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Ambrosio, Luigi:
Problema di trasporto e equazione di Cauchy per campi vettoriali a variazione limitata
Bollettino dell'Unione Matematica Italiana Serie 8 7-B (2004), fasc. n.3, p. 529-543, Unione Matematica Italiana (Italian)
pdf (269 Kb), djvu (190 Kb). | MR2101650 | Zbl 1182.35083

Sunto

In questa conferenza descrivo alcuni recenti sviluppi relativi al problema dell'unicità per l'equazione differenziale ordinaria e per l'equazione di continuità per campi vettoriali debolmente differenziabili. Descrivo infine un'applicazione di questi risultati a un sistema di leggi di conservazione.
Referenze Bibliografiche
[1] M. AIZENMAN, On vector fields as generators of flows: a counterexample to Nelson's conjecture, Ann. Math., 107 (1978), 287-296. | MR 482853 | Zbl 0394.28012
[2] G. ALBERTI, Rank-one properties for derivatives of functions with bounded variation, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 239-274. | MR 1215412 | Zbl 0791.26008
[3] G. ALBERTI-L. AMBROSIO, A geometric approach to monotone functions in $\mathbb{R}^n$, Math. Z., 230 (1999), 259-316. | MR 1676726 | Zbl 0934.49025
[4] F. J. ALMGREN, The theory of varifolds - A variational calculus in the large, Princeton University Press, 1972.
[5] L. AMBROSIO-N. FUSCO-D. PALLARA, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, 2000. | MR 1857292 | Zbl 0957.49001
[6] L. AMBROSIO, Transport equation and Cauchy problem for BV vector fields, In corso di stampa su Inventiones Math.. | Zbl 1075.35087
[7] 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. | MR 2000967 | Zbl 1061.35048
[8] L. AMBROSIO-F. BOUCHUT-C. DE LELLIS, Well-posedness for a class of hyperbolic systems of conservation laws in several space dimensions, Di prossima pubblicazione su Comm. PDE, disponibile su http://cvgmt.sns.it. | Zbl 1072.35116
[9] L. AMBROSIO-N. GIGLI-G. SAVARÉ, Gradient flows in metric spaces and in the Wasserstein space of probability measures, Libro di prossima pubblicazione a cura di Birkhäuser.
[10] F. BOUCHUT-F. JAMES, One dimensional transport equation with discontinuous coefficients, Nonlinear Analysis, 32 (1998), 891-933. | MR 1618393 | Zbl 0989.35130
[11] F. BOUCHUT, Renormalized solutions to the Vlasov equation with coefficients of bounded variation, Arch. Rational Mech. Anal., 157 (2001), 75-90. | MR 1822415 | Zbl 0979.35032
[12] A. BRESSAN, An ill posed Cauchy problem for a hyperbolic system in two space dimensions, Rend. Sem. Mat. Univ. Padova, 110 (2003), 103-117. | fulltext mini-dml | MR 2033003 | Zbl 1114.35123
[13] 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. | MR 1411988 | Zbl 0865.35032
[14] A. CELLINA, On uniqueness almost everywhere for monotonic differential inclusions, Nonlinear Analysis, TMA, 25 (1995), 899-903. | MR 1350714 | Zbl 0837.34023
[15] A. CELLINA-M. VORNICESCU, On gradient flows, Journal of Differential Equations, 145 (1998), 489-501. | MR 1620979 | Zbl 0927.37007
[16] F. COLOMBINI-N. LERNER, Uniqueness of continuous solutions for BV vector fields, Duke Math. J., 111 (2002), 357-384. | fulltext mini-dml | MR 1882138 | Zbl 1017.35029
[17] F. COLOMBINI-N. LERNER, Uniqueness of $L^\infty$ solutions for a class of conormal BV vector fields, Preprint, 2003. | MR 2126467
[18] F. COLOMBINI-T. LUO-J. RAUCH, Uniqueness and nonuniqueness for nonsmooth divergence-free transport, Preprint, 2003. | MR 2030717 | Zbl 1065.35089
[19] C. DAFERMOS, Hyperbolic conservation laws in continuum physics, Springer Verlag, 2000. | MR 1763936 | Zbl 0940.35002
[20] 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. | MR 2009116 | Zbl 1024.35029
[21] R. J. DI PERNA-P. L. LIONS, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547. | MR 1022305 | Zbl 0696.34049
[22] M. HAURAY, On Liouville transport equation with potential in $BV_{\text{loc}}$, (2003) Di prossima pubblicazione su Comm. in PDE.
[23] M. HAURAY, On two-dimensional Hamiltonian transport equations with $L^p_{\text{loc}}$ coefficients, (2003) Di prossima pubblicazione su Ann. Nonlinear Analysis IHP. | fulltext mini-dml | Zbl 1028.35148
[24] L. V. KANTOROVICH, On the transfer of masses, Dokl. Akad. Nauk. SSSR, 37 (1942), 227-229.
[25] B. L. KEYFITZ-H. C. KRANZER, A system of nonstrictly hyperbolic conservation laws arising in elasticity theory, Arch. Rational Mech. Anal., 72 (1980), 219- 241. | MR 549642 | Zbl 0434.73019
[26] P. L. LIONS, Sur les équations différentielles ordinaires et les équations de transport, C. R. Acad. Sci. Paris Sér. I, 326 (1998), 833-838. | MR 1648524 | Zbl 0919.34028
[27] G. PETROVA-B. POPOV, Linear transport equation with discontinuous coefficients, Comm. PDE, 24 (1999), 1849-1873. | MR 1708110 | Zbl 0992.35104
[28] F. POUPAUD-M. RASCLE, Measure solutions to the liner multidimensional transport equation with non-smooth coefficients, Comm. PDE, 22 (1997), 337-358. | MR 1434148 | Zbl 0882.35026
[29] L. C. YOUNG, Lectures on the calculus of variations and optimal control theory, Saunders, 1969. | MR 259704 | Zbl 0177.37801

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