Publication 21-CNA-014

Growth of Sobolev norms and loss of regularity in transport equations

Gianluca Crippa
Department of Mathematics and Computer Science
University of Basel
Switzerland
gianluca.crippa@unibas.ch

Tarek M. Elgindi
Mathematics Department
Duke University
Durham, NC

Gautam Iyer
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
gautam@math.cmu.edu

Anna L. Mazzucato
Mathematics Department
Penn State University
University Park, PA
alm24@psu.edu

Abstract: We consider transport of a passive scalar advected by an irregular divergence free vector field. Given any non-constant initial data $\overline{\rho}$ $\in$ $H^1_{loc}$ ($\mathbb{R}^d$), $d \geqslant 2$, we construct a divergence free advecting velocity field v (depending on $\overline{\rho}$) for which the unique weak solution to the transport equation does not belong to $H^1_{loc}$ ($\mathbb{R}^d$) for any positive positive time. The velocity field v is smooth, except at one point, controlled uniformly in time, and belongs to almost every Sobolev space $W^{s,p}$ that does not embed into the Lipschitz class. The velocity field v is constructed by pulling back and rescaling an initial data dependent sequence of sine/cosine shear flows on the torus. This loss of regularity result complements that in Ann. PDE, 5(1):Paper No. 9, 19, 2019.

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