Norm inflation for the Boussinesq system

Abstract

We prove the norm inflation phenomena for the Boussinesq system on T 3 . For arbitrarily small initial data (u0, ρ0) in the negative-order Besov spaces Ḃ−1 ∞,∞ × Ḃ−1 ∞,∞ , the solution can become arbitrarily large in a short time. Such largeness can be detected in ρ in Besov spaces of any negative order: Ḃ−s ∞,∞ for any s > 0. Notice that our initial data space is scaling critical for u and is scaling subcritical for ρ.