Discrete approximations and stability properties of inertial manifolds for dissipative differential equations

耗散微分方程惯性流形的离散近似和稳定性性质

• 批准号: 10640217
• 负责人: KOBAYASHI Kazuo
• 金额: $ 0.32万
• 依托单位: Waseda University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 无数据
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640217/
• 关键词: inertial manifold; dissipative PDE; evolution equation; finite difference; 差分近似; 構造安定性

We have gotten the following results.1) We proved the existence and uniqueness of renormalized solution to the degenerate quasilinear elliptic equation with the convection termβ(u) - div{a (x, ∇u) + h (u) } ∋ f (x ∈ Ω), u (x) = 0 (x ∈ ∂Ω) .Here, β is a maximal monotone graph in R, h : R → RィイD1dィエD1 is a continuous function, a : Ω×RィイD1dィエD1 → RィイD1dィエD1 is a Caratheodory function satisfying an appropriate monotone and coercive condition, and f ∈ LィイD11ィエD1 (Ω) . Since this problem is not sufficiently known in case Ω is unbounded, we gave a result of it.2) We proved the existence, smoothness and CィイD11ィエD1-approximation of the inertial manifolds for the evolution equation (E) : du/dt = Au + F (t, u) in Banach spaces, and applied the results to studying the asymptotic behavior of Kuramoto-Sivashinsky equation. Namely, if AィイD2nィエD2 → A, FィイD2nィエD2 → F and DFィイD2nィエD2 → DF, then the inertial manifold for the approximate equation (EィイD2nィエD2) : du/dt = AィイD2nィエD2u + FィイD2nィエD2 (t,u) , converges to that of (E).3) By using the above results about initial manifolds, we improved the theorems of the local center unstable manifolds for the evolution equation (E) . Moreover, we try to apply these theorems to the study of the asymptotics of the nonlinear heat equation UィイD2tィエD2 = Δu + F (u, ∇u) on unbounded domains.

我们得到了如下结果：1）证明了带对流项的退化拟线性椭圆方程β（u）- div{a（x，u）+ h（u）}<$f（x ∈ Ω），u（x）= 0（x ∈ <$Ω）重整化解的存在唯一性，其中β是R中的极大单调图，h：R → R <$D 1 d <$D 1是连续函数，a：Ω×R D1 d D1 → R D1 d D1是一个满足适当单调和强制条件的Caratheodory函数，f ∈ L D11 D1（Ω）. 2）在Banach空间中证明了发展方程（E）：du/dt = Au + F（t，u）的惯性流形的存在性、光滑性和C_∞ D_（11）_∞ D_（11）-逼近性，并应用这些结果研究了Kuramoto-Sivashinsky方程的渐近性态.也就是说，如果A D2 n D2 → A，F D2 n D2 → F和DF D2 n D2 → DF，则近似方程的惯性流形（E D2n D2）：du/dt = A ∈ D2 n ∈ D2 u + F ∈ D2 n ∈ D2（t，u），收敛于（E）的结果。3）利用上述关于初始流形的结果，本文改进了发展方程（E）的局部中心不稳定流形定理。此外，我们还尝试将这些定理应用于无界区域上非线性热方程U_u_dD_2 = Δu + F（u，u_dD_2）的渐近性的研究.

项目成果

小林和夫: "C^1 approximations of inertial manifolds via finite differences"Proceedings of Amer.Math.Soc.. 127,4. 1143-1150 (1999)

Kazuo Kobayashi：“通过有限差分的惯性流形的 C^1 近似”Proceedings of Amer.Math.Soc.. 127,4 (1999)。

Kazuo Kobayasi: "Equivalent conditions between renormalized solutions and entropy solutions"Gakujutsu Kenkyu, School of Education, Waseda Univ. Vol.46. 17-23 (1998)

小林和夫：“重正化解与熵解之间的等效条件”Gakujutsu Kenkyu，早稻田大学教育学院。

小林和夫: "C' approximations of inertial manifolds via finite differences and applications" 京都大学数理解析研究所講究録.

Kazuo Kobayashi：“通过有限差分进行惯性流形的 C 近似及其应用”Kokyuroku，京都大学数学科学研究所。

Kazuo Kobayasi: "Existence of renormalized solutions of degenerate quasilinear elliptic equations"早稲田大学教育学部学術研究-数学論-. 47. 29-46 (1999)

小林和夫：“简并拟线性椭圆方程重正化解的存在性”早稻田大学教育学部学术研究-数学理论-47。29-46（1999）。

Kazuo Kobayasi: "C^1 approximations of inertial manifolds via finite differences"Proc. Amer. Math. Soc.. 127. 1145-1150 (1999)

Kazuo Kobayasi：“通过有限差分的惯性流形的 C^1 近似”Proc。