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- W3204070129 abstract "We consider transport of a passive scalar advected by an irregular divergence free vector field. Given any non-constant initial data $bar rho in H^1_text{loc}({mathbb R}^d)$, $dgeq 2$, we construct a divergence free advecting velocity field $v$ (depending on $bar rho$) for which the unique weak solution to the transport equation does not belong to $H^1_text{loc}({mathbb R}^d)$ 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]." @default.
- W3204070129 created "2021-10-11" @default.
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- W3204070129 date "2022-04-25" @default.
- W3204070129 modified "2023-10-17" @default.
- W3204070129 title "Growth of Sobolev norms and loss of regularity in transport equations" @default.
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- W3204070129 doi "https://doi.org/10.1098/rsta.2021.0024" @default.
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