Norm inflation for the Boussinesq system

TL;DR

This paper demonstrates that solutions to the Boussinesq system on the 3D torus can exhibit norm inflation, where arbitrarily small initial data lead to solutions becoming arbitrarily large in a short time, especially in negative-order Besov spaces.

Contribution

It establishes the phenomenon of norm inflation for the Boussinesq system in critical and subcritical Besov spaces, highlighting potential ill-posedness issues.

Findings

Solutions can become arbitrarily large in short time from small initial data.

Norm inflation occurs in negative-order Besov spaces for both velocity and density.

Initial data spaces are scaling critical for velocity and subcritical for density.

Abstract

We prove the norm inflation phenomena for the Boussinesq system on $\mathbb T^3$. For arbitrarily small initial data $(u_0,\rho_0)$ in the negative-order Besov spaces $\dot{B}^{-1}_{\infty, \infty} \times \dot{B}^{-1}_{\infty, \infty}$, the solution can become arbitrarily large in a short time. Such largeness can be detected in $\rho$ in Besov spaces of any negative order: $\dot{B}^{-s}_{\infty, \infty}$ for any $s>0$. Notice that our initial data space is scaling critical for $u$ and is scaling subcritical for $\rho$.