Transport equation and Cauchy problem for $BV$ vector fields and applications

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1. M.Aizenman:On vector fields as generators of flows: a counterexample to Nelson’s conjecture. Ann. Math., 107 (1978), 287–296. Zbl0394.28012MR482853
2. G.Alberti:Rank-one properties for derivatives of functions with bounded variation. Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 239–274. Zbl0791.26008MR1215412
3. G.Alberti & L.Ambrosio:A geometric approach to monotone functions in ${ℝ}^n$. Math. Z., 230 (1999), 259–316. Zbl0934.49025MR1676726
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. Zbl0957.49001MR1857292
7. L.Ambrosio:Transport equation and Cauchy problem for $BV$ vector fields. To appear on Inventiones Math. . Zbl1075.35087MR2096794
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. Zbl1061.35048MR2000967
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. Zbl1072.35116
10. L.Ambrosio, G.Crippa & S.Maniglia:Traces and fine properties of a $BD$ class of vector fields and applications. Preprint, 2004. Zbl1091.35007
11. L.Ambrosio:Lecture Notes on transport equation and Cauchy problem for $BV$ vector fields and applications. Available at http://cvgmt.sns.it. Zbl1075.35087
12. L.Ambrosio, N.Gigli & G.Savaré:Gradient flows in metric spaces and in the Wasserstein space of probability measures. Birkhäuser, to appear. Zbl1090.35002MR2129498
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. Zbl0915.35024MR1627555
14. F.Bouchut & F.James:One dimensional transport equation with discontinuous coefficients. Nonlinear Analysis, 32 (1998), 891–933. Zbl0989.35130MR1618393
15. F.Bouchut:Renormalized solutions to the Vlasov equation with coefficients of bounded variation. Arch. Rational Mech. Anal., 157 (2001), 75–90. Zbl0979.35032MR1822415
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. Zbl1114.35123MR2033003
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. Zbl0865.35032MR1411988
18. A.Cellina:On uniqueness almost everywhere for monotonic differential inclusions. Nonlinear Analysis, TMA, 25 (1995), 899–903. Zbl0837.34023MR1350714
19. A.Cellina & M.Vornicescu:On gradient flows. Journal of Differential Equations, 145 (1998), 489–501. Zbl0927.37007MR1620979
20. G.-Q.Chen & H.Frid:Extended divergence-measure fields and the Euler equation of gas dynamics. Comm. Math. Phys., 236 (2003), 251–280. Zbl1036.35125MR1981992
21. F.Colombini & N.Lerner:Uniqueness of continuous solutions for $BV$ vector fields. Duke Math. J., 111 (2002), 357–384. Zbl1017.35029MR1882138
22. F.Colombini & N.Lerner:Uniqueness of $L^{\infty }$ solutions for a class of conormal $BV$ vector fields. Preprint, 2003. Zbl1064.35033
23. F.Colombini, T.Luo & J.Rauch:Uniqueness and nonuniqueness for nonsmooth divergence-free transport. Preprint, 2003. Zbl1065.35089
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. Zbl0985.76008MR1816477
25. M.Cullen & M.Feldman:Lagrangian solutions of semigeostrophic equations in physical space. To appear. Zbl1097.35004MR2215268
26. C.Dafermos:Hyperbolic conservation laws in continuum physics. Springer Verlag, 2000. Zbl0940.35002MR1763936
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. Zbl1024.35029MR2009116
28. R.J. Di Perna & P.L.Lions:Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math., 98 (1989), 511–547. Zbl0696.34049MR1022305
29. M.Hauray:On Liouville transport equation with potential in $B{V}_{\mathrm{loc}}$. (2003) Di prossima pubblicazione su Comm. in PDE. 
30. M.Hauray:On two-dimensional Hamiltonian transport equations with $L_{\mathrm{loc}}^p$ coefficients. (2003) Di prossima pubblicazione su Ann. Nonlinear Analysis IHP. Zbl1028.35148
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. Zbl0434.73019MR549642
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. Zbl1170.35364MR2044334
34. N.Lerner:Transport equations with partially $BV$ velocities. Preprint, 2004. Zbl1170.35362
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. Zbl0919.34028MR1648524
36. G.Petrova & B.Popov:Linear transport equation with discontinuous coefficients. Comm. PDE, 24 (1999), 1849–1873. Zbl0992.35104MR1708110
37. F.Poupaud & M.Rascle:Measure solutions to the liner multidimensional transport equation with non-smooth coefficients. Comm. PDE, 22 (1997), 337–358. Zbl0882.35026MR1434148
38. L.C.Young:Lectures on the calculus of variations and optimal control theory, Saunders, 1969. Zbl0177.37801MR259704