Morrey-Campanato空間中三維Navier-Stokes方程的正則性準則

張輝

(湘潭大學數學學院,湖南湘潭 411105)

Morrey-Campanato空間中三維Navier-Stokes方程的正則性準則

張輝

(湘潭大學數學學院,湖南湘潭 411105)

利用能量不等式和一些臨界空間中的不等式,在Morrey-Campanato空間獲得了兩個只涉及水平速度場的正則性準則,改進了一些已有的結果.

Navier-Stokes方程;Morrey-Campanato空間;正則性準則

1 引言

在?3中考慮Navier-Stokes方程的Cauchy問題:

這里u(x,t)表示未知的速度場,p(x,t)表示未知的壓力,u0(x)表示給定的初始速度場且在分布意義下滿足?·u0=0.

對于給定的初始速度場u0∈L2(?3),文獻[1]構建了方程(1)的整體弱解.然而,關于弱解的正則性或唯一性仍然是流體力學中最有挑戰性的問題.1962年,文獻[2]證明了:如果弱解u滿足:

則弱解實際上就是[0,T]上唯一的強解.

1995年,文獻[3]給出了關于速度場梯度的正則性準則

在文獻[2-3]工作的基礎上有許多文獻對條件(2)和條件(3)進行了改進.2000年,文獻[4]將條件(2)改進到了只涉及兩個速度分量的正則性準則:

2008年,文獻[5]將條件(3)改進到只涉及兩個速度分量梯度的正則性準則:

則弱解實際上就是[0,T]上的唯一強解.

注1.1相比較條件(2)-條件(7),定理1.1是一個較大的改進,因為包含了上述的結論.

注1.2本文中函數Lp的范數用‖.‖p表示,C表示一般的常數.

2 先驗估計

3 定理的證明

這里C與T?相關.

(26)式給出了速度場的H1估計,對H3范數的估計與第一種情形類似,故省略.

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Regularity criteria for the 3D Navier-Stokes equations in Morrey-Campanato space

ZhangHui
(College of Mathematics,Xiangtan University,Xiangtan411105,China)

By energy estimate method and some inequalities in critical space,we obtained two regularity criteria involving partial components of the velocity in the Morrey-Campanato space.Our results improved some results early.

Navier-Stokes equations,Morrey-Campanato space,regularity criteria

O175.29

A

1008-5513(2013)02-0140-06

10.3969/j.issn.1008-5513.2013.02.005

2012-11-22.

湖南省研究生科研創新項目(CX2011B246).

張輝(1980-),博士生,講師,研究方向：應用偏微分方程.

2010 MSC:35Q35