The Navier-Stokes equations with dual-scale hereditary viscosity: supercritical norm inflation and global well-posedness in critical spaces
This manuscript investigates the Cauchy problem for an incompressible fluid flow governed by a dual-scale hereditary memory, representing a viscoelastic variant of the classical Navier-Stokes equations that captures anomalous momentum transport. The non-local dissipation breaks exact global scale invariance, dictating a pseudo-differential analysis within the H\"{o}rmander class $S^{-2}_{1,0}$ where the spatial gradient induces a fractional temporal penalty. We rigorously establish $L^q-L^p$ decay estimates and identify the critical Lebesgue threshold $p_c = N (\frac{1+\alpha_\infty}{1-\alpha_\infty})$. In the supercritical regime $1 < p < p_c$, we bypass the loss of spatial localization by mapping the frequency-modulated bilinear flow directly into Fourier space; by utilizing Bernstein's inequalities, we prove instantaneous norm inflation at the origin and confirm intrinsic ill-posedness. Conversely, in the topological limit $p \to \infty$, we demonstrate that the dual-scale memory structurally prevents the collapse traditionally observed for classical fluids within the maximal critical Besov space $\dot{B}^{-1}_{\infty, \infty}$. By exploiting an asymmetric interpolation within Bony's para-differential calculus, we prove that the temporal smoothing overpowers the high-high convective resonant cascade. This delicate analytical balance confines ill-posedness to the non-separable high-frequency tail of the Besov topology, thereby establishing global-in-time Hadamard well-posedness for small initial data possessing high-frequency adherence within $\dot{B}^{-\kappa}_{\infty, \infty}(\mathbb{R}^N)$, a well-posedness regime strictly broader than the separable little Besov closure $\dot{b}^{-\kappa}_{\infty, \infty}(\mathbb{R}^N)$, where $\kappa = \frac{1-\alpha_\infty}{1+\alpha_\infty}$.