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 $$\Vert a_0{\Vert }_{{\dot{B}}_{p,1}^{\frac{3}{p}}}+\Vert u_0{\Vert }_{{\dot{B}}_{6,1}^{-\frac{1}{2}}}\le \delta ,p>6$$ or $$\Vert a_0{\Vert }_{{\dot{B}}_{6,1}^{\frac{1}{2}}}+\Vert u_0{\Vert }_{{\dot{B}}_{p,1}^{\frac{3}{p}-1}}\le \delta ,p>6.$$ 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.

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References in corpus (2)

• Ill-posedness of the incompressible Navier-Stokes equations in ${\dot{F}}_{\infty }^{-1,q}(R^3)$
• Sharp well-posedness and ill-posedness of the Navier-Stokes initial value problem in Besov-type spaces