Takeshita's examples for Leray's Inequality
Takeshita's examples for Leray's Inequality
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竹下关于勒雷不等式的例子
DOI：
10.14492/hokmj/1362406641
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发表时间：
2013
影响因子：
0.5
通讯作者：
Teppei Kobayashi
中科院分区：
文献类型：
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作者：
Teppei Kobayashi
我们知道Leray不等式在严格流出条件（SOC）下成立。但Leray不等式在一般流出条件下不成立。本文给出了Leray不等式破裂的直接计算。1问题和主要定理我们考虑不可压缩粘性流体在Dirichlet边界条件下的定常流动。设Ω是R中具有光滑边界的有界区域，Ω的多连通分支为Γ0，Γ1，· · ·，ΓJ . r 0是外边界，其他都是内边界。Ω充满不可压缩的粘性流体。u =（u1（x），u2（x））是流体运动的速度，p = p（x）是以Ω为单位的流体压力。然后由Navier-Stokes方程控制的流体运动是− u+ u ·u+ p = f in Ω（1.1）divu = 0 in Ω（1.2），其中Dirichlet边界条件u = β在Ω上，（1.3）其中f是规定的外力，β是定义在Ω上的给定函数。边界条件β必须是Ω β · ndS = 0，（1.4）其中n是向外垂直于Ω的单位。我们称条件（1.4）为“一般流出条件”（简称GOC）。此外，若β满足<$r j β · ndS = 0（j = 0，· · ·，J），（1.5），则条件（1.5）称为“严格流出条件”，简称SOC。设B是β的无发散扩张。Navier-Stokes方程（1.1）-（1.3）解的收敛性的证明取决于是否存在β的扩张B使得项（（v ·ε）B，v）很小。例如，下面的命题是众所周知的。2000年数学学科分类：35 Q30、76 D 05。
We know that Leray’s Inequality holds under stringent outflow condition (SOC). But Leray’s Inequality does not hold under general outflow condition (GOC). In this paper the direct calculation of the breakdown of Leray’s Inequality appears. 1 Problem and Main Theorem We consider a stationary flow of an incompressible viscous fluid with Dirichlet boundary conditions. Let Ω be a bounded domain in R with a smooth boundary ∂Ω which has multiply connected components Γ0, Γ1, · · · , ΓJ . Γ0 is an outer boundary and the others are inner boundaries. Ω is filled with an incompressible viscous fluid. u = (u1(x), u2(x)) is the velocity of the fluid motion and p = p(x) is the pressure of the fluid in Ω. Then the fluid motion governed by the Navier-Stokes equations is −∆u+ u · ∇u+∇p = f in Ω (1.1) divu = 0 in Ω (1.2) with the Dirichlet boundary conditions u = β on ∂Ω, (1.3) where f is the prescribed external force and β is the given function defined on ∂Ω. The boundary condition β must ∫ ∂Ω β · ndS = 0, (1.4) where n is the unit outward normal to ∂Ω. We call the condition (1.4) “General Outflow Condition”, (GOC) in short. Moreover if β satisfies ∫ Γj β · ndS = 0 (j = 0, · · · , J), (1.5) the condition (1.5) is called “Stringent Outflow Condition”, (SOC) in short. Let b be an extension of β with divergence free. The proof of existense of the solution of the Navier-Stokes equations (1.1)-(1.3) depend on whether there is an extension b of β such that the term ((v ·∇)b,v) is small. For example, the following Proposition is well known. 2000 Mathematics Subject Classification : 35Q30, 76D05.
关于平稳纳维-斯托克斯方程 D 解的 Leray 问题
DOI：
--
发表时间：
2013
期刊：
影响因子：
--
作者：
金田行雄;石原 卓;横川三津夫;板倉憲一;宇野篤;Hideo KOZONO;Hideo KOZONO;小薗 英雄;Hideo KOZONO;Hideo KOZONO;Hideo Kozono;Hideo Kozono
通讯作者：
Hideo Kozono