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(δ2) in squared space-time L2 norm. Consequently, the nonlinear remainder has weighted energy O(δ2), and its pressure and velocity components satisfy the L∞(0,T;L2) bounds CTδ/√ε and CTδ, respectively. In the sufficient parameter window δ2≪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.
Versions
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BibTeX
@misc{wang2026linear,
title = {{Linear Corrector Estimates and Corrected Pressure Convergence for a Two-Parameter Hyperbolic Relaxation of the Incompressible Navier--Stokes Equations}},
author = {Leyang Wang and Wenlong Lin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.13222},
eprint = {2609.13222},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13222},
}
RIS
TY - UNPB
TI - Linear Corrector Estimates and Corrected Pressure Convergence for a Two-Parameter Hyperbolic Relaxation of the Incompressible Navier--Stokes Equations
AU - Leyang Wang
AU - Wenlong Lin
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2609.13222
UR - https://arxiv.org/abs/2609.13222
AB - 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_Tδ/√ε and C_Tδ, 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.
ER -