# American Institute of Mathematical Sciences

June  2016, 11(2): 223-237. doi: 10.3934/nhm.2016.11.223

## Relaxation approximation of Friedrichs' systems under convex constraints

 1 Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France, France 2 Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris

Received  April 2015 Revised  September 2015 Published  March 2016

This paper is devoted to present an approximation of a Cauchy problem for Friedrichs' systems under convex constraints. It is proved the strong convergence in $L^2_{\text{loc}}$ of a parabolic-relaxed approximation towards the unique constrained solution.
Citation: Jean-François Babadjian, Clément Mifsud, Nicolas Seguin. Relaxation approximation of Friedrichs' systems under convex constraints. Networks & Heterogeneous Media, 2016, 11 (2) : 223-237. doi: 10.3934/nhm.2016.11.223
##### References:
 [1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces,, $2^{nd}$ edition, (2003). Google Scholar [2] H. Brézis, Analyse Fonctionnelle,, Masson, (1983). Google Scholar [3] B. Després, F. Lagoutière and N. Seguin, Weak solutions to Friedrichs systems with convex constraints,, Nonlinearity, 24 (2011), 3055. doi: 10.1088/0951-7715/24/11/003. Google Scholar [4] L. C. Evans, Partial Differential Equations,, $2^{nd}$ edition, (2010). doi: 10.1090/gsm/019. Google Scholar [5] K. O. Friedrichs, Symmetric positive linear differential equations,, Comm. Pure Appl. Math., 11 (1958), 333. doi: 10.1002/cpa.3160110306. Google Scholar [6] C. Mifsud, B. Després and N. Seguin, Dissipative formulation of initial boundary value problems for Friedrichs' systems,, Comm. Partial Differential Equations, 41 (2016), 51. doi: 10.1080/03605302.2015.1103750. Google Scholar [7] A. Morando and D. Serre, On the $L^2$-well posedness of an initial boundary value problem for the 3D linear elasticity,, Commun. Math. Sci., 3 (2005), 575. doi: 10.4310/CMS.2005.v3.n4.a7. Google Scholar [8] J.-J. Moreau, Proximité et dualité dans un espace hilbertien,, Bull. Soc. Math. France, 93 (1965), 273. Google Scholar [9] A. Nouri and M. Rascle, A global existence and uniqueness theorem for a model problem in dynamic elastoplasticity with isotropic strain-hardening,, SIAM J. Math. Anal., 26 (1995), 850. doi: 10.1137/S0036141091199601. Google Scholar [10] J. Simon, Compact Sets in the Space $L^p(0,T,B)$,, Annali Mat. Pura Appl., 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar [11] P.-M. Suquet, Evolution problems for a class of dissipative materials,, Quart. Appl. Math., 38 (1980), 391. Google Scholar [12] P.-M. Suquet, Sur les équations de la plasticité: Existence et régularité des solutions,, J. Mécanique, 20 (1981), 3. Google Scholar

show all references

##### References:
 [1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces,, $2^{nd}$ edition, (2003). Google Scholar [2] H. Brézis, Analyse Fonctionnelle,, Masson, (1983). Google Scholar [3] B. Després, F. Lagoutière and N. Seguin, Weak solutions to Friedrichs systems with convex constraints,, Nonlinearity, 24 (2011), 3055. doi: 10.1088/0951-7715/24/11/003. Google Scholar [4] L. C. Evans, Partial Differential Equations,, $2^{nd}$ edition, (2010). doi: 10.1090/gsm/019. Google Scholar [5] K. O. Friedrichs, Symmetric positive linear differential equations,, Comm. Pure Appl. Math., 11 (1958), 333. doi: 10.1002/cpa.3160110306. Google Scholar [6] C. Mifsud, B. Després and N. Seguin, Dissipative formulation of initial boundary value problems for Friedrichs' systems,, Comm. Partial Differential Equations, 41 (2016), 51. doi: 10.1080/03605302.2015.1103750. Google Scholar [7] A. Morando and D. Serre, On the $L^2$-well posedness of an initial boundary value problem for the 3D linear elasticity,, Commun. Math. Sci., 3 (2005), 575. doi: 10.4310/CMS.2005.v3.n4.a7. Google Scholar [8] J.-J. Moreau, Proximité et dualité dans un espace hilbertien,, Bull. Soc. Math. France, 93 (1965), 273. Google Scholar [9] A. Nouri and M. Rascle, A global existence and uniqueness theorem for a model problem in dynamic elastoplasticity with isotropic strain-hardening,, SIAM J. Math. Anal., 26 (1995), 850. doi: 10.1137/S0036141091199601. Google Scholar [10] J. Simon, Compact Sets in the Space $L^p(0,T,B)$,, Annali Mat. Pura Appl., 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar [11] P.-M. Suquet, Evolution problems for a class of dissipative materials,, Quart. Appl. Math., 38 (1980), 391. Google Scholar [12] P.-M. Suquet, Sur les équations de la plasticité: Existence et régularité des solutions,, J. Mécanique, 20 (1981), 3. Google Scholar
 [1] Krešimir Burazin, Marko Vrdoljak. Homogenisation theory for Friedrichs systems. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1017-1044. doi: 10.3934/cpaa.2014.13.1017 [2] Hermano Frid. Invariant regions under Lax-Friedrichs scheme for multidimensional systems of conservation laws. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 585-593. doi: 10.3934/dcds.1995.1.585 [3] Xiantao Xiao, Jian Gu, Liwei Zhang, Shaowu Zhang. A sequential convex program method to DC program with joint chance constraints. Journal of Industrial & Management Optimization, 2012, 8 (3) : 733-747. doi: 10.3934/jimo.2012.8.733 [4] Ingrid Daubechies, Gerd Teschke, Luminita Vese. Iteratively solving linear inverse problems under general convex constraints. Inverse Problems & Imaging, 2007, 1 (1) : 29-46. doi: 10.3934/ipi.2007.1.29 [5] Gaohang Yu, Shanzhou Niu, Jianhua Ma. Multivariate spectral gradient projection method for nonlinear monotone equations with convex constraints. Journal of Industrial & Management Optimization, 2013, 9 (1) : 117-129. doi: 10.3934/jimo.2013.9.117 [6] Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329 [7] Sergio Grillo, Marcela Zuccalli. Variational reduction of Lagrangian systems with general constraints. Journal of Geometric Mechanics, 2012, 4 (1) : 49-88. doi: 10.3934/jgm.2012.4.49 [8] Santiago Capriotti. Dirac constraints in field theory and exterior differential systems. Journal of Geometric Mechanics, 2010, 2 (1) : 1-50. doi: 10.3934/jgm.2010.2.1 [9] Mikhail Gusev. On reachability analysis for nonlinear control systems with state constraints. Conference Publications, 2015, 2015 (special) : 579-587. doi: 10.3934/proc.2015.0579 [10] Ünver Çiftçi. Leibniz-Dirac structures and nonconservative systems with constraints. Journal of Geometric Mechanics, 2013, 5 (2) : 167-183. doi: 10.3934/jgm.2013.5.167 [11] Jinlong Bai, Xuewei Ju, Desheng Li, Xiulian Wang. On the eventual stability of asymptotically autonomous systems with constraints. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4457-4473. doi: 10.3934/dcdsb.2019127 [12] Md. Haider Ali Biswas, Maria do Rosário de Pinho. A nonsmooth maximum principle for optimal control problems with state and mixed constraints - convex case. Conference Publications, 2011, 2011 (Special) : 174-183. doi: 10.3934/proc.2011.2011.174 [13] Yuan Shen, Wenxing Zhang, Bingsheng He. Relaxed augmented Lagrangian-based proximal point algorithms for convex optimization with linear constraints. Journal of Industrial & Management Optimization, 2014, 10 (3) : 743-759. doi: 10.3934/jimo.2014.10.743 [14] Zhi-Bin Deng, Ye Tian, Cheng Lu, Wen-Xun Xing. Globally solving quadratic programs with convex objective and complementarity constraints via completely positive programming. Journal of Industrial & Management Optimization, 2018, 14 (2) : 625-636. doi: 10.3934/jimo.2017064 [15] Nithirat Sisarat, Rabian Wangkeeree, Gue Myung Lee. Some characterizations of robust solution sets for uncertain convex optimization problems with locally Lipschitz inequality constraints. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-25. doi: 10.3934/jimo.2018163 [16] Piernicola Bettiol, Hélène Frankowska. Lipschitz regularity of solution map of control systems with multiple state constraints. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 1-26. doi: 10.3934/dcds.2012.32.1 [17] Jiongmin Yong. Optimality conditions for controls of semilinear evolution systems with mixed constraints. Discrete & Continuous Dynamical Systems - A, 1995, 1 (3) : 371-388. doi: 10.3934/dcds.1995.1.371 [18] Hernán Cendra, María Etchechoury, Sebastián J. Ferraro. An extension of the Dirac and Gotay-Nester theories of constraints for Dirac dynamical systems. Journal of Geometric Mechanics, 2014, 6 (2) : 167-236. doi: 10.3934/jgm.2014.6.167 [19] H. O. Fattorini. The maximum principle for linear infinite dimensional control systems with state constraints. Discrete & Continuous Dynamical Systems - A, 1995, 1 (1) : 77-101. doi: 10.3934/dcds.1995.1.77 [20] Fabio Camilli, Paola Loreti, Naoki Yamada. Systems of convex Hamilton-Jacobi equations with implicit obstacles and the obstacle problem. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1291-1302. doi: 10.3934/cpaa.2009.8.1291

2018 Impact Factor: 0.871