Linear transport equations for vector fields with subexponentially integrable divergence

Albert Clop; Renjin Jiang; Joan Orobitg; Juan Eugenio Mateu Bennassar

doi:https://doi.org/10.1007/s00526-016-0956-0

Linear transport equations for vector fields with subexponentially integrable divergence

Albert Clop, Renjin Jiang, Joan Orobitg, Juan Eugenio Mateu Bennassar

Research output: Contribution to journal › Article › Research › peer-review

3 Citations (Scopus)

Abstract

© 2016, Springer-Verlag Berlin Heidelberg. We face the well-posedness of linear transport Cauchy problems (Formula presented.) under borderline integrability assumptions on the divergence of the velocity field b. For (Formula presented.) vector fields b satisfying (Formula presented.) and (Formula presented.), we prove existence and uniqueness of weak solutions. Moreover, optimality is shown in the following way: for every (Formula presented.) , we construct an example of a bounded autonomous velocity field b with (Formula presented.) for which the associate Cauchy problem for the transport equation admits infinitely many solutions. Stability questions and further extensions to the BV setting are also addressed.

Original language	English
Article number	21
Pages (from-to)	1-30
Journal	Calculus of Variations and Partial Differential Equations
Volume	55
DOIs	https://doi.org/10.1007/s00526-016-0956-0
Publication status	Published - 1 Feb 2016

Keywords

Primary 35F05
Secondary 35F10

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Clop, A., Jiang, R., Orobitg, J., & Mateu Bennassar, J. E. (2016). Linear transport equations for vector fields with subexponentially integrable divergence. Calculus of Variations and Partial Differential Equations, 55, 1-30. [21]. https://doi.org/10.1007/s00526-016-0956-0

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T1 - Linear transport equations for vector fields with subexponentially integrable divergence

AU - Clop, Albert

AU - Jiang, Renjin

AU - Orobitg, Joan

AU - Mateu Bennassar, Juan Eugenio

PY - 2016/2/1

Y1 - 2016/2/1

N2 - © 2016, Springer-Verlag Berlin Heidelberg. We face the well-posedness of linear transport Cauchy problems (Formula presented.) under borderline integrability assumptions on the divergence of the velocity field b. For (Formula presented.) vector fields b satisfying (Formula presented.) and (Formula presented.), we prove existence and uniqueness of weak solutions. Moreover, optimality is shown in the following way: for every (Formula presented.) , we construct an example of a bounded autonomous velocity field b with (Formula presented.) for which the associate Cauchy problem for the transport equation admits infinitely many solutions. Stability questions and further extensions to the BV setting are also addressed.

AB - © 2016, Springer-Verlag Berlin Heidelberg. We face the well-posedness of linear transport Cauchy problems (Formula presented.) under borderline integrability assumptions on the divergence of the velocity field b. For (Formula presented.) vector fields b satisfying (Formula presented.) and (Formula presented.), we prove existence and uniqueness of weak solutions. Moreover, optimality is shown in the following way: for every (Formula presented.) , we construct an example of a bounded autonomous velocity field b with (Formula presented.) for which the associate Cauchy problem for the transport equation admits infinitely many solutions. Stability questions and further extensions to the BV setting are also addressed.

KW - Primary 35F05

KW - Secondary 35F10

U2 - https://doi.org/10.1007/s00526-016-0956-0

DO - https://doi.org/10.1007/s00526-016-0956-0

M3 - Article

VL - 55

SP - 1

EP - 30

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

SN - 0944-2669

M1 - 21

ER -