Analytical Reconstruction of Singular Shear-Strip Boundary Layers via Nonlocal Friedrichs Commutators and Biharmonic Dissipation

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Abstract
The mathematical modeling of singular shear-strip boundary layers in continuous media represents a profound challenge due to the breakdown of smooth differential bounds and the emergence of localized, non-smooth shear displacements. In this paper, we establish a rigorous, closed-form analytical framework to stabilize coupled transport gradients using a nonlocal Friedrichs commutator mapped within an anisotropic Sobolev functional space. By mapping the coupled transport gradients through a gauge-covariant intrinsic tensor Friedrichs mollifier, we demonstrate how high-frequency ultraviolet interactions can be structurally neutralized. We rigorously prove the exact integral formulation of the commutator and demonstrate that its L^p norm vanishes uniformly in the local limit (ε → 0), provided the velocity field maintains W^{1,∞} regularity. To justify the necessity of this regularity condition, we construct an explicit counterexample using a discontinuous shear flow, proving that when velocity gradients contain singular jumps, the commutator does not vanish but instead generates concentrated singular layers (distributions). To control these layers, we introduce a fourth-order biharmonic hydrodynamic dissipative spectral cage (κΔ²) on a bounded domain with Navier-slip boundary conditions. We rigorously establish the energy identity, prove the exponential decay of the energy, and utilize Galerkin approximations and the Aubin-Lions compactness lemma to prove the global existence and uniqueness of weak solutions. This framework provides a mathematically robust foundation for understanding boundary layer singularities, analytically isolating the anomalous dissipation profile without relying on ad-hoc empirical truncations.

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