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On the concentration of entropy for scalar conservation laws

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  • 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.


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  • [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.


    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.


    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.


    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.


    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.


    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.


    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.


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


    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.


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


    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.


    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.


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


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

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