Gradient flows of non convex functionals in Hilbert spaces and applications

Riccarda Rossi; Giuseppe Savaré

ESAIM: Control, Optimisation and Calculus of Variations (2006)

  • Volume: 12, Issue: 3, page 564-614
  • ISSN: 1292-8119

Abstract

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This paper addresses the Cauchy problem for the gradient flow equation in a Hilbert space  ℋ u ' ( t ) + ∂ ℓ φ ( u ( t ) ) ∋ f ( t ) a.e. in ( 0 , T ) , u ( 0 ) = u 0 , where φ : ℋ → ( - ∞ , + ∞ ] is a proper, lower semicontinuous functional which is not supposed to be a (smooth perturbation of a) convex functional and ∂ ℓ 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: in particular, we will focus on quasistationary models, which exhibit highly non convex Lyapunov functionals.

How to cite

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Rossi, Riccarda, and Savaré, Giuseppe. "Gradient flows of non convex functionals in Hilbert spaces and applications." ESAIM: Control, Optimisation and Calculus of Variations 12.3 (2006): 564-614. <http://eudml.org/doc/249707>.

@article{Rossi2006,
abstract = { This paper addresses the Cauchy problem for the gradient flow equation in a Hilbert space $\mathcal\{H\}$\[ \begin\{cases\} u'(t)+ \partial\_\{\ell\}\phi(u(t))\ni f(t) &\text\{\{\it a.e.\}\ in \}(0,T), u(0)=u\_0, \end\{cases\} \] where $\phi: \mathcal\{H\} \to (-\infty,+\infty]$ is a proper, lower semicontinuous functional which is not supposed to be a (smooth perturbation of a) convex functional and $\partial_\{\ell\}\phi$ 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: in particular, we will focus on quasistationary models, which exhibit highly non convex Lyapunov functionals. },
author = {Rossi, Riccarda, Savaré, Giuseppe},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Evolution problems; gradient flows; minimizing movements; Young measures; phase transitions; quasistationary models.; evolution problems; minimizing movements; phase transitions; quasistationary models},
language = {eng},
month = {6},
number = {3},
pages = {564-614},
publisher = {EDP Sciences},
title = {Gradient flows of non convex functionals in Hilbert spaces and applications},
url = {http://eudml.org/doc/249707},
volume = {12},
year = {2006},
}

TY - JOUR
AU - Rossi, Riccarda
AU - Savaré, Giuseppe
TI - Gradient flows of non convex functionals in Hilbert spaces and applications
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2006/6//
PB - EDP Sciences
VL - 12
IS - 3
SP - 564
EP - 614
AB - This paper addresses the Cauchy problem for the gradient flow equation in a Hilbert space $\mathcal{H}$\[ \begin{cases} u'(t)+ \partial_{\ell}\phi(u(t))\ni f(t) &\text{{\it a.e.}\ in }(0,T), u(0)=u_0, \end{cases} \] where $\phi: \mathcal{H} \to (-\infty,+\infty]$ is a proper, lower semicontinuous functional which is not supposed to be a (smooth perturbation of a) convex functional and $\partial_{\ell}\phi$ 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: in particular, we will focus on quasistationary models, which exhibit highly non convex Lyapunov functionals.
LA - eng
KW - Evolution problems; gradient flows; minimizing movements; Young measures; phase transitions; quasistationary models.; evolution problems; minimizing movements; phase transitions; quasistationary models
UR - http://eudml.org/doc/249707
ER -

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Citations in EuDML Documents

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  1. Antonio Segatti, Attrattori globali per alcuni problemi di evoluzione senza unicità
  2. Matteo Negri, Quasi-static rate-independent evolutions: characterization, existence, approximation and application to fracture mechanics
  3. Martin Heida, Existence of solutions for two types of generalized versions of the Cahn-Hilliard equation
  4. Alexander Mielke, Ulisse Stefanelli, Weighted energy-dissipation functionals for gradient flows
  5. Alexander Mielke, Riccarda Rossi, Giuseppe Savaré, BV solutions and viscosity approximations of rate-independent systems
  6. Riccarda Rossi, Alexander Mielke, Giuseppe Savaré, A metric approach to a class of doubly nonlinear evolution equations and applications
  7. Alexander Mielke, Ulisse Stefanelli, Weighted energy-dissipation functionals for gradient flows
  8. Alexander Mielke, Riccarda Rossi, Giuseppe Savaré, BV solutions and viscosity approximations of rate-independent systems

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