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arXiv · 2309.08576

Norm Growth, Non-uniqueness, and Anomalous Dissipation in Passive Scalars

Tarek M. Elgindi·Kyle Liss
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Abstract

We construct a divergence-free velocity field $u:[0,T] \times \mathbb{T}^2 \to \mathbb{R}^2$ satisfying $$u \in C^\infty([0,T];C^α(\mathbb{T}^2)) \quad \forall α\in [0,1)$$ such that the corresponding drift-diffusion equation exhibits anomalous dissipation for every smooth initial data. We also show that, given any $α_0 < 1$, the flow can be modified such that it is uniformly bounded only in $C^{α_0}(\mathbb{T}^2)$ and the regularity of solutions satisfy sharp (time-integrated) bounds predicted by the Obukhov-Corrsin theory. The proof is based on a general principle implying $H^1$ growth for all solutions to the transport equation, which may be of independent interest.

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Tarek M. Elgindi, Kyle Liss. 2023-09-15. Norm Growth, Non-uniqueness, and Anomalous Dissipation in Passive Scalars. https://arxiv.org/abs/2309.08576

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