Holder Continuous Solutions of Active Scalar Equations

TL;DR

This paper investigates the existence and properties of weak solutions to active scalar equations with Fourier multipliers, revealing conditions under which nontrivial solutions exist and discussing the implications for the h-principle.

Contribution

It demonstrates the existence of nontrivial, compactly supported weak solutions with Hölder regularity for active scalar equations when the multiplier is not odd, and analyzes the solution limits when it is odd.

Findings

Nontrivial solutions exist for non-odd multipliers with Hölder regularity $C^{1/9-}_{t,x}$.

Every integral scalar field can be approximated by such solutions.

Weak limits of solutions are solutions when the multiplier is odd.

Abstract

We consider active scalar equations $\partial_t \theta + \nabla \cdot (u \, \theta) = 0$, where $u = T[\theta]$ is a divergence-free velocity field, and $T$ is a Fourier multiplier operator with symbol $m$. We prove that when $m$ is not an odd function of frequency, there are nontrivial, compactly supported solutions weak solutions, with H\"older regularity $C^{1/9-}_{t,x}$. In fact, every integral conserving scalar field can be approximated in ${\cal D}'$ by such solutions, and these weak solutions may be obtained from arbitrary initial data. We also show that when the multiplier $m$ is odd, weak limits of solutions are solutions, so that the $h$-principle for odd active scalars may not be expected.