Non-uniqueness of Regular Solutions for Incompressible Static Euler Equations with Given Boundary Conditions and Turbulent Global Solutions of Incompressible Navier-Stokes Equations (2502.11485v1)

Abstract: The incompressible Navier-Stokes equations and static Euler equations are considered. We find that there exist infinite non-trivial regular solutions of incompressible static Euler equations with given boundary conditions. Moreover there exist random solutions of incompressible static Euler equations. Provided Reynolds number is large enough and time variable $t$ goes to infinity, these random solutions of static Euler equations are the path limits of corresponding Navier-Stokes flows. But the double limits of these Navier-Stokes flows do not exist. These phenomena reveal randomness and turbulence of incompressible fluids. Therefore these solutions are called turbulent solutions. Here some typing models without Prandtl layer are given.

Non-Uniqueness of Regular Solutions and Turbulent Global Solutions in Incompressible Fluid Dynamics

The paper by Yongqian Han presents a sophisticated analysis of the incompressible Navier-Stokes and static Euler equations, detailing the existence of infinite non-trivial solutions and avenues for turbulent solutions to arise. This paper is particularly impactful in understanding the complexity and inherent randomness present in fluid dynamics, which has implications for both theoretical research and practical applications.

The main argument of the paper revolves around demonstrating the non-uniqueness of regular solutions for the incompressible static Euler equations when subjected to specific boundary conditions. These equations, central to fluid dynamics, describe how the velocity field of a fluid evolves over time, factoring in viscosity (Navier-Stokes) or assuming an ideal fluid with zero viscosity (Euler). Han's work critically highlights that given a large enough Reynolds number, random solutions to the Euler equations serve as path limits for corresponding flows predicted by the Navier-Stokes equations.

Technical Approach and Results

The paper leverages a formal mathematical framework using differential operators and Fourier analysis to describe these fluid dynamics equations under various conditions. A significant technical result is encapsulated in Theorem 1.2, which introduces a global turbulent solution to the incompressible Navier-Stokes equations. The theorem articulates a method of constructing solutions using linear operators defined over Fourier transformed domains, emphasizing the classical representation $A(V)$ and function $\varphi(x) = M(x)$.

This approach yields a series of corollaries, particularly Corollaries 1.4 and 1.5. Corollary 1.4 establishes the non-uniqueness of periodic solutions for the static Euler equations, while Corollary 1.5 assures the global well-posedness of the Navier-Stokes equations with periodic boundary conditions. These results suggest that initial conditions derived from eigenfunctions can lead to multiple valid solutions, extending the potential complexity of solutions in bounded fluid systems.

Implications and Future Directions

The implications of these findings are manifold. The demonstration of non-unique solutions compels a reevaluation of certain methods used to predict fluid behavior, potentially affecting computational fluid dynamics (CFD) simulations and models used in engineering and meteorology. Moreover, the observation of randomness and turbulence challenges assumptions often made about the stability and predictability of fluid systems.

Looking ahead, future research may focus on further exploring the boundary conditions and domain constraints under which these non-uniqueness phenomena manifest, as well as incorporating these insights to refine computational models that seek to simulate turbulent flows. Additionally, the absence of Prandtl layers as indicated in Han's work opens new questions about the boundary behavior of fluids and their transition between laminar and turbulent states.

Han's paper contributes a nuanced perspective to the paper of fluid dynamics, emphasizing the complexity inherent in seemingly well-understood equations and motivating continued exploration of the mathematical and physical principles that govern fluid behavior.