Global well-posedness and symmetries for dissipative active scalar equations with positive-order couplings

TL;DR

This paper establishes global well-posedness, decay, and symmetry properties for a class of dissipative active scalar equations with positive-order couplings, extending previous results to larger initial data spaces.

Contribution

It proves global existence, decay, and symmetry results for active scalar equations outside the L^2 space with positive-order couplings, broadening the initial data class.

Findings

Global well-posedness without smallness assumptions

Solutions exhibit time decay

Symmetry properties depend on initial data and operators

Abstract

We consider a family of dissipative active scalar equations outside the $L^{2}$-space. This was introduced in [D. Chae, P. Constantin, J. Wu, to appear in IUMJ (2014)] and its velocity fields are coupled with the active scalar via a class of multiplier operators which morally behave as derivatives of positive order. We prove global well-posedness and time decay of solutions, without smallness assumptions, for initial data belonging to the critical Lebesgue space $L^{\frac{n}{\gamma-\beta}}(\mathbb{R}^{n})$ which is a class larger than that of the above reference. Symmetry properties of solutions are investigated depending on the symmetry of initial data and coupling operators.