Global Well-Posedness for a Family of MHD-Alpha-Like Models

Global well-posedness is proved for a family of 𝑛 -dimensional MHD-alpha-like models.


Introduction
In this paper, we consider a family of MHD-alpha-like models: where v is the fluid velocity field, u is the "filtered" fluid velocity, p is the pressure, H is the magnetic field, and b is the "filtered" magnetic field.α > 0 and α M > 0 are the length scales and for simplicity we will take α α M 1.The parameter θ 1 ≥ 0 affects the strength of the nonlinear term and θ 2 ≥ 0 represents the degree of viscous dissipation satisfying When θ 1 θ 2 1 and n 3, a global well-posedness is proved in 1 .The aim of this paper is to prove a global well-posedness theorem under 1.6 .We will prove the following theorem.
Theorem 1.1.Let u 0 , b 0 ∈ H s with s ≥ 1, div v 0 div u 0 div H 0 div b 0 0 in R n , and 1.6 holding true.Then for any T > 0, there exists a unique strong solution u, b satisfying 1.7 Remark 1.2.For studies on some standard MHD-α or Leray-α models, we refer to 2-7 and references therein.

Proof of Theorem 1.1
Since it is easy to prove that the problem 1.1 -1.5 has a unique local smooth solution, we only need to establish the a priori estimates.Testing 1.1 by u, using 1.3 and 1.4 , and letting Λ : −Δ 1/2 , we see that Testing 1.2 by b and using 1.3 and 1.4 , we find that 1 2 Summing up 2.1 and 2.2 , thanks to the cancellation of the right-hand side of 2.1 and 2.2 , we infer that In the following calculations, we will use the following commutator estimates due to Kato and Ponce 8 : with s > 0 and 1/p 1/p 1 1/q 1 1/p 2 1/q 2 .We will also use the Sobolev inequality: and the Gagliardo-Nirenberg inequality:

2.7
Taking Λ s to 1.1 , testing by Λ s u, and using 1.3 and 1.4 , we infer that 1 2

2.8
Taking Λ s to 1.2 , testing by Λ s b, and using 1.3 and 1.4 , we deduce that 1 2

2.9
Summing up 2.8 and 2.9 , thanks to the cancellation of the right-hand side of 2.8 and 2.9 , and using 2.5 , 2.6 and 2.7 , we conclude that 1 2 2.10 which implies 1.7 .

2.11
Here we have used the Sobolev inequalities

2.12
Similarly, testing 1.2 by H and using 1.4 and 2.12 , we find that 1 2