Two-Speed Solutions to Non-convex Rate-Independent Systems

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F21%3A10441308" target="_blank" >RIV/00216208:11320/21:10441308 - isvavai.cz</a>

Výsledek na webu

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=1JfHAdHoYn" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=1JfHAdHoYn</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s00205-020-01599-z" target="_blank" >10.1007/s00205-020-01599-z</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Two-Speed Solutions to Non-convex Rate-Independent Systems

Popis výsledku v původním jazyce

We consider evolutionary PDE inclusions of the form -λu.λ+Δu-DW0(u)+fCONTAINS AS MEMBERPARTIAL DIFFERENTIALR1(u.)in(0,T)xΩ,where R1 is a positively 1-homogeneous rate-independent dissipation potential and W is a (generally) non-convex energy density. This work constructs solutions to the above system in the slow-loading limit λDOWNWARDS ARROW 0. Our solutions have more regularity both in space and time than those that have been obtained with other approaches. On the &quot;slow&quot; time scale we see strong solutions to a purely rate-independent evolution. Over the jumps, we obtain a detailed description of the behavior of the solution and we resolve the jump transients at a &quot;fast&quot; time scale, where the original rate-dependent evolution is still visible. Crucially, every jump transient splits into a (possibly countable) number of rate-dependent evolutions, for which the energy dissipation can be explicitly computed. This, in particular, yields a global energy equality for the whole evolution process. It also turns out that there is a canonical slow time scale that avoids intermediate-scale effects, where movement occurs in a mixed rate-dependent/rate-independent way. In this way, we obtain precise information on the impact of the approximation on the constructed solution. Our results are illustrated by examples, which elucidate the effects that can occur.

Název v anglickém jazyce

Two-Speed Solutions to Non-convex Rate-Independent Systems

Popis výsledku anglicky

We consider evolutionary PDE inclusions of the form -λu.λ+Δu-DW0(u)+fCONTAINS AS MEMBERPARTIAL DIFFERENTIALR1(u.)in(0,T)xΩ,where R1 is a positively 1-homogeneous rate-independent dissipation potential and W is a (generally) non-convex energy density. This work constructs solutions to the above system in the slow-loading limit λDOWNWARDS ARROW 0. Our solutions have more regularity both in space and time than those that have been obtained with other approaches. On the &quot;slow&quot; time scale we see strong solutions to a purely rate-independent evolution. Over the jumps, we obtain a detailed description of the behavior of the solution and we resolve the jump transients at a &quot;fast&quot; time scale, where the original rate-dependent evolution is still visible. Crucially, every jump transient splits into a (possibly countable) number of rate-dependent evolutions, for which the energy dissipation can be explicitly computed. This, in particular, yields a global energy equality for the whole evolution process. It also turns out that there is a canonical slow time scale that avoids intermediate-scale effects, where movement occurs in a mixed rate-dependent/rate-independent way. In this way, we obtain precise information on the impact of the approximation on the constructed solution. Our results are illustrated by examples, which elucidate the effects that can occur.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2021

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Archive for Rational Mechanics and Analysis

ISSN

0003-9527

e-ISSN

—

Svazek periodika

239

Číslo periodika v rámci svazku

1

Stát vydavatele periodika

DE - Spolková republika Německo

Počet stran výsledku

65

Strana od-do

1667-1731

Kód UT WoS článku

000606170800002

EID výsledku v databázi Scopus

2-s2.0-85098972134