United theory of existence of global solution and its asymptotic behavior to the nonlinear partial differential equations

非线性偏微分方程全局解的存在性及其渐近行为的联合理论

• 批准号: 15104001
• 负责人: KOZONO Hideo
• 金额: $ 50.75万
• 依托单位: Tohoku University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (S)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-15104001/
• 关键词: Navier-Stokes Equations; Leray-Hopf Class; Very weak Solutions; Scaling invariance; Turbulent Solutions; Helmholtz-Weyl decomposition; Non-compact boundary; Stokes operator; ナビエ・ストークス方程式; コーシー問題; 渦度ベクトル; 平均振動; エネルギー不等式; ベゾフ空間; 爆発解; 大域解; 調和

1. Constructions of very weak solutions of the Navier-Stokes equations in exterior domains.We show the unique existence of local very weak solutions to the prescribed non-homogeneous boundary data which belong to the larger class than the usual trace class. Our solutions satisfy the Serrin condition which implies the scaling invariant class.2. New regularity criterion on weak solutions of the Navier-Stokes equations.We prove that every turbulent solution which is α-Hoelder continuous in the kinetic energy in the time interval with α>1/2 necessarily regular.3. Helmholtz-Weyl de composition in unbounded domains with non-compact boundaries of uniformly C^2-class.Despite of a counter example of valiclity of the Helmholtz-Weyl decomposition in L^r, we introduce the space of sum and intersection of L^r and prove the Helmholtz-Weyl decomposition in such spaces. As an application, we can define the Stokes operator.

1.在外部区域上构造Navier-Stokes方程的很弱解.对于给定的非齐次边界数据，我们证明了局部很弱解的唯一存在性，这些解属于比通常的迹类更大的一类.我们的解决方案满足Serrin条件，这意味着标度不变类。给出了Navier-Stokes方程弱解的新的正则性判据，证明了在α>1/2的时间区间内，在动能上α-Hoelder连续的湍流解必然是正则的.本文讨论了具有一致C^2-类非紧边界的无界域上的Helmholtz-Weyl de复合，给出了Helmholtz-Weyl分解在L^r中有效性的反例，引入了L^r的和交空间，证明了Helmholtz-Weyl分解在这类空间中的有效性。作为应用，我们可以定义Stokes算子。

项目成果

A Liouville property and quasiconvergence for a semilinear heat equation

• DOI: 10.1016/j.jde.2003.10.019
• 发表时间: 2005
• 期刊: Journal of Differential Equations
• 影响因子: 2.4
• 作者: P. Polácik;E. Yanagida

Damped wave equation with a critical nonlinearity

• DOI: 10.1090/s0002-9947-05-03818-3
• 发表时间: 2005-04
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: N. Hayashi;E. Kaikina;P. Naumkin

Method of the distance function to the Bence-Merriman-Osher algorithm for motion by mean curvature

Bence-Merriman-Osher 平均曲率运动算法的距离函数方法

• 发表时间: 2005
• 期刊: Communications on Pure and Applied Analysis 4・2
• 作者: Yoko GOTO;Katsuyuki ISHII;Takayoshi OGAWA
• 通讯作者: Takayoshi OGAWA

Kozono, H., Yatsu, N.: "Extension criterion via two-components of vorticity on strong solutions to the 3D Navier-Stokes equations"Math.Z.. 246. 55-68 (2004)

Kozono, H.、Yatsu, N.：“通过涡量的两个分量对 3D 纳维-斯托克斯方程的强解进行扩展准则”Math.Z.. 246. 55-68 (2004)

Bilinear estimates in homogeneous Triebel‐Lizorkin spaces and the Navier‐Stokes equations

• DOI: 10.1002/mana.200310213
• 发表时间: 2004-10
• 期刊: Mathematische Nachrichten
• 影响因子: 1
• 作者: H. Kozono;Yukihiro Shimada