Bounds on the growth of high Sobolev norms of solutions to 2D Hartree Equations

TL;DR

This paper establishes polynomial bounds on the growth of high Sobolev norms for solutions to 2D Hartree equations on the torus and plane, adapting techniques from previous studies on $S^1$ and $ $.

Contribution

It provides the first polynomial bounds for Sobolev norm growth of 2D Hartree solutions, extending methods from 1D and Euclidean settings to two dimensions.

Findings

Polynomial bounds on Sobolev norm growth established

Techniques adapted from 1D and Euclidean cases to 2D

Results apply to equations on both torus and plane

Abstract

In this paper, we consider Hartree-type equations on the two-dimensional torus and on the plane. We prove polynomial bounds on the growth of high Sobolev norms of solutions to these equations. The proofs of our results are based on the adaptation to two dimensions of the techniques we previously used to study analogous problems on $S^1$, and on $\mathbb{R}$.