Transport equations with partially B V velocities

Nicolas Lerner[1]

  • [1] Université de Rennes 1, Irmar, Campus de Beaulieu, 35042 Rennes cedex, France

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2004)

  • Volume: 3, Issue: 4, page 681-703
  • ISSN: 0391-173X

Abstract

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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 + 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.

How to cite

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Lerner, Nicolas. "Transport equations with partially $BV$ velocities." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 3.4 (2004): 681-703. <http://eudml.org/doc/84546>.

@article{Lerner2004,
abstract = {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 $\partial _tu+Xu=f, u_\{\vert _ \{t=0\}\}=g,$ 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.},
affiliation = {Université de Rennes 1, Irmar, Campus de Beaulieu, 35042 Rennes cedex, France},
author = {Lerner, Nicolas},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {4},
pages = {681-703},
publisher = {Scuola Normale Superiore, Pisa},
title = {Transport equations with partially $BV$ velocities},
url = {http://eudml.org/doc/84546},
volume = {3},
year = {2004},
}

TY - JOUR
AU - Lerner, Nicolas
TI - Transport equations with partially $BV$ velocities
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2004
PB - Scuola Normale Superiore, Pisa
VL - 3
IS - 4
SP - 681
EP - 703
AB - 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 $\partial _tu+Xu=f, u_{\vert _ {t=0}}=g,$ 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.
LA - eng
UR - http://eudml.org/doc/84546
ER -

References

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