LIOUVILLE FOR MHD SYSTEM AND ITS APPLICATIONS

. In this paper, we construct Liouville theorem for the MHD system and apply it to study the potential singularities of its weak solution. And we mainly study weak axi-symmetric solutions of MHD system in R 3 × (0 ,T ).

There are lots of works on the solution of the MHD equations (1.1). In particular, Duvaut and Lions [4] constructed a class of global weak solutions and the local strong solutions to the initial boundary value problem, and Sermange and Temam in [17] discussed some properties of such solutions. For the 2-dimensional case, the smoothness and uniqueness of solutions have been shown. But for ndimensional(n ≥ 3), the problem is still open in general case like Navier-Stokes equations v t − ∆v + v · ∇v + ∇q = 0, div v = 0, (1.2) where v is fluid velocity and q is the press. For Navier-Stokes equations, many regularity criteria have been established(e.g. [18,19,1,24,9]), and some of these criteria can be extended to the 3-D MHD equations by assumptions only on u, see

XIAN-GAO LIU AND XIAOTAO ZHANG
He and Xin [8]. We note that 3-D MHD system need assumptions both on u and b for Ladyzhenskaya-Prodi-Serrin class L 3,∞ , see Mahalov, Nicolaenko and Shilkin [14]. But there is still a gap between the case of existence and the case of regularity. Scheffer [16] began to study the partial regularity theory of Navier-Stokes equations. Scheffer's works were improved by Caffarelli-Kohn-Nirenberg [2], Tian-Xin [21]. We mention that, according to [2,21], the 1-D Hausdorff measure of the blow-up set must be zero. For self-similar singularities in the Navier-Stokes equations, the work of Nečas, Růžička andŠverák [15] and Tsai, Tai-Peng [22] showed the trivial solutions with some integration conditions. And then Koch, Nadirashvili, Seregin andŠverák [11] directly studied the potential singularities of weak axi-symmetric solutions in R 3 × (0, T ), by Liouville theorem for the Navier-Stokes equations. For MHD system, He and Xin [7] extend the result of [21] to it. And [14] extend the result of [15] and [22] to the MHD system. Moreover, Lei [13] constructed one kind of smooth axially symmetric solutions(u θ = b r = b z = 0) of MHD in three dimensions. In this paper, we construct Liouville theorem for the MHD system and apply it to study the potential singularities of its weak solution. Because the term b · ∇b in the equations velocity u and equations about magnetic field b, the problem becomes more complicated.

Main result and outline.
Under some conditions about u, b, we prove the Liouville Theorems for MHD system. And, we mainly study weak axi-symmetric solutions of the MHD system in R 3 × (0, T ). For 2-D case, with the condition b 2,1 −b 1,2 = 0 or b 1 = 0 or b 2 = 0 or b 1 u 2 = b 2 u 1 , we have the Liouville Theorems for the bounded weak solutions of MHD system. Meanwhile, with the integral condition |u|+|b| ∈ L s,r x,t (R 2 ×(−∞, 0))(2/s+2/r ≥ 1, s ≥ 3, 3 ≤ r < ∞), we prove the Liouville Theorem for the solutions in C 2,1 x,t . For 3-D case, with the conditions u θ = 0(no swirl), b r = b z = 0 or |u(x, t)| ≤ we have the Liouville Theorems for the weak bounded axi-symmetric solutions of the MHD system. And, when µ 1 = µ 2 , with the conditions u z,z = b z,z = 0, we get the Liouville Theorem for the weak bounded axi-symmetric solutions. Moreover, when u = 0, we obtain that b must be a constant vector, this means b is smoother than u, see section 6. As in [11], by scaling transformations and the Liouville Theorems which we have proved, we proved the regularity for axi-symmetric solutions of MHD systems with the conditions |u(x, t)| ≤ or some other conditions. The paper is organized as follows. In section 2, we introduce the strong maximum principle which is very essential in this article. In section 3, we introduce the mild solution and bounded weak solution and their properties, for Stokes equations and heat equations. In section 4, we study the regularity of mild solution and bounded weak solution of MHD system, and the limit properties of bounded mild solution of MHD system. In section 5, we construct the Liouville theorem for MHD system, we mention that we need conditions of b to prove the Liouville theorem in R 2 ×(−∞, 0). And for 3-dimensional case, we study the axi-symmetric solutions of MHD system. In section 6, we apply the Liouville theorem constructed in section 5, to study the potential singularities of the finite time weak solution of MHD system.
2. maximum principle. The strong maximum principle plays essential role in this article, and we mainly use the form in [11]. For reader's convenient, we write it as Lemma 2.1. Let Ω ⊂ R n be a bounded domain and T > 0. We consider the parabolic equation . And we mention that u is a scalar valued function.
Lemma 2.1. [11, Lemma 2.1] Assume that u is a bounded solution of the equation (2.1). Let K be a compact subset of Ω, and Ω ⊂ Ω ⊂ Ω, and τ > 0. Let M = sup Ω×(0,T ) |u|. Then, for each ε > 0, there exists δ = δ(Ω, 3. mild solution and bounded weak solution for linear case. Let u = (u 1 , · · · , u n ), b = (b 1 , · · · , b n ) : R n × (0, ∞) → R n . We first consider the following linear equations about cauchy problem: Here f k = (f 1k , · · · , f nk ), g k = (g 1k , · · · , g nk ) for k = 1, · · · , n. Let P denote the Helmholtz projection of vector fields on divergence free fields and let S be solution operator of the heat equation. Then we get the formula where u(t), b(t) denote the two functions u(·, t), b(·, t). These can be written as more clearly in terms of some kernels. We first deal with the formula about u. This is similar to the [11], and for reader's convenient, we write it as follows. Let where Φ is defined in terms of the fundamental solution of Laplace operator G and the the heat kernel Γ: then we can rewrite the equality (3.3) as

XIAN-GAO LIU AND XIAOTAO ZHANG
Note that K ij has the following estimates: Now we deal with (3.4). From the theory of heat equation, one can easily rewrite it as Now we give the definition of mild solution and bounded weak solution of the linear system (3.1) and (3.2). Let Then we call (u, b) is a bounded weak solution of the Cauchy problem (3.1).
Let f, g ∈ L ∞ x,t , we have some standard estimates for u, b. In particular, if u 0 = 0, b 0 = 0, then for any α ∈ (0, 1) and p ∈ (0, ∞) is any parabolic ball ball contained in R n × (0, ∞)). And the space C α par is defined by means of the parabolic distance |x − x | 2 + |t − t |.
Taking difference on both sides of the equations, we obtain:
x,t (R n × (0, T )), let u, b ∈ L ∞ x,t (R n × (0, T )) be any weak solution of (3.1) in R n × (0, T ). And let v, e be the mild solution of the Cauchy problem (3.1) and (3.2) where w 1 , w 2 satisfy the heat equations w t −∆w = 0 in R n ×(0, T ) and d 1 is bounded measurable R n -valued functions on (0, T ). Moreover, The proof is similar to the way of G.Koch, N.Nadirashvili, G.A.Seregin, V.Sverák(see [11], Lemma 3.1), so we omit it. 4. Bounded solutions of MHD. Now we consider the Cauchy problem for the MHD system: 2) The considerations of the section 3 can be repeated with x,t (R n × (0, T )), then its definitions of mild solution and bounded weak solution follow as the definition 3.1 and the definition 3.2.
We define two bilinear form as follows: And let U 1 , U 2 be the heat extension of the initial datum u 0 , b 0 . Then the solu- and by heat kernel theory, we also get Now we give some regularity properties of mild solutions in L ∞ x,t (R n × (0, T )).
Proof. We use the method of local existence and uniqueness of the solution to Navier-Stokes equations(see [6,10]). For simplicity, we just give the estimate of the where u 0 , b 0 are the initial data of the MHD system and the definition of operator B 1 and B 2 follows (4.3) and (4.4). Let . By heat kernel theory and the estimates (4.5) and (4.6), we have . We take T 1 small so that max i 4C i T 1/2 1 K 0 < 1 and T 1/2 < 1/2, then it is easy to prove that By direct calculation, we have The uniqueness of solution can be proved by estimating the difference sup 0≤t≤T2 (||u where T 2 is very small(use the equation(3.6) and (3.9)). Finally, by the estimate (4.9), we obtain the result.
x,t (R n × (0, T )) be a sequence of mild solution of (4.1) and (4.2) with initial conditions u Then a subsequence of the sequence u (k) converges locally uniformly in R n × (0, T ) to a mild solution u ∈ L ∞ x,t (R n × (0, T )) with initial datum u 0 , where u 0 is the weak * limit of a suitable subsequence of the sequence u (k) 0 . Proof. It is easy to get this result by lemma 4.1 and the decay estimates (3.8) and heat kernel theory.
We now consider the regularity of bounded weak solutions of (4.1) and (4.2).
. If one of the following conditions is satisfied: Proof. In two dimensions space, the vorticity is a scalar, which is defined by where u k,j = ∂u k /∂x j , that is say, the indices after comma mean derivatives. For magnetic field, we also use the definition of vorticity We first consider the first equation of system (4.1). Then the vorticity ω satisfies can be written as ω t + u · ∇ω − ∆ω = 0. (5.4) This is similar to the Navier-Stokes equations, we know that u(x, t) = d 1 (t), where d 1 is bounded measurable functions from (−∞, 0) to R 2 (see [11], Theorem 5.1).
With ω = b 2,1 − b 1,2 = 0 and div b = 0, we find that b 1 , b 2 are harmonic functions, then b is constant in x for each t by the classical Liouville theorem. Take it into the second equation of the system (4.1), we find that b is constant in x and t.
2. If condition (2) holds, then b 2,2 ≡ 0, thus b · ∇b ≡ 0. Therefore, the equation (5.3) can be written as (5.4), so u(x, t) = d 1 (t), where d 1 is bounded measurable functions from (−∞, 0) to R 2 . Then, the second equation can be written as b t − ∆b + u · ∇b = 0, and assume that M 1 > 0. Applying Lemma 2.1 to b 1,1 − 1 2 (M 1 +M 2 ), we get that there exist arbitrarily large parabolic balls Q ((x,t) ,t), R). For such parabolic balls, we have But, on the other hand, we can obtain where n is the normal to the boundary of B(x, R). When R is big enough, we find that (5.6) contradicts to (5.7), unless M 1 ≤ 0. By the same way, we conclude that M 2 ≥ 0. Therefore, b 1,1 = 0 in R 2 × (−∞, 0). In the same way, we conclude that Take it into the second equation of the system (4.1), we find that b is constant in x and t.
3. If the condition (3) holds, then the proof is similar to (2), we omit it. 4. If the condition (4) holds, then the second equation of (4.1) can be written as and assume that M 1 > 0. Applying Lemma 2.1 to b 1,1 − 1 2 (M 1 +M 2 ), we get that there exist arbitrarily large parabolic balls Q((x,t), But, on the other hand, we can obtain where n is the normal to the boundary of B(x, R). When R is big enough, we find that (5.8) contradicts to (5.9), unless M 1 ≤ 0. By the same way, we conclude that M 2 ≥ 0. Therefore, b 1,1 = 0 in R 2 × (−∞, 0). In the same way, we conclude that b 1,2 = 0, b 2,1 = 0, b 2,2 = 0 in R 2 × (−∞, 0). Therefore, b is constant in x for each t.
Take it into the second equation of the system (4.1), we find that b is constant in x and t. Therefore, b · ∇b ≡ 0, then we have u(x, t) = d 1 (t), where d 1 is bounded measurable functions from (−∞, 0) to R 2 .
Then to next, we prove a Liouville theorem under integration condition. We get the idea from the way of dealing with Steady-state problems, see [5,23,3].
Since the vorticity is no longer a scalar function in three dimensions space, the problem becomes very different. But one can obtain the similar result under the additional assumption when the solutions are axi-symmetric. A vector field u : R 3 → R 3 is called axi-symmetric if it is invariant under rotations about a suitable axis, and here we choose x 3 -coordinate as the "suitable axis". That is to say, the velocity field u is axi-symmetric if u(Rx) = Ru(x) for every rotation R of the form then, in cylindrical coordimates (r, θ, z), the axi-symmetric fields are given by u = u r e r + u θ e θ + u z e z where the coordinate functions u r , u θ and u z depend only on r, z and time t. The axi-symmetric magnetic fields also have the similar representation b = b r e r + b θ e θ + b z e z where the coordinate functions b r , b θ and b z depend only on r, z and time t. Therefore, in these coordinates, the MHD system (4.1) becomes where ∆ = ∂ rr + ∂ r /r + ∂ zz is the scalar Laplacian(expressed in the coordinates (r, θ, z)), and the indices after comma mean derivatives, i.e. u r,z = ∂u r /∂z. Then we consider the case of axi-symmetric flows without swirl in the velocity fields(u θ = 0). As usual, ω = curl u, and in cylindrical coordinates we write ω = ω r e r + ω θ e θ + ω z e z , (5.20) For axi-symmetric flows u without swirl we have ω r = ω z = 0 by direct calculation. Thus we can write ω = ω θ e θ . (5.21) And in the magnetic fields, we assume that b r = b z = 0. where ∆ = ∂ rr + ∂ r /r + ∂ zz is the scalar Laplacian(expressed in the coordinates (r, θ, z)), and the indices after comma mean derivatives, i.e. u r,z = ∂u r /∂z. Proof. With b r = b z = 0, we can write (5.17) as The term on the right side of equation (5.22) can be treated as the 5-D Laplacian acting on SO(4)-invariant function in R 5 . We write r = y 2 1 + y 2 2 + y 2 3 + y 2 4 and y 5 = z, and letf (y 1 , · · · , y 5 ) = f (r, z), then we have ∆ yf (y 1 , · · · , y 5 ) = ∂ 2 f ∂r 2 + 3∂f r∂r + ∂ 2 f ∂z 2 (r, z) Therefore, with a slight abuse of notation, we can write the equation (5.22) as From section 4, we know that ∇ k x b are bounded in R 3 × (−∞, 0), and by the condition, b is axi-symmetric, thus b θ /r is bounded R 3 × (−∞, 0). Let and assume that M 1 > 0. Applying Lemma 2.1 to the solution b θ /r − 1 2 (M 1 + M 2 ) of equation (5.23), considered as an equation in R 5 × (−∞, 0). With suitable centers, we see that b θ /r ≥ 1 2 M 1 in arbitrarily large parabolic balls, this means b θ is unbounded, a contradiction. Therefore, M 1 ≤ 0. In the same way, we find that M 2 ≥ 0. Thus b θ ≡ 0. Therefore, we can write (5.12), (5.14), and (5.15) as This is similar to the Navier-Stokes equations, and ω θ satisfies Therefore, u(x, t) = (0, 0, d 3 (t)) T (see [11], Theorem 5.2).
Next, we will prove other conditions that makes Lioullive theorem hold.  0)(see the proof of Theorem 5.4). Therefore, equation (5.13) can be rewrite as then we use the equations expressed in the cylindrical coordinates (r, θ, z) for the equation (5.26), and we set f = ru θ , we have and f ≤ C. Then we have u θ = 0(this is the result of Theorem 5.3 in [11], we omit the proof).
Proof. With the condition u r b z = u z b r , equations (5.16) and (5.18) can be written as Then we rewrite (5.28) as Like (5.23), the right term of (5.30) also can be treated as Laplacian acting on SO(4)-invariant functions in By the condition, b is axi-symmetric, and from section 4, we know that ∇ k x b are bounded in R 3 × (−∞, 0), thus b r /r is bounded R 3 × (−∞, 0). Let and assume that M 1 > 0. Applying Lemma 2.1 to the solution b r /r − 1 2 (M 1 + M 2 ) of equation (5.23), considered as an equation in R 5 × (−∞, 0). With suitable centers, we see that b r /r ≥ 1 2 M 1 in arbitrarily large parabolic balls, this means b r is unbounded, a contradiction. Therefore, M 1 ≤ 0. In the same way, we find that M 2 ≥ 0. Thus b r ≡ 0, and b z u r ≡ 0. Then, by the equation (5.19), b z,z = 0. Therefore, b z is a function in R 2 × (−∞, 0).
Then, u z,r and b z satisfy Therefore, applying Lemma 2.1 to u z,r /r and b z,r /r, we conclude that u z,r = b z,r ≡ 0. Thus u z and b z are constant functions in x for each t. Take b z into the equation (5.18), we have the result. Now we introduce a special case for MHD system in R n × (−∞, 0).
Proof. By section 4, with the condition, we know that ∇ k x b is bounded in R n × (−∞, 0). Because u = 0 in R n × (−∞, 0), the second equation of (4.1) can be written as b t − ∆b = 0.
and assume that M 1 > 0. Applying Lemma 2.1 to b 1,1 − 1 2 (M 1 +M 2 ), we get that there exist arbitrarily large parabolic balls Q((x,t), ,t), R). For such parabolic balls, we have But, on the other hand, we can obtain where n is the normal to the boundary of B(x, R). When R is big enough, we find that (5.45) contradicts to (5.46), unless M 1 ≤ 0. By the same way, we conclude that M 2 ≥ 0. Therefore, b 1,1 = 0 in R n × (−∞, 0). In the same way, we conclude that b i,j = 0(i, j = 1, 2, · · · , n) in R n × (−∞, 0). Therefore, b is constant in x for each t.
Take it into the second equation of the system (4.1), we obtain the result. 6. Singularities. In this section we will consider the potential singularity in the solutions of the Cauchy problem for the MHD system (4.1) and (4.2). We aim to show that singularities generate bounded ancient solution, which are the solutions defined in R n × (−∞, 0). More precisely, an ancient weak solution of the MHD system is a weak solution defined in R n × (−∞, 0), and (u, b) is an ancient mild solution if there is a sequence T l → −∞ such that (u(·, T l ), b(·, T l )) is well defined and (u, b) is a mild solution of the Cauchy problem in R n × (T l , 0) with initial data (u(·, T l ), b(·, T l )).
Lemma 6.1. Let (u l , b l ) be a sequence of bounded mild solution of the MHD system defined in R n × (−∞, 0)(for some initial data) with a uniform bound |u l | + |b l | ≤ C, and T l → −∞. Then, there a subsequence along which (u l , b l ) converges locally uniformly in R n × (−∞) to an ancient mild solution (u, b) satisfying |u| ≤ C in R n × (−∞).
Proof. It is easy to prove with the regularity results in section 4. Now assume that the mild solution develops a singularity in finite time, and that (0, T ) is its maximal time interval of the existence. There are two situations: 1. there exists a positive number C 0 , such that First, we assume that case (1) holds((6.1) holds). Let h(t) = sup x∈R n |u(x, t)| and H(t) = sup 0≤s≤t h(s). It is easy to see that there exist a sequence t k ↑ T such that h(t k ) = H(t k ). Now we choose a sequence of numbers γ k ↓ 1. For all k, let N k = H(t k ) and choose x k ∈ R n such that M k = |u(x k , t k )| ≥ N k /γ k . Then we set The functions v (k) and e (k) are defined in R n × (A k , B k ), where and satisfy Also, v (k) , e (k) are mild solution of the MHD system in R n × (A k , 0) with initial data v (k) 0 (y) = (1/M k )u 0 (x k + y/M k ), e (k) 0 (y) = (1/M k )b 0 (x k + y/M k ). By Lemma 6.1, there is a subsequence of v (k) , e (k) converging to an ancient mild solution (v, e) of the MHD system. Note that, by the construction, we have |v| ≤ 1, |e| ≤ C 0 in R n × (−∞, 0) and v(0, 0) = 1.
and b z is bounded in R 3 × (0, T ).
is a mild solution of the MHD system (for a suitable initial data).
Proof. We first prove the statement assuming that u is a mild solution (for a suitable initial data). Arguing by contradiction, assume that (u, b) is a mild solution which is bounded in R 3 ×(0, T ) for each T < T and develops a singularity at time T and case (1) is true, that means (6.1) holds. Let v (k) and b (k) be as in the construction (6.3) and (6.4). We write x k = (x k , x 3k ), where x k = (x 1k , x 2k ). With the assumption (6.7), we find that |x k | ≤ C/M k . This implies that the functions v (k) , e (k) are axisymmetric with respect to an axis parallel to the y 3 -axis and at distance at most C from it. Therefore we can assume (by passing to suitable subsequence) that the limit function v is axi-symmetric with respect to a suitable axis. Moreover, since assumption (6.7) is scale-invariant, it will again be satisfied (in suitable coordinates) by v, and in addition e z = 0 by the assumption of "b z is bounded in R 3 × (0, T )". Then applying Theorem 5.5, we see that v = 0. On the other hand, |v(0, 0)| = 1, this is a contradiction.
Recall that Lemma 3.4, weak solution u can be decomposed as u = v + ω 1 + d 1 . Thus applying the condition (6.7), we can obtain that d 1 = 0. Therefore, u, b are mild solutions of the MHD system. Remark 6.3. If the condition "b z is bounded in R 3 × (0, T )" in Theorem 6.2 is replaced by "b z u r = u r b z in R 3 ×(0, T )", the result also holds in the same way(applying Theorem 5.6).