We prove the norm inflation phenomena for the Boussinesq system on <tex-math id="M1">{document}T³{document}</tex-math>. For arbitrarily small initial data <tex-math id="M2">{document}(u₀,ρ₀){document}</tex-math> in the negative-order Besov spaces <tex-math id="M3">{document}Ḃ⁻¹∞, ∞ × Ḃ⁻¹∞, ∞{document}</tex-math>, the solution can become arbitrarily large in a short time. Such largeness can be detected in <tex-math id="M4">{document}ρ{document}</tex-math> in Besov spaces of any negative order: <tex-math id="M5">{document}Ḃ⁻ˢ∞, ∞{document}</tex-math> for any <tex-math id="M6">{document}$ s>0 ${document}</tex-math>.
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Li et al. (2020) studied this question.
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