Geodesic
Lagrangian representations for linear and nonlinear transport
Contemporary Mathematics. Fundamental Directions, Differential and functional differential equations, Tome 63 (2017) no. 3, pp. 418-436.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{CMFD_2017_63_3_a2,
     author = {S. Bianchini and P. Bonicatto and E. Marconi},
     title = {Lagrangian representations for linear and nonlinear transport},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {418--436},
     publisher = {mathdoc},
     volume = {63},
     number = {3},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2017_63_3_a2/}
}
TY  - JOUR
AU  - S. Bianchini
AU  - P. Bonicatto
AU  - E. Marconi
TI  - Lagrangian representations for linear and nonlinear transport
JO  - Contemporary Mathematics. Fundamental Directions
PY  - 2017
SP  - 418
EP  - 436
VL  - 63
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CMFD_2017_63_3_a2/
LA  - ru
ID  - CMFD_2017_63_3_a2
ER  - 
%0 Journal Article
%A S. Bianchini
%A P. Bonicatto
%A E. Marconi
%T Lagrangian representations for linear and nonlinear transport
%J Contemporary Mathematics. Fundamental Directions
%D 2017
%P 418-436
%V 63
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CMFD_2017_63_3_a2/
%G ru
%F CMFD_2017_63_3_a2
S. Bianchini; P. Bonicatto; E. Marconi. Lagrangian representations for linear and nonlinear transport. Contemporary Mathematics. Fundamental Directions, Differential and functional differential equations, Tome 63 (2017) no. 3, pp. 418-436. http://geodesic.mathdoc.fr/item/CMFD_2017_63_3_a2/

[1] Alberti G., Bianchini S., Crippa G., “Structure of level sets and Sard-type properties of Lipschitz maps”, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 12:4 (2013), 863–902 | MR

[2] Alberti G., Bianchini S., Crippa G., “A uniqueness result for the continuity equation in two dimensions”, J. Eur. Math. Soc. (JEMS), 16:2 (2014), 201–234 | DOI | MR

[3] Ambrosio L., “Transport equation and Cauchy problem for BV vector fields”, Invent. Math., 158:2 (2004), 227–260 | DOI | MR

[4] Ambrosio L., Fusco N., Pallara D., Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford, 2000 | MR

[5] Bardos C., le Roux A. Y., Nédélec J.-C., “First order quasilinear equations with boundary conditions”, Commun. Part. Differ. Equ., 4 (1979), 1017–1034 | DOI | MR

[6] Bianchini S., Bonicatto S., A uniqueness result for the decomposition of vector fields in $\mathbb R^d$, Preprint 15/2017/MATE, SISSA

[7] Bianchini S., Bonicatto A., Gusev N. A., “Renormalization for autonomous nearly incompressible BV vector fields in two dimensions”, SIAM J. Math. Anal., 48:1 (2016), 1–33 | DOI | MR

[8] Bianchini S., Gusev N. A., “Steady nearly incompressible vector fields in two-dimension: chain rule and renormalization”, Arch. Ration. Mech. Anal., 222:2 (2016), 451–505 | DOI | MR

[9] Bianchini S., Marconi E., “On the concentration of entropy for scalar conservation laws”, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 73–88 | MR

[10] Bianchini S., Marconi E., “On the structure of $L^\infty$ entropy solutions to scalar conservation laws in one-space dimension”, Arch. Ration. Mech. Anal., 226:1 (2017), 441–493 | DOI | MR

[11] Bianchini S., Marconi E., Bonicatto S., A Lagrangian approach to multidimensional scalar conservation laws, Preprint 36/2017/MATE, SISSA

[12] Bianchini S., Modena S., “Quadratic interaction functional for general systems of conservation laws”, Commun. Math. Phys., 338:3 (2015), 1075–1152 | DOI | MR

[13] Bianchini S., Yu L., “Structure of entropy solutions to general scalar conservation laws in one space dimension”, J. Math. Anal. Appl., 428:1 (2015), 356–386 | DOI | MR

[14] Bressan A., “An ill posed Cauchy problem for a hyperbolic system in two space dimensions”, Rend. Semin. Mat. Univ. Padova, 110 (2003), 103–117 | MR

[15] Cheng K. S., “A regularity theorem for a nonconvex scalar conservation law”, J. Differ. Equ., 61 (1986), 79–127 | DOI | MR

[16] Dafermos C. M., “Continuous solutions for balance laws”, Ric. Mat., 55:1 (2006), 79–92 | DOI | MR

[17] Dafermos C. M., Hyperbolic Conservation Laws in Continuum Physics, Springer, Berlin–Heidelberg, 2010 | MR

[18] de Lellis C., “Notes on hyperbolic systems of conservation laws and transport equations”, Handb. Differ. Equ., 3 (2007), 277–382 | MR

[19] de Lellis C., Riviere T., “Concentration estimates for entropy measures”, J. Math. Pures Appl. (9), 82 (2003), 1343–1367 | DOI | MR

[20] DiPerna R. J., Lions P.-L., “Ordinary differential equations, transport theory and Sobolev spaces”, Invent. Math., 98:3 (1989), 511–547 | DOI | MR

[21] Oleĭnik O. A., “Discontinuous solutions of non-linear differential equations”, Am. Math. Soc. Transl. Ser. 2, 26 (1963), 95–172 | MR

[22] Otto F., “Initial-boundary value problem for a scalar conservation law”, C. R. Math. Acad. Sci. Paris, 322:8 (1996), 729–734 | MR