Algebraic classification of degenerate solutions of the 3D Navier–Stokes equations and its applications

This work is devoted to a systematic analysis of solutions of the three-dimensional Navier–Stokes equations satisfying det(∇V) ≡ 0. Using an algebraic reformulation of the equations of motion, the problem reduces to the study of the linear systemsJV = b and JTω = c with a singular matrix J = ∇V. The compatibility conditions of these systems impose restrictions on the relationship between the velocityfield, pressure and external forces. We obtain a complete classification of smooth solutions with a globally degenerate gradient depending on the rank of J: (i) rank 2 — the kernel is one-dimensional, pressure and external forces satisfy one linear con-dition; (ii) rank 1 — the velocity field is collinear with a fixed direction, which is an eigenvector of JT, and e · ∇e = 0 holds; (iii) rank 0 — the velocity field is constant. For each case we give the general form of solutions and analyse their compatibilitywith the full equations of motion. Locally, any velocity field with constant rank and the incompressibility condition can be supplemented with appropriate pressure and external forces to become a solution of the Navier–Stokes equations; global solv-ability imposes additional integrability conditions. The results are independent of boundary conditions; the latter act as a filter that discards solutions incompatible with the geometry of the domain.In the second part we investigate the evolution of det(∇V) for non-degenerate solutions. An exact transport equation is derived and an integral non-degeneracy criterion is proved, linking the preservation of positivity of the determinant to the integrability of the L∞-norm of the velocity gradient. It is shown that the classification of degenerate solutions is closed: a smooth non-degenerate solution cannotbecome degenerate in finite time while remaining smooth. As applications, we obtain new proofs of two classical results: (a) global regularity for small initial data in L∞; (b) global regularity of axisymmetric flows without swirl. Both proofs use only local existence, the integral criterion and the classification of degenerate solutions, avoiding Sobolev embedding theorems and interpolation inequalities.