Implicit time discretization and global existence for a quasi-linear evolution equation with nonconvex energy

G. Friesecke; G. Dolzmann

doi:10.1137/S0036141095285958

Implicit time discretization and global existence for a quasi-linear evolution equation with nonconvex energy

G. Friesecke, G. Dolzmann

Research output: Contribution to journal › Article › peer-review

50 Scopus citations

Abstract

We establish global existence of weak solutions for the viscoelastic system u_tt = Div(∂Φ/∂F(Du) + Du_t) with nonconvex stored-energy function Φ. Unlike previous methods [P. Rybka, Proc. Roy. Soc. Edinburgh Sect. A, 121 (1992), pp. 101-138], our result does not require that ∂Φ/∂F be globally Lipschitz continuous. Our approach is based on implicit time discretization and a compactness property of the discrete dynamical scheme not shared by energy-minimizing sequences and not known to be shared by approximation schemes of Galerkin type.

Original language	English
Pages (from-to)	363-380
Number of pages	18
Journal	SIAM Journal on Mathematical Analysis
Volume	28
Issue number	2
DOIs	https://doi.org/10.1137/S0036141095285958
State	Published - Mar 1997
Externally published	Yes

Keywords

Evolution equations
Implicit time discretization
Nonconvex functionals
Solid-solid phase transitions
Viscoelasticity

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10.1137/S0036141095285958

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title = "Implicit time discretization and global existence for a quasi-linear evolution equation with nonconvex energy",

abstract = "We establish global existence of weak solutions for the viscoelastic system utt = Div(∂Φ/∂F(Du) + Dut) with nonconvex stored-energy function Φ. Unlike previous methods [P. Rybka, Proc. Roy. Soc. Edinburgh Sect. A, 121 (1992), pp. 101-138], our result does not require that ∂Φ/∂F be globally Lipschitz continuous. Our approach is based on implicit time discretization and a compactness property of the discrete dynamical scheme not shared by energy-minimizing sequences and not known to be shared by approximation schemes of Galerkin type.",

keywords = "Evolution equations, Implicit time discretization, Nonconvex functionals, Solid-solid phase transitions, Viscoelasticity",

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T1 - Implicit time discretization and global existence for a quasi-linear evolution equation with nonconvex energy

AU - Friesecke, G.

AU - Dolzmann, G.

PY - 1997/3

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N2 - We establish global existence of weak solutions for the viscoelastic system utt = Div(∂Φ/∂F(Du) + Dut) with nonconvex stored-energy function Φ. Unlike previous methods [P. Rybka, Proc. Roy. Soc. Edinburgh Sect. A, 121 (1992), pp. 101-138], our result does not require that ∂Φ/∂F be globally Lipschitz continuous. Our approach is based on implicit time discretization and a compactness property of the discrete dynamical scheme not shared by energy-minimizing sequences and not known to be shared by approximation schemes of Galerkin type.

AB - We establish global existence of weak solutions for the viscoelastic system utt = Div(∂Φ/∂F(Du) + Dut) with nonconvex stored-energy function Φ. Unlike previous methods [P. Rybka, Proc. Roy. Soc. Edinburgh Sect. A, 121 (1992), pp. 101-138], our result does not require that ∂Φ/∂F be globally Lipschitz continuous. Our approach is based on implicit time discretization and a compactness property of the discrete dynamical scheme not shared by energy-minimizing sequences and not known to be shared by approximation schemes of Galerkin type.

KW - Evolution equations

KW - Implicit time discretization

KW - Nonconvex functionals

KW - Solid-solid phase transitions

KW - Viscoelasticity

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EP - 380

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

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ER -

Implicit time discretization and global existence for a quasi-linear evolution equation with nonconvex energy

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Keywords

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