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arXiv · 2502.11531

Transport equations for Osgood velocity fields

Ulrik Skre Fjordholm·Ola Isaac Høgåsen Mæhlen
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Abstract

We consider the transport equation with a velocity field satisfying the Osgood condition. The weak formulation is not meaningful in the usual Lebesgue sense, meaning that the usual DiPerna--Lions treatment of the problem is not applicable {(in particular, the divergence of the velocity might be unbounded)}. Instead, we use Riemann--Stieltjes integration to interpret the weak formulation, leading to a well-posedness theory in regimes not covered by existing works. The most general results are for the one-dimensional problem, with generalisations to multiple dimensions in the particular case of log-Lipschitz velocities.

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Ulrik Skre Fjordholm, Ola Isaac Høgåsen Mæhlen. 2025-02-17. Transport equations for Osgood velocity fields. https://doi.org/10.1016/j.jde.2025.113566

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