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 $O(\delta^2)$ in squared space-time $L^2$ norm. Consequently, the nonlinear remainder has weighted energy $O(\delta^2)$, and its pressure and velocity components satisfy the $L^\infty(0,T;L^2)$ bounds $C_T\delta/\sqrt{\epsilon}$ and $C_T\delta$, respectively. In the sufficient parameter window $\delta^2\ll\epsilon\leq\mu_*\delta$, 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.