Geometric Analysis of Singularities in the Chemotaxis–Navier–Stokes System via Lagrangian Flows

We develop a comprehensive Lagrangian framework for the analysis of singularities in the threedimensional chemotaxis–Navier–Stokes system. Focusing on suitable weak solutions, we introduce the notion of Lagrangian singular trajectories 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. More precisely, we prove that the space–time singular set S is contained in a countable union of Lagrangian trajectories associated with the velocity field and satisfies the sharp estimate dimH(S) ≤ 1. 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. The proof combines global energy inequalities, compactness methods, and partial regularity theory with a refined analysis of the Lagrangian flow map in the DiPerna–Lions–Ambrosio setting. A key feature of our approach is the propagation of regularity along particle trajectories, which allows singularities to be tracked dynamically and yields improved dimensional estimates via tools from geometric measure theory. Beyond the dimensional bound, the proposed Lagrangian formulation clarifies the mechanism by which chemotactic forcing interacts with fluid transport 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 PDEs.