Minimal entropy conditions for Burgers equation
Authors:
Camillo De Lellis, Felix Otto and Michael Westdickenberg
Journal:
Quart. Appl. Math. 62 (2004), 687-700
MSC:
Primary 35L65; Secondary 35L45, 35L60, 35L67, 35Q53
DOI:
https://doi.org/10.1090/qam/2104269
MathSciNet review:
MR2104269
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Abstract: We consider uniformly convex, $1 - d$ scalar conservation laws. We show that a single uniformly convex entropy is sufficient to characterize a Kruzhkov solution. The proof uses the concept of viscosity solution for the related Hamilton-Jacobi equation.
- Luigi Ambrosio, Nicola Fusco, and Diego Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. MR 1857292
- Luigi Ambrosio, Myriam Lecumberry, and Tristan Rivière, A viscosity property of minimizing micromagnetic configurations, Comm. Pure Appl. Math. 56 (2003), no. 6, 681–688. MR 1959737, DOI https://doi.org/10.1002/cpa.10070
- Michael G. Crandall and Pierre-Louis Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), no. 1, 1–42. MR 690039, DOI https://doi.org/10.1090/S0002-9947-1983-0690039-8
- Constantine M. Dafermos, Hyperbolic conservation laws in continuum physics, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2000. MR 1763936
- Lorenzo Giacomelli and Felix Otto, New bounds for the Kuramoto-Sivashinsky equation, Comm. Pure Appl. Math. 58 (2005), no. 3, 297–318. MR 2116616, DOI https://doi.org/10.1002/cpa.20031
S. N. Kruzhkov, First order quasilinear equations in several independent variables, Math. Sb. 123 (1970), pp. 228–255; English transl. in Math. USSR Sbornik 10 (1970), pp. 217–243.
- Pierre-Louis Lions, Generalized solutions of Hamilton-Jacobi equations, Research Notes in Mathematics, vol. 69, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR 667669
O. A. Oleinik, Discontinuous solutions of nonlinear differential equations, Usp. Mat. Nauk. 12 (1957), pp. 3–73; English transl. in AMS Transl. 26 (1963), pp. 1155–1163.
- E. Yu. Panov, Uniqueness of the solution of the Cauchy problem for a first-order quasilinear equation with an admissible strictly convex entropy, Mat. Zametki 55 (1994), no. 5, 116–129, 159 (Russian, with Russian summary); English transl., Math. Notes 55 (1994), no. 5-6, 517–525. MR 1296003, DOI https://doi.org/10.1007/BF02110380
B. Riemann, Über die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite, Abh. d. Königl. Ges. d. Wiss. zu Göttingen, Bd. 8 (1858/59) (Math. Cl.), pp. 43–65.
L. Ambrosio, N. Fusco, D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, Clarendon Press, Oxford, 2000.
L. Ambrosio, M. Lecumberry, T. Rivière, A viscosity property of minimizing micromagnetic configurations, Comm. Pure Appl. Math. 56 (2003), pp. 681–688.
M. G. Crandall, P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), pp. 1–42.
C. Dafermos, Hyperbolic conservation laws in continuum physics, volume 325 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 2000.
L. Giacomelli, F. Otto, New bounds for the Kuramoto-Shivashinsky equation, to appear in Comm. Pure Appl. Math.
S. N. Kruzhkov, First order quasilinear equations in several independent variables, Math. Sb. 123 (1970), pp. 228–255; English transl. in Math. USSR Sbornik 10 (1970), pp. 217–243.
P. L. Lions, Generalized Solutions of Hamilton-Jacobi Equations, volume 69 of Research Notes in Mathematics, Pitman Advanced Publishing Program, London, 1982.
O. A. Oleinik, Discontinuous solutions of nonlinear differential equations, Usp. Mat. Nauk. 12 (1957), pp. 3–73; English transl. in AMS Transl. 26 (1963), pp. 1155–1163.
E. Y. Panov, Uniqueness of the Cauchy problem for a first order quasilinear equation with one admissible strictly convex entropy, Mat. Zametki 55 (1994), pp. 116–129, 159; English transl. in Math. Notes 55 (1994), pp. 517–525.
B. Riemann, Über die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite, Abh. d. Königl. Ges. d. Wiss. zu Göttingen, Bd. 8 (1858/59) (Math. Cl.), pp. 43–65.
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