January 3, Abstract. functions with total radially equidistributed energy decay algebraically

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1 Large-time Behaviour of Solutions to the Magneto-Hydrodynamics equations M. E. Schonbek T. P. Schonbek y Endre Suli z January 3, Abstract In this paper we establish lower and upper bounds for the rate of decay of the total energy and of the magnetic energy of solutions to the magneto-hydrodynamics equations on R n, n 4. It is shown that weak solutions subject to large initial data outside a class of functions with total radially equidistributed energy decay algebraically (rather than exponentially). It is also proved that initial data with radially equidistributed energy gives rise to weak solutions which decay exponentially (rather than algebraically). Key words: Magneto-hydrodynamics equations, decay of solutions AMS(MOS) subject classication: 35B4, 76W5 1 Introduction The purpose of this paper is to derive upper and lower bounds on the decay of the total energy and the magnetic energy of a viscous incompressible University of California, Santa Cruz, California. Research partially supported by NSF Grant No. DMS yflorida Atlantic University, Boca Raton, Florida. zoxford University Computing Laboratory, 11 Keble Road, Oxford OX1 3QD, England. e{mail: endre.suli@comlab.ox.ac.uk 1

2 electrically conducting resistive uid. The interactions between the uid motions and the magnetic eld are modelled by the magneto-hydrodynamics equations which, in non-dimensional form, can be written as (cf. [4], u + u +(ur)u ; S(B r)b + r(p + 1 SjBj ) = 1 1 B +(ur)b ; (B r)u = Rm r(rb) ru = rb = : Here u, P and B are non-dimensional quantities corresponding to the velocity of the uid, its pressure, and the magnetic eld f(x t) represents a nondimensional density ofvolume force, and jbj = is the magnetic pressure. The non-dimensional numbers appearing in these equations are the Reynolds number, Re, the magnetic Reynolds number Rm, ands = M =ReRm, where M is the Hartman number. For the sake of notational simplicity, and without restricting generality, we set all of these numbers to unity. Thus, upon such normalisation and letting p = P + 1 SjBj denote the total pressure, we consider the magneto-hydrodynamics (MHD) equations in R n,n4: (MHD) supplemented with the initial u +(ur)u ; (B r)b + rp = u B +(ur)b ; (B r)u = ru = rb = u(x ) = u (x) B(x ) = B (x): There is no loss in generality in assuming that the forcing function is divergence free i.e., that rf(t) = for all t. We shall always make this assumption. We show that weak solutions to the MHD equations, subject to large initial data outside a class of functions with total radially equidistributed energy, decay algebraically (rather than exponentially). In particular we prove that, for such solutions, the total energy (kinetic plus magnetic) and the magnetic energy have slowly decaying algebraic lower bounds. Moreover, we show in which cases the lower bounds are valid for the kinetic energy alone

3 or the magnetic energy alone. Thus, our results reinforce mathematicallythe observation made by Chandrasekhar [3] that \the magnetic eld in systems of large linear dimensions can endure for relatively long periods of time". The following notation will be used throughout the paper: H m (R n ) will denote the Hilbertian Sobolev space on R n of index m, m L p (R n ) will stand for the Lebesgue space equipped with its standard norm kk p, 1 p 1 V = fv [C 1 (R n )] n : ru =g H = closure of V in [L (R n )] n : In addition we introduce the following weighted Lebesgue spaces (and associated norms): W 1 = fv : jxj jv(x)jdx < 1g W = fv : jxjjv(x)j dx < 1g R n R n 1= jvj W1 = jxj jv(x)jdx jvj W = jxjjv(x)j dx : R R n n Suppose that (u B ) H H, and let f L 1 ( 1 L (R n ) n ). By a weak solution of the MHD equations we mean a function (u B) BB, where B = C w ([ 1) H) \ L loc(( 1) [H 1 (R n )] n ) satisfying the integral relations hu(t) (t)i + t (s)i + hru(s) + h(u(s) r)u(s) ; (B(s) r)b(s) (s)ids = hu ()i + hb(t) (t)i + t (s)i + hrb(s) r(s)i + h(u(s) r)b(s) ; (B(s) r)u(s) (s)ids = hb t hf(s) (s)ids for all smooth divergence-free vector elds (x t) with compact support in R n [ 1). Here h i denotes the scalar product of the space [L (R n )] n, and C w ([ 1) H) is the space of weakly continuous functions from [ 1) into H. (1) 3

4 When n =, the existence of a unique strong solution is well known, whereas for n 3 no uniqueness result without restricting the class of solutions is available (see, [5] and [1]). Nevertheless, all solutions to the MHD equations constructed so far do obey the energy inequality, stated in () below, or at least they may be approximated by solutions of this type. This is well known for solutions to the Navier-Stokes equations (see [13], [4]). The methods used to construct solutions to the Navier-Stokes equations ([], [13], []) can be easily extended to a construction for the MHD equations, and the solutions will in the same fashion satisfy the energy inequality () for n 4. Thus, following the approach of Wiegner [4] for the incompressible Navier-Stokes equations, we shall assume in the rest of the paper that the solution (u(t) B(t)) to the MHD equations obeys: t ku(t)k + kb(t)k + (kru(r)k + krb(r)k)dr s () ku(s)k + kb(s)k + t s jhf(r) u(r)ijdr for s =, and almost all s> and all t s. By our assumption that f L 1 ( 1 L (R n ) n ), this implies (in the same fashion as in [4]) the existence of a constant C = C(u B f) such thatfor s =, and almost all s>andallt s, ku(t)k + kb(t)k + t ku(s)k + kb(s)k + C s (kru(r)k + krb(r)k )dr t s kf(r)k dr: (3) Indeed, if f is smooth and kf(t)k >, then h(t) =jhf(t) u(t)ij kf(t)k ;1 is a continuous function, and, by the Cauchy-Schwarz inequality, h (t) ku(t)k ku k + kb k + ku k + kb k + C (u B f)+ t 1 t 4 h(r)kf(r)k dr kf(r)k dr + h (r)kf(r)k dr: t h (r)kf(r)k dr

5 Thence, by virtue of Gronwall's inequality,wehave that h(t) C(u B f), and the required inequality follows from (). The general case of f L 1 ( 1 L (R n ) n ) is dealt with by approximation. In Section we analyze the lower and upper bounds of L decay rates for solutions to the heat equation subject to a forcing term. The upper bounds are included for completeness. In this paper, we improve known upper bounds given in [16] and [9]. The derivation of the lower bounds for weak solutions to the MHD equations presented in this paper hinges on decay properties of solutions (u (x t) B (x t)) to the non-homogeneous heat u =u B =B u (x ) = u (x) B (x ) = B (x) with the same initial data, (u B ), and volume force, f, as in the MHD equations. We will impose certain conditions on the functions f and prove that these conditions guarantee that the solutions of the heat equation with forcing function f decay at the same rate as the corresponding solutions of the \free" heat equation. We need these conditions on comparing solutions of the MHD equations with solutions of the heat equation in our proof that, at least generically, the square L -norms of the solutions to the MHD equations have a rate of decay bounded below by C(t +1) ;n=;1 (with C>). Denition. Let f : R n [ 1)! R n be measurable, and assume rf(t)= for all t. Let R. Wesay a f A if there exists C suchthat kf(t)k C(t +1) ; for t : b f B if there exists C suchthat c f C if there exists C suchthat j ^f( t)jcjj for t R n : kf(t)k 1 C(t +1) ; for t : 5

6 We can make these spaces into Banach spaces dening norms in the obvious way for example, kfk A =sup(t +1) kf(t)k t and similarly for kfk B, kfk C. We also have to assume frequently that f L 1 ( 1 L (R n ) n ) we dene the norm in this space by kfk L 1 (L ) = 1 kf(t)k dt: For our lower bounds, we need to be able to estimate rst order moments of the solution (u B) of the MHD equations (cf. Lemma 6.1). For this purpose we assume f L 1 ( 1 W 1 ). In order to derive the lower bounds, we use the ideas presented in [] for solutions to the Navier-Stokes equations. We compare the decay of solutions of the MHD equations to those of the heat system with the same initial datum. The decay inl -norm of solutions of the heat equation (or system) depends mainly on the presence (or absence) of long waves in the initial datum i.e., on the size of the Fourier transform of the datum near the origin. We show that for initial datum (u B ) H H, thel -norm of the dierence between the solutions of the MHD equations and the heat system (with the same datum) satises an upper bound which decays faster than the L -norm of the solution of the heat system. Specically, weprove the following theorem Theorem. Let the initial datum (u B ) H H and assume f L 1 ( 1 L (R n ) n ) \ A n=4+1 \ B 4 \ C (n+3)= : a) If n =and not all components of ^u () or ^B () are zero (in the sense dened at the beginning of Section ), then ku(t) ; u (t)k + kb(t) ; B (t)k C D (t +1) ;n=;1 (1 + log (t + 1)) b) if n 3, or if n =and (u B ) [H \ L 1 (R n ) n ], then ku(t) ; u (t)k + kb(t) ; B (t)k C D1 (t +1) ;n=;1 : 6

7 Cases a) and b) are mutually exclusive due to the following lemma established by Borchers [1]. Lemma [B]. Let u L 1 (R n ) n \ H. Then R n udx=: Proof. Suppose rst u [C 1 R n u 1 dx = (R n )] n. Then u rx 1 dx = (ru)x 1 dx =: R n R n The general case of u L 1 (R n ) n \ H follows using approximations and passing to the limit. For the sake of completeness, we notice that case a) of the theorem can obtain in fact there do exist functions u H such that ^u() 6= in the sense of Section. For example, let g L \ L 1 loc (Rn ) and dene u by ^u() =( jj g() ; 1 g() ::: ): jj Then u H, but^u() 6= ifg is bounded away from zero near the origin. By showing that in case (^u () ^B ()) 6= ( ), the solution of the heat system satises the lower bounds ku (t)k C (t +1) ;n= kb (t)k C (t +1) ;n= the lower bound for the MHD equations easily follows from the estimates in cases a) and b) above. For the dierence between a weak solution u(t) of the Navier-Stokes equations and a semigroup solution u (t) of the heat system, Wiegner [4] showed that one has the estimate ku(t) ; u (t)k C(t +1) ;n=;1 : It is worth mentioning, however, that whereas for the Navier-Stokes equations this inequality holds regardless whether or not the initial data has zero average, this is no longer true for the MHD equations in two dimensions (cf. Corollary 3.). This distinctive feature stems from dimensional arguments and the fact that the nonlinear terms in the MHD equations involve both 7

8 the velocity and the magnetic eld thus when we estimate the dierence between solutions of the MHD equations and the heat system, the bound involves kru (t)k 1 which decays at a rate depending on whether or not the initial data has zero average. A more subtle situation occurs when the average of the initial data is equal to zero: in this instance there are no long waves present in the initial data and the corresponding solution of the heat equation in Fourier space maycontain only short waves, so that the L -norm of the solution to the heat system decays very fast. However, for the MHD equations the nonlinear terms can produce a mixing of the Fourier modes which slows down the decay. As was shown in [] for the Navier-Stokes equations, solutions with data outside of a class of radially equidistributed functions will immediately produce lowfourier modes. More precisely,itcan be shown that in frequency space solutions to the MHD equations take the form ^u( t ) = ( t )+h( t ) ^B( t ) = ( t )+k( t ) where and are homogeneous functions of degree zero, and h and k are O(jj ). This is proved as in []. The appearance of long waves (i.e., low Fourier modes) does not seem to be sucient to insure that the decay rate will slow down. We need to add a hypothesis on the solutions guaranteeing that these slow Fourier modes keep showing up in a sequence of times tending to innity. We conjecture that this hypothesis is removable whenever a slow Fourier mode has appeared it will persist as time goes to innity. In other words, once long waves are present they will stay indenitely. For the time being, we show that if the slow modes do not persist then the solution decays at a faster rate. We alsoshow that the persistence of the slow modes is generic (Section 4). Next we compare the solution of the MHD equations with solutions of the heat system with initial data (u( t m ) B( t m )), where ft m g is a sequence of times satisfying lim m!1 t m = 1. The form of the initial data given by (4),witht replaced by t m, guarantees that the corresponding solution, (u (t m ) B (t m )), of the heat system satises ku (t m )k + kb (t m )k C 3 (t m +1) ;n=;1 with C 3 independent ofm. The derivation of the lower bound for the total energy of the solution to the MHD equations is now reduced to showing that 8 (4)

9 the upper decay constant, C D1, for the dierence, (u(t);u (t) B(t);B (t)), satises C D1 <C 3 where C 3 is the constant corresponding to the lower bound for the solution of the heat system. A similar analysis yields the lower bound on the decay for the magnetic energy alone. This part of the analysis is performed for initial data giving rise to solutions in M c, the complement of the set M of functions with total radially equidistributed energy. The set M is a generalisation of the class of radially equidistributed functions, M, introduced in [18] to obtain algebraic lower bounds for the Navier-Stokes equations. Given u =(u 1 ::: u n )and B =(B 1 ::: B n )in[l 1 ( 1 L (R n ))] n,let We also set ~A ij = ~C ij = 1 1 hx B i ij = R n (u i u j ; B i B j )dx R n (u i B j ; B i u j )dx: R n x j B i (x ) dx: Then, introducing the matrices ~A =[~A ij ], ~C =[~C ij ], and hx B i =[hx B i ij ], we dene M = f(u B) [L 1 ( 1 L (R n )] n : ~A is scalar and ~ C = hx B ig: We note that M is very small. More precisely,if(u B) [L 1 ( 1 L (R n ))], then generically (u B) M c. This will be established in Section 4 (cf. Lemma 4. and Corollary 4.3). In other words, generically solutions are not in M, the set where slow Fourier modes do not persist. We show that the lower bound holds if and only if the solution is not in M (cf. Section 4, Theorem 4. 3). We strengthen this result in the nal section where we give examples of solutions to the MHD equations not in M which decay exponentially (rather than algebraically). These examples, of solutions which decay exponentially rather than algebraically, are of two types. Solutions where the magnetic eld B is zero and the velocity issimultaneously a solution to the Navier-Stokes and the heat equations secondly, solutions where both the velocity u and the magnetic eld B satisfy heat systems. The example we 9

10 have is valid in all even space dimensions and shows that the classical radial solution of the Navier-Stokes equations is also a solution of the homogenenous heat equation. It extends in a quite technical way the dimensional example given in []. The analysis done here gives a general method for constructing simultaneous solutions to the heat equation and the Navier-Stokes equations. It raises the question whether only solutions which also satisfy the heat equation can decay exponentially. Our results are also a rst step to showing that, in the far eld, the behaviour of solutions to the MHD equations and the heat system are quite dierent. Whereas solutions to the heat system may have very fast algebraic or exponential decay when the initial data is rapidly oscillating, the nonlinear terms in the MHD equations generically alter the decay of solutions to a slow algebraic rate when started from the same initial data. We note that, with minor modications, the analysis presented here can be adapted to the case when the second equation in (MHD) also contains a forcing term. We have the following result. Theorem. Let (u B ) [W \ H], f L 1 ( 1 L (R n ) n \ W 1 ) \ A n=4+1 \ B 4 \ C (n+3)= and let (u(x t) B(x t)) be aweak solution of the MHD equations with initial datum (u(x ) B(x )) = (u (x) B (x)). a) If ^u () 6= or ^B () 6=,then C (t +1) ;n= ku( t)k + kb(t)k C 1 (t +1) ;n= : b) If (u B ) [W \ H \ [L 1 (R n )] n ] (so that ^u () = () ;n= R n u (x) dx = ^B () = () ;n= by Borchers' Lemma) and (u B) =M,then R n B (x) dx = C (t +1) ;n=;1 ku( t)k + kb( t)k C 3 (t +1) ;n=;1 : We note that if < (r^u ()) 6= the solutions stay inm c. In this paper we also obtain a very precise expression for the rst term of the Taylor expansion in frequency space of solutions to the MHD equations. The expansion is valid, as well, for solutions to the Navier-Stokes equations 1

11 setting B. In fact, the MHD equations reduce to the Navier-Stokes equations when B so that the methods presented here are also suitable for the analysis of the decay ofsolutionstothenavier-stokes equations with a forcing term f. The lower bound on the decay rate of the total energy for the MHD equations is the same as that of the kinetic energy for the incompressible Navier-Stokes equations, with the set M replaced by its counterpart M = fu L 1 ( 1) L (R n ) n :[r ij ] is scalar, where r ij = 1 R n u i u j dxg: We note that in this paper we corrected a gap that was found in the corresponding result for the Navier-Stokes equations []. Finally, we mention that upper bounds for the MHD equations have also been derived in [16] and [9]. Bounds for solutions of the heat equation We begin with the homogeneous heat equation. If L 1 loc (or, more generally if is a distribution), we say has a zero of order at the origin if in some neighborhood V of the origin () =()+h() for V where is a non-zero function homogeneous of degree, continuous on R n nfg, andh() =O(jj + )for!, some >. If has a zero of order, we willsay (with some abuse of language) that does not vanish at, write () 6=. This generalisation of the concept of a zero at the origin will be used exclusively in case () =^u(), u L (R n ). Moreover, u will be the solution of a heat equation or part of the system of solutions of the MHD equations for some (xed) time value. If is a non-negative integer and jxj u L 1, then ^u is -times continuously dierentiable (with bounded derivatives) and ^u has a zero of order at in the generalised sense if and only if it has a zero of order in the conventional sense. In particular, if u L 1,then^u() 6= if and ond only if ^u has a zero of order at the origin. Since we also have foru L 1 that ^u() = () ;n= R n u(x) dx 11

12 so that ^u()=ifandonly if u has a zero average. With some abuse of language, we shall say thatafunctionu L (R n ) has zero average if its Fourier transform ^u has a zero of order at least 1 at. Also, with some abuse of language, we shall say that a function u L satises ^u() 6= if^u has a zero of order at the origin i.e., if ^u diers from a homogeneous function of degree by a quantity of order O(jj )for!, where >. If u W 1, these concepts reduce to their usual interpretation. Lemma.1 Let u L (R n ) and let v = v(x t) be the solution of the homogeneous heat equation v t =v with initial datum v() = u. If^u has a zero oforder at the origin, then there exist constants C >, C 1 such that C (t +1) ; n ; kv(t)k C 1 (t +1) ; n ; for all t. Proof. Let >. By Plancherel's Theorem, with Since A (t) = kv(t)k = R n j^u ()j e ;tjj d = A (t)+b (t) jj j^u ()j e ;tjj B (t) = B (t) e ; t ku k jj> j^u ()j e ;tjj : B (t) decays exponentially and it suces to see that A (t) decays at the desired rate. We select >sothat ^u () =()+h 1 () for jj <,where is homogeneous of order and h 1 () is bounded by constjj +, where >. The result follows, since while jj< jj< j()j e ;tjj d t ;;n= j()h 1 ()+h 1 () )je ;tjj d const t ;;n=+ : We notice that we can replace t by t + 1 in the estimates, since kv(t)k is bounded at. In particular, if ^u () 6=,we get 1

13 Corollary. Let u L (R n ) and let v = v(x t) be the solution of the homogeneous heat equation v t =v with initial datum v() = u. Assume ^u () 6=. Then there exist positive constants C C 1,depending only on the norms of the datum, such that for t. C (t +1) ; n kv(t)k C 1 (t +1) ; n We want to explore a case in which the initial datum has zero average in a bit more detail. Lemma.3 Let u [L (R n )] n and let v(x t) be the solution of the homogeneous heat system with initial datum u.suppose in addition that there is > such that the Fourier transform of u admits the representation ^u () =P () + h() for jj, where P and h satisfy the following conditions: i)p is a homogeneous, n n matrix-valued function of degree zero with kpk =supjp ()j < 1 jj=1 (where jp ()j denotes a matrix norm of P ()). ii)jh()jm jj, for some M and all R n, jj. Then kv(t)k = c n( t) jp (!)!j d! S n;1 where for t 1, and! n +1 1 e ; c n ( t) jo(t ; n ; 3 )jct ; n ; 3 with C depending only on M, kpk and. If, in addition, 1 := t ; n ;1 + O(t ; n ; 3 ) (5) n + ;( n + ) (6) S n;1 jp (!)!j d! 6= (7) 13

14 then there exist positive constants C, C 1 with C 1 depending only on M, ku k, kpk, and 1, C depending on all these quantities and on such that C (t +1) ; n ;1 kv(t)k C 1 (t +1) ; n ;1 : Proof. We have kv(t)k ; e ;tjj j^u ()j d jj = jj e ;t ku k : e ;tjj j^u ()j d It suces to show that R jj e;tjj j^u ()j d is of the form c n ( t) 1 t ;n=;1 + O(t ;n=;3= ). For jj, where j^u ()j = jp ()j + () j ()jcjj 3 and C depends only on M and kpk. Equation (5) follows multiplying j^u ()j by e ;tjj and integrating over jj. In fact, jj where C is a constant, while jj Notice that jj 3 e ;tjj d R n jj 3 e ;tjj d = C t ; n ; 3 e ;tjj jp ()j d = = 1 1 ( ) n +1 e ; t! r n+1 e ;tr dr jp (!)!j d! S! n;1 t s n e ;s ds (t) ; n ;1 : s n e ;s ds ;( n + ) for t 1. The nal statement of the lemma is an easy consequence of (5) and the boundedness near of kv(t)k. 14

15 Having derived upper and lower bounds on the decay of solutions of the homogeneous heat equation, wenow consider the heat equation with a forcing function f. Let v = v + f v(x ) = u (x): We split v = u + w, where u solves the free heat equation u t =u, with u() = u, and w solves w t =w + f with initial datum w() =. Then 1 ku(t)k ;kw(t)k kv(t)k ku(t)k +kw(t)k and to see that ku(t)k and kv(t)k have the same rate of decay, it suces to show that kw(t)k decays at a faster rate than ku(t)k.wehave Lemma.4 Let v be the solution of v t =v + f, assume the Fourier transform ^u of the initial datum of v has a zero oforder at and that f A \ B, where > 1 (n + +) and >+. There exist positive constants C 1, C, such that for all t. C 1 (1 + t) ;n=; kv(t)k C (1 + t) ;n=; Proof. In view of Lemma.1 and by the remarks preceding this lemma, it suces to prove kw(t)k decays at a faster rate than t ;n=;,wherev = u+w as above. Writing w(t) = = t t= e ;(t;s) f(s) ds e ;(t;s) f(s) ds + t t= e ;(t;s) f(s) ds = I 1 + I : By Plancherel, ke ;(t;s) f(s)k = e ;(t;s)jj j ^f( s)j d R n kfk B jj R e ;(t;s)jj d = Ckfk B (t ; s) ;n=; n 15

16 so that t= ki 1 k Ckfk B (t ; s) ;n=4;= ds = Ckfk B t ;n=4;=+1 : By Plancherel (or otherwise), ke ;(t;s) f(s)k kf(s)k, hence ki k t t= t kf(s)k ds kfk A (s +1) ; ds Ckfk A (t +1) ;+1 : t= >From these bounds we conclude (after squaring and with a constant C depending on kfk A, kfk B ) that for t 1 kw(t)k C[(t +1) ;n=;+ +(t +1) ;+ ] C(t +1) ;n=;; where = min( ; ; ; ; n= ; ) >. As for the case of the free heat equation, the main cases in which Lemma.4 is applied are contained in the following Lemma. Lemma.5 Let v be the solution of v t =v + f with initial datum v() = u. Then 1. If ^u () 6= and f A \ B with >n=4+1, >, then there exist constants C and C 1, depending on norms of u and f, such that C (t +1) ;n= ku( t)k C 1 (t +1) ;n= :. If ^u () =P () + h() for jj, where > and P, h are as in Lemma.3, and f A \ B where >n=4+3=, >3, then there exist constants C and C 3, depending on norms of u and f, such that C (t +1) ;n=;1 ku( t)k C 3 (t +1) ;n=;1 : To show that the solution (u B) to the MHD equations with initial data (u B ) is asymptotically equivalent to the solution of the heat system with the same initial data, we need the following auxiliary result concerning the decay of the L 1 -norm of the solution of the heat system. 16

17 Lemma.6 Suppose, andletv(x t) denote the solution of the heat system (HS,f) with f C, where = n If Then (a) krv( t)k 1 C(t +1) ;n=;;1 kv( t)k C(t +1) ; t : (b) kv( t)k 1 C(t +1) ;n=; t 1. t 1 Proof. Let D i denote the identity operator if i =, the gradient operator r if i =1. Let K(t)(x) =K(x t) =(4t) ;n= e ; jxj 4t be the heat kernel we can then write so that D i v(t) =D i K(t=) v(t=) + kd i v(t)k 1 kd i K(t=)k kv(t=)k + Standard estimates yield hence t t= t t= kd i K(t)k = Ct ; 1 (n=+i) kd i K(t)k 1 = Ct ; i kd i v(t)k 1 Ct ; 1 ( n +i) kv(t=)k + C D i K(t ; s) f(s) ds t kd i K(t ; s)k 1 kf(s)k 1 ds: t= (t ; s) ;i= kf(s)k 1 ds: Using f C and the hypothesis on the decay ofkv(t)k,weget kd i v(t)k 1 Ct ; 1 ( n +i) t ; 1 + Ct ; 1 (+ n +) t Ct ; 1 (+ n +i) t= (t ; s) ;i= ds since i= < 1 (and (t ; s) ;i= integrated from t= tot is of order t 1;i= ). The proof is complete. 17

18 3 Fourier representation and upper bounds for solutions of the MHD equations In this section, and in the sequel, (u B) will denote a solution of the MHD equations with initial datum (u B )anddivergence free forcing function f. In deriving upper bounds for the decay of(u B) we mimic the methods developed to establish decay of solutions of the Navier Stokes equations in [18] and in [4], most notably the Fourier splitting method (introduced by one of the authors in [17]). The Fourier splitting method has already been used by Mohgooner and Sarayker for the MHD equations (cf. [16]), but their rate is not optimal. Our approach is somewhat formal (if n 3) since we are applying ordinary time derivatives to functions which may only be weakly dierentiable in time. We remark, however, that the rigorous proof follows applying the method to approximating sequences to solutions, similar to those constructed for the Navier-Stokes equations by Leray [1], and by Caarelli, Kohn and Nirenberg[]incasen =3 by Sohr, Wiegner and von Wahl [], and by Kajikiya and Miyakawa [7] if n = 3 4. The decay results established for the approximating sequence will be valid for the limiting weak solution. We begin introducing some notation. Let H =(H 1 ::: H n )=(u r)u ; (B r)b + rp M =(M 1 ::: M n )=(u r)b ; (B r)u so that the MHD equations become u t ; u = ;H + f B t ; B = ;M: Fourier transforming with respect to the space variables and solving the resulting rst order equations in time, we get ^u( t) = e ;tjj^u () ; t e ;(t;s)jj ^H( s) ; ^f( s) ds (8) ^B( t) = e ;tjj ^B () ; t e ;(t;s)jj ^M( s) ds: 18

19 Since ru = rb = rf =, applying the divergence operator to the rst set of MHD equations gives where we used p = ; n X k j=1 r((v r)w) = X j (u k u j ; B k B j (v k w j ) if rv =: Hence Notice also that ^p( t) =; 1 X jj k j uk d u j ; B d k B j : (9) (v r)w = X j j (v j w) if rv =: It follows that ^H( t) = i X j ^M( t) = i X j j duj u ; d B j B ; i X k j j duj B ; d B j u : k j uk d u jj j ; B d k B j Setting a kj = ud k u j b kj = B d k B j c kj = u d j B k we write all this in a somewhat more compact way introducing the n n matrices A =[A kj ], C =[C kj ], =[ kj ], where A kj ( t) =a kj ( t) ; b kj ( t) C kj ( t) =c kj ( t) ; c jk ( t) and Then kj () = k j jj : ^H( t) = i(i ; ())A( t) ^M( t) = ic( t): 19

20 Since ja k j ( t)j + jb k j ( t)j kud k u j k 1 + kb d k B j k 1 () ;n= (ku k u j k 1 + kb k B j k 1 ) () ;n= (ku k k ku j k + kb k k kb j k ) we see that ja( t)j X jakj ( t)j + jb k j ( t)j () ;n ku(t)k + kb(t)k k j (1) for all R n, t since () (hence also I ; ()) is an orthogonal projection matrix for each R n nfg, we get ^H( t) () ;n= ku(t)k + kb(t)k jj (11) for all ( t). Similarly, jc( t)j 4 X k j jc kj ( t)j () ;n ku(t)k + kb(t)k (1) so that ^M( t) () ;n= ku(t)k + kb(t)k jj: (13) for all t. We denote by (v w) the solution of the (HS,f) system with the same initial datum (u B ). We set D =(D 1 D )=(u; v B ; w). We now obtain several auxiliary estimates which will be needed later. First we have d dt kd(t)k = d dt (kd 1(t)k + kd (t)k ) = hd 1 D 1 ; Hi +hd D ; Mi: Recall that the form (u v w) 7! hu (v rw)i is skew symmetric in the rst and third entries (if u v w are divergence free), moreover divergence free vector elds are orthogonal to gradients. We thus get, after some integration by parts, d dt kd(t)k = ;krd(t)k ; hd 1 urvi +hb B rvi;hb u rwi +hu B rwi (14) = ;krd(t)k ; hd 1 urvi +hd 1 Brwi;hD urwi +hd Brvi:

21 We proceed to apply the Fourier splitting method. Let g(t) fort (to be determined later) and let G(t) = exp( R t g(s) ds), so that G =g G. By (14), d dt G(t)kD(t)k = [g(t) kd(t)k ;krd(t)k ;hd 1 urvi + hb 1 Brwi;hD urwi + hd Brvi]G(t): Estimating each of the last four terms on the right handsideby we get d dt jhq 1 (q r)q 3 ijkq 1 k kq k krq 3 k 1 G(t)kD(t)k G(t) g(t) kd(t)k ;krd(t)k (15) +(t) ku(t)k + kb(t)k 1= kd(t)k (krvk 1 + krwk 1 ) For some of our results it suces to use a simpler inequality. Noticing that hd 1 urvi = hu u rvi and using the rst inequality in(14)we get (estimating as before) d dt G(t)kD(t)k G(t) g(t) kd(t)k ;krd(t)k Now, by Plancherel's theorem, (16) +G(t)kD(t)k ku(t)k + kb(t)k 1= (krvk1 + krwk 1 ) g(t) kd(t)k ;krd(t)k = (g(t) ;jj )j ^D( t)j d R n jjg(t)(g(t) ;jj )j ^D( t)j d g(t) j ^D( t)j d: jjg(t) 1

22 This inequality is used to derive both the upper and the lower bounds for the decay ofthel -norm of (u B). Using it in (16), we get d dt G(t)kD(t)k g(t) G(t) jjg(t) j ^D( t)j d +(ku(t)k + kb(t)k )(krvk 1 + krwk 1 )G(t): For the purpose of nding upper bounds, it will suce to bound as follows. Since jjg(t) j ^D( t)j d D 1t =D 1 ; H D t =D ; M (17) and D() = ( ), we have (Fourier transforming and solving the rst-order equation in t) By (11), (13), ^D 1 ( t) = ; t ^D ( t) = ; t e ;(t;s)jj ^H( s) ds e ;(t;s)jj ^M( s) ds: j ^D( t)j = j ^D 1 ( t)j + j ^D ( t)j 1= C n jj t (ku(s)k + kb(s)k ) ds (18) where from now onc n denotes a constant depending only on the dimension n, not always the same in all formulas. We are ready to prove our result on upper bounds. It can be summarised by saying that the solutions of the MHD equation decay at the same rate as the corresponding solutions of the heat system the dierence D(t) decays at a faster rate. Theorem 3.1 Let (u B ) H H, and let f L 1 ( 1 [L (R n )] n ) \ C, = n=4+=+1,and < n=+1. Assume that the solution (v w) of the (HS,f) system satises for all t, some constant K. kv(t)k + kw(t)k K(t +1) ; (19)

23 1. There exists a constant C, depending only on the L -norm of the initial datum (u B ), on the L 1 ( 1 L (R n ) n ) \ C -norm of f, andonk such that ku(t)k + kb(t)k C(t +1) ; for t.. If 1 n=+1, then there isaconstant C, depending only on the L -norm of the initial datum (u B ), on the L 1 ( 1 L (R n ) n ) \ C - norm of f, andonk such that kd(t)k for t. ( C(t +1) ;n=;1 if 1 <n=+1 C(t +1) ;n=;1 (1 + log (t +1)) if =1 : Remark. Note that if f A \B with >(1=)(+) and >;n=+, then (19) holds (see Lemma.4). Proof. All constants C, C 1, C, appearing in this proof depend only on the dimension n,thel -norm of the initial datum (u B ), the L 1 ( 1 L (R n ) n )\ C R -norm of f, and on K. We write(t) =ku(t)k + kb(t)k, (t) = t (s) ds. The energy inequality implies (t) () + 1 kf(t)k dt = C for all t hence (t) Ct for all t. By (18), j ^D( t)jc n (t)jj hence j ^D( t)j d C n (t) jj d C n (t) g(t) n+ : () jjg(t) jjg(t) Using this in (17), we get d h G(t)kD(t)k dt Cn g(t) n+4 (t) +(t)(krv(t)k 1 + krw(t)k i 1 ) G(t): 3

24 We estimate the last term on the right by Lemma.6 obtaining d h i G(t)kD(t)k dt C g(t) n+4 (t) +(t +1) ; n 4 ; ; 1 (t) G(t): (1) We now select g(t) =(=(t +1)) 1=, G(t) =e R t g(s) ds =(t +1) where > n +. The last displayed inequality becomes d (t +1) kd(t)k dt C(t) (t +1) ; n ;+ + C(t)(t +1) ; n 4 ; ; 1 + : () The choice of insures that all powers of t+1 appearing in the last inequality are positive. Integrating from 1 to t, we get ( is increasing and (t) R t 1 (s) ds) (t +1) kd(t)k kd(1)k + C t 1 (s +1) ; n ;+ ds (t) t +C(1 + t) ; n 4 ; ; 1 + (s) ds 1 kd(1)k + C(t +1) ; n ;1+ (t) + C(1 + t) ; n 4 ; ; 1 + (t): By the energy inequality satised by solutions of the MHD equations, and by the corresponding one for solutions of the heat equation, we can estimate the L -norm of D(1) = (u(1) ; v(1) B(1) ; w(1)) in terms of the L -norm of (u B ) and the L 1 ( 1 L (R n ) n ) norm of f. Wethus get, dividing by (t +1), kd(t)k C(t +1) ; + C(t +1) ; n ;1 (t) + C(1 + t) ; n 4 ; ; 1 (t) (3) for t 1, hence also for t sincekd(t)k is bounded for all t (by the energy inequality). Using (u B) =(v w)+d, we get (t) = ku(t)k + kb(t)k kv(t)k +kw(t)k +kd(t)k K(t +1) ; +kd(t)k K(t +1) ; + C(t +1) ; + C(t +1) ; n ;1 (t) + C(1 + t) ; n 4 ; ; 1 (t): 4

25 Since > n +>we can replace (t +1); by(t +1) ; and get (notice that n = 1 (n +1+)) (t) C 1 (t +1) ; + C (t +1) ; n ;1 (t) : (4) We claim that lim t!1 (t) =. Since is bounded, hence (t)isbounded by Ct, this is clear from (4) if n 3. For n =,we proceed as in [19] for the Navier-Stokes equations: We return to (1), taking now g(t) = 3 (t +1)log(t +1) G(t) = exp( t Instead of () we get d log 3 (t +1)kD(t)k dt e;1 g(s) ds) =log 3 (t +1): C(t +1) ;3 (t) + C(t +1) ;1; (t)log 3 (t +1) C t +1 + C (t +1) 1+ log 3 (t +1) where we have used again that (t) is bounded by Ct.Integrating from to t, weget log 3 (t +1)kD(t)k C log(t +1)+C t log 3 (s +1) (s +1) 1+ ds C log(t +1)+C: Dividing by log 3 (t + 1), we see that lim t!1 kd(t)k =. The claim follows since (t) K(t +1) ; +kd(t)k : Moreover, it is clear from all these bounds that the rate of decay of at 1 (as dened in the Appendix, after Lemma 6.), depends only on K, thel - norm of (u B ) and the L 1 ( 1 L (R n ) n )-norm of f. This holds for n = as well as for n>. We can now apply Lemma 6. of the Appendix (with =1+n=, so and minf g = ) to conclude the proof of part 1 of the theorem. To prove part,we assume 1. We use (15) instead of (16), which allows us to replace (17) by the improved version d dt G(t)kD(t)k g(t) G(t) jjg(t) j ^D( t)j d +(t) 1= kd(t)k (krvk 1 + krwk 1 )G(t) 5 (5)

26 Working as before, using () to estimate the rst term on the right hand side of (5) and invoking Lemma.6), with g(t) =(=(t + 1))61=, G(t) = (t +1),we get instead of (), d (t +1) kd(t)k C(t) (t+1) ; n ;+ +C(t) 1= kd(t)k (t+1) ; n 4 ; ; 1 + : dt We integrate from to t, estimating (by part1),(t) by C(t +1) ; and also (as before) t (s) (s +1) ; n ;+ ds (t) t (s +1) ; n ;+ ds = ( ; n ; 1);1 (t) (t +1) ; n ;1+ ; 1 C(t) (t +1) ; n ;1+ : We get, after dividing by (t +1) t kd(t)k C(t) (t +1) ; n ;1 + C(t +1) ; (s +1) ; n 4 ;; 1 + kd(s)k ds: Setting Y (t) = sup st (s +1) n=4+1= the last inequality implies thus Since (t) = Y (t) C(t) + C(t +1) 1; Y (t) Y (t) C(t)+C(t +1) 1; : t (s) ds ( C if >1 C log(t +1) if =1 for all t, part follows at once. The following corollary summarises the consequences of Theorem 3.1 which will be needed in the sequel. The hypothesis f C (n+3)= insures f C for n=4+=+1, = n= or = n=+1. Corollary 3. Let (u B ) H H, and let f L 1 ( 1 L (R n ) n ) \ C (n+3)=. Assume that the solution (v w) of the (HS,f) system satises kv(t)k + kw(t)k K(t +1) ; for all t, some constant K, where = n= or = n=+1. 6

27 1. There exists a constant C such that for t. ku(t)k + kb(t)k C(t +1) ;. If n =and = n= =1, there isaconstant C, such that for t. kd(t)k C(t +1) ; (1 + log (t +1))C(t +1) ;3= 3. If n> or if n =but = n=+1=, there isaconstant C, such that kd(t)k C(t +1) ; n ;1 for t. We would like tomention that the result kd(t)k C(t +1) ;n=;1,in the case in which D(t) is the dierence between a solution of the Navier- Stokes equation and the solution of the heat equation with the same initial datum was rst established by Wiegner [4] for n, using a dierent approach. It is also worth mentioning that hang Linghai has recently made some improvements to the the Fourier splitting method for n = (cf. [5]), which we have used in our proof above. 4 The lower bounds In this section we derive lower bounds for the decay rate of the L -norms of solutions to the MHD equations. Our lower bounds take the form (t) =ku(t)k + kb(t)k C(t +1) ; for t, where C>and = n= or = n= + 1 (see below). We assume from now on that our forcing function is in the set F = L 1 ( 1 L (R n ) n \ W 1 ) \ A n=4+4 \ B 4 \ C (n+3)= : The choice of this set is motivated by the immediately veriable fact that if f F, then f satises the hypotheses of Lemma.5, Lemma.6 and 7

28 Theorem 3.1 with n= + 1, and Corollary 3.. We need something like f L 1 ( 1 L \ W 1 ) to be able to apply Lemma 6.1 of the Appendix. We may assume that our solutions are smooth. In fact, a general solution can be approximated by smooth solutions, and it is an easy matter to verify that such an approximation can be done without aecting the constants appearing in our estimates. It follows that general solutions satisfy the lower bounds for a.e. t. However, we notice that by the results of the last section, since (t) decays in time, we can nd t > such that (t) is arbitrarily small for t t. By the energy inequality (), (t) =kru(t)k + krb(t)k is integrable over t <1, hence lim inf t!1 (t) =. It follows we can nd t for which (t ) (t ) is arbitrarily small. It is a well known classical result that, for n 3, the solution (u B) is smooth for t t if (t ) (t ) is suciently small (see [1], [6] for the case of the Navier-Stokes equations, the MHD case is similar). For n = 4 it is also known that, for t suciently large, the solution is strong. The lower bounds are then valid everywhere for t t. Since we are assuming that our solutions satisfy the strong energy inequality (hence cannot vanish for some t without also vanishing for all t t),the lower bound also has to hold everywhere for t t, possibly with a smaller positive constant. The case = n= is easily dealt with. ^B () 6=,weproved in the last section that Assuming that ^u () 6= or kd(t)k C(t +1) ;n=;1 if n 3 kd(t)k C(t +1) ;n=;1 (1 + log (t +1)) if n = while, by Lemma.5, kv(t)k C(t +1) ;n=.itfollows that and, in fact, if ^u () 6=, ku(t)k + kb(t)k C(t +1) ;n= ku(t)k C(t +1) ;n= kb(t)k C(t +1) ;n= if ^B () 6=. Thus we will assume for the rest of the section that ^u () = ^B () =. The proof of the lower bounds is based on Fourier analysis of the equations satised by the dierence between the solution of the MHD equations and 8

29 the solution of the heat system with the same initial data. We begin with two auxiliary Lemmas. The rst one calculates the value of the constant 1 of Lemma.3 for the solution of a heat equation whose initial value is u(t). Lemma 4.1 Let P () =I ; (), where I is the n n identity matrix and () = 1 jj ( k j ) 1k jn (6) for =( 1 ::: n ) R n nfg and assume S =(s kj ) is a symmetric matrix. Then 1 = S n;1 P (!)S! S! d! = X n(n + );( n ) (s kk ; X 1 s jj ) +n s kj k6=j k6=j Proof. We shall be using the following easily established formulas: Sn;1! k d! = n= n;( n ) A : for k =1 ::: n S n;1! k! j d! = 8 >< >: n= n(n+);( n ) 6 n= n(n+);( n ) if k 6= j if k = j for 1 k j n. Since S is symmetric, we get for! S n;1, (I ; (!))S! X S! = s kp s jp! k! j ; X s kp s jq! p! q! k! j : (7) k j p k j p q We integrate over S n;1 noticing that only terms in which allpowers of the! k 's are even have non-vanishing integrals. In the rst sum appearing in (7), these are the terms with k = j. In the second sum, we have to consider the terms for which all indices are equal and the terms in which the indices appear in pairs. We get S n;1 (I ; (!))S! S! d! 9

30 X = s k j k j = X k j n= s k j S n;1! k d! ; X k n;( n ) ; X k Decomposing now s k k X k:j s k k Sn;1 X 4k! d! ; (s kk s jj +4s kj )! 1k<jn S n;1 k! j d! 6 n= n(n + );( n ) ; X k6=j s kj = X k grouping terms together and using (n ; 1) X k s kk ; X k6=j s kk + X k6=j s kj s kk s jj = 1 X k6=j s kk s jj n= n(n + );( n ) ; X k6=j (s kk ; s jj ) the expression for 1 follows. The Lemma we have justproved will be used to calculate the constant 1 of Lemma.3 for solutions v of the heat equation with forcing function f and initial datum u(t ), for some T>. In this case, we willhave S = A(T ), where, for t, we dene the n n matrix A(t) =[A kj (t)] = We will also need the n n matrix C(t) =[C kj (t)] = t t A( s) ds: C( s) ds: In these formulas, A C are the matrices dened in Section 3. By (1), (1) and Theorem 3.1, we see that A( ) C( ) L 1 ( 1) forevery R n if either n>orifn =andu B haveaverages. In estimating the solution of the heat system with initial value (u(t) B(t)), for some t = T >, the following quantities play a role. We dene for t, 1 (t) = n= n(n + );( n X (A kk (t) ;A X 1 jj (t)) +n A kj (t) A k6=j k6=j s kj n= n(n +);( n ): and (t) =; n= n;( n ) X ^B! k () ; ic kj (t) = j X n;( n ^B j k6=j () ; ic kj (t) : 3

31 The rst quantity, 1, comes directly from applying Lemma 4.1. Both quantities, 1 and, are related to the rst order terms of the Taylor expansion of the Fourier transforms of u B as is made explicit in the proof of Theorem 4.5. Notice that if u B are real valued (as we assume), then the matrices A(t), C(t) are real. In fact, all entries are Fourier transforms evaluated at = of real valued functions. Notice moreover that (@=@ j ) ^Bk () ; ic kj (t) is purely imaginary. We also dene and (t) = 1 (t)+ (t) ~ i = i (1) = lim t!1 i (t) for i =1 ~ =~ 1 +~ : By the remarks following the introduction of A C, the functions A kj (t), C kj (t), are integrable over [ 1), hence ~ is nite (recall that we are assuming that ^u () = ^B () =, which is needed for integrability in case n =). Denition. Assume (u B ) [H \ W 1 (R n ) n ], f Fand let (u B) be the corresponding solution of the MHD equations. We say(u B) M 1 i 1. D ^u () =, and. ~ 1 =. We say(u B) M i k () = for k =1 ::: n,and. ~ =. We set M = M 1 \M. Note that by the denition of ~ 1 and ~,apair (u B) has~ 1 =~ = if and only if the matrices ~A ~C of the Introduction satisfy ~A is scalar and ~C coincides with hx B i.itfollows that the set M is a subset of the set M of the Introduction. We note that M is very small. More precisely,if(u B) [L 1 ( 1 L (R n ) n )], then generically (u B) M c.to see this, we set ; 1 ij = f(u B) h L 1 1 L (R n ) ni : Aii ;A jj =g 31

32 for 1 i 6= j n and set We then have ; i j = f(u B) h L 1 1 L (R n ) ni : Aij =g ; 1 = \ i j ; 1 i j ; = \ i j ; i j: Lemma 4. The manifolds ; 1 and ; are transversal. Proof. To show that ; 1 and ; are transversal it suces to show thatanytwo submanifolds ; 1 i j, ; i j are transversal. This will be established by showing that ; i j can be obtained from ; i j by a double rotation by an angle of 45 (and vice-versa). Let Q 1 = 1 p 1 ;1 1 1!! Q= Q 1 : Q 1 Set (v i v j w i w j ) t = Q(u i u j B i B j ) t. A simple computation yields v i v j ; w i w j = 1 [(u i ; B i ) ; (u j ; B j )]: The pair(u w) withv k = u k w k = B k for k 6= i j, v i, v j, w i, w j dened as above, is thus in ; ij. Since the eigenvalues of the orthogonal matrix Q 1 are equal to 1= p (1 i), we see that Q 1 is a rotation by an angle of 45 and Q is the double rotation we were referring to above. Corollary 4.3 Let (u B) [L 1 ( 1 L (R n ) n )]. Then generically (u B) M c. Proof. Immediate it is just a restatement of Lemma 4. We will also need the following auxiliary lemma. Lemma 4.4 Let V be ann n matrix of complex entries, let B be asymmetric n n matrix of real entries. If for every! S n;1 we have then V =and B is a scalar matrix. V!; i (I ; (!)) B! = 3

33 Proof. Let! be an eigenvector of B. Noticing that (!)! =!, we get V!=. Since B is real symmetric, it has a complete set of eigenvectors and it follows that V =. It is not too hard to see that for every! S n;1,1 is an eigenvalue of multiplicity one of (!) with eigenvector!. Since now B! = (!)B! we conclude that B! =! for some R. Thisproves that every vector is an eigenvector of B, which is only possible if B is a scalar matrix. We now prove our main theorem. Theorem 4.5 Assume n 4 and let f F, >n=+1. Let (u B ) [H \ H 1 (R n ) \ L 1 (R n ) n \ W 1 \ W ] (which, by Borchers' Lemma B implies R n u (x) dx = R n B (x) dx =): Let (u B) be the corresponding solution of the MHD equations. Then 1. If (u B) =M, then there exist positive constants M M 1 such that M (t +1) ;n=;1 ku(t)k + kb(t)k M 1 (t +1) ;n=;1 :. If (u B) M, then for every > there exists T such that for t T. ku(t)k + kb(t)k (t +1) ;n=;1 Proof. We will denote by O t () aquantity depending on t, bounded in jj for each t, where > depends only on f. (If f, then one can assume O t () is bounded for all R n.) Since A C are continuously dierentiable in with bounded partials (cf. Appendix, Lemma 6.1 here is the only place where we need to assume n 4), A( t) =A( t)+o t ()jj, C( t) =C( t)+o t ()jj. It follows that ^H( t) = i (I ; ()) A( t) + O t ()jj (8) ^M( t) = ic( t) + O t ()jj : (9) We use this expansion in (8), as well as the fact that since f Fwe have ^f( t) =O t ()jj 4. In addition, we expand ^u and ^B in Taylor series around 33

34 the origin up to terms of order (the hypothesis (u B ) W 1 implies that ^u ^B are twice continuously dierentiable with bounded partials) and use that e ;tjj =1+O t ()jj.weget ^u( t) = ^u () + D ^u () ; i(i ; ())A(t) + O t ()jj (3) ^B( t) = ^B () + D ^B () ; ic(t) + O t ()jj (31) where, for an R n -valued function g of the variable R n, D g denotes the Jacobian matrix with (k j)-th k =@ j.for part 1, set so that (since ^u () = ^B () = ) P 1 ( t) = D ^u () ; i(i ; ())A(t) P ( t) = P (t) =D ^B () ; ic(t) ^u( t) = P 1 ( t) + O t ()jj ^B( t) = P ( t) + O t ()jj : (3) Observe thatifd ^u () =, then P 1 ( t) = i(i ; ())A(t) sothatby Lemma 4.1 S n;1 jp 1 (! t)!j d! = 1 (t) while a simple computation shows that (t) = S n;1 jp (t)!j d! for all t. Assume rst ~ 1 > ord ^u () 6=. We claim that there exist T >, >suchthat S n;1 jp 1 (! t)!j d! (33) for all t T. In fact, otherwise lim inf t!1 Since A( ) L 1 ( 1), we can dene ~A = 1 S n;1 jp 1 (! t)!j d! =: A( s) ds =lim t!1 A(t) 34

35 and get where S n;1 j ~ P 1 (!)!j d! = (34) ~P 1 () =D ^u () ; i(i ; ()) ~ A: Since ~ P 1 is homogeneous, (34) is possible only if ~ P 1 () =forall R n n. By Lemma 4.4 we conclude that D ^u () = and ~ A is a scalar matrix (i.e., ~ 1 = ), contradicting our assumption. The claim is established. Assume now ~ > then (since P does not depend on ) it is clear that there exist T, > such that S n;1 jp (t)!j d! (35) for t T. It follows that with the hypothesis of part 1, one of (33), (35) holds. Let T T (to be determined later) and let (v(t) w(t)) be the solution of the (HS,f) system with f replaced by f T = f( + T ) and initial datum (v() w()) = (u(t ) B(T )). In view of the representation (3) (with t = T ) of the initial datum of (v w) we get from Lemma.3 that there exists a constant c n >, depending only on n, suchthat kv(t)k + kw(t)k c n t ;n=;1 + O(t ;n=;3= ): We setd(t) =(D 1 (t) D (t)) = (u(t + T ) B(t + T )) ; (v(t) w(t)) so that D satises D 1t (t) =D 1 (t) ; H(t + T ) D t (t) =D 1 (t) ; M(t + T ) and D() =. We will apply the Fourier splitting method, taking this time g(t) =(=(t)) 1=, G(t) =t, t maxf1 =( )g. Inequality (17) is still valid, with u(t), B(t) replaced by u(t + T ), B(t + T ). The squares of the L - norms of u(t + T ), B(t + T ), v(t), w(t) decay at the same rate as t ;n=;1 for t!1 using Lemma.6 to estimate the L 1 -norm of rv and rw, inequality (17) implies d dt t kd(t)k t ;1 j ^D( t)j d + C T t ;n; (36) tjj 35

36 where C T is a constant depending on T. Inequality (18) also holds in this case, once more with u B translated by T i.e., t j ^D( t)jcjj (ku(s+t )k +kb(s+t )k ) ds Cjj 1 (ku(s)k +kb(s)k ) ds: Since ku(t)k,kb(t)k behave like t ;n=;1 for t!1,weget Thus tjj Using this in (36) gives j ^D( t)jcjjt ;n= : j ^D( t)j d CT ;n tjj T jj = CT ;n t ;n=;1 : d t kd(t)k dt CT ;n t ;n=; + C T t ;n; : Taking >n+ large enough so that all powers of t in the last inequality are positive, integrating from 1 to t, and dividing by t gives kd(t)k CT ;n t ;n=;1 + C T t ;n;1 where C C T depend now also on and we collected all terms with powers of t ; into the last term. Taking now T large enough so that the coecient CT ;n of t ;n=;1 in the last inequality is less than 1 c 4 n, we get ku(t + T )k + kb(t + T )k (kv(t)k + kw(t)k ;kd(t)k ) 1 (kv(t)k + kw(t)k ) ;kd(t)k 1 4 c nt ;n=;1 + O(t ;n=;3= ) proving part 1. For the proof of part, it suces to observe thatifweset 1 (t) = S n;1 jp 1 (! t)!j d! + S n;1 jp (t)!j d! then, under the hypotheses of part, lim t!1 1 (t) =. In fact, since P (t) = D ^B () ; ic(t), it is clear that lim t!1 S n;1 jp (t)!j d! =: 36

37 Concerning the other surface integral, notice that, since wearenow assuming D ^u () =, we have (as remarked in the proof of part 1) S n;1 jp 1 (! t)!j d! = 1 (t): The assertion about 1 follows, since we are assuming ~ 1 = lim t!1 1 (t) =. By Lemma.3, taking T large enough so d n 1 (T ) 4 where d n =( 1 )n=+ ;(n= + 1), so that by Lemma.3 (see inequality (6)), letting again (v(t) w(t)) be the solution of the (HS,f) system with initial datum (u(t ) B(T )), we get kv(t)k + kw(t)k 4 t;n=;1 + O(t ;n=;3= ) for t!1.we dene and estimate D(t) asinpart1. Taking T large enough, we get kd(t)k 4 t;n=;1 + O(t ;n;1 ): The result follows applying the elementary inequality (a + b) a +b. Remark. It should be emphasized that while it may be dicult to always decide when a given solution (u B) isnotinm, there is a trivial way of getting solutions not in M and which, therefore, satisfy the conclusions of part 1 of Theorem 4.5. In fact, it suces to select the initial datum (u B ) so that u (x) dx = B (x) dx = R n R n but there exist j k such that R n x j u k (x) dx 6= : In fact, in this case D ^u () 6=. Noticing that in the proof of Theorem 4.5 the roles of u and B can be kept fairly independent, we actually proved 37

38 Corollary 4.6 Let f F.Let (u B ) [H \ W 1 ] and assume that R n u (x) dx = R n B (x) dx =: Let (u B) be the corresponding solution of the MHD equations. Then 1. If u=m 1, then there exist positive constants M M 1 such that M (t +1) ;n=;1 ku(t)k M 1 (t +1) ;n=;1 :. If B=M, then there exist positive constants M M 3 such that M (t +1) ;n=;1 kb(t)k M 3 (t +1) ;n=;1 : Notice that if <(r^u ()) 6=,thenu M c 1. 5 The counterexample. Assuming an even number of space dimensions, we present an example of a solution (u(t) B(t)) to the MHD equations, subject to initial data in the class M of functions with radially equidistributed energy, whichisexponentially decaying in both the velocity and the magnetic eld component. This is achieved by rst constructing functions u which are simultaneously solutions to the MHD equations and the heat system, and are such that (u r)u is a gradient, i.e.: (u r)u = ;rp u t ; u = ;(u r)u ;rp: By letting (u(t) B(t)) = (u(t) u(t)), we obtain a solution to the MHD equations, since then (u r)b ; (B r)u = and B t =B. The problem is, therefore, reduced to constructing a radial solution u(t) of the heat equation which decays exponentially. Theorem 5.1 Suppose that n is even and let 38