On the Cauchy Problem for Mildly Nonlinear and Non-Boussinesq Case-(ABC) System

In this paper, we investigate the local well-posedness, ill-posedness, and Gevrey regularity of the Cauchy problem for Mildly Nonlinear and Non-Boussinesq case-(ABC) system. The local well-posedness of the solution for this system in Besov spaces $B_{p\mathrm{,}r}^{s1}\times B_{p\mathrm{,}r}^s$ with $1\le p\mathrm{,}r\le \infty$ and $s>\max \left\{1\frac{1}{p},\frac{3}{2}\right\}$ was firstly established. Next, we consider the continuity of the solution-to-data map, i.e. the ill-posedness of the solution for this system in Besov space $B_{p\mathrm{,}\infty }^{s1}\times B_{p\mathrm{,}\infty }^s$ was derived. Finally, the Gevrey regularity of the system was presented.