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 Universités, 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 and 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, Elsevier, Amsterdam, 2003.

[2]

H. Brézis, Analyse Fonctionnelle, Masson, Paris, 1983.

[3]

B. Després, F. Lagoutière and N. Seguin, Weak solutions to Friedrichs systems with convex constraints, Nonlinearity, 24 (2011), 3055-3081. doi: 10.1088/0951-7715/24/11/003.

[4]

L. C. Evans, Partial Differential Equations, $2^{nd}$ edition, American mathematical society, Providence, 2010. doi: 10.1090/gsm/019.

[5]

K. O. Friedrichs, Symmetric positive linear differential equations, Comm. Pure Appl. Math., 11 (1958), 333-418. doi: 10.1002/cpa.3160110306.

[6]

C. Mifsud, B. Després and N. Seguin, Dissipative formulation of initial boundary value problems for Friedrichs' systems, Comm. Partial Differential Equations, 41 (2016), 51-78. doi: 10.1080/03605302.2015.1103750.

[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), 575-586. doi: 10.4310/CMS.2005.v3.n4.a7.

[8]

J.-J. Moreau, Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France, 93 (1965), 273-299.

[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), 850-868. doi: 10.1137/S0036141091199601.

[10]

J. Simon, Compact Sets in the Space $L^p(0,T,B)$, Annali Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.

[11]

P.-M. Suquet, Evolution problems for a class of dissipative materials, Quart. Appl. Math., 38 (1980), 391-414.

[12]

P.-M. Suquet, Sur les équations de la plasticité: Existence et régularité des solutions, J. Mécanique, 20 (1981), 3-39.

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, $2^{nd}$ edition, Elsevier, Amsterdam, 2003.

[2]

H. Brézis, Analyse Fonctionnelle, Masson, Paris, 1983.

[3]

B. Després, F. Lagoutière and N. Seguin, Weak solutions to Friedrichs systems with convex constraints, Nonlinearity, 24 (2011), 3055-3081. doi: 10.1088/0951-7715/24/11/003.

[4]

L. C. Evans, Partial Differential Equations, $2^{nd}$ edition, American mathematical society, Providence, 2010. doi: 10.1090/gsm/019.

[5]

K. O. Friedrichs, Symmetric positive linear differential equations, Comm. Pure Appl. Math., 11 (1958), 333-418. doi: 10.1002/cpa.3160110306.

[6]

C. Mifsud, B. Després and N. Seguin, Dissipative formulation of initial boundary value problems for Friedrichs' systems, Comm. Partial Differential Equations, 41 (2016), 51-78. doi: 10.1080/03605302.2015.1103750.

[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), 575-586. doi: 10.4310/CMS.2005.v3.n4.a7.

[8]

J.-J. Moreau, Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France, 93 (1965), 273-299.

[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), 850-868. doi: 10.1137/S0036141091199601.

[10]

J. Simon, Compact Sets in the Space $L^p(0,T,B)$, Annali Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.

[11]

P.-M. Suquet, Evolution problems for a class of dissipative materials, Quart. Appl. Math., 38 (1980), 391-414.

[12]

P.-M. Suquet, Sur les équations de la plasticité: Existence et régularité des solutions, J. Mécanique, 20 (1981), 3-39.

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