math.AP — Analysis of PDEs

Anomalous regularization of passive scalars up to the Obukhov-Corrsin exponent

Contributed by Scott Armstrong, Vlad Vicol

We consider the advection-diffusion equation $\partial_t\theta+\mathbf{b}\cdot\nabla\theta-\kappa\Delta\theta=0$ on the two-dimensional torus $\mathbb{T}^2$, where $\mathbf{b}$ is a divergence-free velocity field constructed as in arxiv:2305.05048, H\"older continuous with an exponent $\alpha<\nf13$ and built from shear flows alternating in direction at infinitely many scales. We prove anomalous regularization: for every $s$ below the Obukhov--Corrsin exponent $\frac12(1-\alpha)$, with $\mathbf{b}$ depending on $s$, the solutions are bounded in~$C^{0,s}(\mathbb{T}^2)$ at positive times by a constant times the $L^2(\mathbb{T}^2)$ norm of the initial data, uniformly in the diffusivity $\kappa$ in a union of bands accumulating at $0$. As a consequence, we obtain anomalous dissipation at a uniform rate: solutions dissipate a fixed fraction of their variance in unit time, so that the variance decays exponentially, with constants independent of $\kappa$, of the initial datum and of the initial time. The proof combines iterated quantitative homogenization with large-scale regularity: at each scale the solution is compared with a solution of the homogenized equation of that scale. Since the derivatives consumed by each comparison are regenerated at every scale, the H\"older norm is carried from the coarsest scale to the finest by induction. Each of our main results, together with its proof, is fully formalized in the Lean 4 proof assistant.