流体动力学问题中Hadamard不适定性的若干问题
Some aspects of the Hadamard's ill-posedness in the hydrodynamical problem
摘要：
纳维-斯托克斯方程根据定义建立流体动力学问题。这些方程的重要性是很自然的理解，如果我们把重点放在他们承担的作用，在一个大频谱的动力学问题，其中涉及'流体'。然而，它们是PDE理论中纯数学问题的不可否认的来源。他们公式化的基本核心主要是基于一个简单的概念，即连续介质的无穷小部分，以某种方式局部流动，必须在一个“有界”域中遵守我们用来描述孤立拉格朗日系统演化的相同基本规则，基本上是动量和质量守恒定律，因此数学植入物的后续架构看起来非常清晰和可理解。展望数值求解器的框架，考虑到它们的微分结构的丰富性和数学上连贯的复杂动力学的相关存在，我试图以最简单的方式阐明当我们必须施加合理的“初始值问题”以模拟数值上众所周知的流体动力学场景时出现的基本困难，同时试图从柯西问题的阿达玛观点出发，为避免这种障碍提供一种可能的方法。
英文摘要：
Navier-Stokes equations establish the hydrodynamical problem by definition. The importance of these equations is quite natural to understand if we focus on the role they assume in a large spectrum of dynamical problems which involve 'fluids'. Neverthless, they are an undeniable source of pure mathematical problems in PDE's theory. The essential core of their formulation was primarily well structured on the simple concept that the infinitesimal portions of a continuous medium, which flows locally in some manner, must obey in a 'bounded' domain to the same fundamental rules we use to describe the evolution of isolated lagrangian systems, basically momentum and mass conservation laws, so that the consequent architecture of the mathematical implant appears very clear and understandable. Looking to the framework of the numerical solvers, taking in care the richness of their differential structure and the correlated existence of complex dynamics which are mathematically coherent, I try to put in light in the most simple way the fundamental difficulty that arises when we have to impose a reasonable 'initial values problem' in order to simulate numerically well known fluid-dynamical scenarios, trying at the same time to offer a possible method to avoid such an obstacle in determining simulation parameters from which starting in respect of the essential Hadamard's point of view of the Cauchy problem.
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