Degenerate regularization of forward-backward parabolic equations: the vanishing viscosity limit

2012 ◽  
Vol 355 (2) ◽  
pp. 551-584 ◽  
Author(s):  
Flavia Smarrazzo ◽  
Alberto Tesei
Keyword(s):  
Parabolic Equations ◽  
Vanishing Viscosity ◽  
Vanishing Viscosity Limit ◽  
Viscosity Limit ◽  
Backward Parabolic Equations

scholarly journals A unilateral L2L^{2}-gradient flow and its quasi-static limit in phase-field fracture by an alternate minimizing movement

2019 ◽  
Vol 12 (1) ◽  
pp. 1-29 ◽  
Author(s):  
Matteo Negri
Keyword(s):  
Energy Balance ◽  
Phase Field ◽  
Gradient Flow ◽  
Field Variable ◽  
Discrete Solution ◽  
Vanishing Viscosity ◽  
Vanishing Viscosity Limit ◽  
Static Limit ◽  
Viscosity Limit ◽  
Minimization Scheme

AbstractWe consider an evolution in phase-field fracture which combines, in a system of PDEs, an irreversible gradient-flow for the phase-field variable with the equilibrium equation for the displacement field. We introduce a discretization in time and define a discrete solution by means of a 1-step alternate minimization scheme, with a quadratic {L^{2}}-penalty in the phase-field variable (i.e. an alternate minimizing movement). First, we prove that discrete solutions converge to a solution of our system of PDEs. Then we show that the vanishing viscosity limit is a quasi-static (parametrized) BV-evolution. All these solutions are described both in terms of energy balance and, equivalently, by PDEs within the natural framework of {W^{1,2}(0,T;L^{2})}.

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