Concerning the Wk,p-Inviscid Limit for 3-D Flows Under a Slip Boundary Condition
Concerning the Wk,p-Inviscid Limit for 3-D Flows Under a Slip Boundary Condition
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DOI：
10.1007/s00021-009-0012-3
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发表时间：
2011-03
影响因子：
1.3
通讯作者：
H. B. Veiga;F. Crispo
中科院分区：
文献类型：
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作者：
H. B. Veiga;F. Crispo
我们考虑具有Navier滑移型边界条件的三维发展Navier-Stokes方程（见（1.2）），并研究当粘性趋于零时，解强收敛于零通量边界条件下Euler方程的解的问题。我们在这里证明，在平坦边界的情况下，在Sobolev空间Wk，p（Ω）中收敛，对于任意大的kandp（对于先前的结果，参见Xiao和Xin in Comm Pure Appl Math 60：1027-1055，2007以及Beirão da Veiga和Crispo in J Math Fluid Mech，2009，doi： 10.1007/s00021-009-0295-4 ).然而，这个问题仍然是开放的非平坦，任意光滑，边界。主要的障碍在于某些边界积分，这些积分在边界的平坦部分消失。然而，如果我们去掉对流项（斯托克斯问题），则无粘的强极限结果成立，如下所示。这种不同行为的原因是相当微妙的。作为一个副产品，我们建立了一个非常初级的方法，正则性理论，在Lp-空间，解的Navier-Stokes方程的滑移型边界条件。
We consider the 3-D evolutionary Navier–Stokes equations with a Navier slip-type boundary condition, see (1.2), and study the problem of the strong convergence of the solutions, as the viscosity goes to zero, to the solution of the Euler equations under the zero-flux boundary condition. We prove here, in the flat boundary case, convergence in Sobolev spacesWk,p(Ω), for arbitrarily largekandp(for previous results see Xiao and Xin in Comm Pure Appl Math 60:1027–1055, 2007 and Beirão da Veiga and Crispo in J Math Fluid Mech, 2009, doi: 10.1007/s00021-009-0295-4 ). However this problem is still open for non-flat, arbitrarily smooth, boundaries. The main obstacle consists in some boundary integrals, which vanish on flat portions of the boundary. However, if we drop the convective terms (Stokes problem), the inviscid, strong limit result holds, as shown below. The cause of this different behavior is quite subtle. As a by-product, we set up a very elementary approach to the regularity theory, inLp-spaces, for solutions to the Navier–Stokes equations under slip type boundary conditions.