# American Institute of Mathematical Sciences

June  2019, 39(6): 3577-3608. doi: 10.3934/dcds.2019147

## Well-posedness of general 1D initial boundary value problems for scalar balance laws

 Inria Sophia Antipolis - Méditerranée, Université Côte d'Azur, Inria, CNRS, LJAD, 2004 route des Lucioles - BP 93, 06902 Sophia Antipolis Cedex, France

* Corresponding author: Elena Rossi

Received  October 2018 Revised  November 2018 Published  February 2019

We focus on the initial boundary value problem for a general scalar balance law in one space dimension. Under rather general assumptions on the flux and source functions, we prove the well-posedness of this problem and the stability of its solutions with respect to variations in the flux and in the source terms. For both results, the initial and boundary data are required to be bounded functions with bounded total variation. The existence of solutions is obtained from the convergence of a Lax–Friedrichs type algorithm with operator splitting. The stability result follows from an application of Kružkov's doubling of variables technique, together with a careful treatment of the boundary terms.

Citation: Elena Rossi. Well-posedness of general 1D initial boundary value problems for scalar balance laws. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3577-3608. doi: 10.3934/dcds.2019147
##### References:
 [1] C. Bardos, A. Y. le Roux and J.-C. Nédélec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations, 4 (1979), 1017-1034. doi: 10.1080/03605307908820117. Google Scholar [2] F. Bouchut and B. Perthame, Kružkov's estimates for scalar conservation laws revisited, Trans. Amer. Math. Soc., 350 (1998), 2847-2870. doi: 10.1090/S0002-9947-98-02204-1. Google Scholar [3] R. M. Colombo and E. Rossi, Stability of the 1D IBVP for a non autonomous scalar conservation law, Proc. Roy. Soc. Edinburgh Sect. A, To appear, arXiv: 1601.05948. doi: 10.1017/prm.2018.39. Google Scholar [4] R. M. Colombo and E. Rossi, Rigorous estimates on balance laws in bounded domains, Acta Math. Sci. Ser. B Engl. Ed., 35 (2015), 906-944. doi: 10.1016/S0252-9602(15)30028-X. Google Scholar [5] R. M. Colombo and E. Rossi, IBVPs for scalar conservation laws with time discontinuous fluxes, Math. Methods Appl. Sci., 41 (2018), 1463-1479. doi: 10.1002/mma.4676. Google Scholar [6] R. M. Colombo and E. Rossi, Nonlocal conservation laws in bounded domains, SIAM J. Math. Anal., 50 (2018), 4041-4065. doi: 10.1137/18M1171783. Google Scholar [7] C. De Filippis and P. Goatin, The initial-boundary value problem for general non-local scalar conservation laws in one space dimension, Nonlinear Anal., 161 (2017), 131-156. doi: 10.1016/j.na.2017.05.017. Google Scholar [8] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of Numerical Analysis, Vol. VII, Handb. Numer. Anal., Ⅶ, North-Holland, Amsterdam, 2000, 713–1020. Google Scholar [9] K. H. Karlsen and N. H. Risebro, On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients, Discrete Contin. Dyn. Syst., 9 (2003), 1081-1104. doi: 10.3934/dcds.2003.9.1081. Google Scholar [10] S. N. Kružhkov, First order quasilinear equations with several independent variables., Mat. Sb. (N.S.), 81 (1970), 228-255. Google Scholar [11] J. Málek, J. Nečas, M. Rokyta and M. Růžička, Weak and Measure-Valued Solutions to Evolutionary PDEs, vol. 13 of Applied Mathematics and Mathematical Computation, Chapman & Hall, London, 1996. Google Scholar [12] S. Martin, First order quasilinear equations with boundary conditions in the $L^\infty$ framework, J. Differential Equations, 236 (2007), 375-406. doi: 10.1016/j.jde.2007.02.007. Google Scholar [13] F. Otto, Initial-boundary value problem for a scalar conservation law, C. R. Acad. Sci. Paris Sér. I Math., 322 (1996), 729-734. Google Scholar [14] E. Rossi, Definitions of solutions to the IBVP for multi-dimensional scalar balance laws, J. Hyperbolic Differ. Equ., 15 (2018), 349-374. doi: 10.1142/S0219891618500133. Google Scholar [15] J. Vovelle, Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains, Numer. Math., 90 (2002), 563-596. doi: 10.1007/s002110100307. Google Scholar

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##### References:
 [1] C. Bardos, A. Y. le Roux and J.-C. Nédélec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations, 4 (1979), 1017-1034. doi: 10.1080/03605307908820117. Google Scholar [2] F. Bouchut and B. Perthame, Kružkov's estimates for scalar conservation laws revisited, Trans. Amer. Math. Soc., 350 (1998), 2847-2870. doi: 10.1090/S0002-9947-98-02204-1. Google Scholar [3] R. M. Colombo and E. Rossi, Stability of the 1D IBVP for a non autonomous scalar conservation law, Proc. Roy. Soc. Edinburgh Sect. A, To appear, arXiv: 1601.05948. doi: 10.1017/prm.2018.39. Google Scholar [4] R. M. Colombo and E. Rossi, Rigorous estimates on balance laws in bounded domains, Acta Math. Sci. Ser. B Engl. Ed., 35 (2015), 906-944. doi: 10.1016/S0252-9602(15)30028-X. Google Scholar [5] R. M. Colombo and E. Rossi, IBVPs for scalar conservation laws with time discontinuous fluxes, Math. Methods Appl. Sci., 41 (2018), 1463-1479. doi: 10.1002/mma.4676. Google Scholar [6] R. M. Colombo and E. Rossi, Nonlocal conservation laws in bounded domains, SIAM J. Math. Anal., 50 (2018), 4041-4065. doi: 10.1137/18M1171783. Google Scholar [7] C. De Filippis and P. Goatin, The initial-boundary value problem for general non-local scalar conservation laws in one space dimension, Nonlinear Anal., 161 (2017), 131-156. doi: 10.1016/j.na.2017.05.017. Google Scholar [8] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of Numerical Analysis, Vol. VII, Handb. Numer. Anal., Ⅶ, North-Holland, Amsterdam, 2000, 713–1020. Google Scholar [9] K. H. Karlsen and N. H. Risebro, On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients, Discrete Contin. Dyn. Syst., 9 (2003), 1081-1104. doi: 10.3934/dcds.2003.9.1081. Google Scholar [10] S. N. Kružhkov, First order quasilinear equations with several independent variables., Mat. Sb. (N.S.), 81 (1970), 228-255. Google Scholar [11] J. Málek, J. Nečas, M. Rokyta and M. Růžička, Weak and Measure-Valued Solutions to Evolutionary PDEs, vol. 13 of Applied Mathematics and Mathematical Computation, Chapman & Hall, London, 1996. Google Scholar [12] S. Martin, First order quasilinear equations with boundary conditions in the $L^\infty$ framework, J. Differential Equations, 236 (2007), 375-406. doi: 10.1016/j.jde.2007.02.007. Google Scholar [13] F. Otto, Initial-boundary value problem for a scalar conservation law, C. R. Acad. Sci. Paris Sér. I Math., 322 (1996), 729-734. Google Scholar [14] E. Rossi, Definitions of solutions to the IBVP for multi-dimensional scalar balance laws, J. Hyperbolic Differ. Equ., 15 (2018), 349-374. doi: 10.1142/S0219891618500133. Google Scholar [15] J. Vovelle, Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains, Numer. Math., 90 (2002), 563-596. doi: 10.1007/s002110100307. Google Scholar
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