# Lower Bounds for the Advection-Hyperdiffusion Equation

## TL;DR

This paper investigates the decay rates of solutions to the advection-hyperdiffusion equation in multiple dimensions, using Fourier analysis and PDE techniques to establish lower bounds on solution norms over time.

## Contribution

It introduces a novel application of the Fourier-splitting method to the advection-hyperdiffusion equation, providing new lower bounds for solution decay in $H^{-1}$ and $L^2$ norms.

## Key findings

- Established lower bounds for $H^{-1}$-norm decay
- Applied Fourier-splitting method to hyperdiffusion with advection
- Combined classical PDE techniques with Fourier analysis

## Abstract

Motivated by [7], we study the advection-hyperdiffusion equation in the whole space in two and three dimensions with the goal of understanding the decay in time of the $H^{-1}$- and $L^2$-norm of the solutions. We view the advection term as a perturbation of the hyperdiffusion equation and employ the Fourier-splitting method first introduced by Schonbek in [8] for scalar parabolic equations and later generalized to a broader class of equations including Navier-Stokes equations and magneto-hydrodynamic systems. This approach consists of decomposing the Fourier space along a sphere with radius decreasing in time. Combining the Fourier-splitting method with classical PDE techniques applied to the hyperdiffusion equation we find a lower bound for the $H^{-1}$-norm by interpolation.