Theory of global well-posedness on the nonlinear partial differential equations

非线性偏微分方程的全局适定性理论

• 批准号: 20224013
• 负责人: KOZONO Hideo
• 金额: $ 93.52万
• 依托单位: Waseda University (2011-2012)Tohoku University (2009-2010)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (S)
• 财政年份: 2008
• 资助国家: 日本
• 起止时间: 2008 至 2012
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-20224013/
• 关键词: 非線形解析学; 偏微分方程式論; 調和解析学; 関数解析学; Keller-Segel 方程式系; Navier-Stokes 方程式; 弱解の一意性; 時間大域的弱解; L＾p-L＾r-型評価; 最大正則性定理; 関数方程式; 非線形偏微分方程式; keller-Segel系; タイプI型爆発解; タイプII型爆発解; 自己相似解; 局所存在定理; Navier-Stokes方程式; 強エネルギー不等式; 非斉次境界値問題

We investigate the local existence of strong solutions and their blow-up within a finite time in arbitrary dimensional domains. The life-span of local solutions is characterized in terms of the L^1 and L^p-norms of the given initial data. Simultaneously, it is clarified that the total mass and the second momentum of the initial data together with the coefficient of the system of equations have a great influence on the blow-up phenomena. As an application, we prove that the blow-up solution either exhibits a definite blow-up rate determined by p, or oscillates in L^1 with the larger amplitude than the absolute constant. Furthermore, in multi-connected domains, it is still an open question whether there does exist a solution of the stationary Navier-Stoeks equations with the inhomogeneous boundary data whose total flux is zero. The relation between the nonlinear structure of the equations and the topological invariance of the domain plays an important role for the solvability of this problem. We prove that if the harmonic part of solenoidal extensions of the given boundary data associated with the second Betti number of the domain is orthogonal to non-trivial solutions of the Euler equations, then there exists a solution for any viscosity constant. The relation between Leary's inequality and the topological type of the domain is also clarified.

研究了任意维域上强解在有限时间内的局部存在性及其爆破性。局部解的寿命用给定初始数据的L^1和L^p范数来表示。同时阐明了初始数据的总质量和二次动量以及方程组的系数对爆破现象有很大的影响。作为一个应用，我们证明了爆破解要么表现出由p决定的确定的爆破率，要么在L^1中以比绝对常数更大的振幅振荡。此外，在多连通域中，是否存在总通量为零的非齐次边界数据的平稳Navier-Stoeks方程的解仍然是一个悬而未决的问题。方程的非线性结构与区域拓扑不变性之间的关系对该问题的可解性起着重要的作用。我们证明了如果给定边界数据与区域的第2 Betti数相关的螺线线扩展的谐波部分与欧拉方程的非平凡解正交，则对于任何黏度常数都存在解。明确了Leary不等式与域拓扑类型的关系。

项目成果

Blow-up for a semilinear parabolic equation with large diffusion on R^n

R^n 上具有大扩散的半线性抛物线方程的放大

• 发表时间: 2009
• 作者: H.Takahashi;M.Takasaki 他;中野 敏行;岩下芳久;Kazuhiro Ishige
• 通讯作者: Kazuhiro Ishige

Global DIV-CURL Lemma in bounded domains in R^3

R^3 有界域中的全局 DIV-CURL 引理

• 发表时间: 2009
• 期刊: Jour.Funct. Anal 256
• 作者: Kozono;H;Yanagisawa;T
• 通讯作者: T

Generalized Lax-Milgram theorem in Banach spaces and its application to the elliptic system of boundary value problems

• DOI: 10.1007/s00229-012-0586-6
• 发表时间: 2013-07
• 期刊: Manuscripta Mathematica
• 影响因子: 0.6
• 作者: H. Kozono;T. Yanagisawa

$L^r$-variational inequality for vector fields and the Helmholtz-Weyl decomposition in bounded domains

• DOI: 10.1512/iumj.2009.58.3605
• 发表时间: 2009-10
• 期刊: Indiana University Mathematics Journal
• 影响因子: 1.1
• 作者: H. Kozono;T. Yanagisawa

Global Div-Curl lemma on bounded domains in R3

• DOI: 10.1016/j.jfa.2009.01.010
• 发表时间: 2009-06
• 期刊: Journal of Functional Analysis
• 影响因子: 1.7
• 作者: H. Kozono;T. Yanagisawa