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

Ping Zhang; Zhifei Zhang

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arXiv · 1409.1649

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

Ping Zhang·Zhifei Zhang

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Abstract

In this paper, we investigate the global well-posedness of 3-D incompressible inhomogeneous Navier-Stokes equations with ill-prepared large initial data which are slowly varying in one space variable, that is, initial data of the form $\bigl(1+\e^{\be}a_0(x_{\rm h},\e x_3),(\ve^{1-\al} v^{\rm h}_0, \ve^{-\al}v_0^3)(x_{\rm h},\e x_3)\bigr)$ for any $\al\in ]0,1/3[,$ $\be>2\al,$ and $\ve$ being sufficiently small. We remark that initial data of this type do not satisfy the smallness conditions in \cite{c-p-z,HPZ3} no matter how small $\e$ is. In particular, this result greatly improves the global well-posedness result in \cite{PZZ3} with the so-called well-prepared initial data.

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Ping Zhang, Zhifei Zhang. 2014-09-05. Global well-posedness of 3-D inhomogeneous Navier-Stokes equations with ill-prepared initial data. https://arxiv.org/abs/1409.1649

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