The 3-D Inviscid Limit Result Under Slip Boundary Conditions. A Negative Answer
The 3-D Inviscid Limit Result Under Slip Boundary Conditions. A Negative Answer
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DOI：
10.1007/s00021-010-0047-5
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发表时间：
2010-10
影响因子：
1.3
通讯作者：
H. Beirão da Veiga;F. Crispo
中科院分区：
文献类型：
--
作者：
H. Beirão da Veiga;F. Crispo
一般情况下，当黏度趋于零时，广泛采用的navier型滑移边界条件下的Navier-Stokes方程的初边值问题的解在初始时间的任意小邻域内不收敛于经典的零通量边界条件下的Euler方程的解，且初始数据相同光滑。收敛性对任何空间拓扑都不成立，这意味着欧拉方程的解继承了完全滑移型边界条件。在我们的反例中Ω是一个球体，初始数据可能是无限可微的。这里的关键点是边界不是平的。事实上（参见bebe<e:1> o da Veiga等人在J Math Anal apple 377:216 - 227,2011），如果收敛成立，对于任意大的kandp。因此，这里给出的否定答案是意料之外的。
We show that,in general, the solutions to the initial-boundary value problem for the Navier-Stokes equations under a widely adopted Navier-type slip boundary condition do not converge, as the viscosity goes to zero, to the solution of the Euler equations under the classical zero-flux boundary condition, and same smooth initial data, in any arbitrarily small neighborhood of the initial time. Convergence does not hold with respect to any space-topology which is sufficiently strong as to imply that the solution to the Euler equations inherits the complete slip type boundary condition. In our counter-example Ω is a sphere, and the initial data may be infinitely differentiable. The crucial point here is that the boundary is not flat. In fact (see Beirão da Veiga et al. in J Math Anal Appl 377:216–227, 2011) ifconvergence holds in, for arbitrarily largekandp. For this reason, the negative answer given here was not expected.