Linear corrector estimates and corrected pressure convergence for a two-parameter hyperbolic relaxation of the incompressible Navier–Stokes equations

Abstract

We study a two-parameter first-order hyperbolic relaxation approximation of the incompressible Navier–Stokes equations on the two-dimensional torus, under the derivative-level large-perturbation assumptions of Huang, Rohde and Zhang. Building on the auxiliary-system approach used in earlier pressure-convergence results, we introduce a non-autonomous linear corrector that carries the complete initial discrepancy, the linearized convection and the smooth consistency forcing. This corrector describes the full linear response; it is not asserted to be a purely acoustic component. An exact compensated energy identity, including a time integration by parts in the linearized convection term, yields an integrated gradient estimate for the velocity corrector. The two-dimensional Ladyzhenskaya inequality then bounds its quadratic self-interaction by O ( δ ² ) in squared space-time L ² norm. Consequently, the nonlinear remainder has weighted energy O ( δ ² ), and its pressure and velocity components satisfy the L ^∞ (0, T ;  L ² ) bounds

and C _T δ , respectively. In the sufficient parameter window δ ² ≪ ϵ ≤ μ _∗ δ , the pressure after subtraction of the linear corrector converges strongly to the incompressible pressure. A mean-zero Fourier-mode construction, transferred to the nonlinear system by the remainder estimate, shows that the uncorrected pressure need not converge even on time intervals bounded away from zero.