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  1. We consider transport of a passive scalar advected by an irregular divergence-free vector field. Given any non-constant initial data ρ ¯ ∈ H loc 1 ( R d ) , d ≥ 2 , we construct a divergence-free advecting velocity field v (depending on ρ ¯ ) for which the unique weak solution to the transport equation does not belong to H loc 1 ( R d ) for any 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 notmore »embed into the Lipschitz class. The velocity field v is constructed by pulling back and rescaling a sequence of sine/cosine shear flows on the torus that depends on the initial data. This loss of regularity result complements that in Ann. PDE , 5(1):Paper No. 9, 19, 2019. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’.« less
    Free, publicly-accessible full text available June 13, 2023
  2. Free, publicly-accessible full text available March 1, 2023
  3. Locomotion on complex terrains often requires vision. However, how vision serves locomotion is not well understood. Here, we asked when visual information necessary for accurate stepping is collected and how its acquisition relates to the step cycle. In cats of both sexes, we showed that a brief (200–400 ms) interruption of visual input can rapidly influence cat’s walking along a horizontal ladder. Depending on the phase within the step cycle, a 200 ms period of darkness could be tolerated fully without any changes to the strides or could lead to minor increases of stride duration. The effects of 300–400 msmore »of visual input denial, which typically prolonged stances and/or swings, also depended on the phase of the darkness onset. The increase of the duration of strides was always shorter than the duration of darkness. We conclude that visual information for planning a swing is collected starting from the middle of the preceding stance until the beginning of the current swing. For a stance (and/or a swing of the other paw), visual information is collected starting from the end of the previous stance and until the middle of the current stance. Acquisition of visual information during these windows is not uniform but depends on the phase of the step cycle. Notably, both the extension of these windows and their non-homogeneity are closely related to the pattern of gaze behaviour in cats, described previously. This new knowledge will help to guide research and understanding of neuronal mechanisms of visuomotor integration and modulation of visual function by strides during locomotion.« less
    Free, publicly-accessible full text available December 7, 2022