Non-uniform dependence for the inviscid Boussinesq equations in Besov spaces

This paper addresses the initial value problem to the inviscid Boussinesq equations in ${\mathbb{R}}^2$. We rigorously show that the data-to-solution map fails to be uniformly continuous within a broad class of nonhomogeneous Besov spaces $B_{p,r}^s({\mathbb{R}}^2)$ cited in ^[20] (i.e., $s>1+\frac{2}{p},1<p<\mathrm{\infty },1\le r\le \mathrm{\infty }$ or $s=1+\frac{2}{p},1<p<\mathrm{\infty },r=1$). This result partially extends the nowhere uniform continuity previously demonstrated by Inci^[9] in the Sobolev spaces $H^m({\mathbb{R}}^2)$ with $m>2$. Our proof leverages the interaction between terms of low and high frequencies. Besides, the linearized system to the inviscid Boussinesq equations plays a pivotal role in the construction of appropriate approximate solutions.