, j =, j =, we have b, t R T ec u,λ dλ + t [,T ] so that by 4., similarly to 4.4, bτ and hence III c c b τ t b τ 4 III i. 6. i= +ec R T u, dλ + + e c R T u,λ dλ + u, λ b λ dλ. u, b dλ 6. u, λ b λ dλ 6. by 6. and.. We take the identically same estimates on III and III as before from 4.6 and 4.7. On III 4, from 4.8, we have t 8 III 4 c b + t, where due to 6. and 4. we have bτ 8 c b τ 8 Thus, by. similarly to 4.9, t bτ 8 + R T ec u, dλ 4 + u, b dλ. t c b + 8 Xdτ + ec R T u, dλ u, b dλ With 6.5 alied to 6.4, along with 4.6, 4.7 and 6. alied to 6., absorbing dissiative and diffusive terms, we have by Hölder s inequalities Xt t t t b Xdτ + R T dλ ec u, + u, b dλ t u 8/ u dτ + b 8 Xdτ

14 4 K. YAMAZAKI EJDE-4/98 + e c R T u, dλ + + t 8 b + u 8/ Xdτ + e ct /4 R T u, 8/ dλ /4 + + t b 8 dλ /4 4 u, b dλ 4 u, 8/ dλ /4 8 b + u 8/ Xdτ + c, N + b 8 dλ. Thus, the roof of Theorem. is comlete because by Hölder s inequalities as in the roof of Theorem., we have 8 b τ + u τ 8/ + b τ 8 dλ <. 7. Proof of Theorem.4 The roof is similar to that of Theorem.; we sketch it for comleteness. For fixed T >, firstly, from. with j =, j =, j =, we obtain by Hölder s inequality b, t N / e 4 T /4 N /4 t [,T ] so that by 4., + 7 e 4 T /4 N /4 N T /4 N /4 bt 8/5 b / + b /, / t [,T ] t [,T ] 8/5 N + N e 4 T /4 N / e T /4 N /4 N T /4 N /4 7. similarly to 5.. Now as in the roof of Theorem., we choose t [, T ] to be secified subsequently and use the revious estimate of Xt + t [,t ] t + from 5.. By., we can choose t [, T ] so that c 8/5 N + N e 4 T /4 N / e T /4 N /4 N T /4 N /4 t /4 8. Firstly, by 7., IIII c 8/5 N + N e 4 T /4 N / IIII i 7. i= e 7 T /4 N /4 5/ 7.. N T /4 N /4 7.4

15 EJDE-4/98 COMPONENT REDUCTION 5 We use the same estimates of 4.6 and 4.7 for IIII and IIII as before. Finally, from., 7. and 7. and Hölder s inequality, t /4 t IIII 4 c b / c t [,T ] 5/ t /4 t bt / c 8/5 N + N e 4 T /4 N /4 8 t t. + 7 e 4 T /4 N /4 N T /4 N /4 5/ 7.5 After absorbing dissiative and diffusive terms, due to 4.6, 4.7, 7.4 and 7.5 alied to 7., Gronwall s inequality gives the bound on t [,t ] Xt + t. Reiterating on [t, t ], after finite times we obtain the bound on the whole interval [, T ] comleting the roof of Theorem Aendix In this section we give details of the decomosition in the h u + h b - estimate, namely 4. cf. [5, ]. The following lemma is useful: emma 8. [7]. Assume that u H R is smooth and u =. Then u i i u j h u j = i u j i u j u u u u + u u u i,j= i,j= Firstly, for I alying this emma and integrating by arts, we have u i i u j h u j u h u h u. Thus, I = i,j= i,j= u i i u j h u j + u u h u. u u j h u j + j= u i i u h u i= Next, we decomose I : integrating by arts I = b i i b j kku j b i i b kku i,j,k= = i,k= b i i b j kku j b i i b kku i,j,k= i,k= j= k= b b j kku j

16 6 K. YAMAZAKI EJDE-4/98 + k b b j k u j + b kb j k u j j= k= = i,j,k= + j= k= b i i b j kku j b i i b kku i,k= kb b j k u j k b b j ku j + b kb j k u j b h b h u + b h u h b. Next, we write I = u i i b j kkb j + u i i b kkb + u b j kkb j i,j,k= i,k= j= k= = I + I + I. Integrating by arts, I = k u i i b j k b j + u i ikb j k b j = i,j,k= i,j,k= = i,j,k= kku i i b j b j + k u i ikb j b j + iu i k b j k b j kku i i b j b j + k u i ikb j b j iku i k b j b j + i u i kkb j b j b h b h u + b h u h b, I = k u i i b k b + u i i k b i,k= = kku i i b b + k u i ikb b + iu i k b k b i,k= = kku i i b b + k u i ikb b iku i k b b + i u i kkb b i,k= b h b h u + b h u h b, I = k u b j k b j + u k b j = j= k= j= k= ku b j k b j + k u b j kb j + u k b j k b j

17 EJDE-4/98 COMPONENT REDUCTION 7 = ku b j k b j + k u b j kb j j= k= = ku b j k b j + k u b j kb j + j= k= b h b h u + b h u h b. i u i k b j k b j i= iku i k b j b j + i u i kkb j b j i= Therefore, I b h b h u + b h u h b. Finally, Hence, I 4 = i,j,k= b i i u j kkb j b i i u kkb i,k= b h b h u + b u h b. j= k= I + I + I 4 b b h u + b u h b. This comletes the decomosition claimed. b u j kkb j Acknowledgments. The author wants to exresses his gratitude to Professors Jiahong Wu and David Ullrich for their teachings, and to the referees for their valuable comments. References [] J. Beale, T. Kato, A. Majda; Remarks on breakdown of smooth solutions for the threedimensional Euler equations, Comm. Math. Phys. 94, 984, [] H. Beirão da Veiga, A new regularity class for the Navier-Stokes equations in R n, Chin. Ann. Math. Ser. B, 6 995, []. C. Berselli, G. Galdi; Regularity criteria involving the ressure for the weak solutions to the Navier-Stokes equations, Proc. Amer. Math. Soc.,,, [4] C. Cao, E. S. Titi; Regularity criteria for the three-dimensional Navier-Stokes equations, Indiana Univ. Math. J., 57, 6 8, [5] C. Cao, E. S. Titi; Global regularity criterion for the D Navier-Stokes equations involving one entry of the velocity gradient tensor, Arch. Ration. Mech. Anal.,,, [6] C. Cao, J. Wu; Two regularity criteria for the D MHD equations, J. Differential Equations, 48, [7] D. Chae, J. ee; Regularity criterion in terms of ressure for the Navier-Stokes equations, 46, [8] D. Fang, C. Qian; The regularity criterion for D Navier-Stokes equations involving one velocity gradient comonent, Nonlinear Anal., 8, 86-. [9] D. Fang, C. Qian; Some new regularity criteria for the D Navier-Stokes equations, arxiv:.5 [math.ap], 6, Dec.. [] G. P. Galdi; An introduction to the mathematical theory of the Navier-Stokes equations, Vol. I, II. Sringer, New York, 994. [] C. He, Z. Xin; On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations,, 5, [] X. Jia, Y. Zhou; Regularity criteria for the D MHD equations involving artial comonents, Nonlinear Analysis: Real World Al.,, [] X. Jia, Y. Zhou; Regularity criteria for the D MHD equations via artial derivatives, Kinet. Relat. Models, 5,,

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