Abstract
In this note we present a unifying approach for two classes of first-order partial differential equations: we introduce the notion of Lagrangian representation in the settings of continuity equation and scalar conservation laws. This yields, on the one hand, the uniqueness of weak solutions to transport equation driven by a two-dimensional BV nearly incompressible vector field. On the other hand, it is proved that the entropy dissipation measure for scalar conservation laws in one space dimension is concentrated on countably many Lipschitz curves.
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G. Alberti, S. Bianchini, and G. Crippa, “Structure of level sets and Sard-type properties of Lipschitz maps,” Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 12, No. 4, 863–902 (2013).
G. Alberti, S. Bianchini, and G. Crippa, “A uniqueness result for the continuity equation in two dimensions,” J. Eur. Math. Soc. (JEMS), 16, No. 2, 201–234 (2014).
L. Ambrosio, “Transport equation and Cauchy problem for BV vector fields,” Invent. Math., 158, No. 2, 227–260 (2004).
L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford (2000).
C. Bardos, A. Y. le Roux, and J.-C. Nédélec, “First order quasilinear equations with boundary conditions,” Commun. Part. Differ. Equ., 4, 1017–1034 (1979).
S. Bianchini and S. Bonicatto, “A uniqueness result for the decomposition of vector fields in ℝd,” Preprint SISSA, 15/2017/MATE.
S. Bianchini, A. Bonicatto, and N. A. Gusev, “Renormalization for autonomous nearly incompressible BV vector fields in two dimensions,” SIAM J. Math. Anal., 48, No. 1, 1–33 (2016).
S. Bianchini and N. A. Gusev, “Steady nearly incompressible vector fields in two-dimension: chain rule and renormalization,” Arch. Ration. Mech. Anal., 222, No. 2, 451–505 (2016).
S. Bianchini and E. Marconi, “On the concentration of entropy for scalar conservation laws,” Discrete Contin. Dyn. Syst. Ser. S, 9, 73–88 (2016).
S. Bianchini and E. Marconi, “On the structure of L∞ entropy solutions to scalar conservation laws in one-space dimension,” Arch. Ration. Mech. Anal., 226, No. 1, 441–493 (2017).
S. Bianchini, E. Marconi, and S. Bonicatto, “A Lagrangian approach to multidimensional scalar conservation laws,” Preprint SISSA, 36/2017/MATE.
S. Bianchini and S. Modena, “Quadratic interaction functional for general systems of conservation laws,” Commun. Math. Phys., 338, No. 3, 1075–1152 (2015).
S. Bianchini and L. Yu, “Structure of entropy solutions to general scalar conservation laws in one space dimension,” J. Math. Anal. Appl., 428, No. 1, 356–386 (2015).
A. Bressan, “An ill posed Cauchy problem for a hyperbolic system in two space dimensions,” Rend. Semin. Mat. Univ. Padova, 110, 103–117 (2003).
K. S. Cheng, “A regularity theorem for a nonconvex scalar conservation law,” J. Differ. Equ., 61, 79–127 (1986).
C. M. Dafermos, “Continuous solutions for balance laws,” Ric. Mat., 55, No. 1, 79–92 (2006).
C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer, Berlin–Heidelberg (2010).
C. de Lellis, “Notes on hyperbolic systems of conservation laws and transport equations,” Handb. Differ. Equ., 3, 277–382 (2007).
C. de Lellis and T. Riviere, “Concentration estimates for entropy measures,” J. Math. Pures Appl. (9), 82, 1343–1367 (2003).
R. J. DiPerna and P.-L. Lions, “Ordinary differential equations, transport theory and Sobolev spaces,” Invent. Math., 98, No. 3, 511–547 (1989).
O. A. Oleĭnik, “Discontinuous solutions of non-linear differential equations,” Am. Math. Soc. Transl. Ser. 2, 26, 95–172 (1963).
F. Otto, “Initial-boundary value problem for a scalar conservation law,” C. R. Math. Acad. Sci. Paris, 322, No. 8, 729–734 (1996).
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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 63, No. 3, Differential and Functional Differential Equations, 2017.
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Bianchini, S., Bonicatto, P. & Marconi, E. Lagrangian Representations for Linear and Nonlinear Transport. J Math Sci 253, 642–659 (2021). https://doi.org/10.1007/s10958-021-05259-9
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DOI: https://doi.org/10.1007/s10958-021-05259-9