Ill-posedness for the 3D inhomogeneous Navier-Stokes equations in the critical Besov space near $L^6$ framework

We prove the ill-posedness for the 3D incompressible inhomogeneous Navier-stokes equations in critical Besov space. In particular, a norm inflation happens in finite time with the initial data satisfying $$\|a_0\|_{\dot{B}_{p,1}^\frac{3}{p}}+\|u_0\|_{\dot{B}_{6,1}^{-\frac{1}{2}}}\le \delta,\ p>6$$ or $$\|a_0\|_{\dot{B}_{6,1}^\frac{1}{2}}+\|u_0\|_{\dot{B}_{p,1}^{\frac{3}{p}-1}}\le \delta,\ p>6.$$ To obtain the norm inflation, we construct a special class of initial data and introduce a modified pressure. Comparing with the classical Navier-Stokes equations in $L^\infty$ framework, we can obtain the ill-posedness for the inhomogeneous case in near $L^6$ framework.