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},
     year = {2004},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 3},
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     zbl = {1170.35362},
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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. https://www.numdam.org/item/ASNSP_2004_5_3_4_681_0/

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