In this paper, we investigate the global well-posedness of 3-D incompressible inhomogeneous Navier–Stokes system with ill-prepared initial data of the form

$$
\bigl(1+\epsilon^{\beta}a_0(x_{\rm h},\epsilon x_3),(\epsilon^{1-\alpha}\,v^{\rm h}_0, \epsilon^{-\alpha}v_0^3)(x_{\rm h},\epsilon x_3)\bigr)
$$

for any $\alpha \in ]0,1/3[, \beta > 2\alpha$, and $\epsilon$ being sufficiently small. This result improves the global well-posedness result for so-called well-prepared initial data, which corresponds to the case of $\alpha=0$.

Global well-posedness of 3-D inhomogeneous Navier–Stokes system with ill-prepared initial data

Ping Zhang

Zhifei Zhang

Abstract

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