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[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[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