A Calder\'on type inverse problem for the active scalar equations with fractional dissipation

TL;DR

This paper investigates an inverse problem for active scalar equations with fractional dissipation on the torus, utilizing second order linearization and properties of the fractional Laplacian to establish unique continuation and approximation results.

Contribution

It introduces a novel approach combining second order linearization and nonlocal properties of the fractional Laplacian to solve inverse problems in active scalar equations.

Findings

Relation to linear fractional diffusion established

Unique continuation property utilized

Inverse problem solvable under divergence-free structure

Abstract

In this paper, we are interested in an inverse problem for the active scalar equations with fractional dissipation on the torus. We perform a second order linearization to relate our model to the linear fractional diffusion equation. Our approach to solving the inverse problem relies on nonlocal phenomena such as the unique continuation property of the fractional Laplacian and its associated Runge approximation property. A remarkable feature of our model is that the divergence-free structure in the nonlinear term plays an important role in both forward and inverse problems.