Linear corrector estimates and corrected pressure convergence for a two-parameter hyperbolic relaxation of the incompressible Navier–Stokes equations
Year: 2026
Type: article
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 C T δ / ϵ ${C}_{T}\delta /\sqrt{{\epsilon}}$ 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.
Source: Transport Phenomena
DOI: 10.1515/tp-2026-0118
Authors: Leyang Wang, Wenlong Lin
Institution: Shanghai Normal University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213304147