This paper addresses the challenge of proving the existence of solutions for nonlinear equations in Banach spaces, focusing on the Navier-Stokes equations and discretizations of thom. Traditional methods, such as monotonicity-based approaches and fixed-point theorems, often face limitations in handling general nonlinear operators or finite element discretizations. A novel concept, mapped coercivity, provides a unifying framework to analyze nonlinear operators through a continuous mapping. We apply these ideas to saddle-point problems in Banach spaces, emphasizing both infinite-dimensional formulations and finite element discretizations. Our analysis includes stabilization techniques to restore coercivity in finite-dimensional settings, ensuring stability and existence of solutions. For linear problems, we explore the relationship between the inf-sup condition and mapped coercivity, using the Stokes equation as a case study. For nonlinear saddle-point systems, we extend the framework to mapped coercivity via surjective mappings, enabling concise proofs of existence of solutions for various stabilized Navier-Stokes finite element methods. These include Brezzi-Pitk\"aranta, a simple variant, and local projection stabilization (LPS) techniques, with extensions to convection-dominant flows. The proposed methodology offers a robust tool for analyzing nonlinear PDEs and their discretizations, bypassing traditional decompositions and providing a foundation for future developments in computational fluid dynamics.

翻译：本文致力于解决Banach空间中非线性方程解的存在性证明难题，重点研究Navier-Stokes方程及其离散化问题。传统方法（如基于单调性的方法和不动点定理）在处理一般非线性算子或有限元离散化时常常面临局限。本文提出的新概念——映射强制性质，通过连续映射为分析非线性算子提供了统一框架。我们将这些思想应用于Banach空间中的鞍点问题，同时关注无限维形式与有限元离散化。我们的分析包含恢复有限维设定中强制性的稳定化技术，从而确保解的稳定性与存在性。对于线性问题，我们以Stokes方程为例探讨inf-sup条件与映射强制性质之间的关系。针对非线性鞍点系统，我们将框架扩展至基于满射的映射强制性质，从而能够为多种稳定化Navier-Stokes有限元方法简洁地证明解的存在性。这些方法包括Brezzi-Pitkäranta格式、其简单变体以及局部投影稳定化（LPS）技术，并可扩展至对流主导流动。所提出的方法论为分析非线性偏微分方程及其离散化提供了强有力的工具，它绕过了传统分解方法，为计算流体力学未来发展奠定了基础。