非自治单调次梯度发展方程的渐近性
Asymptotic behavior of nonautonomous monotone and subgradient evolution equations
摘要：
在Hilbert背景下，研究了非自治发展方程$\dot x(t)+A_t(x(t))\ni 0$的轨道的渐近性态，其中对于每个$t\geq 0$，$A_t:H\tto H$表示一个极大单调算子.我们提供了一般的条件，保证弱遍历收敛的每个轨迹$x(\cdot)$的极限极大单调算子$ A_\infty$的零，作为时间变量$t$趋于$+\infty$。关键点是使用Brézis-Haraux函数，或者等价的Fitzpatrick函数，来表示在什么速率下$\gph A_\infty$超过$\gph A_t$趋于零。这种方法对这个问题提出了一个尖锐而统一的观点。在这种情况下，运营商$A_t= \partial ϕ_t$这是次微分的闭凸函数$ϕ_t$，我们显示的轨迹的收敛结果。然后，我们将所得结果推广到多尺度发展方程，得到了分层极小化的渐近性质和粘性解的选择。在耦合系统和偏微分方程领域给出了例证。
英文摘要：
In a Hilbert setting $H$, we study the asymptotic behavior of the trajectories of nonautonomous evolution equations $\dot x(t)+A_t(x(t))\ni 0$, where for each $t\geq 0$, $A_t:H\tto H$ denotes a maximal monotone operator. We provide general conditions guaranteeing the weak ergodic convergence of each trajectory $x(\cdot)$ to a zero of a limit maximal monotone operator $ A_\infty$, as the time variable $t$ tends to $+\infty$. The crucial point is to use the Brézis-Haraux function, or equivalently the Fitzpatrick function, to express at which rate the excess of $\gph A_\infty$ over $\gph A_t$ tends to zero. This approach gives a sharp and unifying view on this subject. In the case of operators $A_t= \partial ϕ_t$ which are subdifferentials of closed convex functions $ϕ_t$, we show convergence results for the trajectories. Then, we specialize our results to multiscale evolution equations, and obtain asymptotic properties of hierarchical minimization, and selection of viscosity solutions. Illustrations are given in the field of coupled systems, and partial differential equations.
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