Transport equations with partially $BV$ velocities

Lerner, Nicolas

Transport equations with partially

B V

velocities

Lerner, Nicolas ¹

¹ Université de Rennes 1, Irmar, Campus de Beaulieu, 35042 Rennes cedex, France

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 3 (2004) no. 4, pp. 681-703.

Résumé

We prove the uniqueness of weak solutions for the Cauchy problem for a class of transport equations whose velocities are partially with bounded variation. Our result deals with the initial value problem $\partial_{t} u + X u = f, u_{|_{t = 0}} = g,$ where $X$ is the vector fieldwith a boundedness condition on the divergence of each vector field $a_{1}, a_{2}$ . This model was studied in the paper [LL] with a $W^{1, 1}$ regularity assumption replacing our $B V$ hypothesis. This settles partly a question raised in the paper [Am]. We examine the details of the argument of [Am] and we combine some consequences of the Alberti rank-one structure theorem for $B V$ vector fields with a regularization procedure. Our regularization kernel is not restricted to be a convolution and is introduced as an unknown function. Our method amounts to commute a pseudo-differential operator with a $B V$ function.

MR Zbl | 2 citations dans Numdam

Classification : 35F05, 34A12, 26A45

Affiliations des auteurs :

Lerner, Nicolas ¹

¹ Université de Rennes 1, Irmar, Campus de Beaulieu, 35042 Rennes cedex, France

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     title = {Transport equations with partially $BV$ velocities},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {681--703},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 3},
     number = {4},
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Lerner, Nicolas. Transport equations with partially $BV$ velocities. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 3 (2004) no. 4, pp. 681-703. http://archive.numdam.org/item/ASNSP_2004_5_3_4_681_0/

Bibliographie
Cité par

[Ai] M. Aizenman, On vector fields as generators of flows: a counterexample to Nelson's conjecture, Ann. of Math. 107 (1978), 287-296. | MR | Zbl

[Al] G. Alberti, Rank one properties for derivatives of functions with bounded variations, Proc. Roy. Soc. Edinburgh sect. A 123 (1993), 239-274. | MR | Zbl

[Am] L. Ambrosio, Transport equations and Cauchy problem for $B V$ vector fields, Invent. Math. 158 (2003), 227-260. | MR | Zbl

[AFP] L. Ambrosio - N. Fusco - D. Pallara, “Functions of bounded variations and free discontinuity problems”, Oxford Mathematical Monographs, 2000. | MR | Zbl

[Bo] F. Bouchut, Renormalized solutions to the Vlasov equation with coefficients of bounded variation, Arch. Ration. Mech. Anal. 157 (2001), 75-90. | MR | Zbl

[BD] F. Bouchut - L. Desvillettes, On two-dimensional Hamiltonian transport equations with continuous coefficients, Differential Integral Equations 14 (2001), 1015-1024. | MR | Zbl

[BJ] F. Bouchut - F. James, One dimensional transport equations with discontinuous coefficients, Nonlinear Anal. 32 (1998), 891-933. | MR | Zbl

[CP] I. Capuzzo Dolcetta - B. Perthame, On some analogy between different approaches to first-order PDE's with non smooth coefficients, Adv. Math. Sci. Appl. 6 (1996), 689-703. | MR | Zbl

[ChL] J.-Y. Chemin - N. Lerner, Flot de champ de vecteurs non lipschitziens et équations de Navier-Stokes, J. Differential Equations 121 (1995), 314-328. | MR | Zbl

[CL1] F. Colombini - N. Lerner, Uniqueness of continuous solutions for $B V$ vector fields, Duke Math. J. 111 (2002), 357-384. | MR | Zbl

[CL2] F. Colombini - N. Lerner, Uniqueness of $L^{\infty}$ solutions for a class of conormal $B V$ vector fields, to appear in: “Geometric Analysis of PDE and Several Complex Variables”, 2003, S. Chanillo - P. Cordaro - N. Hanges - J. Hounie - A. Meziani (eds.). | MR | Zbl

[CLR] F. Colombini - T. Luo - J. Rauch, Uniqueness and nonuniqueness for nonsmooth divergence free transport, Séminaire XEDP, Ecole Polytechnique (2003-04). | Numdam | MR | Zbl

[De] N. Depauw, Non unicité des solutions bornées pour un champ de vecteurs $B V$ en dehors d’un hyperplan., C.R. Math. Acad. Sci. Paris 337 (2003), 249-252. | MR | Zbl

[DL] R. J. Diperna - P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989), 511-547. | MR | Zbl

[Fe] H. Federer, “Geometric measure theory”, Grund. der Math. Wiss. 153, Springer-Verlag, 1969. | Zbl

[LL] C. Le Bris - P.-L. Lions, Renormalized solutions of some transport equations with partially $W^{1, 1}$ velocities and applications, Ann. Mat. Pura Appl. 183 (2004), 97-130. | MR | Zbl

[Li] P.-L. Lions, Sur les équations différentielles ordinaires et les équations de transport, C.R. Acad. Sc. Paris, Série I, 326 (1998), 833-838. | MR | Zbl

[Tr] F. Treves, “Topological vector spaces, distributions and kernels”, Pure & Appl. Math. Ser., Academic Press, 1967. | MR | Zbl

[Vo] A. I. Vol'Pert, The space $B V$ and quasi-linear equations, Math. USSR Sbornik 2 (1967), 225-267. | Zbl

[Zi] W. P. Ziemer, “Weakly differentiable functions”, Graduate texts in mathematics, Springer-Verlag, Vol. 120, 1989. | MR | Zbl