# 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

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topRossi, 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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- Martin Heida, Existence of solutions for two types of generalized versions of the Cahn-Hilliard equation
- Alexander Mielke, Ulisse Stefanelli, Weighted energy-dissipation functionals for gradient flows
- Alexander Mielke, Riccarda Rossi, Giuseppe Savaré, BV solutions and viscosity approximations of rate-independent systems
- Riccarda Rossi, Alexander Mielke, Giuseppe Savaré, A metric approach to a class of doubly nonlinear evolution equations and applications
- Alexander Mielke, Riccarda Rossi, Giuseppe Savaré, BV solutions and viscosity approximations of rate-independent systems
- Alexander Mielke, Ulisse Stefanelli, Weighted energy-dissipation functionals for gradient flows

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