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 and Continuous Dynamical Systems, 2019, 39 (6) : 3577-3608. doi: 10.3934/dcds.2019147
References:
[1]

C. BardosA. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

S. N. Kružhkov, First order quasilinear equations with several independent variables., Mat. Sb. (N.S.), 81 (1970), 228-255. 

[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.

[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.

[13]

F. Otto, Initial-boundary value problem for a scalar conservation law, C. R. Acad. Sci. Paris Sér. I Math., 322 (1996), 729-734. 

[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.

[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.

show all references

References:
[1]

C. BardosA. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

S. N. Kružhkov, First order quasilinear equations with several independent variables., Mat. Sb. (N.S.), 81 (1970), 228-255. 

[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.

[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.

[13]

F. Otto, Initial-boundary value problem for a scalar conservation law, C. R. Acad. Sci. Paris Sér. I Math., 322 (1996), 729-734. 

[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.

[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.

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