Space-time decay estimates for the incompressible viscous resistive Hall-MHD equations
aa r X i v : . [ m a t h . A P ] J un Space-time decay estimates for the incompressible viscousresistive Hall-MHD equations
Shangkun WENG ∗ Seoul National University
Abstract
In this paper, we address the space-time decay properties for strong solutions to theincompressible viscous resistive Hall-MHD equations. We obtained the same space-timedecay rates as those of the heat equation. Based on the temporal decay results in [9], wefind that one can obtain weighted estimates of the magnetic field B by direct weightedenergy estimate, and then by regarding the magnetic convection term as a forcing term inthe velocity equations, we can obtain the weighted estimates for the vorticity, which yieldsthe corresponding estimates for the velocity field. The higher order derivative estimateswill be obtained by using a parabolic interpolation inequality proved in [24]. It should beemphasized that the the magnetic field has stronger decay properties than the velocityfield in the sense that there is no restriction on the exponent of the weight. The samearguments also yield the sharp space-time decay rates for strong solutions to the usualMHD equations. Mathematics Subject Classifications 2010: 35Q35; 35Q85; 76W05.Key words: Hall-MHD, space-time decay, weighted estimates, parabolic interpola-tion inequality.
In this paper we address the space time decay properties for strong solutions to the incom-pressible viscous resistive Hall-Magnetohydrodynamic equations. The incompressible viscousresistive Hall-MHD equations take the following form: ∂ t u + u · ∇ u + ∇ π = B · ∇ B + ∆ u,∂ t B − ∇ × ( u × B ) + ∇ × (( ∇ × B ) × B ) = ∆ B, div u = div B = 0 , (1.1)where u ( x, t ) = ( u ( x, t ) , u ( x, t ) , u ( x, t )) and B ( x, t ) = ( b ( x, t ) , b ( x, t ) , b ( x, t )), ( x, t ) ∈ R × [0 , ∞ ), are the fluid velocity and magnetic field, π = p + | B | , where p is the pressure.We will consider the Cauchy problem for (1.1), so we prescribe the initial data u ( x,
0) = u ( x ) , B ( x,
0) = B ( x ) . ∗ PDE and Functional Analysis Research Center (PARC), Seoul National University, 151-742, Seoul, Re-public of Korea.
Email: [email protected]. u and B satisfy the divergence free condition,div u ( x ) = div B ( x ) = 0 . Recently, there are many researches on the Hall-MHD equations, concerning global weaksolutions [1, 10], local and global (small) strong solutions [10, 12, 8, 13], and singularityformation in Hall-MHD [11]. The application of Hall-MHD equations is mainly from theunderstanding of magnetic reconnection phenomena [19, 21], where the topology structureof the magnetic field changes dramatically and the Hall effect must be included to get acorrect description of this physical process. The Hall-MHD equations are also derived froma two-fluids Euler-Maxwell system for electrons and ions, through a set of scaling limits, see[1]. They also provided a kinetic formulation for the Hall-MHD.We will address the space-time decay properties of the strong solutions to (1.1). Let usbriefly review the history of the study of the space-time decay properties of the solutions to theincompressible Navier-Stokes equations. Leray [20] proposed the problem whether the weaksolution to the Navier-Stokes decay to zero in the L norm. This was first solved positively byKato [23] in the 2-D case and Scheonbek [29] in 3-D case. See also [4, 5, 14, 15, 22, 28, 30, 34]and the reference therein for more details. The Fourier splitting method introduced byScheonbek was able to give the explicit decay rate, i.e. k u ( · , t ) k L = O ( t − n/ /p − / ) if u ∈ L p ( R n ) ∩ L ( R n )(1 ≤ p < k| x | r D bx u ( · , t ) k L = O ( t − γ + r − b ) , for all 0 ≤ r ≤ a ,where D bx denote all the derivatives of order b , under the assumptions k u ( · , t ) k L = O ( t − γ ) k| x | a u ( · , t ) k L = O ( t − γ + a ) for some a > Theorem 1.1. ( Theorem 1.1 and 1.2 in [9] ) Let ( u , B ) ∈ ( L ( R ) ∩ L ( R )) withdiv u = div B = 0 . Then there exists a weak solution ( u, B ) to (1.1), which satisfies k u ( t ) k L + k B ( t ) k L ≤ C ( t + 1) − . (1.2)2 f in addition, ( u , B ) ∈ H m ( R ) for m ∈ N and m ≥ and k u k H m + k B k H m ≤ K for somesmall constant K , then the solution ( u, B ) will become strong and belong to L ∞ ( R + ; H m ( R )) and also satisfy k D m u ( t ) k L + k D m B ( t ) k L ≤ C m ( t + 1) − m − (1.3) for all t ≥ T ∗ . Here C m depends on m and C . In this paper, we will address the spatial decay properties of the above strong solution( u, B ) in Theorem 1.1. For simplicity, we assume that the initial data ( u , B ) belong to theSchwartz class S , so that for any a ≥ b ∈ N = { , , , · · · } , k| x | a D b u k L < ∞ , k| x | a D b B k L < ∞ . Hence the solution ( u, B ) will satisfy k u ( · , t ) k L + k B ( · , t ) k L = O ( t − γ ) (1.4)and k D b u ( · , t ) k L p + k D b B ( · , t ) k L p = O ( t − γ − b − (1 − p ) ) (1.5)for any 2 ≤ p ≤ ∞ and b ∈ N , where γ = . Note that (1.5) can be easily obtained from(1.3) by interpolation.Our main results is stated as follows. Theorem 1.2.
Let ( u, B ) be the strong solution to (1.1) in Theorem 1.1 with the initial data ( u , B ) belong to the Schwartz class S . Then we have the following weighted estimates for u and B : k| x | a D b u ( · , t ) k L p = O ( t − γ + a − b − (1 − p ) ) (1.6) for any b ∈ N and ≤ a < b + and ≤ p ≤ ∞ ; k| x | a D b B ( · , t ) k L p = O ( t − γ + a − b − (1 − p ) ) (1.7) for all b ∈ N and a ≥ and ≤ p ≤ ∞ . Furthermore, for the vorticity ω ( t, x ) = curl u ( t, x ) ,we have k| x | a D b ω ( · , t ) k L p = O ( t − γ + a − b − − (1 − p ) ) (1.8) for all b ∈ N and a ≥ and ≤ p ≤ ∞ .Remark . We find that the spatial decay property of the magnetic field is stronger thanthat of the velocity field in the sense that there is no restriction on the exponent of the weight.This is basically due to the pressure term in the velocity equations. Note that the spatial decayof the voricity field is also much stronger than the velocity field.Remark . One can relax the conditions on the initial data, i.e. there are constants r > and k ∈ N , such that for all ≤ a ≤ r and ≤ b ≤ k k| x | a D b u ( · ) k L < ∞ , k| x | a D b B ( · ) k L < ∞ . Then the conclusions in Theorem 1.2 also hold with some obvious modification. B under the assumptions (1.4) and (1.5).Then we estimate the weighted norm of the vorticity ω = curl u by regarding curl ( B · ∇ B )as a forcing term. It turns out that we can get the decay rate for the vorticity withoutany restriction on the exponent of the weight. By using the relation between u and ω , wecan estimate the weighted norm of u as in [26, 27]. With the help of these weighted normestimates of u and B , we can use a parabolic type interpolation inequality proved in [24] toget the decay rates for the weighted norms of higher order derivatives. As above, we firstestimate the magnetic field B and then the velocity field regarding the magnetic convectionterm B · ∇ B as a forcing term. Here the existence of the Hall term requires a separatetreatment of k| x | a ∇ B k L , which will be used in the induction for higher order derivatives.Although the Hall term contains the second order derivative, it is a quadratic term, one canput the weight on another B , this is the reason why the Hall term does not affect the decayrates.Our arguments certainly work in the usual MHD case, yielding the sharp space-time decayrates for strong solutions to the incompressible MHD. There are many previous studies onthe time asymptotic behaviors for the solutions to the usual MHD equations [2, 16, 17, 18,31, 35]. In [6], the authors studied the local in time persistence of space decay rates for theincompressible MHD, showing that if the initial magnetic field decays sufficiently fast, thenthe space decay rates of MHD solutions behave as that of Navier-Stokes solutions. On theother hand, if the initial magnetic field is poorly localized, then the magnetic field will governthe decay. Here our estimates also cover the space-time decay of higher order derivatives andthe long time behavior, this result seems to be new, so we also include this as a theorem here. Theorem 1.5.
Let ( u, B ) is a strong solution to the incompressible viscous resistive MHDequations with initial data ( u , B ) ∈ S with div u = div B = 0 . Assume that (1.4) and (1.5)holds. Then we have the following weighted estimates for u and B : k| x | a D b u ( · , t ) k L p = O ( t − γ + a − b − (1 − p ) ) (1.9) for any b ∈ N and ≤ a < b + and ≤ p ≤ ∞ ; k| x | a D b B ( · , t ) k L p = O ( t − γ + a − b − (1 − p ) ) (1.10) for all b ∈ N and a ≥ and ≤ p ≤ ∞ . Furthermore, for the vorticity ω ( t, x ) = curl u ( t, x ) ,we have k| x | a D b ω ( · , t ) k L p = O ( t − γ + a − b − − (1 − p ) ) (1.11) for all b ∈ N and a ≥ and ≤ p ≤ ∞ .Remark . It is well-known that the assumptions (1.4) and (1.5) will be satisfied either forglobal strong solutions to MHD with small data or global weak solutions to MHD equationsafter a large finite time. See [2, 9, 31] for more details. The paper will be organized as follows. In section 2, we present some lemmas which areneeded in the weighted norm estimates. The weighted norm estimates for the solutions andhigher order norms will be treated in section 3 and 4 separately.4
Preliminary
The following lemma is needed in the weighted estimates.
Lemma 2.1.
Let α > , α < , α < and β , β < . Assume that a continuouslydifferentiable function F : [1 , ∞ ) → [0 , ∞ ) satisfies F ′ ( t ) ≤ C t − α F ( t ) + C t − α F ( t ) β + C t − α F ( t ) β + C t γ − , t ≥ F (1) ≤ K where C , C , C , C , K ≥ and γ i = − α i − β i > for i = 1 , . Assume that γ ≥ γ , then thereexists a constant C ∗ depending on α , α , β , α , β , K , C i , i = 1 , · · · , , such that F ( t ) ≤ C ∗ t γ for t ≥ .Proof. This lemma is a simple variant of Lemma 2.2 in [25]. For the reader’s convenience,we present the proof following the idea in [25]. Let t ≥ C t − ( α − = γ . By Young’sinequality, we have F ′ ( t ) ≤ C F ( t ) + C t − α − β + C t − α − β + C t γ − , t ≥ ,F (1) ≤ K . Hence by the standard Gronwall’s inequality, we know F ( t ) ≤ K , where K depending on α , α , β , α , β , K , C i , i = 1 , · · · ,
4. Let
K > K ≥ max (cid:26) ( C β γ − ) − β , ( C β γ − ) − β , K , C γ (cid:27) . Denote R = { t ≥ t : F ( t ) ≤ Kt γ } . Since F ( t ) ≤ K ≤ K , t ∈ R , and by continuity,there exists a maximal interval [ t , b ) ⊂ R . We show that b = ∞ .Suppose t < b < ∞ . Then F ( b ) = 2 Kb γ and F ′ ( b ) ≥ G ′ ( b ), where G ( t ) = 2 Kt γ . Notethat G ′ ( b ) ≤ F ′ ( b ) ≤ C b − α Kb γ + C b − α (2 Kb γ ) β + C b − α (2 Kb γ ) β + C b γ − ≤ Kγ b γ − (cid:18) C γ − b − α + C γ − β K β − b γ β − α − γ +1 + C γ − β K β − b γ β − α − γ +1 + C Kγ (cid:19) ≤ Kγb γ − (cid:18) C γ − β K β − + C γ − β K β − + C Kγ (cid:19) ≤ Kγ b γ − . However, G ′ ( b ) = 2 Kγ b γ − , which is a contradiction. Hence we finish the proof.The following two lemmas are needed in the weighted estimates for higher order norms.Both of them have been proved in [24] and [25]. Here we omit the proof.5 emma 2.2. Let p ∈ [1 , ∞ ] and T > . Assume that u ∈ L ∞ ((0 , T ); L p ( R n )) and t ( u t − ∆ u ) ∈ L ∞ ((0 , T ); L p ( R n )) . Then t / ∇ u ∈ L ∞ ((0 , T ); L p ( R n )) and the inequality sup t/ ≤ τ ≤ t k∇ u ( · , t ) k L p ≤ C ( sup t/ ≤ τ ≤ t k u ( · , τ ) k L p )( sup t/ ≤ τ ≤ t k ( u t − ∆ u )( · , τ ) k L p )+ Ct sup t/ ≤ τ ≤ t k u ( · , τ ) k L p holds for every t ∈ (0 , T ) . Lemma 2.3.
Let τ > and assume that F : [ τ , ∞ ) → [0 , ∞ ) satisfies sup τ ≤ τ ≤ A F ( τ ) < ∞ for all A > τ . If there exist C > and γ ∈ R such that sup t/ ≤ τ ≤ t F ( τ ) ≤ C t − γ + C t − γ sup t/ ≤ τ ≤ t F ( τ ) , t ≥ τ (2.1) then F ( t ) = O ( t − γ ) as t → ∞ . u and B By Theorem 1.1, we may assume there exists a constant γ = , such that k u ( · , t ) k L + k B ( · , t ) k L = O ( t − γ ) as t → ∞ . (3.1)Then by Gagliardo-Nirenberg inequality and (1.3), we have for 2 ≤ p ≤ ∞k ∂ α u ( · , t ) k L p + k ∂ α B ( · , t ) k L p = O ( t − γ − | α | − (1 − p ) ) , α ∈ N . (3.2)First we observe that the weighted estimate for the magnetic field can be obtained directlyunder the assumptions (3.1) and (3.2). Theorem 3.1.
Under the assumption (3.1) and (3.2), we have k| x | a B ( · , t ) k L = O ( t − γ + a/ ) as t → ∞ (3.3) for all a ≥ .Proof. B t + u · ∇ B + ∇ × (( ∇ × B ) × B ) = B · ∇ u + ∆ B. Multiplying the above equations by 2 | x | a B , and setting G ( t ) = R R | x | a | B ( x, t ) | dx , thenwe get ddt G ( t ) + 2 Z R | x | a |∇ B | dx = − Z R | x | a B · ( u · ∇ B ) dx + Z R | x | a B · ( B · ∇ u ) dx − Z R | x | a B · [ ∇ × (( ∇ × B ) × B )] dx − Z R a | x | a − n X i,j =1 B i x j ∂ j B i dx := I + II + III + IV. | I | ≤ Z R | x | a |∇ B || B || u | dx ≤ Z R | x | a |∇ B | dx + C Z R | x | a | B | | u | dx ≤ Z R | x | a |∇ B | dx + C k u k L ∞ G ( t ) , | II | ≤ Z R | x | a |∇ u || B | dx ≤ k∇ u k L ∞ G ( t ) , | III | ≤ C Z R | x | a | B | ( |∇ B || B | + |∇ B | ) dx ≤ C k∇ B k L ∞ G ( t ) + C k∇ B k L ∞ Z R | x | a | B ||∇ B | dx ≤ Z R | x | a |∇ B | dx + C ( k∇ B k L ∞ + k∇ B k L ∞ ) G ( t ) . | IV | ≤ C Z R | x | a − | B ||∇ B | dx ≤ Z R | x | a |∇ B | dx + C Z R | x | a − | B | dx ≤ Z R | x | a |∇ B | dx + CG ( t ) a − a k B k a L . Combining all these estimates, we get G ′ ( t ) ≤ C ( k u k L ∞ + k∇ u k L ∞ + k∇ B k L ∞ + k∇ B k L ∞ ) G ( t ) + CG ( t ) a − a k B k a L ≤ Ct − γ − − G ( t ) + Ct − γ a G ( t ) a − a . If a > γ , then we may apply Lemma 2.1 with α = γ + + > , α = γ a < , β = a − a < , C = C = 0 to get G ( t ) ≤ Ct γ with γ = 1 − α − β == a − γ and the theorem is proved for all a > γ . The conclusion for a ∈ (0 , γ ] follows byinterpolation.Now we turn to the velocity field. Let the vorticity ω ( t, x ) = curl u ( t, x ), then ∂ t ω + u · ∇ ω − ω · ∇ u − ∆ ω = curl( B · ∇ B ) . With the weighted estimates (3.3) of B at hand, we regard curl( B · ∇ B ) as a forcing term,and estimate the weighted norm of the vorticity as above. Theorem 3.2.
Under the assumptions (3.1) and (3.2), we have the following estimate forall a ≥ k| x | a ω ( · , t ) k L = O ( t − γ − + a ) . (3.4)7 roof. Multiplying the vorticity equation by 2 | x | a ω and setting F ( t ) = R R | x | a | ω ( x, t ) | dx ,then we get ddt F ( t ) + 2 Z R | x | a |∇ ω ( x, t ) | dx = − Z R | x | a ω · ( u · ∇ ω ) dx + Z R | x | a ω · ( ω · ∇ u ) dx − a Z R | x | a − n X i,j =1 x j ω i ∂ j ω i dx + Z R | x | a ω · curl( B · ∇ B ) dx := I + II + III + IV.
These four terms will be estimated as follows. | I | ≤ Z R | x | a |∇ ω | dx + C k u k L ∞ F ( t ) , | II | ≤ k∇ u k L ∞ F ( t ) , | III | ≤ C Z R | x | a − | ω ||∇ ω | dx ≤ Z R | x | a |∇ ω | dx + CF ( t ) a − a k ω k a L , | IV | = 2 (cid:12)(cid:12)(cid:12)(cid:12) Z R curl( | x | a ω ) · ( B · ∇ B ) dx (cid:12)(cid:12)(cid:12)(cid:12) = 2 (cid:12)(cid:12)(cid:12)(cid:12) Z R | x | a curl ω · ( B · ∇ B ) dx + Z R a | x | a − ( x × ω ) · ( B · ∇ B ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C Z R | x | a |∇ ω || B ||∇ B | dx + C Z R | x | a − | ω || B ||∇ B | dx ≤ Z R | x | a |∇ ω | dx + C k∇ B k L ∞ Z R | x | a | B | dx + F ( t ) k∇ B k L ∞ k| x | a − | B |k L . Combining all these estimates together, we obtain F ′ ( t ) ≤ C ( k u k L ∞ + k∇ u k L ∞ ) F ( t ) + C F ( t ) a − a k ω k a L + C F ( t ) k∇ B k L ∞ k| x | a − B k L + C k∇ B k L ∞ k| x | a B k L ≤ C t − γ − / F ( t ) + C F ( t ) a − a t − a ( γ + ) + C F ( t ) t − γ − + a + C t − γ − + a = C t − F ( t ) + C F ( t ) a − a t − a + C F ( t ) t − + a + C t a − . Now we can apply Lemma 2.1. Here β = a − a , α = a , β = , α = − a + . To assurethat α < , α <
1, we require a > . Hence γ = − α − β = a − > γ = − α − β = a − . ByLemma 2.1, we obtain F ( t ) ≤ Ct − + a = Ct − γ − a .
8y the relation − ∆ u = curl ω and the Caffarelli-Kohn-Nirenberg inequality [7] k| x | a u k L p ≤ k| x | a ∇ u k L p , one can argue as in [26] and [27] to obtain the weighted estimates for the velocity field u as stated in the following theorem. Since the proof are almost the same, here we omit thedetails. Theorem 3.3.
Under the assumptions (3.1)-(3.2), we have the following weighted estimates k| x | a u ( · , t ) k L = O ( t − γ + a ) (3.5) for all a ∈ [0 , ) . Based on the estimates (3.3) and (3.5), we can apply Lemma 2.2 to get the weighted estimatesfor higher order derivatives of u and B . First we estimate B , here the existence of the hallterm requires a separate treatment of k| x | a ∇ B k L , which will be used for the induction ofhigher order derivatives. Although the Hall term contains the second order derivative, it is aquadratic term, one can put the weight on another B , this is the reason why the Hall termdoes not affect the decay rates. Theorem 4.1.
Under the assumptions (3.1) and (3.2), then the following estimates hold forall a ≥ , b ∈ N and ≤ p ≤ ∞k| x | a D b B ( · , t ) k L p = O ( t − γ − b + a − (1 − p ) ) . (4.1) Proof.
We only need to prove the case a >
2, since we already know (4.1) holds for a = 0, thecase 0 < a ≤ p > p = 2. Indeed, for any f ∈ L ( R ) ∩ ˙ H ( R ), one has k f k L ∞ ( R ) ≤k f k L ( R ) k f k ˙ H ( R ) . Hence one can derive the estimate of k| x | a D b B ( · , t ) k L ∞ ( R ) from thoseof k| x | a D b B ( · , t ) k L ( R ) . The case p ∈ (2 , ∞ ) just follows from interpolation. Therefore, weassume p = 2. For a >
2, we choose the weight φ : φ ( x, t ) = ( | x | + t ) a , t ≥ , then by simple calculations, we get |∇ φ ( x, t ) | ≤ ( | x | + t ) a − , | ( ∂ t − ∆) φ ( x, t ) | ≤ C ( | x | + t ) a − . The case b = 0 has been proved in Theorem 3.1. The following proof will be separated intotwo steps. The first step addresses the case b = 1, which is needed in the weighted estimatefor higher order derivatives in the second step. Step 1.
The case b = 1. To use Lemma 2.2, we need to derive the equation for φB ( ∂ t − ∆)( φB ) = ( ∂ t − ∆) φB − ∇ φ · ∇ B − φ ( u · ∇ B − B · ∇ u + ∇ × (( ∇ × B ) × B )) . t/ ≤ τ ≤ t k∇ ( φB ) k L ≤ C sup t/ ≤ τ ≤ t k φB k L sup t/ ≤ τ ≤ t (cid:18) k ( ∂ t φ − ∆ φ ) B k L + k∇ φ · ∇ B k L + k φu · ∇ B k L + k φ ∇ × (( ∇ × B ) × B ) k L + k φB · ∇ u k L (cid:19) + Ct sup t/ ≤ τ ≤ t k φB ( τ ) k L Note that sup t/ ≤ τ ≤ t k∇ ( φB ) k L ≥
12 sup t/ ≤ τ ≤ t k φ ∇ B k L − C sup t/ ≤ τ ≤ t k∇ φB k L ≥
12 sup t/ ≤ τ ≤ t k φ ∇ B k L − O ( t − γ + a − ) , k ( ∂ t − ∆) φB k L ≤ k ( | x | + t ) a − B k L ≤ k| x | a − B k L + t a − k B k L ≤ O ( t − γ + a − ) , k∇ φ · ∇ B k L ≤ O ( t − ) k φ ∇ B k L , k φu · ∇ B k L ≤ k u k L ∞ k φ ∇ B k L ≤ O ( t − γ − ) k φ ∇ B k L , k φB · ∇ u k L ≤ k φB k L k∇ u k L ∞ ≤ O ( t − γ + a − − ) . k φ ∇ × (( ∇ × B ) × B ) k L ≤ k φ |∇ B || B |k L + k φ |∇ B | k L ≤ k∇ B k L ∞ k φB k L + k∇ B k L ∞ k φ ∇ B k L ≤ O ( t − γ + a − − ) + O ( t − γ − − ) k φ ∇ B k L . hence we obtainsup t/ ≤ τ ≤ t k φ ∇ B k L ≤ O ( t − γ + a − ) + O ( t − γ + a ) (cid:18) O ( t − γ + a − ) + O ( t − γ + a − − )+ ( O ( t − ) + O ( t − γ − )) sup t/ ≤ τ ≤ t k φ ∇ B k L (cid:19) + Ct O ( t − γ + a ) ≤ O ( t − γ + a − ) + O ( t − γ + a − ) sup t/ ≤ τ ≤ t k φ ∇ B k L . By applying Lemma 2.3, we get k| x | a ∇ B ( · , t ) k L = O ( t − γ − + a ) . Step 2.
The case b ≥
2. Assume that the conclusion holds for all the derivatives up to order b ≥
1, we want to show that it also holds for b + 1. Take any α ∈ N with | α | = b , then( ∂ t − ∆)( φ∂ α B ) = ( ∂ t φ − ∆ φ ) ∂ α B − ∂ j φ∂ j ∂ α B − X ≤ β ≤ α C α,β φ (cid:18) ∂ β u j ∂ j ∂ α − β B − ∂ β B j ∂ j ∂ α − β u (cid:19) − X ≤ β ≤ α C α,β ∇ × (cid:18) ∂ β ( ∇ × B ) × ( ∂ α − β B ) (cid:19) . t/ ≤ τ ≤ t k∇ ( φ∂ α B ) k L ≤ C sup t/ ≤ τ ≤ t k φ∂ α B k L sup t/ ≤ τ ≤ t (cid:20) k ( ∂ t φ − ∆ φ ) ∂ α B k L + k∇ φ · ∇ ∂ α B k L + X ≤ β ≤ α C α,β (cid:18) k φ∂ β u j ∂ j ∂ α − β B k L + k φ∂ β B j ∂ j ∂ α − β u k L + k φ ∇ × (( ∇ × ∂ β B ) × ∂ α − β B ) k L (cid:19)(cid:21) + Ct sup t/ ≤ τ ≤ t k φ∂ α B ( τ ) k L By induction assumptions, we havesup t/ ≤ τ ≤ t k∇ ( φ∂ α B )( τ ) k L ≥
12 sup t/ ≤ τ ≤ t k φ ∇ ∂ α B k L − O ( t − γ − b + a − ) , sup t/ ≤ τ ≤ t k φ∂ α B k L ≤ O ( t − γ − b + a ) , sup t/ ≤ τ ≤ t k ( ∂ t − ∆) φ∂ α B k L ≤ O ( t − γ − b + a − ) , sup t/ ≤ τ ≤ t k∇ φ ∇ ∂ α B k L ≤ O ( t − ) sup t/ ≤ τ ≤ t k φ ∇ ∂ α B k L . For the other three terms, we estimate as follows k φ∂ β u j ∂ j ∂ α − β B k L ≤ ( k ∂ β u j k L ∞ k φ∂ j ∂ α − β B k L , if | β | > k u k L ∞ k φ ∇ ∂ α B k L , if β = 0 ≤ ( O ( t − γ − b + a − − ) , if | β | > O ( t − γ − ) k φ ∇ ∂ α B k L , if β = 0 k φ∂ β B j ∂ j ∂ α − β u k L ≤ k φ∂ β B j k L k ∂ j ∂ α − β u k L ∞ ≤ O ( t − γ − b + a − − ) . k φ ∇ × (( ∇ × ∂ β B ) × ∂ α − β B ) k L ≤ k φ | ∂ β ∇ B || ∂ α − β B |k L + k φ | ∂ β ∇ B || ∂ α − β ∇ B |k L := J + J .J ≤ ( k φ∂ β ∇ B k L k ∂ α − β B k L ∞ , if | β | ≤ b − k ∂ β ∇ B k L ∞ k φ∂ α − β B k L , if | β | = b − b ≤ O ( t − γ − b + a − − ) ,J ≤ ( k φ∂ β ∇ B k L k ∂ α − β ∇ B k L ∞ , if | β | ≤ b − k ∂ α ∇ B k L ∞ k φ ∇ B k L , if β = α ≤ O ( t − γ − b + a − − ) . Note that in the estimate of J , we have used the result from Step 1 .Combining all these estimates together, we getsup t/ ≤ τ ≤ t k φ ∇ ∂ α B k L ≤ O ( t − γ − b + a − ) + O ( t − γ − b + a − ) sup t/ ≤ τ ≤ t k φ ∇ ∂ α B k L . k φ ∇ ∂ α B k L = O ( t − γ − b + a − ) . Theorem 4.2.
Under the assumptions (3.1) and (3.2), we have k| x | a D b u ( · , t ) k L p = O ( t − γ − b − (1 − p )+ a ) for any a ∈ [0 , ) and b ∈ N and p ∈ [2 , ∞ ] .Proof. As before, we only consider a ≥
2. By the Gagliardo-Nirenberg inequality, the case p > p = 2. Therefore, we assume p = 2. Denote φ ( x, t ) = ( | x | + t ) a . The case b = 0 has been proved in Lemma 3.3. Assume that the conclusion holds for b ∈ N + ,and we shall establish it for b + 1. Fix α ∈ N such that | α | = b . Then we have( ∂ t − ∆)( φ∂ α u ) = ( ∂ t φ − ∆ φ ) ∂ α u − ∂ j φ∂ j ∂ α u − φ ∇ ∂ α π − X ≤ β ≤ α C α,β φ (cid:18) ∂ β u j ∂ j ∂ α − β u − ∂ β B j ∂ j ∂ α − β B (cid:19) , Then Lemma 2.2 yieldssup t/ ≤ τ ≤ t k∇ ( φ∂ α u ) k L ≤ sup t/ ≤ τ ≤ t k φ∂ α u k L sup t/ ≤ τ ≤ t (cid:18) k ( ∂ t − ∆) φ∂ α u k L + k∇ φ · ∇ ∂ α u k L + k φ ∇ ∂ α π k L + X ≤ β ≤ α C ( α, β )( k φ∂ β u j ∂ j ∂ α − β u k L + k φ∂ β B j ∂ j ∂ α − β B k L ) (cid:19) + Ct sup t/ ≤ τ ≤ t k φ∂ α u k L . As above, we havesup t/ ≤ τ ≤ t k∇ ( φ∂ α u )( τ ) k L ≥
12 sup t/ ≤ τ ≤ t k φ ∇ ∂ α u k L − O ( t − γ − b + a − ) , sup t/ ≤ τ ≤ t k φ∂ α u k L ≤ O ( t − γ − b + a ) , sup t/ ≤ τ ≤ t k ( ∂ t − ∆) φ∂ α u k L ≤ O ( t − γ − b + a − ) , sup t/ ≤ τ ≤ t k∇ φ ∇ ∂ α u k L ≤ O ( t − ) sup t/ ≤ τ ≤ t k φ ∇ ∂ α u k L . And k φ∂ β u j ∂ j ∂ α − β u k L ≤ ( k ∂ β u j k L ∞ k φ∂ j ∂ α − β u k L , if | β | > k u k L ∞ k φ ∇ ∂ α u k L , if β = 0 ≤ ( O ( t − γ − b + a − − ) , if | β | > O ( t − γ − ) k φ ∇ ∂ α u k L , if β = 0 k φ∂ β B j ∂ j ∂ α − β B k L ≤ k φ∂ β B j k L k ∂ j ∂ α − β B k L ∞ ≤ O ( t − γ − b + a − − ) . k φ ∇ ∂ α π k L , note that π = R i R j ( u i u j − B i B j ) , we need to use the weighted estimates for Riesz operator in Lemma 4.2 in [24]. Hence k φ ∇ ∂ α π k L ≤ k| x | a ∇ ∂ α π k L + t a k∇ ∂ α π k L ≤ k| x | a ∇ ∂ α ( u i u j − B i B j ) k L + k| x | a − ∂ α ( u i u j − B i B j ) k L + t a k∇ ∂ α π k L := H + H + H ,H ≤ X ≤ β ≤ α (cid:18) k φ |∇ ∂ β u || ∂ α − β u |k L + k φ |∇ ∂ β B || ∂ α − β B |k L (cid:19) ≤ k φu k L k∇ ∂ α u k L ∞ + k φB k L k∇ ∂ α B k L ∞ + X ≤ β<α (cid:18) k φ ∇ ∂ β u k L k ∂ α − β u k L ∞ + k φ ∇ ∂ β B k L k ∂ α − β B k L ∞ (cid:19) ≤ O ( t − γ − b + a − − ) ,H ≤ X ≤ β ≤ α (cid:18) k| x | a − ∂ β u k L k ∂ α − β u k L ∞ + k| x | a − ∂ β B k L k ∂ α − β B k L ∞ (cid:19) ≤ O ( t − γ − b + a − − ) ,H ≤ t a k∇ ∂ α ( u i u j − B i B j ) k L ≤ t a X ≤ β ≤ α (cid:18) k∇ ∂ β u k L k ∂ α − β u k L ∞ + k∇ ∂ β B k L k ∂ α − β B k L ∞ (cid:19) ≤ O ( t − γ − b + a − − ) . Hence we obtainsup t/ ≤ τ ≤ t k φ ∇ ∂ α u k L ≤ O ( t − γ − b + a − ) + O ( t − γ − b + a − ) sup t/ ≤ τ ≤ t k φ ∇ ∂ α u k L . Applying Lemma 2.3, we get k φ ∇ ∂ α u k L = O ( t − γ − b − + a ) . Now we show that the vorticity field has much stronger decay properties than the velocityfield in the sense that there is no restriction on the exponent of the weight. Indeed, we have
Theorem 4.3.
Under the assumptions (3.1) and (3.2), the following estimates k| x | a D b ω ( x, t ) k L p = O ( t − γ − b − + a − (1 − p ) ) (4.2) hold for any a ≥ and b ∈ N and ≤ p ≤ ∞ . roof. We choose same weight function as before. The conclusion is true for b = 0 as showedin Theorem 3.2. We assume that the conclusion holds for any derivatives up to order b , wewant to show that it also holds for b + 1. Take any α ∈ N with | α | = b , then( ∂ t − ∆)( φ∂ α ω ) = ( ∂ t − ∆) φ∂ α ω − ∇ φ · ∇ ) ∂ α ω − X ≤ β ≤ α C ( α, β ) (cid:18) φ∂ β u j ∂ j ∂ α − β ω − φ∂ β ω j ∂ j ∂ α − β u − φ ∇ × ( ∂ β B j ∂ j ∂ α − β B ) (cid:19) . Then applying Lemma 2.2, we obtainsup t/ ≤ τ ≤ t k∇ ( φ∂ α ω ) k L ≤ C sup t/ ≤ τ ≤ t k φ∂ α ω k L sup t/ ≤ τ ≤ t (cid:20) k ( ∂ t φ − ∆ φ ) ∂ α ω k L + k∇ φ · ∇ ∂ α ω k L + X ≤ β ≤ α C α,β (cid:18) k φ∂ β u j ∂ j ∂ α − β ω k L + k φ∂ β ω j ∂ j ∂ α − β u k L + k φ ∇ × ( ∂ β B j ∂ j ∂ α − β B ) k L (cid:19)(cid:21) + Ct sup t/ ≤ τ ≤ t k φ∂ α ω k L As above, we havesup t/ ≤ τ ≤ t k∇ ( φ∂ α ω ) k L ≥
12 sup t/ ≤ τ ≤ t k φ ∇ ∂ α ω k L − O ( t − γ − b − a − ) , sup t/ ≤ τ ≤ t k φ∂ α ω k L ≤ O ( t − γ − b − + a ) , sup t/ ≤ τ ≤ t k ( ∂ t − ∆) φ∂ α ω k L ≤ O ( t − γ − b − + a − ) , sup t/ ≤ τ ≤ t k∇ φ ∇ ∂ α ω k L ≤ O ( t − ) sup t/ ≤ τ ≤ t k φ ∇ ∂ α ω k L . For the other three terms, we estimate as follows k φ∂ β u j ∂ j ∂ α − β ω k L ≤ ( k ∂ β u j k L ∞ k φ∂ j ∂ α − β ω k L , if | β | > k u k L ∞ k φ ∇ ∂ α ω k L , if β = 0 ≤ ( O ( t − γ − b + a − − ) , if | β | > O ( t − γ − ) k φ ∇ ∂ α ω k L , if β = 0 k φ∂ β ω j ∂ j ∂ α − β u k L ≤ k φ∂ β ω j k L k ∂ j ∂ α − β u k L ∞ ≤ O ( t − γ − b + a − − ) . k φ ∇ × ( ∂ β B j ∂ j ∂ α − β B ) k L ≤ k φ | ∂ β ∇ B || ∂ j ∂ α − β B |k L + k φ | ∂ β B || ∂ α − β ∇ B |k L ≤ k φ ∇ ∂ β B j k L k∇ ∂ α − β B k L ∞ + k φ∂ β B k L k∇ ∂ α − β B k L ∞ ≤ O ( t − γ − b + a − − ) . Combining all these estimates together, we getsup t/ ≤ τ ≤ t k φ ∇ ∂ α ω k L ≤ O ( t − γ − b + a − ) + O ( t − γ − b + a − ) sup t/ ≤ τ ≤ t k φ ∇ ∂ α ω k L . Hence k φ∂ α ω k L = O ( t − γ − b +12 − + a ). 14n particular, we have showed that k| x | a u ( · , t ) k L = O ( t − γ + a ) for all a ∈ [0 , ). Now onecan argue as in Theorem 3.2 of [25] to get the following theorem. Theorem 4.4.
Assume that (3.1) and (3.2) hold, then k| x | a D bx u ( · , t ) k L p = O ( t − γ − b + a − (1 − p ) ) (4.3) for every ≤ p ≤ ∞ , a ∈ [0 , b + ) and b ∈ N . Indeed, the proof of Theorem 3.2 in [25] only produce the estimate for p = 2. The othercases can be derived from p = 2 by Gagliardo-Nirenberg inequality and interpolation theory. Proof of Theorem 1.2 and 1.5.
Theorem 4.1, 4.3 and 4.4 imply Theorem 1.2. From theproof of Theorem 4.1, 4.3 and 4.4, one can see that same arguments certainly work for theusual incompressible viscous, resisitve MHD equations, hence Theorem 1.5 holds.
Acknowledgements.
The author would like to thank Prof. Dongho Chae for his interestand encouragement. Special thanks also go to Prof. Brandolese for his interest in this paperand kindly notifying the author of his important works on space-time decay for Navier-Stokesand MHD.
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