227, Mostaganem 27000, Algeria b Department of Mathematics, Kyoto University, Kyoto

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1 This article was downloaded by: [Polska Akademia Nauk Instytut Matematyczny] On: 9 June 4, At: :4 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 7954 Registered office: Mortimer House, 7-4 Mortimer Street, London WT JH, UK Applicable Analysis: An International Journal Publication details, including instructions for authors and subscription information: On the uniqueness of weak solutions of the D MHD equations in the Orlicz Morrey space Sadek Gala a, Yoshihiro Sawano b & Hitoshi Tanaka c a Department of Mathematics, University of Mostaganem, Box 7, Mostaganem 7, Algeria b Department of Mathematics, Kyoto University, Kyoto 66-85, Japan c Department of Mathematical Science, The University of Tokyo, Tokyo 5-894, Japan Published online: 9 Feb. To cite this article: Sadek Gala, Yoshihiro Sawano & Hitoshi Tanaka () On the uniqueness of weak solutions of the D MHD equations in the Orlicz Morrey space, Applicable Analysis: An International Journal, 9:4, , DOI:.8/ To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &

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3 Applicable Analysis, Vol. 9, No. 4, , On the uniqueness of weak solutions of the D MHD equations in the Orlicz Morrey space Sadek Gala a *, Yoshihiro Sawano b and Hitoshi Tanaka c Downloaded by [Polska Akademia Nauk Instytut Matematyczny] at :4 9 June 4 a Department of Mathematics, University of Mostaganem, Box 7, Mostaganem 7, Algeria; b Department of Mathematics, Kyoto University, Kyoto 66-85, Japan; c Department of Mathematical Science, The University of Tokyo, Tokyo 5-894, Japan Communicated by R.P. Gilbert (Received August ; final version received 5 October ) In this article, we show that if a Leray solution (u, b) to the D magnetohydrodynamic equation belongs to L r ðð, T Þ; Mr ðr ÞÞ with r (, ) and P 4, then (u, b) is the unique Leray solution on (, T). Keywords: uniqueness; weak solutions; Orlicz Morrey spaces AMS Subject Classifications: 5Q5; 5Q8; 4B5; 76N. Introduction We consider the following D incompressible magneto-hydrodynamic (MHD) equations: t u Du þðurþuþrpþ rjj b ðb rþb ¼ t b Db þðurþb ðbrþu ¼, ðþ divðuþ ¼divðbÞ ¼ >: uð,þ ¼ u, bð,þ ¼ b, where u ¼ u(x, t) is the velocity field, b is the magnetic field, p ¼ p(x, t) is the scalar pressure while u and b are given initial velocity and initial magnetic field with div(u ) ¼ div(b ) ¼ in the sense of distribution. For simplicity, we assume that the external force has a scalar potential and is included into the pressure gradient. It is well-known [] that the problem () is local well-posed for any given initial datum u, b H s ( ), s. But whether the unique local solution can exist globally or the weak solution is regular and unique is an outstanding challenging problem, just as the situation for the Navier Stokes equations. The purpose of this article is to establish Serrin s uniqueness result of the Leray weak solution for the D incompressible MHD equations in Orlicz Morrey spaces. *Corresponding author. sadek.gala@gmail.com ß Taylor & Francis

4 Applicable Analysis 777 More precisely, we prove if ðu, bþl r, T, M =r R, for r ð, Þ and P 4, then (u, b) is the unique Leray weak solution on [, T ], where the function space is given by the norm (). M =r Downloaded by [Polska Akademia Nauk Instytut Matematyczny] at :4 9 June 4. Orlicz Morrey spaces Before stating our result, let us recall some definitions and properties of the spaces that we are going to use (see, e.g. [,] and references therein). Definition defined by For P R and 5 u 5 v 5, the Orlicz Morrey space M v L u log P L is n o k f k M v :¼ sup r n=v k f k L u log P Bðx,rÞ,Lu log P L : x Rn, r 4 L, ðþ where k f k Bðx,rÞ,L u log P L denotes the tu log P ( þ t) average given by j f ðxþj u inf 4 : log þ j f ðxþj P dx jbðx, RÞj : Bðx,RÞ Meanwhile, the definition of classical Morrey ( Campanato) spaces are as follows. Definition For 5 p q þ, the Morrey Campanato space M p q is defined by ( M p q ¼ f Lq loc R : f M p ¼ sup supjbj =p =q f ) q x R4 Lp ðbðx, RÞÞ 5, ðþ where B(x, R) denotes the closed ball in with centre x and radius R. In view of () and (), the definition () covers () as a special case when P ¼. Here and below we write (log a) P :¼ log P a. With our new Morrey spaces (), we prove and apply the following result. THEOREM. [] Let 5 5 /. The fractional integral operator I is defined by If P 4, then we have I f ðxþ ¼ f ð yþ dy: jx yj ð4þ k g I f k L C k gk M = k f k L : Additionally, we have the following inclusion. PROPOSITION. [] Let P 4 and 5 u 5 ~u 5 v. Then ð5þ L v,! L v,,!m v ~u,!mv L u log P L,!Mv u ð6þ in the sense of continuous embedding and the inclusion is proper.

5 778 S. Gala et al. Note that Theorem. dates back to [4] and its subsequent papers [ 8]. In [7], we have dealt with the model case. In [6], we compared a result which is a prototype of Theorem. with an estimate obtained by using the Ho lder inequality. For a result related to fractional maximal operators, we refer to [5]. In [], we refined the results in [5,6] in terms of the Orlicz Morrey spaces, which were further refined as Theorem.. Downloaded by [Polska Akademia Nauk Instytut Matematyczny] at :4 9 June 4. Serrin s uniqueness theorem Before turning our attention to uniqueness issues, we start with some prerequisites for our main result. Let ¼ C : div ¼ C : C, R The subspace L R ¼ C, kk : L ¼ u L : divðuþ ¼ is obtained as the closure of C, with respect to L -norm kk L. H r denotes the closure of C, with respect to the norm kuk H r:¼ kuk L þð DÞ r u L, for r : Now, we recall the definition of weak solutions to (). Definition A measurable vector pair (u, b) is called a weak solution to MHD Equations (), if (u, b) satisfies the following properties: (i) u L ([, T ); L ) \ L ([, T ); H ), b L ([, T ); L ) \ L ([, T ); H ) (ii) (u, b) verifies () in the sense of distribution; that is, T þððurþþ u dx dt þ u ðx,þdx ¼ ður þðbrþbþdx R T þððurþþ b dx dt þ b ðx,þdx ¼ ðbr þðbrþuþdx for all C ðr ½, T ÞÞ with div() ¼, and T T ðu rþdx dt ¼, ðb rþdx dt ¼ for every C ðr ½, T ÞÞ. (iii) The energy inequality, that is, for t T. kuðtþk L kbðtþk L t þ kruðsþk L ds ku k L, t þ krbðsþk L ds kb k L,

6 Applicable Analysis 779 Downloaded by [Polska Akademia Nauk Instytut Matematyczny] at :4 9 June 4 Remark The definition of weak solution for () is similar to that for the Navier Stokes equations. Wu [9] defined the weak solutions without (iii), the energy inequality. However, from the existence proof (by Galerkin method), we can find the existence of weak solutions possessing the energy inequality. Our main theorem is as follows. THEOREM. Let (u, b ) L ( ) with div(u ) ¼ div(b ) ¼. Assume that there exists a solution (u, b) of the MHD equations on (, T ) ( for some T (, þ]) with some initial data (u, b ) such that ðu, bþl ð, T Þ; L R \ L ð, T Þ; _H R, and for some r (, ), ðu, bþl r ð, T Þ; M =r ðr Þ : Then, (u, b) is the unique Leray solution associated with (u, b ) on (, T ). Remark that this result is a further extension of the fundamental works on regularity criteria for the MHD equations, and also some works on this kind of spaces [ 5]. Proof Let ðeu, bþ e be another weak Leray solution associated with (u, b ) on (, T ) such that eu, b e L ð, T Þ; L R \ L ð, T Þ; _H R : In the same way as in [6], for the Navier Stokes equation case, we have the following equalities in D (, t u eu dx þ rðu rþeu dx ¼ u ðurþeu dx u ðeu rþeu dx R þ eu ðbrþbdx þ u ðb e rþb e dx t b b R e dx þ rðb rþb e dx ¼ b ðurþb e dx b ðeu rþb e dx R þ b e ðbrþudx þ b ðb e rþeu dx, which implies t ðu eu þ b bþdx R e þ ru reu þrbr e b dx R ¼ u ðu ½ euþršeu dx þ b ðu ½ euþrš e h b dx þ eu b e i b :r b dx R þ b e ½ðb bþršu e dx:

7 78 S. Gala et al. Since (u, b) and ðeu, e b Þ are the Leray Hopf weak solutions, we have Downloaded by [Polska Akademia Nauk Instytut Matematyczny] at :4 9 June 4 and T kuk L þkb k L þ kruk L þrb k k L d k u k L þ k b k ð7þ L eu þ e b L T þ reu þ r e b L L d k u k L L þ k b k L: ð8þ Next we consider the difference w ¼ u eu, z ¼ b b e and write t t I ¼ u ðwrþwdx d, I ¼ b ðwrþzdx d R t t I ¼ b ðzrþwdx d, I 4 ¼ u ðzrþzdx d: We obtain kwk L þ z kk L ¼ ku k L þkb k L þ eu þkbk e L L R u eu dx b b e dx t t jrwj dx d jrzj dx d þ X4 I j : Moreover, we have the antisymmetry property: t t u ðwrþudx d ¼ ¼ b ðwrþbdx d: The first term I in (9) is dominated by the Ho lder inequality. We proceed as follows: t ji j ku krwk kl r, t, M =r L d t r : kwk r d H _ ðþ L log P r L It follows immediately from the definition of the norms that kwk H _ r If we insert () into (), then we have ji j Cu k kl r, t, M =r Cu k kl r, t, M =r kwk r L krw kr L j¼ ð9þ : ðþ t krwk L d t r kwk kwk ð rþ r d _H L kwk r L ð, t, L Þ krw k þr L ð, t, L Þ : By Young s inequality a b a þð Þb a þ b with a, b and,

8 Applicable Analysis 78 we find ji j Cu krwk kkl r, t, M =r L ð, t, L Þ þ w k k L ð, t, L Þ : Similarly, we have Downloaded by [Polska Akademia Nauk Instytut Matematyczny] at :4 9 June 4 ji j ku krzkl kl r, t, M =r ð, t, L Þ kwk r Cu k kl r, t, M =r L ð, t, L Þ t r kwk r d H _ r krwk r L ð, t, L Þkrz k L ð, t, L Þ Cu k kwk kl r, t, M =r L ð, t, L Þ þrw k k L ð, t, L Þ þrz k k L ð, t, L Þ where the last inequality follows from the following convex inequality: for all s i, p i 4, i ¼,..., n, we have!! s :s...:s n ¼ exp Xn ln s i ¼ exp Xn ln s p i i Xn expðln s p i i Þ¼ Xn s p i i, p i¼ i¼ i p i¼ i p i¼ i whenever P n i¼ p i ¼. For the third term I, we can also be dominated as ji j Cu k kwk kl r, t, M =r L ð, t, L Þ þrw k k L ð, t, L Þ þrz k The last term I 4 can be dealt with a similar manner: ji 4 j Cu kwk kkl r, t, M =r L ð, t, L Þ þrw k k L ð, t, L Þ þrz k From the above inequalities and (9), we obtain t t kwk L ð,t,l Þ þkk z L ð,t,l Þ jrwj dxd jrzj dxd R þ C@ ku þkbk kl L r,t,m =r k L ð, t, L Þ k L ð, t, L Þ r,t,m =r : A kwk L ð,t,l Þ þkk z L ð,t,l Þ þkwk L ð,t,l Þ þkk z L ð,t,l Þ, : ðþ : Since t (, T ) can be taken arbitrary, we choose t small enough so that C@ ku þkbk A 5 : kl L So we see that for T, r, t, M =r r, t, M =r w ¼ z ¼ on ½,Š:

9 78 S. Gala et al. Thus, we see that ft 4 : w ¼ z ¼ on ½,Šg Downloaded by [Polska Akademia Nauk Instytut Matematyczny] at :4 9 June 4 is an non-empty open set in (, ), from which we obtain the desired uniqueness. This completes the proof of Theorem.. g The following corollary, which is an immediate consequence of Theorem., gives a simpler sufficient condition in term of Lorentz spaces. COROLLARY. Let (u, b ) L ( ) with div(u ) ¼ div(b ) ¼. Assume that there exists a solution u for the MHD equations on (, T ) ( for some T (, þ] with some initial data (u, b ) so that ðu, bþl ð, T Þ; L R \ L ð, T Þ; _H R, and for some r (, ), ðu, bþl r ð, T Þ; L r,, where L p, denotes the usual Lorentz (weak L p ) space. Then, (u, b) is the unique Leray- Hopf solution associated with (u, b ) on (, T ). Acknowledgements This work is partially supported by Grant-in-Aid for Young Scientists (B) No. 744, Japan Society for the Promotion of Science. References [] M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure appl. Math 6 (98), pp [] S. Gala, Y. Sawano, and H. Tanaka, A new regularity criteria for the D magnetomicropolar fluid equations in the Orlicz Morrey space, to appear in Math. Method Appl. Sci. (), Submitted. [] Y. Sawano, S. Sugano, and H. Tanaka, Orlicz Morrey spaces and fractional operators, Potential Anal., online, in press, doi:.7/s [4] H. Tanaka, Morrey spaces and fractional operators, J. Aust. Math. Soc. 88() (), pp [5] Y. Sawano, S. Sugano, and H. Tanaka, Generalized fractional integral operators and fractional maximal operators in the framework of Morrey spaces, Trans. Amer. Math. Soc. 6() (), pp [6] Y. Sawano, S. Sugano, and H. Tanaka, A note on generalized fractional integral operators on generalized morrey spaces, boundary value problems, Vol. 9, p. 8, Article ID 85865, doi:.55/9/ [7] Y. Sawano, S. Sugano, and H. Tanaka, Olsen s inequality and its applications to Schrodinger equations, RIMS Koˆkyuˆroku Bessatsu B6 (), pp [8] S. Sugano, Some inequalities for generalized fractional integral operators on generalized Morrey spaces, Math. inequal. Appl. 4(4) (), pp [9] J. Wu, Viscous and inviscid magneto-hydrodynamics equations, J. Anal. Math. 7 (997), pp

10 Applicable Analysis 78 Downloaded by [Polska Akademia Nauk Instytut Matematyczny] at :4 9 June 4 [] J. Fan, S. Jiang, G. Nakamura, and Y. hou, Logarithmically improved regularity criteria for the Navier Stokes and MHD equations, J. Math. Fluid Mech. (in press) doi:.7/ s [] Y. hou, Remarks on regularities for the D MHD equations, Discr. Contin. Dyn. Syst. (5) (5), pp [] Y. hou, Regularity criteria for the D MHD equations in terms of the pressure, Int. J. Non-linear Mech. 4() (6), pp [] Y. hou, Regularity criteria for the generalized viscous MHD equations, Ann. Inst. H. Poincare Anal. Non-lineaire 4() (7), pp [4] Y. hou and S. Gala, Regularity criteria for the solutions to the D MHD equations in the multiplier space,. Angew. Math. Phys. 6() (), pp [5] Y. hou and J. Fan, Logarithmically improved regularity criteria for the D viscous MHD equations, Forum Math., in press, doi:.55/form..79. [6] P.G. Lemarie -Rieusset, Recent Developments in the Navier Stokes Problem, Research Notes in Mathematics, Chapman & Hall, CRC Press, Boca Raton,.