-
Previous Article
On the spectral theory of positive operators and PDE applications
- DCDS Home
- This Issue
-
Next Article
A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic fisher kpp equation
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 |
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.
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. |
[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. |
[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. |
[4] |
M. Bertsch, F. Smarrazzo, A. Terracina and A. Tesei, Discontinuous viscosity solutions of first order Hamilton-Jacobi equations, Preprint, (2019), arXiv: 1906.05625. |
[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. |
[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. |
[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. |
[8] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Qquasi-linear equations of parabolic type, Amer. Math. Soc., (1991). |
[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. |
[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. |
[11] |
F. Otto,
Initial-boundary value problem for a scalar conservation law, Comptes Rendus Acad. Sci. Paris Sér. I Math., 322 (1996), 729-734.
|
[12] |
D. Serre, Systems of Conservation Laws, Vol. 1: Hyperbolicity, Entropies, Shock Waves,
Cambridge University Press, Cambridge, 1999.
doi: 10.1017/CBO9780511612374. |
[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. |
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. |
[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. |
[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. |
[4] |
M. Bertsch, F. Smarrazzo, A. Terracina and A. Tesei, Discontinuous viscosity solutions of first order Hamilton-Jacobi equations, Preprint, (2019), arXiv: 1906.05625. |
[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. |
[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. |
[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. |
[8] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Qquasi-linear equations of parabolic type, Amer. Math. Soc., (1991). |
[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. |
[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. |
[11] |
F. Otto,
Initial-boundary value problem for a scalar conservation law, Comptes Rendus Acad. Sci. Paris Sér. I Math., 322 (1996), 729-734.
|
[12] |
D. Serre, Systems of Conservation Laws, Vol. 1: Hyperbolicity, Entropies, Shock Waves,
Cambridge University Press, Cambridge, 1999.
doi: 10.1017/CBO9780511612374. |
[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. |
[1] |
Alberto Bressan, Marta Lewicka. A uniqueness condition for hyperbolic systems of conservation laws. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 673-682. doi: 10.3934/dcds.2000.6.673 |
[2] |
Stefano Bianchini. A note on singular limits to hyperbolic systems of conservation laws. Communications on Pure and Applied Analysis, 2003, 2 (1) : 51-64. doi: 10.3934/cpaa.2003.2.51 |
[3] |
Xavier Litrico, Vincent Fromion, Gérard Scorletti. Robust feedforward boundary control of hyperbolic conservation laws. Networks and Heterogeneous Media, 2007, 2 (4) : 717-731. doi: 10.3934/nhm.2007.2.717 |
[4] |
Eitan Tadmor. Perfect derivatives, conservative differences and entropy stable computation of hyperbolic conservation laws. Discrete and Continuous Dynamical Systems, 2016, 36 (8) : 4579-4598. doi: 10.3934/dcds.2016.36.4579 |
[5] |
Darko Mitrovic. New entropy conditions for scalar conservation laws with discontinuous flux. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 1191-1210. doi: 10.3934/dcds.2011.30.1191 |
[6] |
Tatsien Li, Bopeng Rao, Zhiqiang Wang. Exact boundary controllability and observability for first order quasilinear hyperbolic systems with a kind of nonlocal boundary conditions. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 243-257. doi: 10.3934/dcds.2010.28.243 |
[7] |
Mapundi K. Banda, Michael Herty. Numerical discretization of stabilization problems with boundary controls for systems of hyperbolic conservation laws. Mathematical Control and Related Fields, 2013, 3 (2) : 121-142. doi: 10.3934/mcrf.2013.3.121 |
[8] |
Stefano Bianchini, Elio Marconi. On the concentration of entropy for scalar conservation laws. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 73-88. doi: 10.3934/dcdss.2016.9.73 |
[9] |
Tai-Ping Liu, Shih-Hsien Yu. Hyperbolic conservation laws and dynamic systems. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 143-145. doi: 10.3934/dcds.2000.6.143 |
[10] |
Yanning Li, Edward Canepa, Christian Claudel. Efficient robust control of first order scalar conservation laws using semi-analytical solutions. Discrete and Continuous Dynamical Systems - S, 2014, 7 (3) : 525-542. doi: 10.3934/dcdss.2014.7.525 |
[11] |
Tatsien Li (Daqian Li). Global exact boundary controllability for first order quasilinear hyperbolic systems. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1419-1432. doi: 10.3934/dcdsb.2010.14.1419 |
[12] |
K. T. Joseph, Philippe G. LeFloch. Boundary layers in weak solutions of hyperbolic conservation laws II. self-similar vanishing diffusion limits. Communications on Pure and Applied Analysis, 2002, 1 (1) : 51-76. doi: 10.3934/cpaa.2002.1.51 |
[13] |
Gui-Qiang Chen, Monica Torres. On the structure of solutions of nonlinear hyperbolic systems of conservation laws. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1011-1036. doi: 10.3934/cpaa.2011.10.1011 |
[14] |
Constantine M. Dafermos. A variational approach to the Riemann problem for hyperbolic conservation laws. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 185-195. doi: 10.3934/dcds.2009.23.185 |
[15] |
Fumioki Asakura, Andrea Corli. The path decomposition technique for systems of hyperbolic conservation laws. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 15-32. doi: 10.3934/dcdss.2016.9.15 |
[16] |
Dirk Hartmann, Isabella von Sivers. Structured first order conservation models for pedestrian dynamics. Networks and Heterogeneous Media, 2013, 8 (4) : 985-1007. doi: 10.3934/nhm.2013.8.985 |
[17] |
Dominic Veconi. SRB measures of singular hyperbolic attractors. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3415-3430. doi: 10.3934/dcds.2022020 |
[18] |
Christophe Prieur. Control of systems of conservation laws with boundary errors. Networks and Heterogeneous Media, 2009, 4 (2) : 393-407. doi: 10.3934/nhm.2009.4.393 |
[19] |
Darko Mitrovic, Ivan Ivec. A generalization of $H$-measures and application on purely fractional scalar conservation laws. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1617-1627. doi: 10.3934/cpaa.2011.10.1617 |
[20] |
Zhi-Qiang Shao. Lifespan of classical discontinuous solutions to the generalized nonlinear initial-boundary Riemann problem for hyperbolic conservation laws with small BV data: shocks and contact discontinuities. Communications on Pure and Applied Analysis, 2015, 14 (3) : 759-792. doi: 10.3934/cpaa.2015.14.759 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]