JournalsaihpcVol. 37, No. 5pp. 1075–1108

Convex integration solutions to the transport equation with full dimensional concentration

  • Stefano Modena

    Technische Universität Darmstadt, Fachbereich Mathematik, D-64289 Darmstadt, Germany
  • Gabriel Sattig

    Universität Leipzig, Mathematisches Institut,D-04109 Leipzig, Germany
Convex integration solutions to the transport equation with full dimensional concentration cover
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Abstract

We construct infinitely many incompressible Sobolev vector fields u \in C_{t}W_{x}^{1,p\limits^{˜}} on the periodic domain Td\mathbb{T}^{d} for which uniqueness of solutions to the transport equation fails in the class of densities ρCtLxp\rho \in C_{t}L_{x}^{p}, provided 1/ p + 1/ p\limits^{˜} > 1 + 1/ d. The same result applies to the transport-diffusion equation, if, in addition, p<dp^{′} < d.

Cite this article

Stefano Modena, Gabriel Sattig, Convex integration solutions to the transport equation with full dimensional concentration. Ann. Inst. H. Poincaré Anal. Non Linéaire 37 (2020), no. 5, pp. 1075–1108

DOI 10.1016/J.ANIHPC.2020.03.002