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doi: 10.3934/dcds.2020041

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

 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.

Citation: Michiel Bertsch, Flavia Smarrazzo, Andrea Terracina, Alberto Tesei. Signed Radon measure-valued solutions of flux saturated scalar conservation laws. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020041
##### References:
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##### 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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