Title: Symplectic and variational formulations of compressible and incompressible Navier-Stokes equation Show Chinese title

Title: 辛与变分形式化的可压缩和不可压缩Navier-Stokes方程

Abstract: In a previous paper, we proposed a symplectic version of Brezis-Ekeland-Nayroles principle based on the concepts of Hamiltonian inclusions and symplectic polar functions. We illustrated it by application to the standard plasticity in small deformations. The object of this work is to generalize the previous formalism to dissipative media in large deformations and Eulerian description. This aim is reached in three steps. Firstly, we develop a Lagrangian formalism for the reversible media based on the calculus of variation by jet theory. Next, we propose a corresponding Hamiltonian formalism for such media. Finally, we deduce from it a symplectic minimum principle for dissipative media and we show how to obtain a minimum principle for unstationary compressible and incompressible Navier-Stokes equation. Show Chinese abstract

Abstract: 在之前的一篇文章中，我们基于哈密顿包含和辛极化函数的概念，提出了一个关于Brezis-Ekeland-Nayroles原理的辛版本。我们通过将其应用于小变形下的标准塑性来举例说明这一点。本工作的目的是将之前的表述推广到大变形和欧拉描述下的耗散介质。这一目标分三步实现。 首先，我们基于喷射理论的变分演算，为可逆介质发展了一种拉格朗日表述。 接下来，我们为这类介质提出相应的哈密顿表述。 最后，从中推导出耗散介质的辛最小原理，并展示如何获得非定常可压缩和不可压缩Navier-Stokes方程的最小原理。