Pointwise wave behavior and  L ^p -estimate for a P1-approximation model arising in radiation hydrodynamics

Abstract

We study the L ^p -estimate ( p ≥ 1) and wave propagation properties of smooth solutions to three dimensional P1-approximation system. Using the space-time characterization of Green function across frequency regimes, we establish L ^p -estimate for initial perturbations in H ² ∩ L ¹ ; concurrently, the derived space-time estimates show the solution’s time-asymptotic profile comprises a stationary diffusion wave superimposed on a traveling one with speed

. This work has three novelties: First, we overcome the key challenge of the non-conservative nature – distinct from L ^p -estimates in Hoff and Zumbrun ( Multi-dimensional diffusion waves for the Navier–Stokes equations of compressible flow , Indiana Univ. Math. J. 44 (1995), 603–676) and wave propagation results

speed  p ′ ( ρ ∞ ) M a 2 $\left(\text{speed\,}\sqrt{\frac{{p}^{\prime }\left({\rho }_{\infty }\right)}{{M}_{a}^{2}}}\right)$

in Liu and Wang ( The pointwise estimates of diffusion wave for the Navier–Stokes systems in odd multi-dimension , Comm. Math. Phys. 196 (1998), 145–173) for the conservative Navier–Stokes equations. Second, we verify that thermal radiation effectively enhances the bulk propagation speed since b ′( ρ _∞ ) > 0, which is theoretically consistent with Fick’s laws of diffusion in radiation theory. Third, we refine the L ^p -estimate ( p ∈ (2, ∞]) of the density ρ under the same assumptions as Wang ( Global well-posedness and decay rates of strong solutions to a P1-approximation model arising in radiation hydrodynamics , SIAM J. Math. Anal. 53 (2021), 5361–5404).