Norm inflation for the Boussinesq system

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 ${\dot{B}}_{\infty ,\infty }^{-1}\times {\dot{B}}_{\infty ,\infty }^{-1}$, the solution can become arbitrarily large in a short time. Such largeness can be detected in $\rho$ in Besov spaces of any negative order: ${\dot{B}}_{\infty ,\infty }^{-s}$ for any $s>0$. Notice that our initial data space is scaling critical for $u$ and is scaling subcritical for $\rho$.

16 pages