Mathematical Fluid Dynamics: Theory, Analysis and Emerging Trends

Dear Colleagues,

This Special Issue explores the mathematical essence of fluid motion through rigorous analysis, geometric reasoning, and modern data-driven methodologies. It seeks to illuminate how the foundational equations of continuum mechanics—most notably the Navier–Stokes and Euler equations—continue to pose profound challenges concerning existence, regularity, and singularity formation, while also inspiring new analytical and computational paradigms.

The Issue highlights advances in the mathematical theory of turbulence, the energy cascade across scales, and the analysis of free-boundary, multiphase, and magnetohydrodynamic (MHD) systems. Emphasis is placed on frameworks that integrate functional and harmonic analysis, stochastic formulations, and variational or geometric principles, thereby extending the reach of classical fluid theory. Moreover, the Issue welcomes mathematically grounded approaches to physics-informed and machine-learning-assisted modeling, which reconcile data adaptability with physical and analytical integrity. Collectively, these contributions aim to redefine the modern landscape of mathematical fluid dynamics, revealing how timeless equations give rise to new structures, symmetries, and pathways toward a deeper understanding of fluid motion in both nature and computation.

Prof. Dr. Shin'ichi Inage
Guest Editor

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