Space-time localized nonuniqueness for the inhomogeneous Navier-Stokes equations

We prove space-time localized nonuniqueness for the inhomogeneous incompressible Navier–Stokes equations on the three-dimensional torus. For every $0<s<3/2$ and $p>1$ sufficiently close to one, we construct weak solutions with strictly positive density in $C_tL^p$ and velocity in $C_tH^s$. These solutions approximate prescribed smooth density and velocity fields satisfying the necessary mass and spatial first moment conditions, while agreeing with a given smooth solution outside a compact subset of a prescribed cylinder. In particular, there are infinitely many nontrivial solutions which coincide with the rest state outside that cylinder and whose density is not renormalized. The proof uses convex integration with concentrated density profiles transported exactly by the perturbed velocity. Nonnegative coefficients cancel the continuity defect and the stress simultaneously. A stationary density corrector preserves positivity through successive corrections, and compactly supported inverse divergence operators preserve spatial localization.