Vortex Conservation in Critical Besov Spaces

This work develops a comprehensive functional-analytic framework for the study of viscous fluid flows within the setting of critical Besov spaces and Littlewood--Paley theory. We establish optimal global well-posedness for the incompressible Navier--Stokes equations in the scale-invariant spaces \(B^{n/p-1}_{p,q}\) for all \(1\le p<\infty\), \(1\le q\le\infty\), providing a complete proof of the bilinear estimates via Bony's paraproduct decomposition and demonstrating the requisite commutator estimates within this framework. We then prove a sharp version of the energy conservation criterion in the inviscid limit: any sequence of weak solutions with regularity \(L^3(0,T;B^{1/3}_{3,\infty})\) has vanishing energy dissipation anomaly, resolving Onsager's conjecture for the Navier--Stokes equations. The core of the paper is devoted to the compressible Navier--Stokes--Fourier system, for which we identify the critical regularity indices \(s_\rho = n/p\) for the density and \(s_u = n/p-1\) for the velocity, and establish global well-posedness using Strichartz estimates for the acoustic operator. We then study the singular high-Mach number limit for arbitrary large initial data (under the technical condition \(p\le n\) for the compactness argument), proving that the acoustic modes decouple completely and the velocity field converges strongly to a solution of the incompressible Navier--Stokes equations. As a direct consequence, we derive the limiting vorticity equation and show that the baroclinic torque vanishes, leading to the conservation of circulation in the limit. This rigorously justifies the persistence of vortex structures in highly compressible flows. Finally, we incorporate a quaternionic bifurcation analysis for rotating flows, illustrating the versatility of the framework for studying symmetry-breaking instabilities. This work establishes new mathematical foundations for turbulence modeling, large-eddy simulation, and vortex dynamics, contributing to the broader effort toward resolving the global regularity problem for the Navier--Stokes equations.