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

Abstract

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

1. Introduction

In this paper, we consider a family of MHD-alpha-like models:𝜕_𝑡𝑣+(−Δ)^𝜃2𝑣+𝑢⋅∇𝑢+∇𝑝+1𝜕2𝑏²=𝑏⋅∇𝑏,(1.1)_𝑡𝑣=1+−𝛼₂𝐻+(−Δ)^𝜃2𝐻+𝑢⋅∇𝑏−𝑏⋅∇𝑢=0,(1.2)Δ(𝑣d,iv𝐻^𝜃1𝑣)𝑢(=,0)d𝐻=iv=𝑢𝑣=₀1d+iv𝐻−𝛼=^2𝑀Δdiv^𝜃1𝑏=𝑏,0,𝛼>0,𝛼^𝑀>0,((11..34)),𝐻₀inℝ^𝑛(𝑛≥3),(1.5) 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 𝜃₁≥0 affects the strength of the nonlinear term and 𝜃₂≥0 represents the degree of viscous dissipation satisfying3𝜃₁+2𝜃₂=𝑛+22.(1.6)

When 𝜃₁=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)∈𝐻^𝑠 with 𝑠≥1, div𝑣₀=div𝑢₀=div𝐻₀=div𝑏₀=0 in ℝ^𝑛, and (1.6) holding true. Then for any 𝑇>0, there exists a unique strong solution (𝑢,𝑏) satisfying (𝑢,𝑏)∈𝐿^∞0,𝑇;𝐻^𝑠+𝜃1∩𝐿²0,𝑇;𝐻^{𝑠+𝜃1+𝜃2}.(1.7)

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 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 that12𝑑𝑑𝑡𝑢²+||Λ^𝜃1𝑢||²𝑑𝑥+||Λ^𝜃2𝑢||²+||Λ^𝜃1+𝜃2𝑢||²𝑑𝑥=(𝑏⋅∇)𝑏⋅𝑢𝑑𝑥.(2.1)

Testing (1.2) by 𝑏 and using (1.3) and (1.4), we find that12𝑑𝑑𝑡𝑏²+||Λ^𝜃1𝑏||²𝑑𝑥+||Λ^𝜃2𝑏||²+||Λ^𝜃1+𝜃2𝑏||²𝑑𝑥=(𝑏⋅∇)𝑢⋅𝑏𝑑𝑥.(2.2)

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 12𝑑𝑑𝑡(𝑢,𝑏)²+||Λ^𝜃1(𝑢,𝑏)||²𝑑𝑥+||Λ^𝜃2(𝑢,𝑏)||²+||Λ^𝜃1+𝜃2(𝑢,𝑏)||²𝑑𝑥=0,(2.3) whence‖(𝑢,𝑏)‖_𝐿²_(0,𝑇;𝐻^𝜃12+𝜃₎≤𝐶.(2.4)

Case 1. 𝜃₁+𝜃₂>1.
In the following calculations, we will use the following commutator estimates due to Kato and Ponce [8]: ‖Λ^𝑠(𝑓𝑔)−𝑓Λ^𝑠𝑔‖_𝐿^𝑝≤𝐶‖∇𝑓‖_𝐿^𝑝1‖‖Λ^𝑠−1𝑔‖‖_𝐿𝑞1+‖Λ^𝑠𝑓‖_𝐿^𝑝2‖𝑔‖_𝐿^𝑞2,(2.5) with 𝑠>0 and 1/𝑝=1/𝑝₁+1/𝑞₁=1/𝑝₂+1/𝑞₂.We will also use the Sobolev inequality: ‖∇𝑢‖_𝐿^𝑝≤𝐶‖‖Λ^𝜃1+𝜃2𝑢‖‖_𝐿21−𝑛𝑝=𝜃¹+𝜃₂−𝑛2,(2.6) and the Gagliardo-Nirenberg inequality: ‖Λ^𝑠𝑢‖^2𝐿_{2𝑝/𝑝−1}≤𝐶‖‖Λ^𝑠+𝜃1𝑢‖‖_𝐿2‖‖Λ^{s+𝜃1+𝜃2}𝑢‖‖_𝐿2.(2.7)
Taking Λ^𝑠 to (1.1), testing by Λ^𝑠𝑢, and using (1.3) and (1.4), we infer that 12𝑑𝑑𝑡||Λ^𝑠=−𝑢||²Λ+_𝑠||Λ^𝑠+𝜃1𝑢||²𝑑𝑥+||Λ^𝑠+𝜃2𝑢||²+||Λ^{𝑠+𝜃1+𝜃2}𝑢||²𝑑𝑥+𝑏⋅∇Λ(𝑢_𝑠⋅∇𝑢)−𝑢⋅∇Λ^𝑠𝑢Λ^𝑠𝑢𝑑𝑥+Λ^𝑠(𝑏⋅∇𝑏)−𝑏⋅∇Λ^𝑠𝑏Λ^𝑠𝑢𝑑𝑥𝑏⋅Λ^𝑠𝑢𝑑𝑥.(2.8)
Taking Λ^𝑠 to (1.2), testing by Λ^𝑠𝑏, and using (1.3) and (1.4), we deduce that 12𝑑𝑑𝑡||Λ^𝑠=−𝑏||²Λ+_𝑠||Λ^𝑠+𝜃1𝑏||²𝑑𝑥+||Λ^𝑠+𝜃2𝑏||²+||Λ^{𝑠+𝜃1+𝜃2}𝑏||²𝑑𝑥+𝑏⋅∇Λ(𝑢_𝑠⋅∇𝑏)−𝑢⋅∇Λ^𝑠𝑏Λ^𝑠𝑏𝑑𝑥+Λ^𝑠(𝑏⋅∇𝑢)−𝑏⋅∇Λ^𝑠𝑢Λ^𝑠𝑏𝑑𝑥𝑢⋅Λ^𝑠𝑏𝑑𝑥.(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 12𝑑𝑑𝑡||Λ^𝑠≤𝐶‖(∇𝑢𝑢,𝑏‖)||²_𝐿𝑝+||Λ^𝑠+𝜃1(𝑢,𝑏)||²𝑑𝑥+||Λ^𝑠+𝜃2(𝑢,𝑏)||²+||Λ^{𝑠+𝜃1+𝜃2}(𝑢,𝑏)||²𝑑𝑥‖Λ^𝑠𝑢‖^2𝐿_{2𝑝/𝑝−1}+𝐶‖∇𝑏‖_𝐿^𝑝‖Λ^𝑠𝑏‖_𝐿^{2𝑝/𝑝−1}‖Λ^𝑠𝑢‖_𝐿^{2𝑝/𝑝−1}+𝐶‖∇𝑢‖_𝐿^𝑝‖Λ^𝑠𝑏‖^2𝐿_{2𝑝/𝑝−1}≤𝐶‖∇(𝑢,𝑏)‖_𝐿^𝑝‖Λ^𝑠(𝑢,𝑏)‖^2𝐿_{2𝑝/𝑝−1}≤𝐶‖‖Λ^𝜃1+𝜃2(𝑢,𝑏)‖‖_𝐿2‖‖Λ^𝑠+𝜃1(𝑢,𝑏)‖‖_𝐿2‖‖Λ^{𝑠+𝜃1+𝜃2}(𝑢,𝑏)‖‖_𝐿2≤12‖‖Λ^{𝑠+𝜃1+𝜃2}(𝑢,𝑏)‖‖^2𝐿₂+𝐶‖‖Λ^𝜃1+𝜃2(𝑢,𝑏)‖‖^2𝐿₂‖‖Λ^𝑠+𝜃1(𝑢,𝑏)‖‖^2𝐿₂,(2.10) 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 12𝑑𝑑𝑡𝑣²𝑑𝑥+||Λ^𝜃2𝑣||²𝑑𝑥=≤‖(𝑏𝑏‖⋅∇_𝐿𝑝1𝑏−𝑢⋅∇𝑢)𝑣𝑑𝑥‖∇𝑏‖_𝐿^{112𝑝/𝑝−2}+‖𝑢‖_𝐿^𝑝1‖∇𝑢‖_𝐿^{112𝑝/𝑝−2}‖𝑣‖_𝐿²≤‖(𝑢,𝑏)‖_𝐿^𝑝1‖∇(𝑢,𝑏)‖_𝐿^{112𝑝/𝑝−2}‖𝑣‖_𝐿²≤𝐶‖(𝑢,𝑏)‖_𝐻^𝜃12+𝜃‖‖Λ^𝜃2(𝑣,𝐻)‖‖_𝐿2‖𝑣‖_𝐿².(2.11) Here we have used the Sobolev inequalities ‖(𝑢,𝑏)‖_𝐿^𝑝1≤𝐶‖(𝑢,𝑏)‖_𝐻^𝜃12+𝜃−𝑝3₁=𝜃₁+𝜃₂−3‖∇(𝑢,𝑏)‖_{𝐿112𝑝/𝑝−2}2,≤𝐶‖‖Λ^𝜃2(𝑣,𝐻)‖‖_𝐿21−3𝑝¹−22𝑝₁=𝜃₂+2𝜃₁−32.(2.12) Similarly, testing (1.2) by 𝐻 and using (1.4) and (2.12), we find that 12𝑑𝑑𝑡𝐻²𝑑𝑥+||Λ^𝜃2𝐻||²𝑑𝑥=≤‖((𝑢𝑏,𝑏⋅)∇‖𝑢−_𝐿𝑝1𝑢⋅∇𝑏)𝐻𝑑𝑥‖∇(𝑢,𝑏)‖_𝐿^{112𝑝/𝑝−2}‖𝐻‖_𝐿²≤𝐶‖(𝑢,𝑏)‖_𝐻^𝜃12+𝜃‖‖Λ^𝜃2(𝑣,𝐻)‖‖_𝐿2‖𝐻‖_𝐿².(2.13) Combining (2.11) and (2.13) and using (2.4) and the Gronwall inequality, we have ‖(𝑢,𝑏)‖_𝐿²_(0,𝑇;𝐻^𝜃21+2𝜃₎≤𝐶.(2.14) Similarly to (2.10), we have 12𝑑𝑑𝑡||Λ^𝑠≤𝐶‖(∇𝑢,(𝑏𝑢),||𝑏²)+‖||Λ^𝑠+𝜃1_𝐿𝑝2(𝑢,𝑏)||²𝑑𝑥+||Λ^𝑠+𝜃2(𝑢,𝑏)||²+||Λ^{𝑠+𝜃1+𝜃2}(𝑢,𝑏)||²𝑑𝑥‖Λ^𝑠(𝑢,𝑏)‖^2𝐿_{222𝑝/𝑝−1}≤𝐶‖(𝑢,𝑏)‖_𝐻^𝜃21+2𝜃‖‖Λ^𝑠+𝜃1(𝑢,𝑏)‖‖^{2(1−𝛼1)𝐿}₂‖‖Λ^{𝑠+𝜃1+𝜃2}(𝑢,𝑏)‖‖^2𝛼1_𝐿2≤12‖‖Λ^{𝑠+𝜃1+𝜃2}(𝑢,𝑏)‖‖^2𝐿₂+𝐶‖(𝑢,𝑏)‖^1/1−𝛼1_{𝐻𝜃21+2𝜃}‖‖Λ^𝑠+𝜃1(𝑢,𝑏)‖‖^2𝐿₂,(2.15) which implies (1.7) by 1/(1−𝛼₁)≤2. Here we have used the Sobolev inequality: ‖∇(𝑢,𝑏)‖_𝐿^𝑝2≤𝐶‖(𝑢,𝑏)‖_𝐻^𝜃21+2𝜃1−𝑝𝑛₂<𝜃₂+2𝜃₁−𝑛2(2.16) and the Gagliardo-Nirenberg inequality: ‖Λ^𝑠(𝑢,𝑏)‖_𝐿^{222𝑝/(𝑝−1)}≤𝐶‖‖Λ^𝑠+𝜃1(𝑢,𝑏)‖‖^1−𝛼1_𝐿2‖‖Λ^{𝑠+𝜃1+𝜃2}(𝑢,𝑏)‖‖^𝛼1_𝐿2,(2.17) with −((𝑝₂−1)/2𝑝₂)𝑛=𝛼₁𝜃₂+𝜃₁−𝑛/2 and 𝑝₂≥2≥3/(2𝜃₁+𝜃₂). This completes the proof.