GLOBAL REGULARITY FOR THE 3D AXISYMMETRIC MHD EQUATIONS WITH HORIZONTAL DISSIPATION AND VERTICAL MAGNETIC DIFFUSION

. Whether or not classical solutions of the 3D incompressible MHD equations with full dissipation and magnetic diﬀusion can develop ﬁnite-time singularities is a long standing open problem of ﬂuid dynamics and PDE theory. In this paper, we investigate the Cauchy problem for the 3D axisymmetric MHD equations with horizontal dissipation and vertical magnetic diﬀusion. We get a unique global smooth solution under the assumption that u θ and b r are trivial. In absence of some viscosities, there is no smoothing eﬀect on the derivatives of that direction. However, we take full advantage of the structures of MHD system to make up this shortcoming.


(Communicated by Chongchun Zeng)
Abstract. Whether or not classical solutions of the 3D incompressible MHD equations with full dissipation and magnetic diffusion can develop finite-time singularities is a long standing open problem of fluid dynamics and PDE theory. In this paper, we investigate the Cauchy problem for the 3D axisymmetric MHD equations with horizontal dissipation and vertical magnetic diffusion. We get a unique global smooth solution under the assumption that u θ and br are trivial. In absence of some viscosities, there is no smoothing effect on the derivatives of that direction. However, we take full advantage of the structures of MHD system to make up this shortcoming.
1. Introduction. In this paper, we consider the 3D axisymmetic incompressible Magneto-hydrodynamics (MHD) equations u t − ν x ∂ xx u − ν y ∂ yy u − ν z ∂ zz u + u · ∇u = −∇(p + 1 2 We will prove the global regularity for the 3D axisymmetric MHD system of (1.4)-(1.6) with u θ and b r being trivial in the cylindrical coordinate systems. It should be noted that in [10] the 3D axisymmetric MHD system was studied with the full dissipations on the velocity and magnetic fields and with u θ , b r and b z being trivial. To obtain the existence and uniqueness of regular solution, we will make the L ∞ ([0, ∞); H 2 ) estimates of the velocity and magnetic fields. To his end, we will apply for the equations satisfied by the vorticity ∇ × u and the current of magnetic fields ∇ × b. New difficulties will be encountered in this paper. First, because of lacking smoothing effect from the full dissipation terms, the system is degenerate along some directions and this leads to more difficulties in the a priori estimates. Second, since b z is not zero, the current of the magnetic fields will have three non-trivial components j r , j θ and j z and hence this deduces that the estimates of (u, b) in L ∞ ([0, ∞); H 2 ) become much more difficult due to the strong coupling in the nonlinearities. However, we notice that, since b r is trivial, it follows from the incompressible condition which is that b z does not depend on z-variable. This is one of advantages and will play an important role in our high-order estimates. Now we are in the position to state the main results of this article.
Theorem 1.1. Let (u 0 , b 0 ) ∈ H 2 are axisymmetric divergence free vector fields such that u θ 0 = b r 0 = 0 and ∇b 0 ∈ L ∞ . Then the system (1.4)-(1.6) with the initial data (u 0 , b 0 ) has a unique global classical solution (u, b) satisfying where u θ 0 = u θ (x, 0), b r 0 = b r (x, 0), w = ∇ × u and j = ∇ × b. This paper is organized as follows. In section 2, we introduce some notations and technical lemmas used for our estimates in the following sections. In section 3, we will concentrate on doing H 2 estimates. Section 4 is devoted to proof of the main theorem.

Preliminary.
2.1. Notations. In this section, we introduce some definitions and notations for the axisymmetic solutions.
Remark. In the rest of this article, for convenience, we denote u r (r, z, t) as u r and others are similar.
In the cylindrical coordinate systems, the gradient operator ∇ is given by Thus, some simple calculations can lead to Moreover, under the assumptions that u θ and b r are trivial and the incompressible condition, one can get that 304 QUANSEN JIU AND JITAO LIU Therefore, the system (1.4)-(1.6) can be rewritten as Now we deduce the equations of the vorticity and current. In the cylindrical coordinate systems, w = ∇ × u can be written as Thus, set j = ∇ × b, one can rewrite the equations of vorticity and current as By use of the condition that ∂ z b z = 0, it follows that therefore the equations of vorticity and current can be written as 14) Axisymmetric estimates. In this subsection, we present some estimates in the axisymmetric case.
Lemma 2.2. Suppose that u = u(r, z) ∈ H 1 (R 3 ) be an axisymmetric field with zero divergence, then there holds

16)
whereũ = u r e r + u z e z and C 0 is an absolute constant.
be an axisymmetric field with zero divergence, then there holds where C 0 is an absolute constant.
The following estimates and proofs can be found in [10,13] and we present them here for completeness.
be an axisymmetric field with zero divergence, then there holds where C 0 is an absolute constant.
Proof. Similar to [8], by incompressible constraint, one can set the angular stream function φ θ such that Since ∂ rr + 3 r ∂ r + ∂ zz can be viewed as a five-dimension Laplace operator in the axisymmetric form, we can write φ θ r as Moreover, in the 3D axisymmetric case, thus some simple calculations give that QUANSEN JIU AND JITAO LIU +∂ zr φ θ r (e z ⊗ e r + e r ⊗ e z ).
which implies Here ≈ means equivalence. Similarly, in the 5D axisymmetric case, we also have Thus we have where w(r) = 1 r 2 is a weight function. Consequently, for 1 < p < ∞, by Lemma 2 in [8], the inequality holds.
Lemma 2.5. Suppose that u = u(r, z) ∈ H 1 (R 3 ) be an axisymmetric field with zero divergence, then there holds where C 0 is an absolute constant.
Proof. It is clear that thus by Riesz theorem, imbedding theorem and Proposition 2.9 in [7], ∀p ∈ (1, 3), one can reach that u r r L 2.3. Partial derivative estimates. Now we list some inequalities needed later, the proof of which can be found in [5] or [13].
be an vector field, then there holds where C 0 is an absolute constant.
Lemma 2.7. Let f, g, h be smooth functions in R 3 , then there exists an absolute constant C 0 such that the following inequality where C 0 is an absolute constant.
where C 0 is an absolute constant.
Proof. We can get the conclusion by setting p = 4 in Lemma 2.7.

A priori estimates.
3.1. H 1 estimates. In this subsection, we intend to get H 1 estimates of (u, b).
The following is the usual energy estimates: Proposition 1. If (u,b) solves the system (2.7)-(2.11) with u = u r e r +u z e z and b = b θ e θ + b z e z , there holds where the constant C 1 depends only on u 0 L 2 and b 0 L 2 .
To achieve H 1 estimates, we first get the estimates of b θ r L ∞ and where the constant C 2 depends only on b0 r L p . Proof. Considering that b θ satisfies the following equation one can easily deduce that the quantity Ω = b θ r solves ∂ t Ω + u · ∇Ω − ∂ zz Ω = 0.

QUANSEN JIU AND JITAO LIU
It follows that letting p → ∞, then there holds that Lemma 3.2. If (u,b) solves the system (1.4)-(1.6) with u = u r e r + u z e z and b = b θ e θ + b z e z , then the following estimate holds where the constant C 3 depends only on w0 r L 2 and b0 r L 2 ∩L ∞ . Proof. Let Γ = w θ r , then (Γ, Ω) solves the following system   Taking inner product of (3.27) with Γ, Ω and integrating on R 3 , we have by use of Gronwall inequality, one can reach that where the constant C 4 depends only on T, w0 r L 2 , b0 r L 2 ∩L ∞ and b 0 L ∞ . Proof. By Lemma 2.3 and Lemma 3.2, one can get that In addition, as b θ satisfies the following equation For p > 1, taking inner product with |b θ | p−2 b θ and integrating on R 3 , we finally obtain It follows from Gronwall inequality that Letting p → ∞, then one can derive that The proof of the lemma is finished.
The H 1 estimates of (u, b) are as follows.
where the constant C 5 depends only on T, w0 Proof. Taking inner product of (2.12) with w θ and integrating on R 3 lead to Using Lemma 2.9, one has Thus combing with Gronwall inequality, it follows that Taking inner product of (2.13)-(2.15) with j r , j θ , j z and integrating on R 3 respectively, it is easy to get In the following context, we do the estimate for each term respectively. By use of Lemma 2.6, one can reach that Thus, summing up all the estimates I − V II, there holds that Since w = w θ e θ and j = j r e r + j θ e θ + j z e z , combining the estimates (3.31) and (3.32), one can derive that

H 2 estimates.
This subsection is devoted to getting H 2 estimates of (u, b), to begin with, we do estimate of ∇( b θ r ) L ∞ ([0,T ],L 2 ) . Lemma 3.4. If (u,b) solves the system (1.4)-(1.6) with u = u r e r + u z e z and b = b θ e θ + b z e z , it holds that
Taking inner product of (3.34) with −∆Ω and integrating on R 3 yields Now, we estimate the terms I −V respectively. Making use of Lemma 2.6-Lemma 2.9, one can reach that Thus, summing up all above estimates and making use of Gronwall inequality, there will hold that We continue to pursue H 2 estimates.
Proposition 3. If (u,b) solves the system (1.4)-(1.6) with u = u r e r + u z e z , b = b θ e θ + b z e z and denote w = ∇ × u = w θ e θ , j = ∇ × b = j r e r + j θ e θ + j z e z , then there holds that where the constant C 7 depends only on T, w0 r L 2 , b0 r L 2 ∩L ∞ , b 0 L ∞ , w 0 H 1 and j 0 H 1 .
Proof. By (2.12), w = w θ e θ solves Taking inner product of this equation with −∆w and integrating on R 3 , we find 1 2 Before estimating term I, we rewrite it as follows.
Similarly, by Lemma 2.6-Lemma 2.9, one can reach that To estimate II, we rewrite it as follows, Making use of Lemma 2.8 and 3.1, one has Adding up all above estimates {1}-{8} yields Combining with Gronwall inequality, it is clear that Similarly, by (2.13)-(2.15), j = j r e r + j θ e θ + j z e z solves Taking inner product of this equation with −∆j and integrating on R 3 , it follows that 1 2 Using Lemma 2.6-Lemma 2.9, one has Noting that ∂ z j θ = −∂ r ∂ z b z = 0, one can deduce that As for the last term IV , we split it into three parts and do estimates separately, Adding up all above estimates, there holds that 1 2 . Combining with Gronwall inequality, one can achieve that where the constant C 8 depends only on T, b0 r L ∞ , w 0 H 1 and j 0 H 1 .
Since u solves then it follows that ∇ × u solves  It follows from Proposition 3 that (4.42) Step 2. To deduce T 0 ∇b L ∞ dt ≤ C.
Then for any p > 1, taking inner product with |∇b θ | p−2 ∇b θ , |∇b z | p−2 ∇b z and integrating on R 3 respectively, it isn't hard to derive that d dt ∇b θ L p + ∇b z L p ≤ ∇u L ∞ ∇b θ L p + ∇b z L p By Gronwall inequality, there holds that Letting p → ∞, this gives that together with estimates (4.42) and Lemma 3.1, it can be reached that The proof of this proposition is finished.

4.2.
Proof of Theorem 1.1. By Propositions 1-3, the proof can be achieved through a parabolic regularization process. Let δ > 0 be a small parameter and consider a family solutions (u δ , b δ ) satisfying the regularized system ∂ t u δ − δ∆ h u δ + u δ · ∇u δ = −∇π δ + b δ · ∇b δ , (4.43) where φ δ is a standard mollifier. Since u δ (x, 0) and b δ (x, 0) are smooth, the standard theory on the 3D viscous MHD equations ensures that (4.43)-(4.46) has a unique global smooth solution (u δ , b δ ) which obeys the a priori bounds in Propositions 1-3 uniformly in δ. By standard compactness arguments and Lions-Aubin Lemma, we can show that this family (u δ , b δ ) converges to (u, b) which satisfies in turn our initial problem and obeys the bounds in Propositions 1-3. Using T 0 ( ∇u L ∞ + ∇b L ∞ )dt < ∞, the uniqueness can be proved by the standard method and we omit the details here (see [16,1] for instance). The proof of the theorem is finished.