Duke Mathematical Journal

Uniqueness of continuous solutions for BV vector fields

Ferruccio Colombini and Nicolas Lerner

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We consider a vector field whose coefficients are functions of bounded variation, with a bounded divergence. We prove the uniqueness of continuous solutions for the Cauchy problem.

Article information

Source
Duke Math. J., Volume 111, Number 2 (2002), 357-384.

Dates
First available in Project Euclid: 18 June 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1087575044

Digital Object Identifier
doi:10.1215/S0012-7094-01-11126-5

Mathematical Reviews number (MathSciNet)
MR1882138

Zentralblatt MATH identifier
1017.35029

Subjects
Primary: 35F10: Initial value problems for linear first-order equations
Secondary: 26A45: Functions of bounded variation, generalizations

Citation

Colombini, Ferruccio; Lerner, Nicolas. Uniqueness of continuous solutions for BV vector fields. Duke Math. J. 111 (2002), no. 2, 357--384. doi:10.1215/S0012-7094-01-11126-5. https://projecteuclid.org/euclid.dmj/1087575044


Export citation

References

  • H. Bahouri and J.-Y.Chemin, Équations de transport relatives à des champs de vecteurs non-lipschitziens et mécanique des fluides, Arch. Rational Mech. Anal. 127 (1994), 159--181.
  • F. Bouchut, On transport equations and the chain rule, preprint, 1999.
  • F. Bouchut and L. Desvillettes, On two-dimensional Hamiltonian transport equations with continuous coefficients, Differential Integral Equations 14 (2001), 1015--1024.
  • F. Bouchut and F. James, One-dimensional transport equations with discontinuous coefficients, Nonlinear Anal. 32 (1998), 891--933.
  • J.-Y. Chemin and N. Lerner, Flot de champ de vecteurs non lipschitziens et équations de Navier-Stokes, J. Differential Equations 121 (1995), 314--328.
  • F. Colombini and N. Lerner, Hyperbolic equations with non-Lipschitz coefficients, Duke Math. J. 77 (1995), 657--698.
  • B. Desjardins, Global existence results for the incompressible density-dependent Navier-Stokes equations in the whole space, Differential Integral Equations 10 (1995), 587--598.
  • --. --. --. --., Linear transport equations with initial values in Sobolev spaces and application to the Navier-Stokes equations, Differential Integral Equations 10 (1995), 577--586.
  • --. --. --. --., A few remarks on ordinary differential equations, Comm. Partial Differential Equations 21 (1996), 1667--1703.
  • R. J. DiPerna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989), 511--547.
  • H. Federer, Geometric Measure Theory, Grundlehren Math. Wiss. 153, Springer, New York, 1969.
  • T. M. Flett, Differential Analysis: Differentiation, Differential Equations, and Differential Inequalities, Cambridge Univ. Press, Cambridge, 1980.
  • L. Hörmander, The Analysis of Linear Partial Differential Operators, I, II; III, IV, Grundlehren Math. Wiss. 256, 257; 274, 275, Springer, Berlin, 1983; 1985., ;,
  • --------, Lectures on Nonlinear Hyperbolic Differential Equations, Math. Appl. 26, Springer, Berlin, 1997.
  • P. L. Lions, Sur les équations différentielles ordinaires et les équations de transport, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), 833--838.
  • G. Petrova and B. Popov, Linear transport equations with discontinuous coefficients, Comm. Partial Differential Equations 24 (1999), 1849--1873.
  • F. Poupaud and M. Rascle, Measure solutions to the linear multi-dimensional transport equation with non-smooth coefficients, Comm. Partial Differential Equations 22 (1997), 337--358.
  • F. Trèves, Topological Vector Spaces, Distributions and Kernels, Pure Appl. Math. 25, Academic Press, New York, 1967.
  • A. I. Vol'pert, Spaces $\BV$ and quasilinear equations (in Russian), Mat. Sb (N.S.) 73 (115) (1967), 255--302.; English translation in Math. USSR-Sb. 2 (1967), 225--267.
  • W. P. Ziemer, Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation, Grad. Texts in Math. 120, Springer, New York, 1989.