Transport equation and Cauchy problem for BV vector fields and applications
Journées équations aux dérivées partielles (2004), article no. 1, 11 p.

Voir la notice de l'acte provenant de la source Numdam

DOI : 10.5802/jedp.1

Ambrosio, Luigi 1

1 Scuola Normale Superiore, Pisa
@incollection{JEDP_2004____A1_0,
     author = {Ambrosio, Luigi},
     title = {Transport equation and {Cauchy} problem for $BV$ vector fields and applications},
     booktitle = {},
     series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {1},
     pages = {1--11},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2004},
     doi = {10.5802/jedp.1},
     mrnumber = {2135356},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.5802/jedp.1/}
}
TY  - JOUR
AU  - Ambrosio, Luigi
TI  - Transport equation and Cauchy problem for $BV$ vector fields and applications
JO  - Journées équations aux dérivées partielles
PY  - 2004
SP  - 1
EP  - 11
PB  - Groupement de recherche 2434 du CNRS
UR  - http://geodesic.mathdoc.fr/articles/10.5802/jedp.1/
DO  - 10.5802/jedp.1
LA  - en
ID  - JEDP_2004____A1_0
ER  - 
%0 Journal Article
%A Ambrosio, Luigi
%T Transport equation and Cauchy problem for $BV$ vector fields and applications
%J Journées équations aux dérivées partielles
%D 2004
%P 1-11
%I Groupement de recherche 2434 du CNRS
%U http://geodesic.mathdoc.fr/articles/10.5802/jedp.1/
%R 10.5802/jedp.1
%G en
%F JEDP_2004____A1_0
Ambrosio, Luigi. Transport equation and Cauchy problem for $BV$ vector fields and applications. Journées équations aux dérivées partielles (2004), article  no. 1, 11 p. doi : 10.5802/jedp.1. http://geodesic.mathdoc.fr/articles/10.5802/jedp.1/

[1] M.Aizenman: On vector fields as generators of flows: a counterexample to Nelson’s conjecture. Ann. Math., 107 (1978), 287–296. | Zbl | MR

[2] G.Alberti: Rank-one properties for derivatives of functions with bounded variation. Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 239–274. | Zbl | MR

[3] G.Alberti & L.Ambrosio: A geometric approach to monotone functions in n . Math. Z., 230 (1999), 259–316. | Zbl | MR

[4] G.Alberti: Personal communication.

[5] F.J.Almgren: The theory of varifolds – A variational calculus in the large. Princeton University Press, 1972.

[6] L.Ambrosio, N.Fusco & D.Pallara: Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs, 2000. | Zbl | MR

[7] L.Ambrosio: Transport equation and Cauchy problem for BV vector fields. To appear on Inventiones Math. . | Zbl | MR

[8] L.Ambrosio & C.De Lellis: Existence of solutions for a class of hyperbolic systems of conservation laws in several space dimensions. International Mathematical Research Notices, 41 (2003), 2205–2220. | Zbl | MR

[9] L.Ambrosio, F.Bouchut & C.De Lellis: Well-posedness for a class of hyperbolic systems of conservation laws in several space dimensions. To appear on Comm. PDE, and available at http://cvgmt.sns.it. | Zbl

[10] L.Ambrosio, G.Crippa & S.Maniglia: Traces and fine properties of a BD class of vector fields and applications. Preprint, 2004. | Zbl | mathdoc-id

[11] L.Ambrosio: Lecture Notes on transport equation and Cauchy problem for BV vector fields and applications. Available at http://cvgmt.sns.it.

[12] L.Ambrosio, N.Gigli & G.Savaré: Gradient flows in metric spaces and in the Wasserstein space of probability measures. Birkhäuser, to appear. | Zbl | MR

[13] J.-D.Benamou & Y.Brenier: Weak solutions for the semigeostrophic equation formulated as a couples Monge-Ampere transport problem. SIAM J. Appl. Math., 58 (1998), 1450–1461. | Zbl | MR

[14] F.Bouchut & F.James: One dimensional transport equation with discontinuous coefficients. Nonlinear Analysis, 32 (1998), 891–933. | Zbl | MR

[15] F.Bouchut: Renormalized solutions to the Vlasov equation with coefficients of bounded variation. Arch. Rational Mech. Anal., 157 (2001), 75–90. | Zbl | MR

[16] A.Bressan: An ill posed Cauchy problem for a hyperbolic system in two space dimensions. Rend. Sem. Mat. Univ. Padova, 110 (2003), 103–117. | mathdoc-id | Zbl | MR | EuDML

[17] I.Capuzzo Dolcetta & B.Perthame: On some analogy between different approaches to first order PDE’s with nonsmooth coefficients. Adv. Math. Sci Appl., 6 (1996), 689–703. | Zbl | MR

[18] A.Cellina: On uniqueness almost everywhere for monotonic differential inclusions. Nonlinear Analysis, TMA, 25 (1995), 899–903. | Zbl | MR

[19] A.Cellina & M.Vornicescu: On gradient flows. Journal of Differential Equations, 145 (1998), 489–501. | Zbl | MR

[20] G.-Q.Chen & H.Frid: Extended divergence-measure fields and the Euler equation of gas dynamics. Comm. Math. Phys., 236 (2003), 251–280. | Zbl | MR

[21] F.Colombini & N.Lerner: Uniqueness of continuous solutions for BV vector fields. Duke Math. J., 111 (2002), 357–384. | Zbl | MR

[22] F.Colombini & N.Lerner: Uniqueness of L solutions for a class of conormal BV vector fields. Preprint, 2003.

[23] F.Colombini, T.Luo & J.Rauch: Uniqueness and nonuniqueness for nonsmooth divergence-free transport. Preprint, 2003.

[24] M.Cullen & W.Gangbo: A variational approach for the 2-dimensional semi-geostrophic shallow water equations. Arch. Rational Mech. Anal., 156 (2001), 241–273. | Zbl | MR

[25] M.Cullen & M.Feldman: Lagrangian solutions of semigeostrophic equations in physical space. To appear. | Zbl | MR

[26] C.Dafermos: Hyperbolic conservation laws in continuum physics. Springer Verlag, 2000. | Zbl | MR

[27] N.De Pauw: Non unicité des solutions bornées pour un champ de vecteurs BV en dehors d’un hyperplan. C.R. Math. Sci. Acad. Paris, 337 (2003), 249–252. | Zbl | MR

[28] R.J. Di Perna & P.L.Lions: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math., 98 (1989), 511–547. | Zbl | MR | EuDML

[29] M.Hauray: On Liouville transport equation with potential in BV loc . (2003) Di prossima pubblicazione su Comm. in PDE.

[30] M.Hauray: On two-dimensional Hamiltonian transport equations with L loc p coefficients. (2003) Di prossima pubblicazione su Ann. Nonlinear Analysis IHP. | Zbl | mathdoc-id

[31] L.V.Kantorovich: On the transfer of masses. Dokl. Akad. Nauk. SSSR, 37 (1942), 227–229.

[32] B.L.Keyfitz & H.C.Kranzer: A system of nonstrictly hyperbolic conservation laws arising in elasticity theory. Arch. Rational Mech. Anal. 1980, 72, 219–241. | Zbl | MR

[33] C.Le Bris & P.L.Lions: Renormalized solutions of some transport equations with partially W 1,1 velocities and applications. Annali di Matematica, 183 (2004), 97–130. | Zbl | MR

[34] N.Lerner: Transport equations with partially BV velocities. Preprint, 2004.

[35] P.L.Lions: Sur les équations différentielles ordinaires et les équations de transport. C. R. Acad. Sci. Paris Sér. I, 326 (1998), 833–838. | Zbl | MR

[36] G.Petrova & B.Popov: Linear transport equation with discontinuous coefficients. Comm. PDE, 24 (1999), 1849–1873. | Zbl | MR

[37] F.Poupaud & M.Rascle: Measure solutions to the liner multidimensional transport equation with non-smooth coefficients. Comm. PDE, 22 (1997), 337–358. | Zbl | MR

[38] L.C.Young: Lectures on the calculus of variations and optimal control theory, Saunders, 1969. | Zbl | MR

Cité par Sources :