• 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

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.
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]

Math. Models Methods Appl. Sci., 18 (2008), 1027-1071. doi: 10.1142/S0218202508002942.  Google Scholar

[2]

Arch. Ration. Mech. Anal., 189 (2008), 469-544. doi: 10.1007/s00205-008-0117-5.  Google Scholar

[3]

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]

Netw. Heterog. Media, 5 (2010), 97-132.  Google Scholar

[6]

J. Convex Anal., 13 (2006), 151-167.  Google Scholar

[7]

Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1983.  Google Scholar

[8]

Pure and Applied Mathematics, XX1, Krieger, Florida, 1980. Google Scholar

[9]

Graduate Texts in Mathematics, 33, Springer-Verlag, New York, 1976. Google Scholar

[10]

Phys. D, 239 (2010), 1470-1484. doi: 10.1016/j.physd.2009.02.008.  Google Scholar

[11]

Discrete Contin. Dyn. Syst., 25 (2009), 585-615. doi: 10.3934/dcds.2009.25.585.  Google Scholar

[12]

ESAIM Control Optim. Calc. Var., 2011. doi: 10.1051/cocv/2010054.  Google Scholar

[13]

Discrete Contin. Dyn. Syst., 27 (2010), 1189-1217. doi: 10.3934/dcds.2010.27.1189.  Google Scholar

[14]

Boll. Unione Mat. Ital. (9), 2 (2009), 1-35.  Google Scholar

[15]

Discrete Contin. Dyn. Syst., 18 (2007), 657-675. doi: 10.3934/dcds.2007.18.657.  Google Scholar

show all references

References:
[1]

Math. Models Methods Appl. Sci., 18 (2008), 1027-1071. doi: 10.1142/S0218202508002942.  Google Scholar

[2]

Arch. Ration. Mech. Anal., 189 (2008), 469-544. doi: 10.1007/s00205-008-0117-5.  Google Scholar

[3]

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]

Netw. Heterog. Media, 5 (2010), 97-132.  Google Scholar

[6]

J. Convex Anal., 13 (2006), 151-167.  Google Scholar

[7]

Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1983.  Google Scholar

[8]

Pure and Applied Mathematics, XX1, Krieger, Florida, 1980. Google Scholar

[9]

Graduate Texts in Mathematics, 33, Springer-Verlag, New York, 1976. Google Scholar

[10]

Phys. D, 239 (2010), 1470-1484. doi: 10.1016/j.physd.2009.02.008.  Google Scholar

[11]

Discrete Contin. Dyn. Syst., 25 (2009), 585-615. doi: 10.3934/dcds.2009.25.585.  Google Scholar

[12]

ESAIM Control Optim. Calc. Var., 2011. doi: 10.1051/cocv/2010054.  Google Scholar

[13]

Discrete Contin. Dyn. Syst., 27 (2010), 1189-1217. doi: 10.3934/dcds.2010.27.1189.  Google Scholar

[14]

Boll. Unione Mat. Ital. (9), 2 (2009), 1-35.  Google Scholar

[15]

Discrete Contin. Dyn. Syst., 18 (2007), 657-675. doi: 10.3934/dcds.2007.18.657.  Google Scholar

[1]

Beixiang Fang, Qin Zhao. Uniqueness of steady 1-D shock solutions in a finite nozzle via vanishing viscosity aguments. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021066

[2]

Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881

[3]

Alexander Tolstonogov. BV solutions of a convex sweeping process with a composed perturbation. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021012

[4]

Bruno Premoselli. Einstein-Lichnerowicz type singular perturbations of critical nonlinear elliptic equations in dimension 3. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021069

[5]

Tianyu Liao. The regularity lifting methods for nonnegative solutions of Lane-Emden system. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021036

[6]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258

[7]

Meiqiang Feng, Yichen Zhang. Positive solutions of singular multiparameter p-Laplacian elliptic systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021083

[8]

Maoding Zhen, Binlin Zhang, Vicenţiu D. Rădulescu. Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2653-2676. doi: 10.3934/dcds.2020379

[9]

Qiao Liu. Partial regularity and the Minkowski dimension of singular points for suitable weak solutions to the 3D simplified Ericksen–Leslie system. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021041

[10]

Prasanta Kumar Barik, Ankik Kumar Giri, Rajesh Kumar. Mass-conserving weak solutions to the coagulation and collisional breakage equation with singular rates. Kinetic & Related Models, 2021, 14 (2) : 389-406. doi: 10.3934/krm.2021009

[11]

Horst R. Thieme. Remarks on resolvent positive operators and their perturbation. Discrete & Continuous Dynamical Systems, 1998, 4 (1) : 73-90. doi: 10.3934/dcds.1998.4.73

[12]

Christopher Bose, Rua Murray. Minimum 'energy' approximations of invariant measures for nonsingular transformations. Discrete & Continuous Dynamical Systems, 2006, 14 (3) : 597-615. doi: 10.3934/dcds.2006.14.597

[13]

Alberto Bressan, Carlotta Donadello. On the convergence of viscous approximations after shock interactions. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 29-48. doi: 10.3934/dcds.2009.23.29

[14]

Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2021, 20 (2) : 933-954. doi: 10.3934/cpaa.2020298

[15]

Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2677-2698. doi: 10.3934/dcds.2020381

[16]

Daniele Cassani, Luca Vilasi, Jianjun Zhang. Concentration phenomena at saddle points of potential for Schrödinger-Poisson systems. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021039

[17]

Nikolaz Gourmelon. Generation of homoclinic tangencies by $C^1$-perturbations. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 1-42. doi: 10.3934/dcds.2010.26.1

[18]

Qigang Yuan, Jingli Ren. Periodic forcing on degenerate Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208

[19]

Philippe Jouan, Ronald Manríquez. Solvable approximations of 3-dimensional almost-Riemannian structures. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021023

[20]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (46)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]