Pointwise convergence over fractals for dispersive equations with homogeneous symbol

TL;DR

This paper investigates fractal pointwise convergence for dispersive equations with homogeneous symbols, establishing conditions for convergence and divergence on fractal sets, and demonstrating the sharpness of these results with counterexamples.

Contribution

It provides new convergence criteria depending on dispersive strength and constructs divergence examples using fractal geometry and Diophantine approximation techniques.

Findings

Solutions converge to initial data on fractals under certain regularity conditions.

Constructs divergence sets of specific Hausdorff dimension using Talbot-like effects.

Shows the optimality of convergence results with counterexamples for quadratic symbols.

Abstract

We study the fractal pointwise convergence for the equation $i\hbar\partial_tu + P(D)u = 0$, where the symbol $P$ is real, homogeneous and non-singular. We prove that for initial data $f\in H^s(\mathbb{R}^n)$ with $s>(n-\alpha+1)/2$ the solution $u$ converges to $f$ $\mathcal{H}^\alpha$-a.e, where $\mathcal{H}^\alpha$ is the $\alpha$-dimensional Hausdorff measure. We improve upon this result depending on the dispersive strength of $P$. On the other hand, for a family of polynomials $P$ and given $\alpha$, we exploit a Talbot-like effect to construct initial data whose solutions $u$ diverge in sets of Hausdorff dimension $\alpha$. To compute the dimension of the sets of divergence, we adopt the Mass Transference Principle from Diophantine approximation. We also construct counterexamples for quadratic symbols like the saddle to show that our positive results are sometimes best possible.