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February  2016, 9(1): 73-88. doi: 10.3934/dcdss.2016.9.73

On the concentration of entropy for scalar conservation laws

Stefano Bianchini 1, and  Elio Marconi 1,

1. 

SISSA, via Bonomea 265, Trieste, I-34163, Italy, Italy

Received  September 2014 Revised  February 2015 Published  December 2015

We prove that the entropy for an $L^\infty$-solution to a scalar conservation laws with continuous initial data is concentrated on a countably $1$-rectifiable set. To prove this result we introduce the notion of Lagrangian representation of the solution and give regularity estimates on the solution.
Keywords: Conservation laws, shocks, entropy solutions, Lagrangian representation., concentration.
Mathematics Subject Classification: Primary: 35L6.
Citation: Stefano Bianchini, Elio Marconi. On the concentration of entropy for scalar conservation laws. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 73-88. doi: 10.3934/dcdss.2016.9.73
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

A. Bressan, Hyperbolic Systems of Conservation Laws, Oxford Lecture Series in Mathematics and its Applications, vol. 20, Oxford University Press, Oxford, 2000.  Google Scholar

[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.  Google Scholar

[10]

S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

A. Bressan, Hyperbolic Systems of Conservation Laws, Oxford Lecture Series in Mathematics and its Applications, vol. 20, Oxford University Press, Oxford, 2000.  Google Scholar

[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.  Google Scholar

[10]

S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

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

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