On the Decay and Stability of Global Solutions to the 3D Inhomogeneous MHD system

In this paper, we investigative the large time decay and stability to any given global smooth solutions of the $3$D incompressible inhomogeneous MHD systems. We proved that given a solution $(a, u, B)$ of (\ref{mhd_a}), the velocity field and magnetic field decay to $0$ with an explicit rate, for $u$ which coincide with incompressible inhomogeneous Navier-Stokes equations \cite{zhangping}. In particular, we give the decay rate of higher order derivatives of $u$ and $B$ which is useful to prove our main stability result. For a large solutions of (\ref{mhd_a}) denoted by $(a, u, B)$, we proved that a small perturbation to the initial data still generates a unique global smooth solution and the smooth solution keeps close to the reference solution $(a, u, B)$. Due to the coupling between $u$ and $B$, we used elliptic estimates to get $\|(u, B)\|_{L^{1}(\mathbb{R}^{+};\dot{B}_{2,1}^{5/2})}

If there is no magnetic field, i.e., B = 0, MHD system turns to be nonhomogeneous Navier-Stokes system. Since the second equation and third equation of (1.1) are similar, the study about MHD system has been along with that for Navier-Stokes one. Let us first recall some results about Navier stokes equations. When ρ 0 is bounded away from zero, the global existence of weak solutions was established by Kazhikov [4]. Moreover, Antontsev, Kazhikov and Monakhov [5] gave the first result on local existence and uniqueness of strong solutions. For the two dimensional case, they even proved that the strong solution is global. But the global existence of strong or smooth solutions in 3D is still an open problem.
Recently, R.Danchin proved the global existence in the Besov space framework [6]. His result states the global in time existence of regular solutions to the inhomogeneous Navier Stokes equations in R n in the optimal Besov setting, under suitable smallness of the data. In particular his results allows the initial densities have a jump at the interface. At the same time, H. Abidi, G. Gui, P. Zhang [7] proved the local well-posedness of three-dimensional incompressible inhomogeneous Navier-Stokes equations with initial data in the critical Besov spaces, without assumptions of small density variation. And they also proved the global well-posedness when the initial velocity is small inḂ 1/2 2,1 (R 3 ). For more results in this direction, see [8,9,10]and reference therein. Now, let us go back to the MHD system (1.1). As said before, the research for MHD goes along with that for Naiver-Stokes equations. The results are similar. When we assume ρ is a constant, which means the fluid is homogeneous, the MHD system has been extensively studied. Duraut and Lions [11] constructed a class of weak solutions with finite energy and a class of local strong solutions. Recently, C. Cao, J. Wu [12] proved global regularity of classical solutions for the MHD equations with mixed partial dissipation and magnetic diffusion. And they also give the global existence, conditional regularity and uniqueness of a weak solution for 2D MHD equations with only magnetic diffusion. For more results in this direction, see [13,14]and reference therein.
When the fluid is nonhomogeneous. H. Abidi and M. Paicu [3] proved that the magneto-hydrodynamic system in R N with variable density, variable viscosity and variable conductivity has a local weak solution in suitable Besov space if the initial density approaches a constant. And they also proved that the constructed solution exist globally in time if the initial data are small enough. X. Huang and Y. Wang [2] proved the global existence of strong solution with vacuum to the 2D nonhomogeneous incompressible MHD system, as long as the initial data satisfies some compatibility condition. In this paper, we only consider non-vacuum case.
Our first result concerns the global stability of the given solution of (1.2) when the initial density ρ 0 is close to a positive constant. This is a simple generalization of Theorem 1.1 in [21].
In order to get the stability of large solutions of system (1.2), here, we need to investigate the decay properties of the velocity field u and magnetic field B. Compared to the (INS) case, our case is more complex and we need to use the coupling between the equations of u and B. Due to the coupling between u and B, we can not get the estimate of (u, B) L 1 (R + ;Ḃ 5/2 2,1 ) < C by using propositions like Proposition 3.6 in [21], which is the methods used in (INS) case. In order to overcome this difficulty, we need to give the estimate of ∇a(t) L ∞ , so that in addition to get the higher order decay properties of u and B. The rigorous statement is the following Theorem.
At the above theorem in hand, we can use the elliptic estimates and various interpolation theorems in Besov space to get the estimate of (u, B) L 1 (R + ;Ḃ 5/2 2,1 ) < C which is completely different to (INS) case. Then after complex calculations, we can get the global estimates of the reference solution (ā,ū,B). At last, similar to Theorem 1.2 but need more complex calculations, we obtain the decay properties of the perturbed solution (a −ā, u −ū, B −B). Using the decay properties of the reference solution and perturbed solution, we finally got the following theorem. . We assume thatā ∈ C([0, ∞); B 7/2 2,1 (R 3 )),ū ∈ C([0, ∞); B 2 2,1 (R 3 ))∩L 1 loc (R + ;Ḃ 4 2,1 ),B ∈ C([0, ∞); B 2 2,1 (R 3 )) ∩ L 1 loc (R + ;Ḃ 4 2,1 ) is a given global solution of (1.2) with initial data (ā 0 ,ū 0 ,B 0 ). Then there exists a constant c so that if for any s ∈ [ 1 2 , 2]. Remark 1.4. The above Theorems may not be obtained by regarding the term B · ∇B as a source term in the velocity equation. The reason is that if we regard this term as a source term, we will encounter terms likeB · ∇B andB · ∇ũ in (3.16) and (6.43). For the appearance of these terms, the Bootstrap argument will not work. So we consider the linear system (2.12) is necessary and the higher order decay estimates is also necessary to get the results for MHD system. The paper is organized as follows. In section 2, we will give some notations, a brief introduction to the Besov space and some useful Lemmas. In section 3, as a warm up, we give the proof of Theorem 1.1. Then, in section 4, we proved Theorem 1.2 in a series of propositions. Using Theorem 1.2, we got the global estimates of the reference solutions in section 5. At last, we proved the decay properties of the perturbed solutions and Theorem 1.3 in section 6. In the appendix, we proved some simple Lemmas which will be used in the above sections.

Preliminaries
Throughout this paper we will use the following notations.
• For any tempered distribution u both u and F u denote the Fourier transform of u. • The norm in the mixed space-time Lebesgue space L p ([0, T ]; L r (R d )) is denoted by · L p T L r (with the obvious generalization to · L p T X for any normed space X).
• For X a Banach space and I an interval of R, we denote by C(I; X) the set of continuous functions on I with values in X, and by C b (I; X) the subset of bounded functions of C(I; X). • For any pair of operators P and Q on some Banach space X, the commutator [P, Q] is given by P Q − QP . • C stands for a "harmless" constant, and we sometimes use the notation A B as an equivalent of A ≤ CB. The notation A ≈ B means that A B and B A.
• {c j,r } j∈Z a generic element of the sphere of ℓ r (Z), and (c k ) k∈Z (respectively, (d j ) j∈Z ) a generic element of the sphere of ℓ 2 (Z) (respectively, ℓ 1 (Z)). • Denote γ − be any number smaller than γ. Then, we give a short introduction to the Besov type space. Details about Besov type space can be found in [17] or [18]. There exist two radial positive functions The homogeneous operators are defined bẏ One can easily verifies that with our choice of ϕ, As in Bony's decomposition, we split the product uv into three parts Let us now define inhomogeneous Besov spaces. For (p, r) ∈ [1, +∞] 2 and s ∈ R we define the inhomogeneous Besov space B s p,r as the set of tempered distributions u such that u B s p,r := (2 js ∆ j u L p ) ℓ r < +∞.
The homogeneous Besov spaceḂ s p,r is defined as the set of u ∈ S ′ (R d ) up to polynomials such that u Ḃs p,r := (2 js ∆ j u L p ) ℓ r < +∞.
Notice that the usual Sobolev spaces H s coincide with B s 2,2 for every s ∈ R and that the homogeneous spacesḢ s coincide withḂ s 2,2 . We shall need some mixed space-time spaces. Let T > 0 and ρ ≥ 1, we denote by L ρ T B s p,r the space of distribution u such that We say that u belongs to the space L ρ T B s p,r if u L ρ TḂ s p,r := (2 js ∆ j u L ρ T L p ) ℓ r < +∞, which appeared firstly in [19]. Through a direct application of the Minkowski inequality, the following links between these spaces is true [25]. Let ε > 0, then Lemma 2.1. [18] Let B be a ball and C be a ring of R 3 . A constant C exists so that for any positive real number λ, any nonnegative integer k, any smooth homogeneous function σ of degree m, and any couple of read number Then there hold the following: Let v be a divergence-free vector field with ∇v ∈ L 1 ([0, T ];Ḃ 3/2 2,1 ). For s ∈ (− 5 2 , 5 2 ], given f 0 ∈Ḃ s 2,1 , F ∈ L 1 ([0, T ];Ḃ s 2,1 ), the transport equation has a unique solution f ∈ C([0, T ];Ḃ s 2,1 ). Moreover, there holds for all t ∈ [0, T ] If s ∈ (0, 5 2 ], there also holds Then for the transport equation: there exists a constant C depending only on N , p, p 1 , r and σ, such that the following estimates hold true: and r = 1 . Proof. We apply the operator ∆ q to (2.12); then a standard commutator process gives 14) Thanks to the fact that divu = divv = divw = divB = 0 and 1 + a ≥ c, we get by taking the L 2 inner product of (2.14), (2.15) with ∆ q u and ∆ q B separately that It follows from the product law that for all s ∈ (− 3 2 , 1) that Π∇a(t) Ḃs . Similar to the proof in Proposition 3.6 in [21], we have Using Lemma 2.2, we have the following four inequalities: Combining the above four estimates, we arrive at Through Gronwall's inequality, we complete the proof.
Remark 2.7. It is easy to observe from the following . (2.16) Remark 2.8. Note that divu = 0, taking div to the first equation of (2.12), we obtain div (1 + a)∇Π = div f + a∆u + w · ∇B − v · ∇u , then is follows from Lemma 2.5 that for s ∈ (− 3 2 , 3 2 ), (2.17) Remark 2.9. If the parameter s = 1, then we have the following estimation The proof is similar, so we only give the main estimations in the appendix.

Stability of Global Solutions with Densities Close to 1
The aim of this section is to investigate the global stability of the given solution of (1.2) with the initial density of which is close to 1, namely Theorem 1.1. Next, we give the detailed statement.
Proof. To deal with the global well-posedness of (1.2) with initial data (a 0 , u 0 , B 0 ) given by the theorem, we need some global-in-time control of the reference solution (ā,ū,B). In what follows, we shall always denoteρ := 1 1+ā . Then we get by a standard energy estimate to (1.3) that Combining the above two energy equality, we have After integration, we have The transport equation in (1.3) gives Using the interpolation inequality and from (3.2), we deduce that On the other hand, applying Lemma 2.3 to the transport equation in (1.2) gives for any t ≥ 0. Then applying Gronwall's inequality yields Following the same idea, it is easy to obtain that for t ≥ T 0 Note that forā small, we can rewrite the momentum equation and magnetic field equation in (1.2) as  From the product law, we get Then we get by summing up (3.5), (3.6) and the above estimates that With no loss of generality, we can assume that Then, ifT < ∞, for T 0 < t <T , we havē Taking ǫ in (3.4) small enough so that ǫ ≤ 1 108C , then if c 1 in Theorem 1.1 is so small that c 1 ≤ 1 108C , we havē This contradicts with the definition of (3.7), and thereforeT = ∞. Moreover, there hold which along with (3.10) and the Gronwall's inequality ensures Reformulate equation (3.11) as follows Using standard estimates, we easily obtain that . Summing up (3.13) and (3.15), and then applying Gronwall's inequality, we obtaiñ (3.16) Using similar arguments used in proving (3.9), we can show that if (ũ 0 , With (3.17), we can prove the propagation of regularity for smoother initial data. Applying Lemma 2.3, we obtain for any t > 0, This along with (3.10) and (3.17) implies Noticing that a(t) L 2 = a 0 L 2 , this along with the above inequality shows that a ∈L ∞ (R + ; B 5/2 2,1 (R 3 )). Applying standard energy estimate to the second and third equation of (1.2) gives 3 2 ] and t > 0, this together with (3.10) and (3.17) gives rise to With (3.18) and (3.19), we can easily prove by a classical argument that Now, the proof of Theorem 1.1 is completed.

Decay in Time Estimates for the Reference Solutions
In this section, we will give the decay estimates, namely Theorem 1.2. The main ingredient of the proof will be H. Abidi, G. Gui and P. Zhang's approach in [21]. The difference is that we need to give the decay estimates for more higher order derivatives of momentum and magnetic fields, which is required for the global in time estimates proved in the next section.
In what follows, we shall always denote ρ(t, x) := 1 1+a(t,x) so that we can use both (1.2) and (1.3) just according to our convenience. In order to make our presentation clearly, we divided the proof of Theorem 1.2 into the following propositions: Proof. Taking L 2 inner product of the momentum equation of (1.3) with 1 ρ ∆u that 1 2 Taking L 2 inner product of the magnetic fields equation of (1.3) with ∆B that 1 2 Thanks to the equation of (1.3) and divu = 0, we have for some positive constant c.
On the other hand, taking L 2 inner product of the momentum equations in (1.3) with ∂ t u, we obtain Taking L 2 inner product of the magnetic field equations in ( The above two equalities gives (4.7) The above inequalities along with (4.6) ensures a positive constant e 1 such that Now choosing η > 0 small enough such that We claim that τ * = ∞. Indeed, if τ * < ∞, (4.8) and (4.9) imply that This is a contradiction to the definition of τ * , and thus τ * = ∞. Then the proof is completed. Proof. Multiplying the u i equation in (1.3) by |u i | p−1 sgn(u i ) for i = 1, 2, 3 and integrating the resulting equation over R 3 , we obtain that Using similar ideas, we have Using Hólder's inequalities, we obtain Considering p ∈ (1, 6 5 ), we can easily obtain that (4.11) Also by simple calculation, we have (4.12) Summing up (4.11) and (4.12), we easily obtain that On the other hand, applying the operator div to the first equation in (1.2), we get which together with the classical elliptic estimates implies (4.14) Due to (4.7), we have which together with (4.14) gives rise to , together with (4.14), we get Proof.
Step 1 : Rough decay estimate of u(t) L 2 and B(t) L 2 . We split the phase space R 3 into two time-dependent regions so that and g(t) satisfies g(t) t −1/2 which will be chosen later on. Here and in what follows, we shall always denote 1 + t by t . Then, thanks to (3.1), we obtain (4.17) Next, we need to deal with the low-frequency part of u and B, we rewrite the momentum equation and magnetic field equation as follows where P denote the Leray projection operator ant t 0 is the positive time determined by Proposition 4.1. Taking Fourier transform with respect to x-variables leads to which implies that Thanks to Proposition 4.1 and (3.2), we have By Proposition 4.2, we know that u(t 0 ), B(t 0 ) ∈ L p (R 3 ) for 1 < p < 6 5 , and for and where we used the Hausforff-Young inequality [24]. Then since g(t) t −1/2 , we deduce from (4.18) that where we used the fact 1 2 < β(p) < 3 4 . Substituting (4.21) and (4.22) into (4.17) results in which gives which gives Step 2 : Rough decay estimate of ∇u(t) L 2 and ∇B(t) L 2 . We split the phase space R 3 into two time-dependent regions so that where S(t) := ξ : |ξ| ≤ 1 e2 g(t) and g(t) t − 1 2 , which will be chosen later on. Then we deduce from Proposition 4.1 that which gives In particular, for α > 3 2 , we get (∇u, ∇B) 2 which implies (4.27) Step 3 : Improved decay estimates of u(t) L 2 , ∇u(t) L 2 and B(t) L 2 , ∇B(t) L 2 . We shall utilize an iteration argument. Thanks to (4.23), we obtain (4.28) and similar we have Same argument as in [21], we have Using (4.23) and (4.26), we obtain where ǫ > 0 is a small enough constant. From this and (4.17), we have which implies which in particular gives From (4.32), (4.33) and (4.24), we infer which implies (4.34) In particular, taking α > 1 + 2β(p) in (4.34), we get Combining the following fact where we used (4.35) and (4.2).
Step 1 : Estimate of u t (t) L 2 , u(t) Ḣ2 and B t (t) L 2 , B(t) Ḣ2 . We get by first applying ∂ t to the momentum equation of (1.3) and then taking L 2 inner product of the resulting equation with ∂ t u that (4.41) Applying ∂ t to the magnetic field equation of (1.3) and then taking L 2 inner product of the resulting equation with ∂ t B that (4.42) Using similar methods in [21], we obtain Substituting the above five inequalities into (4.41), (4.42), we easily obtain Integrating the above inequality from t 0 to t, we get Applying Gronwall's inequality gives It then follows from (3.2) and Proposition 4.1 that Applying Remark 4.4 and the standard energy estimate to the momentum and magnetic field equation of (1. Notice from the momentum equation of (1.3) that which along with Proposition 4.1 and (4.43) implies that Step 2 : Estimate of then we have ∇q = −∇(−∆) −1 divf , which implies that for any r ∈ (1, ∞), From this and the momentum equation of (1.3), we infer Taking the L 2 norm for the time variables on [t 0 , t], we get by using (4.43) and (4.44) that (4.45) Using similar methods, we can easily obtain  Summing up (4.45) and (4.46), we get (4.47) Now, let us turn to the estimate of Thanks to (4.36), we get t t0 ∆u(t ′ ) This together with (4.48) proves (4.40).
At last, using (4.3), we can get that which along with (4.40) and p ∈ (1, Hence, the proof is completed. Proof. According to the transport equation in (1.3) and divu = 0, we know for q ∈ [1, ∞]. Then Gronwall's inequality and Proposition 4.5 implies On the other hand, differentiating ∂ t a + u · ∇a = 0 twice with respect to the spatial variables, we get by a standard enery estimate that Applying the Gronwall's inequality, Proposition 4.3, Proposition 4.5 and (4.47), we obtain Hence, the proof is completed.
Proof. Taking spatial derivative to the momentum equation in (1.3), we have with i = 1, 2, 3. Standard energy estimates yields (4.52) Similar argument gives (4.54) On the other hand, taking partial derivative to the momentum equation in (1.3) and multiplying 1 ρ ∆∇u on both side, we can get 1 2 by integrate over R 3 . Through similar estimate, we obtain (4.56) Combining (4.55) and (4.56), we have (4.57) Next, we give the estimation of each term on the right hand side of (4.57), Moreover, using divu = 0, we can get Summing up all the above estimations in (4.57), then after a long and tedious calculations, we have (4.58) We perform (4.54) + 1 2 (4.58) so that (4.59) Note that we can take η > 0 in the proof of Proposition 4.1 smaller so that where C is the constant on the right hand side of (4.59), c satisfies From the proof of Proposition 4.1, we know that τ * = ∞ as in Position 4.1. Hence, there exists two constant c 1 and c 2 such that d dt Integrating the above inequality form t 0 to ∞, we then obtain At last by Proposition 4.3, the proof is completed.

Global in Time Estimates for Reference Solutions
In this section, we prove the global in time estimates for the reference solution of (1.3). The proof will be based mainly on Theorem 1.2. for some C depending on the initial data.

Rewrite the momentum equation in (1.3) as follows
From the elliptic estimate, we have Next, we estimate the three terms on the right hand side of the above inequality. Simple calculations yields Now we give the estimates about ū L 1 (R + ;Ḣ 3 ) , Now, we give the estimates of the right hand side. According to Proposition 4.3 to Proposition 4.7, we have Similar to the above estimates ofū, we can obtain ∞ t0 ∇B · ∇B L 2 dt ≤ C and ∞ t0 B · ∇(∇B) L 2 dt ≤ C.
Summing up all the above estimates, we finally obtain ū L 1 (R + ;Ḃ 5/2 2,1 ) ≤ C. Using similar ideas as forū, we have So we also obtain that B L 1 (R + ;Ḃ 5/2 2,1 ) ≤ C. Hence, finally the proof of Proposition 5.1 is completed. and Proof. Thanks to Proposition 5.1, we get by applying (2.3) to the transport equation in (1.2) that Next, let us turn to the estimates ofū andB. Indeed, from (4.13), we know that . Then by (4.14), Proposition 4.6, Theorem 1.2 and the above inequality, we deduce that On the other hand, applying Proposition 2.6 to the momentum and magnetic field equation of (1.2) ensures that We know that Combining the above estimation, (5.5) and Proposition 4.5, we obtain Due to just a minor change of the proof in [21], we can get the following estimation. So, we postpone the proof in the appendix and give the estimation first Differentiating the momentum equation and magnetic field equation of (1.2) with respect to the spatial variables gives rise to Using Remark 2.9 and Gronwall's inequality, we will obtain Using Proposition 5.1, (5.5), (5.8) and product laws in Besov space, we get
Then the proof of Theorem 1.3 is equivalent to the proof of the global well posedness of (6.1) with small enough initial data (ã 0 ,ũ 0 ). Indeed, according to the coupled parabolic hyperbolic theory [22], it is standard to prove that there exists a positive timeT * such that (1.2) with initial data (ā 0 +ã 0 ,ū 0 +ũ 0 ,B 0 +B 0 ) has a unique solution (a, u, B) with a ∈ C([0,T * ); B Then (ã,ũ,B) with solves (6.1) on [0,T * ). Without loss of generality, we may assume thatT * is the maximal time of the existence to this solution. The aim of what follows is to prove thatT * = ∞ and (ã,ũ,B) remains small for all t > 0.
for someT ≤T * and some sufficiently small positive constant ν, then under the assumptions of Theorem 1.3, we have for t <T . (6.10) Proof. Taking the L 2 inner product between the second equation of (6.3) and ∂ tũ , we obtain Notice that (6.12) By taking the L 2 inner product between (6.3) and ∆ũ, we get Thanks to the momentum equation of (6.3) and divũ = 0, we have This gives (6.14) Combining (6.12) and (6.14), we arrive at for t <T * . Taking the L 2 inner product between the third equation of (6.3) and ∂ tB , we have 1 2 By taking the L 2 inner product between (6.3) and ∆B, we get Combining this and (6.16), we arrive at (6.17) Hence, by (6.15) and (6.17), we obtain that for t <T * . For small enough ν, we obtain Then simple calculations leads to for t <T .
From this and div u = 0, we deduce that for any q ∈ [1, ∞]. From the above and (5.2), we get Plugging (6.36) into (6.34) and integrating the resulting equation over [0, t], we obtain (6.37) Applying the Gronwall's inequality gives From this and (6.36), we deduce that Thanks to (6.29), (6.30) and (6.32), we arrive at (6.18) and (6.19) for t ≤T . Moreover, (6.20) and (6.38) ensures that ≤ Cδ 0 for t ≤T . Now, let ν be the small constant determined in (6.9). We take δ 0 sufficiently small such that for t ≤T . This shows thatT can be any time smaller thanT * . This completes the proof of the proposition.
for all t <T * .
Proof. Firstly, let us give the equation ofã Using basic estimates about transport equations, we have By the definition of A 0 and Proposition 5.2, we get Similarly, we give the equation of ∇ã Then classical estimates about transport equations, Proposition 4.5 and Proposition 6.3 gives , where we used (5.8). From this, we easily obtain ∇ã(t) L ∞ ≤ C ∇ã 0 L ∞ + ā 0 B 7/2 2,1 . Proposition 6.6. Under the assumptions of Theorem 1.3, there holds where C depends on the initial data.
Proof. Taking derivative to the second and third equation of (6.3), we obtain Multiplying (6.50) with ∂ t ∂ jũ i and integrating over R 3 , we obtain (6.52) ForB, using same process, we have (6.53) Combining (6.52) and (6.53), we obtain

Multiplying 1
ρ ∆ũ and ∆ũ to (6.50) and (6.51) separately and doing some basic energy estimates, we obtain 1 2 At this point, we got two completed inequalities. The second inequality times a small number then plus the first inequality yieds Integrating the above inequality, using decay estimates about reference solution and perturbed solution, we obtain Hence, the proof of Proposition 6.6 is completed. Now, we can complete the proof of Theorem 1.3 as following.
We need only prove the maximal existence timeT * = ∞. Indeed, according to all the decay estimates for reference solution and perturbed solution, we repeat the argument used in the proof of Proposition 5.1 and Proposition 5.2 to prove that T * = ∞. Then a standard interpolation argument gives (1.10) and (1.11). This completes the proof of Theorem 1.3.

Appendix
For completeness, in this section, we give the proof of some Lemmas and Propositions. Firstly, we give the proof of Remark 2.9.
Proof. Using the same method as in the proof of Lemma 2.6, we can get the result. The only difference is that we allow the parameter s = 1, so we only give the estimates related to s < 1 which is required in the proof of Lemma 2.6. Notice that div[∆ q , a]∇u = ∆ q div R(a, ∇u) + ∆ q div T ∇u a − divR(a, ∆ q ∇u) − div[T a , ∆ q ]∇u.
Applying Lemma 2.1, we have ∆ q div R(a, ∇u)(t) L 2   , and div R(a, ∆ q ∇u)(t) L 2 Summing up all the above estimates, we arrive at div [∆ q , a]∇u(t) L 2 d q (t)2 −q a(t) Ḃ2 .
Then following the same line of reasoning as above, we obtain [∇a · ∇, ∆ q ]u(t) L 2 d q (t)2 −q a(t) Ḃ2 .
At this point, we list all the different estimates related to the restriction of s.
Next, we will give the proof of (5.8).
Substituting the above inequality into (7.1) and then taking η sufficiently small, we arrive at ∇ū L∞ t (Ḃ