Ordinary differential equations with rough coefficients and the renormalization theorem of Ambrosio [after Ambrosio, DiPerna, Lions]

De Lellis, Camillo

Séminaire Bourbaki - Volume 2006/2007 - Exposés 967-981, Astérisque, no. 317 (2008), Exposé no. 972, 29 p.

MR Zbl

@incollection{AST_2008__317__175_0,
     author = {De Lellis, Camillo},
     title = {Ordinary differential equations with rough coefficients and the renormalization theorem of {Ambrosio} [after {Ambrosio,} {DiPerna,} {Lions]}},
     booktitle = {S\'eminaire Bourbaki - Volume 2006/2007  - Expos\'es 967-981},
     series = {Ast\'erisque},
     note = {talk:972},
     pages = {175--203},
     year = {2008},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {317},
     mrnumber = {2487734},
     zbl = {1169.35060},
     language = {en},
     url = {https://www.numdam.org/item/AST_2008__317__175_0/}
}

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TI  - Ordinary differential equations with rough coefficients and the renormalization theorem of Ambrosio [after Ambrosio, DiPerna, Lions]
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%A De Lellis, Camillo
%T Ordinary differential equations with rough coefficients and the renormalization theorem of Ambrosio [after Ambrosio, DiPerna, Lions]
%B Séminaire Bourbaki - Volume 2006/2007  - Exposés 967-981
%A Collectif
%S Astérisque
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%D 2008
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%N 317
%I Société mathématique de France
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%G en
%F AST_2008__317__175_0

De Lellis, Camillo. Ordinary differential equations with rough coefficients and the renormalization theorem of Ambrosio [after Ambrosio, DiPerna, Lions], dans Séminaire Bourbaki - Volume 2006/2007  - Exposés 967-981, Astérisque, no. 317 (2008), Exposé no. 972, 29 p.. https://www.numdam.org/item/AST_2008__317__175_0/

Bibliographie
Cité par

[1] M. Aizenman - "On vector fields as generators of flows: a counterexample to Nelson's conjecture", Ann. Math. (2) 107 (1978), p. 287-296. | MR | Zbl | DOI

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

[3] G. Alberti, S. Bianchini & C. Gianluca - In préparation.

[4] L. Ambrosio - "Lecture notes on transport equation and Cauchy problem for bv vector fields and applications", Lectures of a course given in Luminy, October 2003.

[5] L. Ambrosio, "Transport equation and Cauchy problem for non-smooth vector fields", Lecture Notes of the CIME Summer school in Cetrary, June 27-July 2, 2005. | Zbl | MR

[6] L. Ambrosio, "Transport equation and Cauchy problem for $B V$ vector fields", Invent. Math. 158 (2004), p. 227-260. | MR | Zbl | DOI

[7] L. Ambrosio, F. Bouchut & C. De Lellis - "Well-posedness for a class of hyperbolic systemes of conservation laws in several space dimensions", Comm. Partial Differential Equations 29 (2004), p. 1635-1651. | MR | Zbl | DOI

[8] L. Ambrosio & G. Crippa - "Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields", in Transport Equations and Multi- $D$ Hyperbolic Conservation Laws, Lecture Notes of the Unione Matematica Italiana, Springer Verlag, Berlin-Heidelberg, 2008. | MR | Zbl | DOI

[9] L. Ambrosio, G. Crippa & S. Maniglia - "Traces and fine properties of a $B D$ class of vector fields and applications", Ann. Fac. Sci. Toulouse Math. (6) 14 (2005), p. 527-561. | MR | Zbl | EuDML | Numdam | DOI

[10] L. Ambrosio & C. De Lellis - "Existence of solutions for a class of hyperbolic Systems of conservation laws in several space dimensions", Int. Math. Res. Not. 41 (2003), p. 2205-2220. | MR | Zbl | DOI

[11] L. Ambrosio, C. De Lellis & J. Maly - "On the chain rule for the divergence of vector fields: applications, partial results, open problems", in Perspectives in nonlinear partial differential équations, 2007, p. 31-67, | MR | Zbl | DOI

[11] L. Ambrosio, C. De Lellis & J. Maly - "On the chain rule for the divergence of vector fields: applications, partial results, open problems", Contemp. Math. 446, Amer. Math. Soc., Providence, RI. | MR | Zbl

[12] L. Ambrosio, N. Fusco & D. Pallara - Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, 2000. | MR | Zbl

[13] L. Ambrosio, M. Lecumberry & S. Maniglia - "Lipschitz regularity and approximate differentiability of the DiPerna-Lions flow", Rend. Sem. Mat. Univ. Padova 114 (2005), p. 29-50 (2006). | MR | EuDML | Zbl | Numdam

[14] L. Ambrosio & J. Malý - "Very weak notions of differentiability", Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), p. 447-455. | MR | Zbl | DOI

[15] F. Bouchut - "Renormalized solutions to the Vlasov equation with coefficients of bounded variation", Arch. Ration. Mech. Anal. 157 (2001), p. 75-90. | MR | Zbl | DOI

[16] F. Bouchut & L. Desvillettes - "On two-dimensional Hamiltonian transport equations with continuous coefficients", Differential Intégral Equations 14 (2001), p. 1015-1024. | MR | Zbl

[17] A. Bressan - "An ill posed Cauchy problem for a hyperbolic System in two space dimensions", Rend. Sem. Mat. Univ. Padova 110 (2003), p. 103-117. | MR | Zbl | EuDML | Numdam

[18] A. Bressan, "A lemma and a conjecture on the cost of rearrangements", Rend. Sem. Mat. Univ. Padova 110 (2003), p. 97-102. | MR | Zbl | EuDML | Numdam

[19] 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), p. 689-703. | MR | Zbl

[20] F. Colombini, G. Crippa & J. Rauch - "A note on two-dimensional transport with bounded divergence", Comm. Partial Differential Equations 31 (2006), p. 1109-1115. | MR | Zbl | DOI

[21] F. Colombini & N. Lerner - "Uniqueness of continuous solutions for $B V$ vector fields", Duke Math. J. 111 (2002), p. 357-384. | MR | Zbl | DOI

[22] F. Colombini, T. Luo & J. Rauch - "Uniqueness and nonuniqueness for nonsmooth divergence free transport", in Séminaire: Équations aux Dérivées Partielles, 2002-2003, Sémin. Équ. Dériv. Partielles, École polytech., 2003, p. 21. | MR | Zbl | EuDML | Numdam

[23] G. Crippa & C. De Lellis - "Oscillatory solutions to transport equations", Indiana Univ. Math. J. 55 (2006), p. 1-13. | MR | Zbl | DOI

[24] G. Crippa & C. De Lellis, "Estimates and regularity results for the DiPerna-Lions flow", J. reine angew. Math. 616 (2008), p. 15-46. | MR | Zbl

[25] C. De Lellis - "Notes on hyperbolic Systems of conservation laws and transport equations", Handbook of evolutionary differential équations, Vol. III, 2006. | MR | Zbl

[26] C. De Lellis, "A note on Alberti's rank-one theorem", in Transport Equations and Multi- $D$ Hyperbolic Conservation Laws, Lecture Notes of the Unione Matematica Italiana, Springer Verlag, Berlin-Heidelberg, 2008. | MR | Zbl | DOI

[27] N. Depauw - "Non-unicité du transport par un champ de vecteurs presque $B V$ ", in Séminaire: Équations aux Dérivées Partielles, 2002-2003, Sémin. Equ. Dériv. Partielles, École polytech., 2003, p. 9. | MR | Zbl | EuDML | Numdam

[28] R. J. Diperna & P.-L. Lions - "Ordinary differential equations, transport theory and Sobolev spaces", Invent. Math. 98 (1989), p. 511-547. | MR | Zbl | EuDML | DOI

[29] L. C. Evans - Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, 1998. | MR | Zbl

[30] L. C. Evans & R. F. Gariepy - Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, 1992. | MR | Zbl

[31] M. Hauray - "On two-dimensional Hamiltonian transport equations with $L \overset{p}{l o c}$ coefficients", Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003), p. 625-644. | MR | Zbl | EuDML | Numdam | DOI

[32] C. Le Bris & P.-L. Lions - "Renormalized solutions of some transport equations with partially $W^{1, 1}$ velocities and applications", Ann. Mat. Pura Appl. (4) 183 (2004), p. 97-130. | MR | Zbl | DOI

[33] P.-L. Lions - "Sur les équations différentielles ordinaires et les équations de transport", C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), p. 833-838. | MR | Zbl | DOI