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June  2020, 40(6): 3143-3169. doi: 10.3934/dcds.2020041

Signed Radon measure-valued solutions of flux saturated scalar conservation laws

Michiel Bertsch 1, , Flavia Smarrazzo 2, , Andrea Terracina 3, and  Alberto Tesei 4,,

1. 

Dipartimento di Matematica, Università di Roma "Tor Vergata", Via della Ricerca Scientifica, 00133 Roma, Italy, and, Istituto per le Applicazioni del Calcolo "M. Picone", CNR, Roma, Italy
2. 

Facoltà Dipartimentale di Ingegneria, Università Campus Bio-Medico di Roma, Via Alvaro del Portillo 21, 00128 Roma, Italy
3. 

Dipartimento di Matematica "G. Castelnuovo", Università "Sapienza" di Roma, P.le A. Moro 5, I-00185 Roma, Italy
4. 

Istituto per le Applicazioni del Calcolo "M. Picone", CNR, Roma, Italy

* Corresponding author: Alberto Tesei

Received  February 2019 Published  October 2019

We prove existence and uniqueness for a class of signed Radon measure-valued entropy solutions of the Cauchy problem for a first order scalar hyperbolic conservation law in one space dimension. The initial data of the problem is a finite superposition of Dirac masses, whereas the flux is Lipschitz continuous and bounded. The solution class is determined by an additional condition which is needed to prove uniqueness.

Keywords: First order hyperbolic conservation laws, signed Radon measures, singular boundary conditions, entropy inequalities, uniqueness.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Michiel Bertsch, Flavia Smarrazzo, Andrea Terracina, Alberto Tesei. Signed Radon measure-valued solutions of flux saturated scalar conservation laws. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3143-3169. doi: 10.3934/dcds.2020041
References:
[1]

C. Bardos, A. Y. le Roux and J. C. Nédélec, First order quasilinear equations with boundary condition, Comm. Partial Differential Equations, 4 (1979), 1017-1034.  doi: 10.1080/03605307908820117.  Google Scholar

[2]

M. Bertsch, F. Smarrazzo, A. Terracina and A. Tesei, Radon measure-valued solutions of first order hyperbolic conservation laws, Adv. in Nonlinear Anal., 9 (2020), 65-107.  doi: 10.1515/anona-2018-0056.  Google Scholar

[3]

M. Bertsch, F. Smarrazzo, A. Terracina and A. Tesei, A uniqueness criterion for measure-valued solutions of scalar hyperbolic conservation laws, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 30 (2019), 137-168.  doi: 10.4171/RLM/839.  Google Scholar

[4]

M. Bertsch, F. Smarrazzo, A. Terracina and A. Tesei, Discontinuous viscosity solutions of first order Hamilton-Jacobi equations, Preprint, (2019), arXiv: 1906.05625. Google Scholar

[5]

F. Demengel and D. Serre, Nonvanishing singular parts of measure valued solutions for scalar hyperbolic equations, Comm. Partial Differential Equations, 16 (1991), 221-254.  doi: 10.1080/03605309108820758.  Google Scholar

[6]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.  Google Scholar

[7]

A. Friedman, Mathematics in Industrial Problems, Part 8, The IMA Volumes in Mathematics and its Applications, 83. Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-1858-6.  Google Scholar

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O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Qquasi-linear equations of parabolic type, Amer. Math. Soc., (1991). Google Scholar

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T.-P. Liu and M. Pierre, Source-solutions and asymptotic behavior in conservation laws, J. Differential Equations, 51 (1984), 419-441.  doi: 10.1016/0022-0396(84)90096-2.  Google Scholar

[10]

J. Málek, J. Nečas, M. Rokyta and M. R${{\rm{\dot u}}}$žička, Weak and Measure-Valued Solutions of Evolutionary PDEs, Applied Mathematics and Mathematical Computation, 13. Chapman & Hall, London, 1996. doi: 10.1007/978-1-4899-6824-1.  Google Scholar

[11]

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

[12]

D. Serre, Systems of Conservation Laws, Vol. 1: Hyperbolicity, Entropies, Shock Waves, Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9780511612374.  Google Scholar

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A. Terracina, Comparison properties for scalar conservation laws with boundary conditions, Nonlinear Anal., 28 (1997), 633-653.  doi: 10.1016/0362-546X(95)00172-R.  Google Scholar

show all references

References:
[1]

C. Bardos, A. Y. le Roux and J. C. Nédélec, First order quasilinear equations with boundary condition, Comm. Partial Differential Equations, 4 (1979), 1017-1034.  doi: 10.1080/03605307908820117.  Google Scholar

[2]

M. Bertsch, F. Smarrazzo, A. Terracina and A. Tesei, Radon measure-valued solutions of first order hyperbolic conservation laws, Adv. in Nonlinear Anal., 9 (2020), 65-107.  doi: 10.1515/anona-2018-0056.  Google Scholar

[3]

M. Bertsch, F. Smarrazzo, A. Terracina and A. Tesei, A uniqueness criterion for measure-valued solutions of scalar hyperbolic conservation laws, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 30 (2019), 137-168.  doi: 10.4171/RLM/839.  Google Scholar

[4]

M. Bertsch, F. Smarrazzo, A. Terracina and A. Tesei, Discontinuous viscosity solutions of first order Hamilton-Jacobi equations, Preprint, (2019), arXiv: 1906.05625. Google Scholar

[5]

F. Demengel and D. Serre, Nonvanishing singular parts of measure valued solutions for scalar hyperbolic equations, Comm. Partial Differential Equations, 16 (1991), 221-254.  doi: 10.1080/03605309108820758.  Google Scholar

[6]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.  Google Scholar

[7]

A. Friedman, Mathematics in Industrial Problems, Part 8, The IMA Volumes in Mathematics and its Applications, 83. Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-1858-6.  Google Scholar

[8]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Qquasi-linear equations of parabolic type, Amer. Math. Soc., (1991). Google Scholar

[9]

T.-P. Liu and M. Pierre, Source-solutions and asymptotic behavior in conservation laws, J. Differential Equations, 51 (1984), 419-441.  doi: 10.1016/0022-0396(84)90096-2.  Google Scholar

[10]

J. Málek, J. Nečas, M. Rokyta and M. R${{\rm{\dot u}}}$žička, Weak and Measure-Valued Solutions of Evolutionary PDEs, Applied Mathematics and Mathematical Computation, 13. Chapman & Hall, London, 1996. doi: 10.1007/978-1-4899-6824-1.  Google Scholar

[11]

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

[12]

D. Serre, Systems of Conservation Laws, Vol. 1: Hyperbolicity, Entropies, Shock Waves, Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9780511612374.  Google Scholar

[13]

A. Terracina, Comparison properties for scalar conservation laws with boundary conditions, Nonlinear Anal., 28 (1997), 633-653.  doi: 10.1016/0362-546X(95)00172-R.  Google Scholar

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