Article Contents
Article Contents

# On the concentration of entropy for scalar conservation laws

• 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.
Mathematics Subject Classification: Primary: 35L65.

 Citation:

•  [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.