Journal
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
Volume 55, Issue 1, Pages -Publisher
SPRINGER HEIDELBERG
DOI: 10.1007/s00526-016-0956-0
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Funding
- Generalitat de Catalunya [2014SGR75]
- Ministerio de Economia y Competitividad [MTM2013-44699]
- Programa Ramon y Cajal
- National Natural Science Foundation of China (NSFC) [11301029]
- 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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