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On the regularity up to the boundary for certain nonlinear elliptic systems
On the concentration of entropy for scalar conservation laws
| 1. | SISSA, via Bonomea 265, Trieste, I-34163, Italy, Italy |
References:
| [1] |
L. Ambrosio and C. De Lellis, A note on admissible solutions of 1d scalar conservation laws and 2d Hamilton-Jacobi equations, J. Hyperbolic Diff. Equ., 1 (2004), 813-826.
doi: 10.1142/S0219891604000263. |
| [2] |
D. Amadori, Initial-boundary value problems for nonlinear systems of conservation laws, NoDEA Nonlinear Differential Equations Appl., 4 (1997), 1-42.
doi: 10.1007/PL00001406. |
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C. Bardos, A. 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. |
| [4] |
G. Bellettini, L. Bertini, M. Mariani and M. Novaga, $\Gamma$-entropy cost for scalar conservation laws, Archive for Rational Mechanics and Analysis, 195 (2010), 261-309.
doi: 10.1007/s00205-008-0197-2. |
| [5] |
S. Bianchini and L. Caravenna, SBV regularity for genuinely nonlinear, strictly hyperbolic systems of conservation laws in one space dimension, Communications in Mathematical Physics, 313 (2012), 1-33.
doi: 10.1007/s00220-012-1480-5. |
| [6] |
S. Bianchini and L. Yu, Structure of entropy solutions to general scalar conservation laws in one space dimension, J. Math. Anal. Appl., 428 (2015), 356-386.
doi: 10.1016/j.jmaa.2015.03.006. |
| [7] |
A. Bressan and P. G. LeFloch, Structural stability and regularity of entropy solutions to hyperbolic systems of conservation laws, Indiana Univ. Math. J., 48 (1999), 43-84.
doi: 10.1512/iumj.1999.48.1524. |
| [8] |
A. Bressan, Hyperbolic Systems of Conservation Laws, Oxford Lecture Series in Mathematics and its Applications, vol. 20, Oxford University Press, Oxford, 2000. |
| [9] |
C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Third edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-642-04048-1. |
| [10] |
S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255. |
| [11] |
C. De Lellis, F. Otto and M. Westdickenberg, Structure of entropy solutions for multi-dimensional scalar conservation laws, Archive for Rational Mechanics and Analysis, 170 (2003), 137-184.
doi: 10.1007/s00205-003-0270-9. |
| [12] |
C. De Lellis and T. Rivière, Concentration estimates for entropy measures, Journal de Mathématiques Pures et Appliquées, 82 (2003), 1343-1367.
doi: 10.1016/S0021-7824(03)00061-8. |
| [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] |
D. Serre, Systems of Conservation Laws. 1, Cambridge University Press, Cambridge, 1999.
doi: 10.1017/CBO9780511612374. |
show all references
References:
| [1] |
L. Ambrosio and C. De Lellis, A note on admissible solutions of 1d scalar conservation laws and 2d Hamilton-Jacobi equations, J. Hyperbolic Diff. Equ., 1 (2004), 813-826.
doi: 10.1142/S0219891604000263. |
| [2] |
D. Amadori, Initial-boundary value problems for nonlinear systems of conservation laws, NoDEA Nonlinear Differential Equations Appl., 4 (1997), 1-42.
doi: 10.1007/PL00001406. |
| [3] |
C. Bardos, A. 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. |
| [4] |
G. Bellettini, L. Bertini, M. Mariani and M. Novaga, $\Gamma$-entropy cost for scalar conservation laws, Archive for Rational Mechanics and Analysis, 195 (2010), 261-309.
doi: 10.1007/s00205-008-0197-2. |
| [5] |
S. Bianchini and L. Caravenna, SBV regularity for genuinely nonlinear, strictly hyperbolic systems of conservation laws in one space dimension, Communications in Mathematical Physics, 313 (2012), 1-33.
doi: 10.1007/s00220-012-1480-5. |
| [6] |
S. Bianchini and L. Yu, Structure of entropy solutions to general scalar conservation laws in one space dimension, J. Math. Anal. Appl., 428 (2015), 356-386.
doi: 10.1016/j.jmaa.2015.03.006. |
| [7] |
A. Bressan and P. G. LeFloch, Structural stability and regularity of entropy solutions to hyperbolic systems of conservation laws, Indiana Univ. Math. J., 48 (1999), 43-84.
doi: 10.1512/iumj.1999.48.1524. |
| [8] |
A. Bressan, Hyperbolic Systems of Conservation Laws, Oxford Lecture Series in Mathematics and its Applications, vol. 20, Oxford University Press, Oxford, 2000. |
| [9] |
C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Third edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-642-04048-1. |
| [10] |
S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255. |
| [11] |
C. De Lellis, F. Otto and M. Westdickenberg, Structure of entropy solutions for multi-dimensional scalar conservation laws, Archive for Rational Mechanics and Analysis, 170 (2003), 137-184.
doi: 10.1007/s00205-003-0270-9. |
| [12] |
C. De Lellis and T. Rivière, Concentration estimates for entropy measures, Journal de Mathématiques Pures et Appliquées, 82 (2003), 1343-1367.
doi: 10.1016/S0021-7824(03)00061-8. |
| [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] |
D. Serre, Systems of Conservation Laws. 1, Cambridge University Press, Cambridge, 1999.
doi: 10.1017/CBO9780511612374. |
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