# Wellposedness and convergence of solutions to a class of forced non-diffusive equations with applications

## TL;DR

This paper studies the well-posedness and convergence of solutions for a class of non-diffusive active scalar equations with applications to geophysical and porous media flows, analyzing the effects of viscosity parameters.

## Contribution

It establishes Gevrey-class local well-posedness and solution convergence as viscosity vanishes for a family of singular non-diffusive equations, with applications to physical models.

## Key findings

- Proved local well-posedness in Gevrey class for the equations.
- Demonstrated convergence of solutions as viscosity parameter approaches zero.
- Applied theoretical results to models of Earth's core turbulence and porous media flow.

## Abstract

This paper considers a family of non-diffusive active scalar equations where a viscosity type parameter enters the equations via the constitutive law that relates the drift velocity with the scalar field. The resulting operator is smooth when the viscosity is present but singular when the viscosity is zero. We obtain Gevrey-class local well-posedness results and convergence of solutions as the viscosity vanishes. We apply our results to two examples that are derived from physical systems: firstly a model for magnetostrophic turbulence in the Earth's fluid core and secondly flow in a porous media with an "effective viscosity".