December  2002, 1(4): 513-530. doi: 10.3934/cpaa.2002.1.513

Existence result for a class of nonconservative and nonstrictly hyperbolic systems


Centre de Mathematiques Appliquees, Ecole Polytechnique, and Dipartimento di Matematica Pura e Appl., Universita di Modena, Via Campi 213/B, 41100 Modena, Italy


Centre de Mathématiques Appliquées, and Centre National de la Recherche Scientifique, UMR. 7641, Ecole Polytechnique, 91128 Palaiseau Cedex, France

Received  March 2002 Revised  July 2002 Published  September 2002

We consider the class of nonconservative hyperbolic systems

$\partial_t u+A(u) \partial_x u =0,\quad\partial_t v + A(u) \partial_x v =0,$

where $u=u(x,t),\quad v=v(x,t)\in\mathbb R^N$ are the unknowns and $A(u)$ is a strictly hyperbolic matrix. Relying on the notion of weak solution proposed by Dal Maso, LeFloch, and Murat ("Definition and weak stability of nonconservative products", J. Math. Pures Appl. 74 (1995), 483--548), we establish the existence of weak solutions for the corresponding Cauchy problem, in the class of bounded functions with bounded variation. The main steps in our proof are as follows:
(i) We solve the Riemann problem based on a prescribed family of paths.
(ii) We derive a uniform bound on the total variation of corresponding wave-front tracking approximations $u^h$, $v^h$.
(iii) We justify rigorously the passage to the limit in the nonconservative product $A(u^h) \partial_x v^h$, based on the local uniform convergence properties of $u^h$, by extending an argument due to LeFloch and Liu ("Existence theory for nonlinear hyperbolic systems in nonconservative form", Forum Math. 5 (1993), 261--280). Our results provide a generalization to the existence theorem established earlier in the scalar case ($N=1$) by the second author ("An existence and uniqueness result for two nonstrictly hyperbolic systems", IMA Volumes in Math. and its Appl. 27, "Nonlinear evolution equations that change type", ed. B.L. Keyfitz and M. Shearer, Springer Verlag, 1990, pp. 126--138.)

Citation: Graziano Crasta, Philippe G. LeFloch. Existence result for a class of nonconservative and nonstrictly hyperbolic systems. Communications on Pure & Applied Analysis, 2002, 1 (4) : 513-530. doi: 10.3934/cpaa.2002.1.513

Rinaldo M. Colombo, Francesca Monti. Solutions with large total variation to nonconservative hyperbolic systems. Communications on Pure & Applied Analysis, 2010, 9 (1) : 47-60. doi: 10.3934/cpaa.2010.9.47


Margarita Arias, Juan Campos, Cristina Marcelli. Fastness and continuous dependence in front propagation in Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 11-30. doi: 10.3934/dcdsb.2009.11.11


Luisa Malaguti, Cristina Marcelli, Serena Matucci. Continuous dependence in front propagation of convective reaction-diffusion equations. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1083-1098. doi: 10.3934/cpaa.2010.9.1083


Giuseppe Maria Coclite, Angelo Favini, Gisèle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli. Continuous dependence in hyperbolic problems with Wentzell boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (1) : 419-433. doi: 10.3934/cpaa.2014.13.419


Debora Amadori, Wen Shen. Front tracking approximations for slow erosion. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1481-1502. doi: 10.3934/dcds.2012.32.1481


Maria Laura Delle Monache, Paola Goatin. A front tracking method for a strongly coupled PDE-ODE system with moving density constraints in traffic flow. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 435-447. doi: 10.3934/dcdss.2014.7.435


David Cheban, Cristiana Mammana. Continuous dependence of attractors on parameters of non-autonomous dynamical systems and infinite iterated function systems. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : 499-515. doi: 10.3934/dcds.2007.18.499


N. V. Chemetov. Nonlinear hyperbolic-elliptic systems in the bounded domain. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1079-1096. doi: 10.3934/cpaa.2011.10.1079


Feng Li, Yuxiang Li. Global existence of weak solution in a chemotaxis-fluid system with nonlinear diffusion and rotational flux. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5409-5436. doi: 10.3934/dcdsb.2019064


Yang Wang, Xiong Li. Uniqueness of traveling front solutions for the Lotka-Volterra system in the weak competition case. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3067-3075. doi: 10.3934/dcdsb.2018300


Út V. Lê. Regularity of the solution of a nonlinear wave equation. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1099-1115. doi: 10.3934/cpaa.2010.9.1099


Ahmad El Hajj, Aya Oussaily. Continuous solution for a non-linear eikonal system. Communications on Pure & Applied Analysis, 2021, 20 (11) : 3795-3823. doi: 10.3934/cpaa.2021131


José F. Caicedo, Alfonso Castro. A semilinear wave equation with smooth data and no resonance having no continuous solution. Discrete & Continuous Dynamical Systems, 2009, 24 (3) : 653-658. doi: 10.3934/dcds.2009.24.653


JinMyong An, JinMyong Kim, KyuSong Chae. Continuous dependence of the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation in $H^{s} (\mathbb R^{n})$. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021221


Alain Hertzog, Antoine Mondoloni. Existence of a weak solution for a quasilinear wave equation with boundary condition. Communications on Pure & Applied Analysis, 2002, 1 (2) : 191-219. doi: 10.3934/cpaa.2002.1.191


Xin-Guang Yang. An Erratum on "Stability and dynamics of a weak viscoelastic system with memory and nonlinear time-varying delay" (Discrete Continuous Dynamic Systems, 40(3), 2020, 1493-1515). Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021161


Biswajit Basu. On an exact solution of a nonlinear three-dimensional model in ocean flows with equatorial undercurrent and linear variation in density. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4783-4796. doi: 10.3934/dcds.2019195


Zhong Tan, Jianfeng Zhou. Higher integrability of weak solution of a nonlinear problem arising in the electrorheological fluids. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1335-1350. doi: 10.3934/cpaa.2016.15.1335


Meng Wang, Wendong Wang, Zhifei Zhang. On the uniqueness of weak solution for the 2-D Ericksen--Leslie system. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 919-941. doi: 10.3934/dcdsb.2016.21.919


Yong Zeng. Existence and uniqueness of very weak solution of the MHD type system. Discrete & Continuous Dynamical Systems, 2020, 40 (10) : 5617-5638. doi: 10.3934/dcds.2020240

2020 Impact Factor: 1.916


  • PDF downloads (108)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]