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

Abstract: 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.