Navier-Stokes-Landau-Lifshitz方程组正则性准则研究

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Navier-Stokes-Landau-Lifshitz 方程组正则性准则研究 Navier-Stokes-Landau-Lifshitz 方程组是一种复杂的非线性偏微分方程组，它将 Navier-Stokes 方程和 Landau-Lifshitz 方程组合，用于描述复杂的流体运动和磁场相互作用。研究 Navier-Stokes-Landau-Lifshitz 方程组的正则性准则是数学物理和应用数学中非常重要的一个问题。本文研究了 Navier-Stokes-Landau-Lifshitz 方程组的正则性准则，得到了 Navier-Stokes-Landau-Lifshitz 方程组的光滑解在 Besov 空间和乘子空间中的存在性和唯一性。 Besov 空间是一种常用的函数空间，它可以用来描述函数的光滑性和近似的性质。乘子空间则是指在某个函数空间中，满足一定条件的函数所组成的子空间。 Navier-Stokes-Landau-Lifshitz 方程组的正则性准则是指在一定条件下， Navier-Stokes-Landau-Lifshitz 方程组的解具有光滑性和连续性的准则。研究 Navier-Stokes-Landau-Lifshitz 方程组的正则性准则对解 Navier-Stokes-Landau-Lifshitz 方程组具有重要的理论和实际意义。本文的结果推广了 Navier-Stokes 方程和 Landau-Lifshitz 方程组的相关结果到 Navier-Stokes-Landau-Lifshitz 方程组。 Navier-Stokes 方程是描述流体运动的基本方程，而 Landau-Lifshitz 方程组是描述磁场的基本方程。 Navier-Stokes-Landau-Lifshitz 方程组则是这两个方程组的耦合，用于描述流体运动和磁场相互作用。 Besov 空间和乘子空间是函数分析和偏微分方程理论中非常重要的概念。 Besov 空间是一种基于 Littlewood-Paley 理论的函数空间，它可以用来描述函数的光滑性和近似的性质。乘子空间则是指在某个函数空间中，满足一定条件的函数所组成的子空间。研究 Navier-Stokes-Landau-Lifshitz 方程组在 Besov 空间和乘子空间中的性质，对解 Navier-Stokes-Landau-Lifshitz 方程组具有重要的理论和实际意义。本文的结果对 Navier-Stokes-Landau-Lifshitz 方程组的研究具有重要的理论和实际意义，对后续研究 Navier-Stokes-Landau-Lifshitz 方程组的性质和应用具有重要的指导意义。

Here

p,q

stands for the homogeneous Morrey space. We point out here that the point-

wise multipliers between diﬀerent spaces of diﬀerentiable functions have been studied

by Maz’ya and co-workers [4, 6]. They are useful tools for stating minimal regularity

requirements on the coeﬃcients of partial diﬀerential operators for proving regularity or

uniqueness of solutions. Our ﬁrst result reads:

Theorem 1.1

Assume that the initial data satisfy

0 < m ≤ ρ

≤ M < ∞, ∇ρ

∈ L

∩L

(3 < q ≤ 6), u

∈ H

, ∇d

∈ H

, div u

= 0 and |d

| = 1.

Let (ρ, u, p, d) be a strong solution to the problem (1.1)-(1.5). If

u ∈ L

1−r

(0, T ;

) with r ∈ (0, 1), (1.9)

and

∇d ∈ L

(0, T ;

∞,∞

), (1.10)

∆d ∈ L

(0, T ;

∞,∞

), (1.11)

then the solution (ρ, u, p, d) can be extended beyond T .

Remark 1.1 If (ρ, u, p, d) is a solution to the problem (1.1)-(1.5), then so does (ρ

, u

, p

, d

) :=

(ρ, λu, λ

p, d)(λx, λ

t). In this sense, our conditions (1.9)-(1.11) are optimal.

Remark 1.2 When ρ = 1 and u ≡ 0, our condition (1.10) or (1.11) improves (1.7) in

[2, 3].

When ρ = 1, the system (1.1)-(1.5) reduces to

div u = 0, (1.12)

+ u · ∇u + ∇p − ∆u = −∇ · (∇d  ∇d), (1.13)

+ u · ∇d − ∆d = d|∇d|

+ d × ∆d, |d| = 1, (1.14)

(u, d)|

t=0

= (u

, d

), |d

| = 1 in R

. (1.15)

When the term d × ∆d is omitted, the system was studied in the context of liquid crystals

[7]. For the Navier-Stokes equations, Kozono-Ogawa-Taniuchi [2] proved the following

regularity condition:

u ∈ L

(0, T ;

∞,∞

)

ω := curl u ∈ L

(0, T ;

∞,∞

http://www.paper.edu.cn

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