Second order approximations of quasistatic evolution problems in finite dimension

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April 2012, 32(4): 1125-1167. doi: 10.3934/dcds.2012.32.1125

Second order approximations of quasistatic evolution problems in finite dimension

1.	via Bonomea 265, 34136 Trieste, Italy

Received November 2010 Revised April 2011 Published October 2011

Abstract
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In this paper, we study the limit, as $\epsilon$ goes to zero, of a particular solution of the equation $\epsilon^2A\ddot u^{\epsilon}(t)+\epsilon B\dot u^{\epsilon}(t)+\nabla_xf(t,u^{\epsilon}(t))=0$, where $f(t,x)$ is a potential satisfying suitable coerciveness conditions. The limit $u(t)$ of $u^{\epsilon}(t)$ is piece-wise continuous and verifies $\nabla_xf(t,u(t))=0$. Moreover, certain jump conditions characterize the behaviour of $u(t)$ at the discontinuity times. The same limit behaviour is obtained by considering a different approximation scheme based on time discretization and on the solutions of suitable autonomous systems.

Keywords: heteroclinic solutions., discrete approximations, saddle-node bifurcation, perturbation methods, vanishing viscosity, Singular perturbations.

Mathematics Subject Classification: Primary: 74H10, 34K18, 34C37; Secondary: 34K2.

Citation: Virginia Agostiniani. Second order approximations of quasistatic evolution problems in finite dimension. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1125-1167. doi: 10.3934/dcds.2012.32.1125

References:

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[9]	M. W. Hirsch, "Differential Topology,", Graduate Texts in Mathematics, 33 (1976). Google Scholar
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[11]	A. Mielke, R. Rossi and G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces,, Discrete Contin. Dyn. Syst., 25 (2009), 585. doi: 10.3934/dcds.2009.25.585. Google Scholar
[12]	A. Mielke, R. Rossi and G. Savaré, BV solutions and viscosity approximations of rate-independent systems,, ESAIM Control Optim. Calc. Var., (2011). doi: 10.1051/cocv/2010054. Google Scholar
[13]	F. Solombrino, Quasistatic evolution for plasticity with softening: the spatially homogeneous case,, Discrete Contin. Dyn. Syst., 27 (2010), 1189. doi: 10.3934/dcds.2010.27.1189. Google Scholar
[14]	R. Toader and C. Zanini, An artificial viscosity approach to quasistatic crack growth,, Boll. Unione Mat. Ital. (9), 2 (2009), 1. Google Scholar
[15]	C. Zanini, Singular perturbation of finite dimensional gradient flows,, Discrete Contin. Dyn. Syst., 18 (2007), 657. doi: 10.3934/dcds.2007.18.657. Google Scholar

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References:

[1]	F. Cagnetti, A vanishing viscosity approach to fracture growth in a cohesive zone model with prescribed crack path,, Math. Models Methods Appl. Sci., 18 (2008), 1027. doi: 10.1142/S0218202508002942. Google Scholar
[2]	G. Dal Maso, A. DeSimone, M. G. Mora and M. Morini, A vanishing viscosity approach to quasistatic evolution in plasticity with softening,, Arch. Ration. Mech. Anal., 189 (2008), 469. doi: 10.1007/s00205-008-0117-5. Google Scholar
[3]	G. Dal Maso, A. DeSimone and F. Solombrino, Quasistatic evolution for Cam-Clay plasticity: a weak formulation via viscoplastic regularization and time rescaling,, Calc. Var. Partial Differential Equations, 40 (2011), 125. Google Scholar
[4]	G. Dal Maso, A. DeSimone and F. Solombrino, Quasistatic evolution for Cam-Clay plasticity: properties of the viscosity solution,, SISSA preprint 46/2010/M., (). Google Scholar
[5]	G. Dal Maso and F. Solombrino, Quasistatic evolution for Cam-Clay plasticity: the spatially homogeneous case,, Netw. Heterog. Media, 5 (2010), 97. Google Scholar
[6]	M. A. Efendiev and A. Mielke, On the rate-independent limit of systems with dry friction and small viscosity,, J. Convex Anal., 13 (2006), 151. Google Scholar
[7]	J. Guckenheimer and P. Holme, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations on Vector Fields,", Applied Mathematical Sciences, 42 (1983). Google Scholar
[8]	J. K. Hale, "Ordinary Differential Equations,", Pure and Applied Mathematics, XX1 (1980). Google Scholar
[9]	M. W. Hirsch, "Differential Topology,", Graduate Texts in Mathematics, 33 (1976). Google Scholar
[10]	D. Knees, A. Mielke and C. Zanini, Crack growth in polyconvex materials,, Phys. D, 239 (2010), 1470. doi: 10.1016/j.physd.2009.02.008. Google Scholar
[11]	A. Mielke, R. Rossi and G. Savaré, Modeling solutions with jumps for rate-independent systems on metric spaces,, Discrete Contin. Dyn. Syst., 25 (2009), 585. doi: 10.3934/dcds.2009.25.585. Google Scholar
[12]	A. Mielke, R. Rossi and G. Savaré, BV solutions and viscosity approximations of rate-independent systems,, ESAIM Control Optim. Calc. Var., (2011). doi: 10.1051/cocv/2010054. Google Scholar
[13]	F. Solombrino, Quasistatic evolution for plasticity with softening: the spatially homogeneous case,, Discrete Contin. Dyn. Syst., 27 (2010), 1189. doi: 10.3934/dcds.2010.27.1189. Google Scholar
[14]	R. Toader and C. Zanini, An artificial viscosity approach to quasistatic crack growth,, Boll. Unione Mat. Ital. (9), 2 (2009), 1. Google Scholar
[15]	C. Zanini, Singular perturbation of finite dimensional gradient flows,, Discrete Contin. Dyn. Syst., 18 (2007), 657. doi: 10.3934/dcds.2007.18.657. Google Scholar

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