Acoustic Filtering and Pressure Recovery for a Two-Parameter Hyperbolic Relaxation of the Incompressible Navier--Stokes Equations

RESUMO

We study a two-parameter first-order hyperbolic relaxation approximation of the incompressible Navier--Stokes equations on the two-dimensional torus. Existing derivative-level large-perturbation estimates recover the velocity but control the pressure only after multiplication by the square root of the artificial-compressibility parameter. We isolate the corresponding acoustic oscillation by introducing a non-autonomous acoustic--stress corrector. Two compensated variables reveal a physical-space cancellation which yields an integrated gradient estimate for the velocity corrector. The two-dimensional Ladyzhenskaya inequality then gives a quadratic bound for its self-interaction. After the corrector is removed, the remaining nonlinear error has zero initial data and is controlled at the next order. If the relaxation parameters $\epsilon$ and $\delta$ satisfy $\delta^2\ll\epsilon\leq\mu_*\delta$, we prove strong recovery of the filtered pressure in $L^\infty(0,T;L^2(\mathbb T^2))$. Quantitatively, the filtered pressure and velocity errors are bounded by $C_T\delta/\sqrt{\epsilon}$ and $C_T\delta$, respectively. The argument also clarifies why strong convergence of the unfiltered pressure cannot in general be expected for ill-prepared acoustic data.