American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control
  • DCDS Home
  • This Issue
  • Next Article
    A convex-concave elliptic problem with a parameter on the boundary condition
April  2012, 32(4): 1125-1167. doi: 10.3934/dcds.2012.32.1125

Second order approximations of quasistatic evolution problems in finite dimension

Virginia Agostiniani 1,

1. 

via Bonomea 265, 34136 Trieste, Italy

Received  November 2010 Revised  April 2011 Published  October 2011

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, 2012, 32 (4) : 1125-1167. doi: 10.3934/dcds.2012.32.1125
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), 1027-1071. 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), 469-544. 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), 125-181. 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), 97-132.  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), 151-167.  Google Scholar

[7]

J. Guckenheimer and P. Holme, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations on Vector Fields," Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1983.  Google Scholar

[8]

J. K. Hale, "Ordinary Differential Equations," Pure and Applied Mathematics, XX1, Krieger, Florida, 1980. Google Scholar

[9]

M. W. Hirsch, "Differential Topology," Graduate Texts in Mathematics, 33, Springer-Verlag, New York, 1976. Google Scholar

[10]

D. Knees, A. Mielke and C. Zanini, Crack growth in polyconvex materials, Phys. D, 239 (2010), 1470-1484. 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), 585-615. 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), 1189-1217. 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), 1-35.  Google Scholar

[15]

C. Zanini, Singular perturbation of finite dimensional gradient flows, Discrete Contin. Dyn. Syst., 18 (2007), 657-675. doi: 10.3934/dcds.2007.18.657.  Google Scholar

show all references

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), 1027-1071. 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), 469-544. 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), 125-181. 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), 97-132.  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), 151-167.  Google Scholar

[7]

J. Guckenheimer and P. Holme, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations on Vector Fields," Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1983.  Google Scholar

[8]

J. K. Hale, "Ordinary Differential Equations," Pure and Applied Mathematics, XX1, Krieger, Florida, 1980. Google Scholar

[9]

M. W. Hirsch, "Differential Topology," Graduate Texts in Mathematics, 33, Springer-Verlag, New York, 1976. Google Scholar

[10]

D. Knees, A. Mielke and C. Zanini, Crack growth in polyconvex materials, Phys. D, 239 (2010), 1470-1484. 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), 585-615. 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), 1189-1217. 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), 1-35.  Google Scholar

[15]

C. Zanini, Singular perturbation of finite dimensional gradient flows, Discrete Contin. Dyn. Syst., 18 (2007), 657-675. doi: 10.3934/dcds.2007.18.657.  Google Scholar

[1]

Flaviano Battelli. Saddle-node bifurcation of homoclinic orbits in singular systems. Discrete & Continuous Dynamical Systems, 2001, 7 (1) : 203-218. doi: 10.3934/dcds.2001.7.203
[2]

Rui Dilão, András Volford. Excitability in a model with a saddle-node homoclinic bifurcation. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 419-434. doi: 10.3934/dcdsb.2004.4.419
[3]

Ping Liu, Junping Shi, Yuwen Wang. A double saddle-node bifurcation theorem. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2923-2933. doi: 10.3934/cpaa.2013.12.2923
[4]

Xiao-Biao Lin, Changrong Zhu. Saddle-node bifurcations of multiple homoclinic solutions in ODES. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1435-1460. doi: 10.3934/dcdsb.2017069
[5]

Victoriano Carmona, Soledad Fernández-García, Antonio E. Teruel. Saddle-node of limit cycles in planar piecewise linear systems and applications. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5275-5299. doi: 10.3934/dcds.2019215
[6]

Ale Jan Homburg, Todd Young. Intermittency and Jakobson's theorem near saddle-node bifurcations. Discrete & Continuous Dynamical Systems, 2007, 17 (1) : 21-58. doi: 10.3934/dcds.2007.17.21
[7]

W.-J. Beyn, Y.-K Zou. Discretizations of dynamical systems with a saddle-node homoclinic orbit. Discrete & Continuous Dynamical Systems, 1996, 2 (3) : 351-365. doi: 10.3934/dcds.1996.2.351
[8]

Majid Gazor, Mojtaba Moazeni. Parametric normal forms for Bogdanov--Takens singularity; the generalized saddle-node case. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 205-224. doi: 10.3934/dcds.2015.35.205
[9]

Kie Van Ivanky Saputra, Lennaert van Veen, Gilles Reinout Willem Quispel. The saddle-node-transcritical bifurcation in a population model with constant rate harvesting. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 233-250. doi: 10.3934/dcdsb.2010.14.233
[10]

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero, Michael Z. Zgurovsky. Strong attractors for vanishing viscosity approximations of non-Newtonian suspension flows. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1155-1176. doi: 10.3934/dcdsb.2018146
[11]

Alexandre A. P. Rodrigues. Moduli for heteroclinic connections involving saddle-foci and periodic solutions. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 3155-3182. doi: 10.3934/dcds.2015.35.3155
[12]

Giuseppe Maria Coclite, Carlotta Donadello. Vanishing viscosity on a star-shaped graph under general transmission conditions at the node. Networks & Heterogeneous Media, 2020, 15 (2) : 197-213. doi: 10.3934/nhm.2020009
[13]

Boris P. Andreianov, Giuseppe Maria Coclite, Carlotta Donadello. Well-posedness for vanishing viscosity solutions of scalar conservation laws on a network. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5913-5942. doi: 10.3934/dcds.2017257
[14]

Chaoqun Huang, Nung Kwan Yip. Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part II. Networks & Heterogeneous Media, 2015, 10 (4) : 897-948. doi: 10.3934/nhm.2015.10.897
[15]

Chaoqun Huang, Nung Kwan Yip. Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part I. Networks & Heterogeneous Media, 2013, 8 (4) : 1009-1034. doi: 10.3934/nhm.2013.8.1009
[16]

Diogo A. Gomes. Viscosity solution methods and the discrete Aubry-Mather problem. Discrete & Continuous Dynamical Systems, 2005, 13 (1) : 103-116. doi: 10.3934/dcds.2005.13.103
[17]

Stefano Bianchini, Alberto Bressan. A case study in vanishing viscosity. Discrete & Continuous Dynamical Systems, 2001, 7 (3) : 449-476. doi: 10.3934/dcds.2001.7.449
[18]

Umberto Mosco, Maria Agostina Vivaldi. Vanishing viscosity for fractal sets. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1207-1235. doi: 10.3934/dcds.2010.28.1207
[19]

Julián López-Gómez, Marcela Molina-Meyer, Paul H. Rabinowitz. Global bifurcation diagrams of one node solutions in a class of degenerate boundary value problems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 923-946. doi: 10.3934/dcdsb.2017047
[20]

Hua Chen, Jian-Meng Li, Kelei Wang. On the vanishing viscosity limit of a chemotaxis model. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1963-1987. doi: 10.3934/dcds.2020101

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (54)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences