Existence and approximation results for gradient flows

Riccarda Rossi; Giuseppe Savaré

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2004)

  • Volume: 15, Issue: 3-4, page 183-196
  • ISSN: 1120-6330

Abstract

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This note addresses the Cauchy problem for the gradient flow equation in a Hilbert space H u ' ( t ) + ∂ φ ( u ( t ) ) ∋ 0 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 ∂ φ is (a suitable limiting version of) its subdifferential. The interest for this kind of equations is motivated by a number of examples, which show that several mathematical models describing phase transitions phenomena and leading to systems of evolutionary PDEs have a common gradient flow structure. In particular, when quasi-stationary models are considered, highly non-convex functionals naturally arise. We will present some existence results for the solution of the gradient flow equation by exploiting a variational approximation technique, featuring some ideas from the theory of Minimizing Movements.

How to cite

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Rossi, Riccarda, and Savaré, Giuseppe. "Existence and approximation results for gradient flows." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 15.3-4 (2004): 183-196. <http://eudml.org/doc/252318>.

@article{Rossi2004,
abstract = {This note addresses the Cauchy problem for the gradient flow equation in a Hilbert space $H$\[ \{\left\lbrace \begin\{array\}\{ll\} u^\{\prime \}(t) + \partial \phi (u(t)) \ni 0 \quad \text\{a.e. in\} \, (0,T),\\ u(0) = u\_\{0\}, \end\{array\}\right.\} \] where $\phi : H \rightarrow (-\infty , +\infty \,]$ is a proper, lower semicontinuous functional which is not supposed to be a (smooth perturbation of a) convex functional and $\partial \phi $ is (a suitable limiting version of) its subdifferential. The interest for this kind of equations is motivated by a number of examples, which show that several mathematical models describing phase transitions phenomena and leading to systems of evolutionary PDEs have a common gradient flow structure. In particular, when quasi-stationary models are considered, highly non-convex functionals naturally arise. We will present some existence results for the solution of the gradient flow equation by exploiting a variational approximation technique, featuring some ideas from the theory of Minimizing Movements.},
author = {Rossi, Riccarda, Savaré, Giuseppe},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Phase transitions; Evolution problems; Gradient flows; Minimizing Movements; Minimizing movements},
language = {eng},
month = {12},
number = {3-4},
pages = {183-196},
publisher = {Accademia Nazionale dei Lincei},
title = {Existence and approximation results for gradient flows},
url = {http://eudml.org/doc/252318},
volume = {15},
year = {2004},
}

TY - JOUR
AU - Rossi, Riccarda
AU - Savaré, Giuseppe
TI - Existence and approximation results for gradient flows
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2004/12//
PB - Accademia Nazionale dei Lincei
VL - 15
IS - 3-4
SP - 183
EP - 196
AB - This note addresses the Cauchy problem for the gradient flow equation in a Hilbert space $H$\[ {\left\lbrace \begin{array}{ll} u^{\prime }(t) + \partial \phi (u(t)) \ni 0 \quad \text{a.e. in} \, (0,T),\\ u(0) = u_{0}, \end{array}\right.} \] where $\phi : H \rightarrow (-\infty , +\infty \,]$ is a proper, lower semicontinuous functional which is not supposed to be a (smooth perturbation of a) convex functional and $\partial \phi $ is (a suitable limiting version of) its subdifferential. The interest for this kind of equations is motivated by a number of examples, which show that several mathematical models describing phase transitions phenomena and leading to systems of evolutionary PDEs have a common gradient flow structure. In particular, when quasi-stationary models are considered, highly non-convex functionals naturally arise. We will present some existence results for the solution of the gradient flow equation by exploiting a variational approximation technique, featuring some ideas from the theory of Minimizing Movements.
LA - eng
KW - Phase transitions; Evolution problems; Gradient flows; Minimizing Movements; Minimizing movements
UR - http://eudml.org/doc/252318
ER -

References

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