Gradient flows of non convex functionals in Hilbert spaces and applications
Gradient flows of non convex functionals in Hilbert spaces and applications
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希尔伯特空间中非凸泛函的梯度流及其应用
DOI：
10.1051/cocv:2006013
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
Giuseppe Savaré
中科院分区：
文献类型：
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作者：
Riccarda Rossi;Giuseppe Savaré
本文讨论了Hilbert空间H � u �（t）+H�φ（u（t））� f（t）a.e.中梯度流方程的Cauchy问题.在（0，T）中，u（0）= u0，其中φ：H →（−∞，+∞）是一个正常的下连续泛函，它不应该是凸泛函的光滑扰动，φ φ是它的次微分.我们将提出一些新的存在性结果的方程的解决方案，利用变分逼近技术，具有一些想法，从理论的最小化运动和杨措施。我们的分析也是出于一些模型描述相变现象，导致系统的进化偏微分方程有一个共同的底层梯度流结构：特别是，我们将专注于准平稳模型，表现出高度非凸的李雅普诺夫泛函。
This paper addresses the Cauchy problem for the gradient flow equation in a Hilbert space H � u � (t )+ ∂φ(u(t)) � f (t) a.e. in (0 ,T ), u(0) = u0, where φ : H → (−∞, +∞) is a proper, lower semicontinuous functional which is not supposed to be a (smooth perturbation of a) convex functional and ∂φ is (a suitable limiting version of) its subdifferential. We will present some new existence results for the solutions of the equation by exploiting a variational approximation technique, featuring some ideas from the theory of Minimizing Movements and of Young measures. Our analysis is also motivated by some models describing phase transitions phenomena, leading to systems of evolutionary PDEs which have a common underlying gradient flow structure :i n particular, we will focus on quasistationary models, which exhibit highly non convex Lyapunov functionals.