Uniqueness and Nonuniqueness for Nonsmooth Divergence Free Transport

Ferrucio Colombini[1]; Tao Luo[2]; Jeffrey Rauch[3]

  • [1] Dipartimento di Matematica, Università di Pisa, Pisa, Italia
  • [2] Department of Mathematics, Georgetown University, Washington DC 20057,USA
  • [3] Department of Mathematics, University of Michigan, Ann Arbor 48104 MI, USA

Séminaire Équations aux dérivées partielles (2002-2003)

  • Volume: 2002-2003, page 1-21

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Colombini, Ferrucio, Luo, Tao, and Rauch, Jeffrey. "Uniqueness and Nonuniqueness for Nonsmooth Divergence Free Transport." Séminaire Équations aux dérivées partielles 2002-2003 (2002-2003): 1-21. <http://eudml.org/doc/11065>.

@article{Colombini2002-2003,
affiliation = {Dipartimento di Matematica, Università di Pisa, Pisa, Italia; Department of Mathematics, Georgetown University, Washington DC 20057,USA; Department of Mathematics, University of Michigan, Ann Arbor 48104 MI, USA},
author = {Colombini, Ferrucio, Luo, Tao, Rauch, Jeffrey},
journal = {Séminaire Équations aux dérivées partielles},
keywords = {linear transport equation; classical solutions; Aizenman's example},
language = {eng},
pages = {1-21},
publisher = {Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Uniqueness and Nonuniqueness for Nonsmooth Divergence Free Transport},
url = {http://eudml.org/doc/11065},
volume = {2002-2003},
year = {2002-2003},
}

TY - JOUR
AU - Colombini, Ferrucio
AU - Luo, Tao
AU - Rauch, Jeffrey
TI - Uniqueness and Nonuniqueness for Nonsmooth Divergence Free Transport
JO - Séminaire Équations aux dérivées partielles
PY - 2002-2003
PB - Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 2002-2003
SP - 1
EP - 21
LA - eng
KW - linear transport equation; classical solutions; Aizenman's example
UR - http://eudml.org/doc/11065
ER -

References

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  1. M. Aizenman, On vector fields as generators of flows: a counterexample to Nelson’s conjecture, Ann. Math. 107 (1978), 287-296. Zbl0394.28012
  2. G. Alberti, Rank-one properties for derivatives of functions with bounded variation, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), 239-274. Zbl0791.26008MR1215412
  3. L. Ambrosio, Transport equation and Cauchy problem for B V vector fields, preprint Scuola Norm. Sup. Pisa, March 2003. Zbl1075.35087MR2096794
  4. M. Beals, Propagation and Interaction of Singularities in Nonlinear Hyperbolic Problems, Birkhäuser, Boston, 1989. Zbl0698.35100MR1033737
  5. F. Bouchut, Renormalized solutions to the Vlasov equation with coefficients of bounded variation, Arch. Rational Mech. Anal. 157 (2001), 75-90. Zbl0979.35032MR1822415
  6. F. Bouchut, F. James, One-dimensional transport equations with discontinuous coefficients, Nonlinear Anal. 32 (1998), 891-933. Zbl0989.35130MR1618393
  7. F. Colombini, N. Lerner, Uniqueness of continuous solutions for B V vector fields, Duke Math. J. 111 (2002), 357-384. Zbl1017.35029MR1882138
  8. F. Colombini, N. Lerner, Sur les champs de vecteurs peu réguliers, Séminaire E.D.P., Ecole Polytechnique, 2000-2001, XIV 1-15. MR1860686
  9. F. Colombini, N. Lerner, Uniqueness of L solutions for a class of conormal B V vector fields, Preprint Univ. Rennes 1, February 2003. Zbl1064.35033MR2126467
  10. N. Depauw, Non unicité des solutions bornées pour un champ de vecteurs B V en dehors d’un hyperplan, C.R.Acad.Sci.Paris Sér.I Math. to appear. Zbl1024.35029
  11. R.J. Di Perna, P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989), 511-547. Zbl0696.34049MR1022305
  12. P.-L. Lions, Sur les équations différentielles ordinaires et les équations de transport, C.R.Acad.Sci.Paris Sér.I Math. 326 (1998), 833-838. Zbl0919.34028MR1648524
  13. E. Nelson, Les écoulements incompressibles d’énergie finie, Les Equations aux Dérivées Partielles (Paris 1962), Colloques Internationaux du C.N.R.S., 117 (1963), 159-165. Zbl0237.35015
  14. G. Petrova, B. Popov, Linear transport equations with discontinuous coefficients, Comm. PDE 24 (1999), 1849-1873. Zbl0992.35104MR1708110
  15. F. Poupaud, M. Rascle, Measure solutions to the linear multi- dimensional transport equation with non-smooth coefficients, Comm. PDE 22 (1997), 337-358. Zbl0882.35026MR1434148

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