# American Institute of Mathematical Sciences

June  2020, 40(6): 3143-3169. 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, 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 [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

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
 [1] Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217 [2] Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045 [3] Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301 [4] Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248 [5] Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267 [6] Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, 2021, 20 (1) : 405-425. doi: 10.3934/cpaa.2020274 [7] Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $p$ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442 [8] Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340 [9] Xuefeng Zhang, Yingbo Zhang. Fault-tolerant control against actuator failures for uncertain singular fractional order systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 1-12. doi: 10.3934/naco.2020011 [10] Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048 [11] José Madrid, João P. G. Ramos. On optimal autocorrelation inequalities on the real line. Communications on Pure & Applied Analysis, 2021, 20 (1) : 369-388. doi: 10.3934/cpaa.2020271 [12] Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380 [13] Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434 [14] Shipra Singh, Aviv Gibali, Xiaolong Qin. Cooperation in traffic network problems via evolutionary split variational inequalities. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020170 [15] Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384 [16] Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 507-523. doi: 10.3934/dcds.2020266 [17] João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138 [18] Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345 [19] Craig Cowan, Abdolrahman Razani. Singular solutions of a Lane-Emden system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 621-656. doi: 10.3934/dcds.2020291 [20] Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453

2019 Impact Factor: 1.338