Quadrupole-Depletion Route to $L^3$ Regularity for 3D Incompressible Navier--Stokes

This upload contains a self-contained research manuscript proposing a single, unified proof program for global regularity of the three-dimensional incompressible Navier--Stokes equations, organized around the critical $L^3$ dissipation identity and an explicit algebraic cancellation in the pressure Hessian. We consider the incompressible Navier--Stokes system (on $\mathbb{T}^3$ or $\mathbb{R}^3$ with decay)\[\partial_t v + (v\cdot\nabla)v = -\nabla p + \nu \Delta v,\qquad \nabla\cdot v = 0,\]with viscosity $\nu>0$. The pressure is determined by\[-\Delta p = \partial_j\partial_k(v_j v_k),\qquadp = \mathcal{R}_j\mathcal{R}_k(v_j v_k),\]where $\mathcal{R}_j$ are Riesz transforms. The starting point is the critical $L^3$ identity\[\frac{d}{dt}\|v(t)\|_3^3 + 3\nu D_3(v(t)) = -3\int |v|\, v\cdot\nabla p\,dx,\]with\[D_3(v) := \int\left(|v|\,|\nabla v|^2 + \frac{(v\cdot\nabla v)^2}{|v|}\right)\,dx.\]Global regularity follows if the pressure work satisfies\[\left|\int |v|\, v\cdot\nabla p\,dx\right| \le c\,D_3(v)\quad\text{for some } c