Norm Inflation for Inviscid and Fully Dissipative Boussinesq Systems in Supercritical Spaces

We prove norm inflation, in the sense of strong ill-posedness, for the two-dimensional Boussinesq system in supercritical Besov spaces. For the inviscid system, norm inflation holds in ${\dot{B}}_{p,q}^{\beta }({\mathbb{R}}^2)\times {\dot{B}}_{p,r}^{\beta }({\mathbb{R}}^2)$ for $\beta \mathrm{\ne }0$, $1<p\le \mathrm{\infty }$, $1\le q,r\le \mathrm{\infty }$, and $-2<\beta -\frac{2}{p}<1$. For the fully dissipative system, the same conclusion holds in the range $-2<\beta -\frac{2}{p}<-1$. In both cases, the results cover almost all supercritical Besov spaces satisfying the local integrability condition. Norm inflation occurs in the density component $\rho$, while the velocity component $u$ remains bounded. In the fully dissipative case, the inflation space is supercritical for $u$, but subcritical for $\rho$ with respect to its own scaling. This is not a contradiction: the density is transported by a velocity field in a supercritical regime, and this transport mechanism is precisely what produces norm inflation in $\rho$.