动态能量稳定出口边界条件下畸变管道中的演化Navier-Stokes方程
On the evolutionary Navier-Stokes equations in distorted pipes under dynamic and energy-stable outflow boundary conditions
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中文总结 AI 辅助
本文研究动态能量稳定出口边界条件下畸变管道的Navier-Stokes方程，引入适配动态边界条件与不可压缩约束的泛函框架，建立弱解存在性，在数据小性假设下证明整体强解的存在唯一性，通过Galerkin方法求解完整系统。
中文摘要 AI 辅助
我们考虑三维有限长畸变管道中粘性不可压缩流体的演化，该流体由带有混合边界条件的Navier-Stokes方程建模。具体而言，入口由任意给定数据确定，出口受包括定向无作用边界条件在内的动态条件约束，计算域其余壁面施加标准无滑移假设。我们引入一种新的泛函框架，以适配该动态边界条件与不可压缩约束。在该框架内，我们建立了弱解的存在性；此外，在数据的小性假设下，我们证明了整体强解的存在性与唯一性。新引入的泛函设置使我们能够先处理相关的Stokes问题，再通过采用适当修改的Stokes算子特征函数基实现的Galerkin方法处理完整的Navier-Stokes系统。
英文摘要
We consider the evolution of a viscous incompressible fluid in three-dimensional distorted pipes, of finite length, modeled through the Navier-Stokes equations with mixed boundary conditions. Specifically, the inflow is given by an arbitrary datum, the outflow is subject to a dynamic condition including a directional do-nothing boundary condition; standard no-slip assumptions are imposed on the remaining walls of the domain. We introduce a new functional framework that accommodates the dynamic boundary condition and the incompressibility constraint. Within this framework, we establish the existence of weak solutions. Moreover, under a smallness assumption on the data, we prove the existence and uniqueness of global strong solutions. The newly introduced functional setting enables us to deal, at first, with the associated Stokes problem, and then with the full Navier-Stokes system, via the Galerkin method implemented with a basis of eigenfunctions of a suitably modified Stokes operator.
