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.

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 ${\partial }_{t}u+Xu=f,{u}_{{|}_{t=0}}=g,$ 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 $BV$ 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 $BV$ 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 $BV$ function.

Classification : 35F05,  34A12,  26A45
@article{ASNSP_2004_5_3_4_681_0,
author = {Lerner, Nicolas},
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},
year = {2004},
zbl = {1170.35362},
mrnumber = {2124585},
language = {en},
url = {http://archive.numdam.org/item/ASNSP_2004_5_3_4_681_0/}
}
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/

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