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

Authors: Xiaowei He
Journal: Journal of Applied Mathematics
 (2011)

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## Abstract

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


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## Body

## 1. Introduction

In this paper, we consider a family of MHD-alpha-like models:(1.1)∂tv+(-Δ)θ2v+u⋅∇u+∇(p+12b2)=b⋅∇b,(1.2)∂tH+(-Δ)θ2H+u⋅∇b-b⋅∇u=0,(1.3)v=[1+(-α2Δ)θ1]u,H=[1+(-αM2Δ)θ1]b,α>0,αM>0,(1.4)div⁡v=div⁡u=div⁡H=div⁡b=0,(1.5)(v,H)(0)=(v0,H0)inRn(n≥3),
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 αM>0 are the length scales and for simplicity we will take α=αM=1. The parameter θ1≥0 affects the strength of the nonlinear term and θ2≥0 represents the degree of viscous dissipation satisfying(1.6)3θ1+2θ2=n+22.Whenθ1=θ2=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(u0,b0)∈Hs with s≥1, divv0=divu0=divH0=divb0=0 in ℝn, and (1.6) holding true. Then for any T>0, there exists a unique strong solution (u,b) satisfying
(1.7)(u,b)∈L∞(0,T;Hs+θ1)∩L2(0,T;Hs+θ1+θ2).Remark 1.2.
For studies on some standard MHD-α or Leray-α models, we refer to [2–7] and references therein.


## 2. Proof of Theorem1.1

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(2.1)12ddt∫u2+|Λθ1u|2dx+∫|Λθ2u|2+|Λθ1+θ2u|2dx=∫(b⋅∇)b⋅udx.Testing (1.2) by b and using (1.3) and (1.4), we find that(2.2)12ddt∫b2+|Λθ1b|2dx+∫|Λθ2b|2+|Λθ1+θ2b|2dx=∫(b⋅∇)u⋅bdx.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 (2.3)12ddt∫(u,b)2+|Λθ1(u,b)|2dx+∫|Λθ2(u,b)|2+|Λθ1+θ2(u,b)|2dx=0,
whence(2.4)‖(u,b)‖L2(0,T;Hθ1+θ2)≤C.Case 1.
θ1+θ2>1.
In the following calculations, we will use the following commutator estimates due to Kato and Ponce [8]:
(2.5)‖Λs(fg)-fΛsg‖Lp≤C(‖∇f‖Lp1‖Λs-1g‖Lq1+‖Λsf‖Lp2‖g‖Lq2),
with s>0 and 1/p=1/p1+1/q1=1/p2+1/q2.We will also use the Sobolev inequality:
(2.6)‖∇u‖Lp≤C‖Λθ1+θ2u‖L2(1-np=θ1+θ2-n2),
and the Gagliardo-Nirenberg inequality:
(2.7)‖Λsu‖L2p/p-12≤C‖Λs+θ1u‖L2‖Λs+θ1+θ2u‖L2.
TakingΛs to (1.1), testing by Λsu, and using (1.3) and (1.4), we infer that
(2.8)12ddt∫|Λsu|2+|Λs+θ1u|2dx+∫|Λs+θ2u|2+|Λs+θ1+θ2u|2dx=-∫[Λs(u⋅∇u)-u⋅∇Λsu]Λsudx+∫[Λs(b⋅∇b)-b⋅∇Λsb]Λsudx+∫b⋅∇Λsb⋅Λsudx.
TakingΛs to (1.2), testing by Λsb, and using (1.3) and (1.4), we deduce that
(2.9)12ddt∫|Λsb|2+|Λs+θ1b|2dx+∫|Λs+θ2b|2+|Λs+θ1+θ2b|2dx=-∫[Λs(u⋅∇b)-u⋅∇Λsb]Λsbdx+∫[Λs(b⋅∇u)-b⋅∇Λsu]Λsbdx+∫b⋅∇Λsu⋅Λsbdx.
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
(2.10)12ddt∫|Λs(u,b)|2+|Λs+θ1(u,b)|2dx+∫|Λs+θ2(u,b)|2+|Λs+θ1+θ2(u,b)|2dx≤C‖∇u‖Lp‖Λsu‖L2p/p-12+C‖∇b‖Lp‖Λsb‖L2p/p-1‖Λsu‖L2p/p-1+C‖∇u‖Lp‖Λsb‖L2p/p-12≤C‖∇(u,b)‖Lp‖Λs(u,b)‖L2p/p-12≤C‖Λθ1+θ2(u,b)‖L2‖Λs+θ1(u,b)‖L2‖Λs+θ1+θ2(u,b)‖L2≤12‖Λs+θ1+θ2(u,b)‖L22+C‖Λθ1+θ2(u,b)‖L22‖Λs+θ1(u,b)‖L22,
which implies (1.7).Case 2.
0<θ1+θ2≤1 only when n=3.
Testing (1.1) by v, using (1.4), we see that
(2.11)12ddt∫v2dx+∫|Λθ2v|2dx=∫(b⋅∇b-u⋅∇u)vdx≤(‖b‖Lp1‖∇b‖L2p1/p1-2+‖u‖Lp1‖∇u‖L2p1/p1-2)‖v‖L2≤‖(u,b)‖Lp1‖∇(u,b)‖L2p1/p1-2‖v‖L2≤C‖(u,b)‖Hθ1+θ2‖Λθ2(v,H)‖L2‖v‖L2.
Here we have used the Sobolev inequalities
(2.12)‖(u,b)‖Lp1≤C‖(u,b)‖Hθ1+θ2(-3p1=θ1+θ2-32),‖∇(u,b)‖L2p1/p1-2≤C‖Λθ2(v,H)‖L2(1-3(p1-2)2p1=θ2+2θ1-32).
Similarly, testing (1.2) by H and using (1.4) and (2.12), we find that
(2.13)12ddt∫H2dx+∫|Λθ2H|2dx=∫(b⋅∇u-u⋅∇b)Hdx≤‖(u,b)‖Lp1‖∇(u,b)‖L2p1/p1-2‖H‖L2≤C‖(u,b)‖Hθ1+θ2‖Λθ2(v,H)‖L2‖H‖L2.
Combining (2.11) and (2.13) and using (2.4) and the Gronwall inequality, we have
(2.14)‖(u,b)‖L2(0,T;Hθ2+2θ1)≤C.
Similarly to (2.10), we have
(2.15)12ddt∫|Λs(u,b)|2+|Λs+θ1(u,b)|2dx+∫|Λs+θ2(u,b)|2+|Λs+θ1+θ2(u,b)|2dx≤C‖∇(u,b)‖Lp2‖Λs(u,b)‖L2p2/p2-12≤C‖(u,b)‖Hθ2+2θ1‖Λs+θ1(u,b)‖L22(1-α1)‖Λs+θ1+θ2(u,b)‖L22α1≤12‖Λs+θ1+θ2(u,b)‖L22+C‖(u,b)‖Hθ2+2θ11/1-α1‖Λs+θ1(u,b)‖L22,
which implies (1.7) by 1/(1-α1)≤2. Here we have used the Sobolev inequality:
(2.16)‖∇(u,b)‖Lp2≤C‖(u,b)‖Hθ2+2θ1(1-np2<θ2+2θ1-n2)
and the Gagliardo-Nirenberg inequality:
(2.17)‖Λs(u,b)‖L2p2/(p2-1)≤C‖Λs+θ1(u,b)‖L21-α1‖Λs+θ1+θ2(u,b)‖L2α1,
with -((p2-1)/2p2)n=α1θ2+θ1-n/2 and p2≥2≥3/(2θ1+θ2). This completes the proof.



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*Source: 408732-2011-10-09.xml*