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 \(\dot B^\beta_{p,q}(\mathbb R^2)\times \dot B^\beta_{p,r}(\mathbb R^2)\) for \(\beta\neq0\), \(1<p\leq\infty\), \(1\leq q,r\leq\infty\), 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 \(\rho\).