×

Uniqueness of continuous solutions for BV vector fields. (English) Zbl 1017.35029

The authors study the Cauchy problem for a transport equation \(Xu=cu\) associated with a vector field \(X=\partial_t+\sum_{j=1}^d a_j(t,x)\partial_j\), in which coefficients \(a_j\) are assumed to be functions of bounded variation. Under some additional assumptions on growth of the coefficients the authors prove the uniqueness of continuous solutions.

MSC:

35F10 Initial value problems for linear first-order PDEs
26A45 Functions of bounded variation, generalizations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] 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. · Zbl 0821.76012
[2] F. Bouchut, On transport equations and the chain rule , preprint, 1999.
[3] F. Bouchut and L. Desvillettes, On two-dimensional Hamiltonian transport equations with continuous coefficients , Differential Integral Equations 14 (2001), 1015–1024. · Zbl 1028.35042
[4] F. Bouchut and F. James, One-dimensional transport equations with discontinuous coefficients , Nonlinear Anal. 32 (1998), 891–933. · Zbl 0989.35130
[5] 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. · Zbl 0878.35089
[6] F. Colombini and N. Lerner, Hyperbolic equations with non-Lipschitz coefficients , Duke Math. J. 77 (1995), 657–698. · Zbl 0840.35067
[7] B. Desjardins, Global existence results for the incompressible density-dependent Navier-Stokes equations in the whole space , Differential Integral Equations 10 (1995), 587–598. · Zbl 0902.76027
[8] –. –. –. –., Linear transport equations with initial values in Sobolev spaces and application to the Navier-Stokes equations , Differential Integral Equations 10 (1995), 577–586. · Zbl 0902.76028
[9] –. –. –. –., A few remarks on ordinary differential equations , Comm. Partial Differential Equations 21 (1996), 1667–1703. · Zbl 0899.35022
[10] R. J. DiPerna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces , Invent. Math. 98 (1989), 511–547. · Zbl 0696.34049
[11] H. Federer, Geometric Measure Theory , Grundlehren Math. Wiss. 153 , Springer, New York, 1969. · Zbl 0176.00801
[12] T. M. Flett, Differential Analysis: Differentiation, Differential Equations, and Differential Inequalities , Cambridge Univ. Press, Cambridge, 1980. · Zbl 0442.34002
[13] L. Hörmander, The Analysis of Linear Partial Differential Operators, I, II; III, IV , Grundlehren Math. Wiss. 256 , 257 ; 274 , 275 , Springer, Berlin, 1983; 1985., ;, Mathematical Reviews (MathSciNet): Mathematical Reviews (MathSciNet): MR87d:35002a Mathematical Reviews (MathSciNet): MR87d:35002b
[14] ——–, Lectures on Nonlinear Hyperbolic Differential Equations , Math. Appl. 26 , Springer, Berlin, 1997. · Zbl 0881.35001
[15] 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. · Zbl 0919.34028
[16] G. Petrova and B. Popov, Linear transport equations with discontinuous coefficients , Comm. Partial Differential Equations 24 (1999), 1849–1873. · Zbl 0992.35104
[17] 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. · Zbl 0882.35026
[18] F. Trèves, Topological Vector Spaces, Distributions and Kernels , Pure Appl. Math. 25 , Academic Press, New York, 1967. · Zbl 0171.10402
[19] 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. · Zbl 0168.07402
[20] W. P. Ziemer, Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation , Grad. Texts in Math. 120 , Springer, New York, 1989. · Zbl 0692.46022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.