THE COMPLEX GINZBURG-LANDAU EQUATION

复杂的 GINZBURG-LANDAU 方程

• 批准号: 11640185
• 负责人: OKAZAWA Noboru
• 金额: $ 0.9万
• 依托单位: SCIENCE UNIVERSITY OF TOKYO
• 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 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640185/
• 关键词: THE COMPLEX GINZBURG-LANDAU EQUATION; INITIAL-BOUNDARY VALUE PROBLEMS; GLOBAL IN TIME SOLVABILITY; NONLINEAR EVOLUTION EQUATIONS; MAXIMAL MONOTONE OPERATORS; NONLINEAR SEMIGROUPS; PROPER LOWER SEMI-CONTINUOUS CONVEX FUNCTIONS; SUBDIFFERENTIAL OPERA

We have considered initial value problem for abstract evolution equations of the formdu/dt + (λ+ iα)∂φ(u) + (κ + iβ)∂ψ(u) - γu = 0, t>0 ; u(0) =u_0,where ∂φ, ∂ψ are subdifferentials of convex functions φ, ψ on a Hilbert space X.Let Ω ⊂ R^N be a bounded domain. Setting X : = L^2(Ω), φ (u) : = (1/p)‖∇u‖^p_<L^p>, u ∈ W^<1, p>_0(Ω) ; ψ(v) : =(1/q)‖v‖^q_<L^q>, v ∈ L^q(Ω), we have ∂ψ(u) = |u|^<q-2>u, u ∈ D(∂ψ) : = L^<2(q-1)>(Ω), ∂φ(u) = -Δ_pu : =-div(|∇u|^<p-2>∇u), u ∈ D(∂φ) : = {u ∈ W^<1, p>_0(Ω) ; Δ_pu∈ L^2(Ω)} and hence(CGL)_p ∂u/∂t - (λ+iα)・Δ_pu + (κ+iβ)|u|^<q-2>u - γu = 0, t>0 ; u(O) : u_0.Here we have assumed that p 【greater than or equal】 2, q 【greater than or equal】 2. We have revealed that the global in time solvability of (CGL)_p is classified as follows according to the pair (α/λ, β/κ) ∈ R^2 and the initial value. In any case we assume that |α|/λ 【less than or equal】 1/c_p : = 2√<p-1>/(p-2).1. Existence of weak solutions for any (α/λ, β/κ), u_0 ∈ L^2(Ω)(no uniqueness, in general)2. Existence of strong solutions for (α/λ, β/κ) belonging to "(CGL) region" : = {(x, y) ∈ R^2 ; xy 【greater than or equal】 0 or |xy| - 1 < (|x| + |y|)/c_q} and u_0 ∈ W^<1, p>_0(Ω) ∩ L^q(Ω) (no uniqueness, in general)3. Unique existence of strong solutions for (α/λ, β/κ) with |β|/κ 【less than or equal】 1/c_q and u_0 ∈ L^2(Ω) (smoothing effect of solution operators).

我们考虑了形式为Du/dt+(λ+Iα)∂φ(U)+(κ+Iβ)∂ψ(U)-γu=0，t&gt；0；u(0)=u_0)的抽象发展方程的初值问题，其中∂φ，∂ψ是φ，ψ在Hilbert空间X上的次微分。设Ω⊂R^N是有界域。设置X：=L^2(Ω)，φ(U)：=(1/p)‖∇u‖^p_&lt；L^p&gt；，u∈W^&lt；1，p&gt；_0(Ω)；ψ(V)：=(1/q)‖v‖^q&lt；L^q&gt；，v∈L^q(Ω)，我们有∂ψ(U)=|u|^&lt；q-2&gt；u，u∈D(∂ψ)：=L^&lt；2(q-1)&gt；(Ω)，∂φ(U)=-Δ_PU：=-div(|∇u|^&lt；p-2&gt；∇u)，u∈D(∂φ)：={u∈W^&lt；1，p&gt；_0(Ω)；Δ_PU∈L^2(Ω)}因此(Cgl)_p∂u/∂t-(λ+iα)·Δ_PU+(κ+iβ)|u|^&lt；q-2&gt；u-γu=0，t&gt；0；U(O)：u_0，这里我们假设p[大于或等于]2，q[大于或等于]2。我们揭示了(Cgl)_p的全局时间可解性根据对(α/λ，β/κ)∈R^2和初值)被分类如下。在任何情况下，我们都假设|α|/λ[小于或等于]1/c_p：=2√&lt；p-1&gt；/(p-2).任意(α/λ，β/κ)，u_0∈L^2(Ω)(一般无唯一性)的弱解的存在性2.属于“(Cgl)区域”的(α/λ，β/κ)的强解的存在性：={(x，y)∈R^2；xy[大于或等于]0或|xy|-1&lt；(|x|+|y|)/c_q}和u_0∈W^&lt；1，p&gt；_0(Ω)∩L^q(Ω)(一般不存在唯一性)3.具有|α/λ，β/κ[小于或等于]1/c_q且u_0β|/κL^2(∈)的(Ω)强解的唯一存在性(解算子的光滑化效应)。

项目成果

N.Okazawa and T.Yokota: "Monotonicity method for the complox Ginzburg-Landau equation, including smoothing effect"Journal of Nonlinear Analysis : Series A Theory and Methods. special issue. (2001)

N.Okazawa 和 T.Yokota：“复数 Ginzburg-Landau 方程的单调性方法，包括平滑效应”非线性分析杂志：A 系列理论和方法。

N.Okazawa and T.Yokota: "Smoothing effect for generalized complex Ginzburg-Landau equations in unbounded domains"Discrete and Continuous Dynamical Systems. Added Volume. 280-288 (2001)

N.Okazawa 和 T.Yokota：“无界域中广义复杂 Ginzburg-Landau 方程的平滑效应”离散和连续动力系统。