Global regularity of 2D MHD equations with almost Laplacian velocity dissipation

We obtain the global existence and global regularity for the 2D MHD equations with almost Laplacian velocity dissipation, which require the dissipative operators weaker than any power of the fractional Laplacian. The result can be regarded as a further improvement and generalization of previous works.


Introduction and main results
This paper focuses on the two-dimensional magnetohydrodynamic (MHD) system where u = (u 1 (x, t), u 2 (x, t)) denotes the velocity, p = p(x, t) the scalar pressure, and B = (B 1 (x, t), B 2 (x, t)) the magnetic field of the fluid.u 0 (x) and B 0 (x) are the given initial data satisfying ∇ • u 0 = 0 and ∇ • B 0 = 0.The operators £ 1 , £ 2 are general negative-definite multipliers with symbol h(ξ ), namely (1.2) We note the convention that by ν = 0 we mean that there is no velocity dissipation in (1.1) 1 and similarly κ = 0 represents that there is no magnetic diffusion in (1.1) 2 .It is well known that the 2D MHD equations with both Laplacian dissipation u and magnetic diffusion B have the global smooth solution [1].In the case without velocity dissipation and magnetic diffusion (ν = κ = 0), the question of whether a smooth solution of the 2D MHD equations develops a singularity in finite time is open [2,3].Therefore, more researches focus on the MHD equations with fractional dissipation and partial dissipation (see e.g., [4][5][6][7]) and the references cited therein, the issue of global regularity for 2D MHD (ν = 0, κ = 1 or ν = 1, κ = 0) has attracted much interest from many mathematicians and has motivated a large number of research papers concerning various generalizations and improvements (see [8][9][10][11][12]). Recent important progress has been obtained by Ren et al. [13], who prove the global existence and the decay estimates of a small smooth solution for the 2D MHD equations without magnetic diffusion, and these results confirm the numerical observation that the energy of the MHD equations is dissipated at a rate independent of the ohmic resistivity, the main tools used in [13] are the anisotropic Littlewood-Paley decomposition and the anisotropic Besov spaces.More recently, Yuan and Ye [11,14] studied separately the global regularity of solutions to the 2D incompressible MHD equations with almost Laplacian magnetic diffusion in the whole space.Furthermore, Lei [15] studied the global regularity for the axially symmetric MHD equations with nontrivial magnetic fields.
Inspired by the previous works, the aim of this paper is to weaken the operator $ as possible as one can, then the global H 1 -bound of the solution can be achieved as long as the fractional dissipation power of the velocity field is positive without losing the global regularity of the system (1.1), we obtained the global regularity of solutions requiring the velocity dissipative operators to be weaker than any power of the fractional Laplacian, the result can be regarded as a further improvement and generalization of previous works [3].More precisely, the main result in this article reads as follows.
for any s > 2 and satisfy ∇ • u 0 = ∇ • B 0 = 0, where ν > 0 and κ > 0, assume that h(ξ ) = h(|ξ |) is a radial nondecreasing smooth function satisfying the following conditions: (1) h is a nonnegative function for all ξ = 0; (2) there exists a constant C > 0 such that (3) for any given T ∈ (0, ∞) such that where B T > 0 is the unique solution of the following equation then for the MHD equations (1.1) there exists a unique global solution such that Remark 1.2 By virtue of the expression in (1.2), the dissipative operator £ in Theorem 1.1 is weaker than any power of the fractional Laplacian.Thus, we improve the results in [3] for system (1.1) in which ν > 0 and κ > 0.
Remark 1.3 For the 2D generalized MHD equations, it remains an open problem whether there exists a global smooth solution without the dissipative operator £.

The proof of Theorem 1.1
In this section, we will give the global existence and uniqueness of the smooth solution to the system (1.1).By the classical hyperbolic method, there exists a finite time T 0 such that the system (1.1) is local well-posed in the interval [0, T 0 ] in H s with s > 2. Therefore, it is sufficient to establish a priori estimates in the interval [0, T] for the given T > T 0 .We shall prove Theorem 1.1 by adapting the elaborate nonlinear energy method.Furthermore, throughout this paper, we use C to denote the positive constants that may vary from line to line.

L 2 estimate for (u, B)
Lemma 2.1 Assume that the conditions in Theorem 1.1 hold, then it holds that u(t) Proof Taking the inner products of (1.1) 1 with u and (1.1) 2 with B, adding the results and utilizing the following cancelation identity: we have Thanks to the Plancherel theorem then integrating the above inequality we obtain (2.6)

H 1 estimate for (u, B)
Lemma 2.2 Assume that the conditions in Theorem 1.1 hold, then there exists a positive constant C dependent on T, u 0 and B 0 , such that Proof First, we obtain the vorticity equation by applying ∇× to the MHD equations (1.1), where the vorticity the corresponding vorticity equation: ) where Taking the inner products of the first equation in (2.8) with ω, the second equation in (2.8) with j, adding them up and using the incompressible condition as well as the following fact then we easily obtain T(∇u, ∇B)j dx. (2.11) For the arbitrary function we obtain finally Utilizing the estimate of ∇u L 2 in (2.13) and the interpolation inequality, we obtain Hence, combined with (2.14) and (2.11) this implies that (2.15) The Gronwall lemma provides us with Hence, by the embedding theorem, Thus, it is sufficient to establish that an a priori estimate (2.19) is valid for t > T 0 .We rewrite the first equation in system (1.1) as According to the solution of the linear inhomogeneous equation, the solution of (2.20) can be explicitly given by Taking L ∞ norm in terms of space variables and using the Young inequality, we obtain where we have used the inequality K(t) L 1 ≤ C(1 + t -1 + t 3 ), which is the property of the operator £ proved in Lemma 2.5 [14], hence we obtain

.26)
Proof We write the first equation of (2.8) as According to the solution of the linear inhomogeneous equation, the solution of (2.27) can be explicitly written as Taking L ∞ norm in terms of space variables and utilizing the Young inequality, we have

.29)
In order to obtain the L 2 t L ∞ estimate on ω, we take the L 2 norm on the time variable and use the convolution Young inequality as well as the estimate (2.25) to obtain where we have used the following known estimates due to the linear inhomogeneous equation: On the other hand,we also used the fact that and

.33)
As a result, one has

.37)
Proof By (2.28), we can check that Taking the operation ∇ to both sides of the above equality, and then taking the L ∞ norm in terms of space variables and using the Young inequality, we have Then, we obtain by taking the L 1 norm in terms of time variables and utilizing the convolution Young inequality By the previous estimates on ∇K L 1 t L 2 , it is easy to show that ∇K L 1 t L 2 j 0 L 2 are nondecreasing functions satisfying Multiplying both sides of the vorticity equation by |ω| p-2 ω and integrating over R 2 , we obtain where we have used the nonnegativity of the almost Laplacian velocity dissipation, then we have Due to the previous estimate, we obtain Now, suppose that The Gronwall lemma provides us with to the original equation u and B, and take the L 2 inner product of the resulting equations with ( s u, s B) to obtain the energy inequality.
Then, we estimate the energy terms as follows: Finally,we obtain

.62)
We need to estimate the bounds on ∇u L ∞ and ω L ∞ , according to the following Sobolev extrapolation inequality with logarithmic correction [17].

.63)
Collecting the above estimate, we have According to the six steps priori estimates in previous results, we give the proof of Theorem 1.1.We introduce the mollification of N f given by η N(xy) f (y) dy, (2.67) where 0 < η(|x|) ∈ C ∞ 0 (R 2 ) satisfies R 2 η(y) dy = 1.We regularize the system (1.1) as follows: u N (x, 0) = N u 0 (x), B N (x, 0) = ρ N B 0 (x), where P denotes the Leray projection operator.For any fixed N > 0, using the properties of mollifiers and the priori estimates obtained in proving the global bounds, for any t ∈ (0, ∞), u N  which are uniform in N , according to the standard Alaogu theorem and compactness arguments, we can extract a subsequence (u N i , B N i ) and pass to the limit as N → ∞ to obtain the fact that the limit function (u, B) is indeed a global classical solution of the problem (1.1).The uniqueness can also be easily obtained.This completes the proof of Theorem 1.1.

Lemma 2 . 5
Assume that the conditions in Theorem 1.1 hold, then it holds that Proof To obtain the global bound for (u, B) in H s (s > 2), we apply s with = (I -) (t) H s + B N (t) H s +