On the second iterate for active scalar equations

TL;DR

This paper analyzes the boundedness of a bilinear operator arising from an iterative scheme for active scalar equations with fractional Laplacians, focusing on super-critical regimes and minimal regularity conditions.

Contribution

It provides new continuity results of the bilinear operator in fractional BMO spaces and identifies minimal regularity requirements for solutions with even symbols.

Findings

Proves continuity of the bilinear operator in BMO^{1-2eta} spaces.

Shows minimal regularity in B^{-eta}_{ ext{infty},q} for solutions with even symbols.

Analyzes boundedness properties in super-critical regimes.

Abstract

We consider an iterative resolution scheme for a broad class of active scalar equations with a fractional power \gamma of the Laplacian and focus our attention on the second iterate. The main objective of our work is to analyze boundedness properties of the resulting bilinear operator, especially in the super-critical regime. Our results are two-fold: we prove continuity of the bilinear operator in BMO^{1-2\gamma} - a fractional analogue of the Koch-Tataru space; for equations with an even symbol we show that the B^{-\gamma}_{\infty,q} -regularity, where q > 2, is in a sense a minimal necessary requirement on the solution.