Nonstationary plane flow of viscous and ideal fluids
Nonstationary plane flow of viscous and ideal fluids
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DOI：
10.1007/bf00251436
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发表时间：
1968
影响因子：
2.5
通讯作者：
F. J. McGrath
中科院分区：
文献类型：
--
作者：
F. J. McGrath
本文在全平面R2上考虑了以粘性v为参数的理想不可压缩流体的非定常欧拉方程的经典解和粘性不可压缩流体的非定常N-S方程的经典解。证明了在任意时刻0<；T<；或当粘性趋于零时，粘性速度和涡量在QR=R2 x[0，T]上分别一致收敛到理想速度和涡量。在全平面上，WOLmNEg[6]证明了欧拉速度方程经典解的存在唯一性，LBRAY[5]证明了Navier-Stokes速度方程经典解的存在唯一性。勒雷的证明没有给出关于他的粘性溶液行为的信息。在本文中，我们利用这些方程的流函数形式和Schauder不动点定理得到这些结果的新的，我们相信更简单的证明。流函数方程的使用导致了对数据更强的可微性要求。在这篇文章中，以及在WOLmNER和Leray的文章中，速度被要求在无穷远处趋于零。在准备发表这篇论文时，作者注意到最近的一篇论文，其中Golov~N[2]建立了一个与下面类似的收敛结果，但初速场和外力场所满足的条件不同，证明方法也不同。
In this paper we consider in the entire plane R 2 a classical solution of the nonstationary Euler equation for an ideal incompressible fluid and a classical solution of the nonstationary Navier-Stokes equation for a viscous incompressible fluid depending on viscosity v as parameter. We prove that, for any time 0< T< or, as viscosity tends to zero the viscous velocity and vorticity converge uniformly on Qr= R 2 x [0, T] to the ideal velocity and vorticity respectively. In the entire plane the existence and uniqueness of classical solutions have been proved by WOLmNEg [6] for the Euler velocity equation and by LBRAY [5] for the Navier-Stokes velocity equation. LERAY'S proof does not yield information on the behavior of his solution with viscosity. In this paper we use the stream function form of these equations and the Schauder fixed-point theorem to get new and, we believe, simpler proofs of these results. The use of the stream function equations results in stronger differentiability requirements on the data. In this paper, as well as in those of WOLmNER and LERAY, velocity is required to tend to zero at infinity. While preparing this paper for publication, a recent paper came to the attention of the author in which GOLOV~ N [2] established a convergence result similar to that below, but with different conditions to be met by the initial-velocity and external-force field and a different method of proof.