A Calderón type inverse problem for the active scalar equations with fractional dissipation

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.