張輝

(湘潭大學(xué)數(shù)學(xué)學(xué)院,湖南湘潭 411105)

Morrey-Campanato空間中三維Navier-Stokes方程的正則性準(zhǔn)則

張輝

(湘潭大學(xué)數(shù)學(xué)學(xué)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(zhǔn)則,改進(jìn)了一些已有的結(jié)果.

Navier-Stokes方程;Morrey-Campanato空間;正則性準(zhǔn)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(xiàn)[1]構(gòu)建了方程(1)的整體弱解.然而,關(guān)于弱解的正則性或唯一性仍然是流體力學(xué)中最有挑戰(zhàn)性的問題.1962年,文獻(xiàn)[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(xiàn)[3]給出了關(guān)于速度場梯度的正則性準(zhǔn)則

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

2008年,文獻(xiàn)[5]將條件(3)改進(jìn)到只涉及兩個速度分量梯度的正則性準(zhǔn)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(jìn),因為包含了上述的結(jié)論.

注1.2本文中函數(shù)Lp的范數(shù)用‖.‖p表示,C表示一般的常數(shù).

2 先驗估計

3 定理的證明

這里C與T?相關(guān).

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

[1]Leray J.Sur le mouvement d′un liquide visqueux emplissant l′espace[J].Acta.Math.,1934,63(1):193-248.

[2]Serrin J.On the interior regularity of weak solutions of the Navier-Stokes equations[J].Arch Rational Mech. Anal.,1962,9(1):187-195.

[3]Beirao da Veiga.A new regularity class for the Navier-Stokes equations in Rn[J].Chinese.Ann.Math. 1995,16:407-412.

[4]Beirao da Veiga.On the smoothness of a class of weak solutions to the Navier-Stokes equations[J].J.Math. Fluid Mech.,2000,2:315-323.

[5]Dong B Q,Chen Z M.Regularity criterion of weak solutions to the 3D Navier-Stokes equations via two velocity components[J].J.Math.Anal.Appl.,2008,338:1-10.

[6]Zhou Y,Gala S.Logarithmically improved regularity criteria for the Navier-Stokes in multiplier space[J].J. Math.Anal.Appl.,2009,356:498-501.

[7]Lemarie P G.Rieusset.The Navier-Stokes equations in the critical Morrey-Campanato space[J].Rev.Mat. Iberoam.,2007,3:897-930.

[8]Chen X C,Gala S.Remarks on logarithmically regularity criteria for the 3D viscous MHD equations[J].J. Korean.Math.Soc.,2011,48(3):465-474.

[9]Gala S.On the regularity criteria for the three-dimensional micropolar fluid equations in the critical Morrey-Campanato space[J].Nonlinear Analysis:Real World Applications,2011,12:2142-2150.

[10]Kato T,Ponce G.Commutator estimate and the Euler and Navier-Stokes equations[J].Commun.Pure. Appl.Math.,1988,41:891-907.

[11]Fujita H,Kato T.On the nonstationary Navier-Stokes initial value problem[J].Arch.Rational.Mech. Anal.,1964,16:269-315.

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(chuàng)新項目(CX2011B246).

張輝(1980-),博士生,講師,研究方向：應(yīng)用偏微分方程.

2010 MSC:35Q35