4.6 Article

Linear transport equations for vector fields with subexponentially integrable divergence

Publisher

SPRINGER HEIDELBERG
DOI: 10.1007/s00526-016-0956-0

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Funding

  1. Generalitat de Catalunya [2014SGR75]
  2. Ministerio de Economia y Competitividad [MTM2013-44699]
  3. Programa Ramon y Cajal
  4. National Natural Science Foundation of China (NSFC) [11301029]
  5. Marie Curie Initial Training Network MAnET [FP7-607647]

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We face the well-posedness of linear transport Cauchy problems {partial derivative u/partial derivative t + b . del u + cu = f (0, T) x R-n u(0, .) = u(0) is an element of L-infinity R-n under borderline integrability assumptions on the divergence of the velocity field b. For W-loc(1,1) vector fields b satisfying vertical bar b(x,t)vertical bar/1+vertical bar x vertical bar is an element of L1 (0, T; L-1) + L-1(0, T; L-infinity) and div b is an element of L-1(0, T; L-infinity) + L-1 (0,T; Exp (L/log L)), we prove existence and uniqueness of weak solutions. Moreover, optimality is shown in the following way: for every gamma > 1, we construct an example of a bounded autonomous velocity field b with div (b) is an element of Exp (L/log(gamma) L) 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.

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