Problem of Second Grade Fluids in Convex Polyhedrons
Problem of Second Grade Fluids in Convex Polyhedrons
复制标题
凸多面体中的二级流体问题
DOI：
10.1137/110852735
复制
发表时间：
2012
期刊：
影响因子：
--
通讯作者：
J. Bernard
中科院分区：
文献类型：
--
作者：
J. Bernard
本文研究了一个多面体中具有切向边界条件的三维二级流体模型的解。我们开始将问题分解成一个包含广义Stokes问题和输运方程的系统，就像Girault和Scott在二维情况下所做的那样。但是，与二维问题相比，我们有一个难以约束的附加项，这需要解的正则性，并且我们必须证明输运方程的解不再是标量的，是无发散的。为了解决这三个维度的具体困难，我们建立了一个新的系统，它包含了前一个系统而不是等价的。本文的主要部分是证明输运方程在L^2（\Omega）^3$中的任何解都是无发散的，如果右边是无发散的，并且速度在W^{1，\infty}（\Omega）^3$中足够小。存在证明在凸多面体有足够的限制.
This paper studies the solutions of a three-dimensional grade-two fluid model with a tangential boundary condition in a polyhedron. We begin to split the problem into a system with a generalized Stokes problem and a transport equation, as Girault and Scott have done in the two-dimensional case. But, compared to the two-dimensional problem, we have an additional term that is difficult to bound which requires regularity of the solutions and we have to prove that the solutions of the transport equation, which is no longer scalar, are divergence-free. In order to deal with these specific difficulties of the three dimensions, we establish a new system which implies the previous one without being equivalent. A substantial part of the article is devoted to proving that any solution in $L^2(\Omega)^3$ of the transport equation is divergence-free if the right-hand side is divergence-free and the velocity small enough in $W^{1,\infty}(\Omega)^3$. Existence is proven in a convex polyhedron with adequate restrictions...