Strong ill-posedness of the 2D Boussinesq equations in supercritical Besov spaces

In this paper, we prove that the 2D Boussinesq equations are strongly ill-posed in the supercritical Besov spaces $B_{p,q}^s$ and Sobolev spaces $W^{s,p}$ with $(p,q)\in (1,\mathrm{\infty })\times [1,\mathrm{\infty }]$ and $s\in (0,1+2\mathrm{/}p)$ by constructing an initial data with arbitrarily small norm for which the solution of the system exhibits norm inflation almost instantaneously. As a further application, we prove the instability of perturbations near the hydrostatic equilibrium for the 2D Boussinesq equations in the same $B_{p,q}^s$ and $W^{s,p}$.