Space-Time Estimates in the Besov Spaces and the Navier-Stokes Equations
Space-Time Estimates in the Besov Spaces and the Navier-Stokes Equations
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DOI：
10.4310/maa.2006.v13.n1.a6
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发表时间：
2006-03
影响因子：
0.3
通讯作者：
Qionglei Chen;Zhifei Zhang
中科院分区：
文献类型：
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作者：
Qionglei Chen;Zhifei Zhang
对于给定的u0∈L2(R)且∇·u0=0，J.Leray和E.Hopf[17，20](另见[21，30])构造了一个整体弱解u(t，x)∈L∞(0，∞；L2(R))∩L2(0，∞；H1(R))。众所周知，弱解在二维空间中是唯一且正则的[11，30]。然而，在维n≥3中，尽管数学家们做了很多努力，弱解的正则性问题仍然是悬而未决的。发展Navier-Stokes方程正则性理论有两种方法。一是给出弱解的正则性判据，二是研究已知存在的弱解的一个较好的局部正则性。对于后者，L.Caffarelli，R.Kohn，L.Nirenberg[4]证明了奇异集的一维Hausdorff测度为零，这是目前最好的结果。J.Serrin[24，25]是前者的开拓者，后来，Fabes，Jones和Riviere[13]，SHR[26]，Giga[16]，Struwe[27]和Takahashi[28]推广和改进了Serrin的正则性准则。为了说明他们的结果，我们首先回顾了Navier-Stokes方程弱解的定义。定义1.1。设u0∈L2(R)且∇·u0=0。函数u(t，x)称为(0，T)上的(NS)的弱解，如果u满足下列性质：
for vector functions u, v. For given u0 ∈ L2(R) with ∇ · u0 = 0, J. Leray and E. Hopf [17, 20](see also [21, 30]) constructed a global weak solution u(t, x) ∈ L∞(0,∞;L2(R)) ∩ L2(0,∞;H1(R)). It is well known that the weak solution is unique and regular in two spatial dimensions[11, 30]. In dimensions n ≥ 3, however, the question of regularity of weak solutions has remained open in spite of much effort of mathematicians. There are two ways to develop the regularity theory for the Navier-Stokes equations. One is to give a regularity criterion on weak solutions and the other is to study a better partial regularity of weak solutions which are known to exist. For the latter, L. Caffarelli, R. Kohn,L. Nirenberg[4] proved that the one dimensional Hausdorff measure of the singular set is zero, which is the best result at present. J. Serrin[24, 25] is the pioneer in the former research, later on, Fabes, Jones and Riviere[13], Sohr[26], Giga[16], Struwe[27] and Takahashi[28] extended and improved Serrin’s regularity criterion. In order to state their result, we first recall the definition of weak solutions to the Navier-Stokes equations. Definition 1.1. Let u0 ∈ L2(R) with ∇ · u0 = 0. The function u(t, x) will be called a weak solution of (NS) on (0, T ) if u satisfies the following properties: