Boussinesq Equation with Fractional Dissipation Exhibits Non-Uniqueness for Dimensions And, Demonstrating Singularities of Arbitrarily Small Hausdorff Dimension

The behaviour of fluids forms a cornerstone of physics, yet predicting their movements remains a complex challenge, particularly when dealing with turbulent flows. Zipeng Chen and Zhaoyang Yin, from the School of Science at Sun Yat-sen University, investigate the mathematical foundations of fluid dynamics by examining the Boussinesq equation, a model frequently used to describe convection in fluids. Their work demonstrates a surprising lack of uniqueness in solutions to this equation under certain conditions, meaning multiple distinct fluid behaviours can satisfy the same governing rules. This finding challenges existing assumptions about predictability in fluid dynamics and highlights the subtle ways in which energy loss, or dissipation, influences the stability of fluid flows, with implications for modelling everything from weather patterns to industrial processes.

Non-Uniqueness of Weak Solutions Demonstrated for Boussinesq Equation

This work establishes a rigorous understanding of non-uniqueness for weak solutions to the Boussinesq equation, a model describing fluid motion with temperature-driven buoyancy. Researchers demonstrated that solutions are not always unique, even with identical initial conditions, challenging established assumptions about fluid dynamics. The study centers on a detailed analysis of the equation with fractional dissipation on a torus, investigating conditions under which multiple weak solutions can coexist. To achieve this, scientists constructed a modified equation, the Boussinesq-Reynolds equation, which served as a crucial tool for their investigation.

The team proved that, for specific parameters, weak solutions can exist alongside intervals of regularity, meaning the solutions exhibit predictable behavior for certain periods. This proposition forms the foundation for proving the non-uniqueness of solutions. Specifically, they showed that infinitely many solutions can exist with the same initial data when dimensions d are greater than or equal to two, and a fractional dissipation parameter α is less than d plus one half. Furthermore, they extended this result to show non-uniqueness for solutions in a broader range of function spaces. This work represents the first sharp non-uniqueness results for the Boussinesq system, and the methods developed can also be applied to the Navier-Stokes equation. Researchers also established a theorem demonstrating unique solutions under specific conditions, confirming that non-uniqueness arises when these criteria are not met.

Non-Uniqueness of Weak Solutions in Fluid Dynamics

This work presents a significant breakthrough in understanding the behavior of solutions to the Boussinesq equation, a model used to describe fluid motion with temperature variations. Researchers have demonstrated that for dimensions greater than or equal to two, uniqueness of weak solutions breaks down under specific conditions, revealing a fundamental limit to predictability in this system. The team proved that if a weak solution exists within a particular function space, then infinitely many solutions exist with the same initial data when the fractional dissipation exponent, α, is less than a specific value determined by the dimension of the system. Further analysis established a similar result for solutions in a broader range of function spaces, demonstrating the robustness of this non-uniqueness property.

The team also proved a corresponding uniqueness result, showing that under certain conditions, a single solution exists. These findings represent the first sharp non-uniqueness results for the Boussinesq system, and the methods developed can also be applied to the more fundamental Navier-Stokes equation. The research involved a complex iterative scheme and a procedure to concentrate errors, ultimately demonstrating the existence of multiple valid solutions under specific conditions.

Singularities and Weak Solutions in Fluid Flow

This research establishes a rigorous mathematical framework for understanding the behavior of fluid dynamics governed by the Boussinesq equation with fractional dissipation, specifically on a torus domain. The work demonstrates that uniqueness of solutions breaks down under certain conditions, revealing a fundamental limit to predictability. Importantly, the researchers constructed weak solutions that, while not unique, exhibit smoothness except on a set of negligible irregularity, indicating a controlled level of irregularity. The achievement lies in developing a method to carefully balance high and low frequency components within the fluid flow, employing temporal correctors and spatial oscillation errors to manage the interactions between different flow regimes. Through a detailed analysis of stresses and the introduction of correction and error terms, the team derived a modified equation that accounts for the complexities arising from fractional dissipation and provides a pathway for analyzing the stability and long-term behavior of these fluid flows.