Relaxation approximation of Friedrichs' systems under convex constraints

  • Received: 01 April 2015 Revised: 01 September 2015
  • Primary: 35L45, 35L60; Secondary: 35A35.

  • 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 Lloc2 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[J]. Networks and Heterogeneous Media, 2016, 11(2): 223-237. doi: 10.3934/nhm.2016.11.223

    Related Papers:

    [1] 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
    [2] Jan Friedrich, Oliver Kolb, Simone Göttlich . A Godunov type scheme for a class of LWR traffic flow models with non-local flux. Networks and Heterogeneous Media, 2018, 13(4): 531-547. doi: 10.3934/nhm.2018024
    [3] John D. Towers . The Lax-Friedrichs scheme for interaction between the inviscid Burgers equation and multiple particles. Networks and Heterogeneous Media, 2020, 15(1): 143-169. doi: 10.3934/nhm.2020007
    [4] Mengyuan Dai, Chunyan Zhang, Yingli Zhang, Lichao Feng . Note on adaptive prescribed-time stabilization of nonlinear systems with uncertainty. Networks and Heterogeneous Media, 2025, 20(3): 798-817. doi: 10.3934/nhm.2025034
    [5] Martin Gugat, Rüdiger Schultz, Michael Schuster . Convexity and starshapedness of feasible sets in stationary flow networks. Networks and Heterogeneous Media, 2020, 15(2): 171-195. doi: 10.3934/nhm.2020008
    [6] Caterina Balzotti, Simone Göttlich . A two-dimensional multi-class traffic flow model. Networks and Heterogeneous Media, 2021, 16(1): 69-90. doi: 10.3934/nhm.2020034
    [7] Boris P. Andreianov, Carlotta Donadello, Ulrich Razafison, Julien Y. Rolland, Massimiliano D. Rosini . Solutions of the Aw-Rascle-Zhang system with point constraints. Networks and Heterogeneous Media, 2016, 11(1): 29-47. doi: 10.3934/nhm.2016.11.29
    [8] Raimund Bürger, Harold Deivi Contreras, Luis Miguel Villada . A Hilliges-Weidlich-type scheme for a one-dimensional scalar conservation law with nonlocal flux. Networks and Heterogeneous Media, 2023, 18(2): 664-693. doi: 10.3934/nhm.2023029
    [9] Ciro D'Apice, Peter I. Kogut, Rosanna Manzo . On relaxation of state constrained optimal control problem for a PDE-ODE model of supply chains. Networks and Heterogeneous Media, 2014, 9(3): 501-518. doi: 10.3934/nhm.2014.9.501
    [10] Shenglun Yan, Wanqian Zhang, Weiyuan Zou . Multi-cluster flocking of the thermodynamic Cucker-Smale model with a unit-speed constraint under a singular kernel. Networks and Heterogeneous Media, 2024, 19(2): 547-568. doi: 10.3934/nhm.2024024
  • 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 Lloc2 of a parabolic-relaxed approximation towards the unique constrained solution.


    [1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd 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, 2nd 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 L2-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 Lp(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.
  • This article has been cited by:

    1. Jean-François Babadjian, Vito Crismale, Dissipative boundary conditions and entropic solutions in dynamical perfect plasticity, 2021, 148, 00217824, 75, 10.1016/j.matpur.2021.02.001
    2. Jean-François Babadjian, Clément Mifsud, Hyperbolic Structure for a Simplified Model of Dynamical Perfect Plasticity, 2017, 223, 0003-9527, 761, 10.1007/s00205-016-1045-4
  • Reader Comments
  • © 2016 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3798) PDF downloads(142) Cited by(2)

Article outline

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog