In this paper we prove the uniqueness of weak solutions and the global-in-time existence of smooth solutions of the 2D generalized MHD system with fractional diffusion with power.
MSC:35Q30, 76D03, 76D09.
MHDfractional diffusionuniquenesssmooth solution
In this paper, we consider the following 2D generalized MHD system with :
Here, u is the fluid velocity field, π is the pressure and b is the magnetic field.
Very recently, Ji  used the Fourier series analysis motivated in  to prove the global-in-time existence of smooth solutions of problem (1.1)-(1.4) when , and Ji  pointed out that his result did not seem to come directly from the method like energy estimates. In this paper, we use the standard energy method to deal with the case ; of course, our method also works when . We will prove the following.
Theorem 1.1Let . Letwithin . Then problem (1.1)-(1.4) has a unique global-in-time weak solutionsatisfying
for any .
Theorem 1.2Let . Letwithandin . Then problem (1.1)-(1.4) has a unique global-in-time smooth solutionsatisfying
for any .
For 3D case and other related problems, we refer to [3, 4].
Our proof will use the following commutator estimates due to Kato and Ponce :
with , and .
2 Proof of Theorem 1.1
This section is devoted to the proof of Theorem 1.1. The global-in-time existence of weak solutions satisfying (1.5) was proved in [1, 6], we only need to show the uniqueness. Let () be two weak solutions of problem (1.1)-(1.4). We define
Then it follows from (1.1)-(1.3) that
Testing (2.2) by δu and using (1.1) and (2.1), we see that
Testing (2.3) by δb and using (1.1) and (2.1), we find that
In the following calculations, we use the Sobolev embedding and the Gagliardo-Nirenberg inequalities
Using (1.1), (2.1), (1.5), (2.6) and (2.7), we bound , , and as follows:
Adding up (2.4) and (2.5) and using the above estimates, we conclude that
This completes the proof.
3 Proof of Theorem 1.2
This section is devoted to the proof of Theorem 1.2. We only need to prove a priori estimates (1.6) for simplicity.
First, we have (1.5).
Applying to (1.2), testing by and using (1.1), we see that
Applying to (1.3), testing by and using (1.1), we find that
Using (1.7), (2.6), (2.7) and (1.5), we bound , , , and as follows:
Adding up (3.1) and (3.2) and using the above estimates, we arrive at
which yields (1.6).
This completes the proof.
School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power
Department of Mathematics, Zhejiang Normal University
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