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 $$\mathrm{\parallel }{a}_0{\mathrm{\parallel }}_{{\dot{B}}_{p,1}^{\frac{3}{p}}}+\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{{\dot{B}}_{6,1}^{-\frac{1}{2}}}\le \delta ,p>6$$ or $$\mathrm{\parallel }{a}_0{\mathrm{\parallel }}_{{\dot{B}}_{6,1}^{\frac{1}{2}}}+\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{{\dot{B}}_{p,1}^{\frac{3}{p}-1}}\le \delta ,p>6.$$ 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 $L^{\mathrm{\infty }}$ framework, we can obtain the ill-posedness for the inhomogeneous case in near $L^6$ framework.