Transport equations with partially BV velocities
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Volume 3 (2004) no. 4, p. 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 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},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 3},
     number = {4},
     year = {2004},
     pages = {681-703},
     zbl = {1170.35362},
     mrnumber = {2124585},
     language = {en},
     url = {http://www.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, Volume 3 (2004) no. 4, pp. 681-703. http://www.numdam.org/item/ASNSP_2004_5_3_4_681_0/

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