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We prove the ill-posedness for the 3D incompressible inhomogeneousNavier-stokes equations in critical Besov space. In particular, a norminflation 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 ofinitial data and introduce a modified pressure. Comparing with the classicalNavier-Stokes equations in $L^\infty$ framework, we can obtain theill-posedness for the inhomogeneous case in near $L^6$ framework.

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