Compressible Rotating Chemotaxis–Navier–Stokes Systems: Lagrangian Singularity Analysis and Dimension Bounds

<jats:p>We develop a comprehensive Lagrangian framework for the analysis of singularities in the three-dimensional compressible rotating chemotaxis–Navier–Stokes system, with particular emphasis on the high Mach number regime. Focusing on suitable weak solutions that satisfy the entropy inequality, we introduce the notion of Lagrangian singular trajectories adapted to the compressible setting and establish a geometric characterization of the space–time blow-up set. Our main theoretical advance shows that singularities are confined to a low-dimensional Lagrangian structure transported by the flow, even in the presence of strong acoustic waves and rotational effects. More precisely, we prove that the space–time singular set is contained in a countable union of Lagrangian trajectories associated with the velocity field and satisfies the sharp estimate that its Hausdorff dimension is at most one. This result constitutes a substantial refinement of classical Eulerian partial regularity bounds of Caffarelli–Kohn–Nirenberg type and provides a genuinely geometric interpretation of singularity formation in coupled fluid–chemotaxis models under extreme compressibility and rotation. The proof combines global entropy inequalities, compactness methods, and partial regularity theory with a refined analysis of the Lagrangian flow map in the DiPerna–Lions–Ambrosio setting for transport equations with variable density. A key feature of our approach is the propagation of regularity along particle trajectories weighted by the density, which allows singularities to be tracked dynamically and yields improved dimensional estimates via tools from geometric measure theory. Additionally, we establish a Lagrangian regularity criterion expressed solely in terms of the integrability of the velocity along particle trajectories, providing a sufficient condition for global smoothness. Beyond the dimensional bound, the proposed Lagrangian formulation clarifies the mechanism by which chemotactic forcing interacts with compressible fluid transport and rotation to produce potential blow-up and establishes a direct connection between singularity formation and low-dimensional invariant structures. These results open new perspectives for the geometric analysis of singularities in active fluid systems and related nonlinear partial differential equations.</jats:p>