Some aspects of the Hadamard's ill-posedness in the hydrodynamical problem

TL;DR

This paper explores the ill-posedness of the hydrodynamical problem related to the Navier-Stokes equations, highlighting mathematical challenges and proposing methods to address initial value problems in fluid dynamics simulations.

Contribution

It analyzes Hadamard's ill-posedness in hydrodynamics and suggests approaches to mitigate issues in numerical initial value problems for fluid equations.

Findings

Identifies fundamental ill-posedness issues in Navier-Stokes-based problems.

Proposes a method to select simulation parameters avoiding ill-posedness.

Highlights the mathematical complexity of initial value problems in fluid dynamics.

Abstract

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.