Abstract
We provide in this article a new proof of the uniqueness of the flow solution to ordinary differential equations with BV vector fields that have divergence in L ∞ (or in L 1), when the flow is assumed nearly incompressible (see the text for the definition of this term). The novelty of the proof lies in the fact it does not use the associated transport equation.
Article PDF
Similar content being viewed by others
References
Alberti G.: Rank one property for derivatives of functions with bounded variation. Proc. R. Soc. Edinburgh Sect. A 123(2), 239–274 (1993)
Ambrosio L.: Transport equation and Cauchy problem for BV vector fields. Invent. Math. 158(2), 227–260 (2004)
Ambrosio, L.: Transport Equation and Cauchy Problem for Non-smooth Vector Fields. Lecture Notes of the CIME Summer school in Cetrary, June 27–July 2, 2005, available at http://cvgmt.sns.it/papers/amb05/Cetrarotext.pdf (2005)
Ambrosio L., Bouchut F., De Lellis C.: Well-posedness for a class of hyperbolic systems of conservation laws in several space dimensions. Comm. Partial. Differ. Equ. 29(9–10), 1635–1651 (2004)
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000)
Bressan A.: An ill posed Cauchy problem for a hyperbolic system in two space dimensions. Rend. Sem. Mat. Univ. Padova 110, 103–117 (2003)
Capuzzo-Dolcetta I., Perthame B.: On some analogy between different approaches to first order PDE’s with nonsmooth coefficients. Adv. Math. Sci. Appl. 6(2), 689–703 (1996)
Champagnat, N., Jabin, P.-E.: Well-posedness in any dimension for hamiltonian flows with non BV force terms. Accepted for publication in Comm. P.D.E., available at http://arxiv.org/abs/0904.1119 (2009)
Crippa G., De Lellis C.: Estimates and regularity results for the DiPerna-Lions flow. J. Reine Angew. Math. 616, 15–46 (2008)
DiPerna R.J., Lions P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98(3), 511–547 (1989)
Hauray M., De Bris C., Lions P.-L.: Deux remarques sur les flots généralisés d’équations différentielles ordinaires. C. R. Acad. Sci. Paris 344, 759–764 (2007)
Jabin, P.-E.: Differential equations with singular fields. Available at http://math1.unice.fr/~jabin/odefinal.pdf (2009)
Lerner N.: Transport equations with partially BV velocities. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3(4), 681–703 (2004)
Lions P.-L.: Sur les équations différentielles ordinaires et les équations de transport. C. R. Acad. Sci. Paris Sér. I Math. 326(7), 833–838 (1998)
Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, NJ (1970)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hauray, M., Bris, C.L. A new proof of the uniqueness of the flow for ordinary differential equations with BV vector fields. Annali di Matematica 190, 91–103 (2011). https://doi.org/10.1007/s10231-010-0140-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10231-010-0140-7