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arXiv 1301.2371math.AP
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二维变粘性与密度相关的Navier-Stokes方程组的整体适定性

On the global well-posedness of 2-D density-dependent Navier-Stokes system with variable viscosity

Hammadi Abidi, Ping Zhang

备注：
arXiv admin note: text overlap with arXiv:1212.3918 by other authors
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摘要：

给定螺线管矢量$u_0\in H^{-2\d}\cap H^1(\R^2),$ $\r_0-1\in L^2(\R^2),$和$\r_0 \in L^\infty\cap\dot{W}^{1,r}(\R^2)$，且$\d\in (0,\f12)$和$2<r<\f{2}{1-2\d},$的下界为正，证明了二维不可压非齐次Navier-Stokes方程组\eqref{1.1}具有唯一的整体解，条件是粘性系数$μ(\r_0)$在$L^\infty$范数中与$\d$的大小和初始数据的范数相比足够接近1.有了更平滑的初始数据，我们可以证明这种解的传播。此外，对于$1<p<4,$，如果$(\r_0-1,u_0)$属于临界Besov空间$\dB^{\f2p}_{p,1}(\R^2)\times \bigl(\dB^{-1+\f2p}_{p,1}\cap L^2(\R^2)\bigr)$，且$\r_0-1$的$\dB^{\f2p}_{p,1}(\R^2)$范数与$\|u_0\|_{L^2}^2+\|u_0\|_{\dB^{-1+\f2p}_{p,1}},$的指数相比足够小，则证明了\eqref{1.1}在标度不变空间中的全局适定性.最后，对于几乎临界Besov空间中的初始数据，在$\r_0-1$的$L^\infty$范数足够小的假设下，证明了\eqref{1.1}的整体适定性.

英文摘要：

Given solenoidal vector $u_0\in H^{-2\d}\cap H^1(\R^2),$ $\r_0-1\in L^2(\R^2),$ and $\r_0 \in L^\infty\cap\dot{W}^{1,r}(\R^2)$ with a positive lower bound for $\d\in (0,\f12)$ and $2<r<\f{2}{1-2\d},$ we prove that 2-D incompressible inhomogeneous Navier-Stokes system \eqref{1.1} has a unique global solution provided that the viscous coefficient $μ(\r_0)$ is close enough to 1 in the $L^\infty$ norm compared to the size of $\d$ and the norms of the initial data. With smoother initial data, we can prove the propagation of regularities for such solutions. Furthermore, for $1<p<4,$ if $(\r_0-1,u_0)$ belongs to the critical Besov spaces $\dB^{\f2p}_{p,1}(\R^2)\times \bigl(\dB^{-1+\f2p}_{p,1}\cap L^2(\R^2)\bigr)$ and the $\dB^{\f2p}_{p,1}(\R^2)$ norm of $\r_0-1$ is sufficiently small compared to the exponential of $\|u_0\|_{L^2}^2+\|u_0\|_{\dB^{-1+\f2p}_{p,1}},$ we prove the global well-posedness of \eqref{1.1} in the scaling invariant spaces. Finally for initial data in the almost critical Besov spaces, we prove the global well-posedness of \eqref{1.1} under the assumption that the $L^\infty$ norm of $\r_0-1$ is sufficiently small.

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