Global estimates without the ellipticity assumption (H6)

Establish global estimates for the hyperbolic component V^1 in the L^p framework under the Shizuta–Kawashima condition without imposing the additional assumption (H6), which requires the matrix B(ω) to be a constant diagonal matrix with positive entries.

Background

The paper introduces a parabolic mode W to obtain improved high-frequency regularity for the diffusive component V2 and then transfers this regularity to the hyperbolic component V1. After a change of variables, the equation for the transformed hyperbolic variable z contains the operator B(D), whose symbol B(ω) is assumed in (H6) to be a constant diagonal matrix with positive entries.

The authors explain that their main Lp global-existence estimates rely on this strong structural assumption. Although the preceding work required only a Shizuta–Kawashima condition for an associated matrix pair, the paper does not establish whether comparable global Lp estimates can be obtained under that weaker condition alone. Resolving this issue would broaden the framework to partially diffusive systems that do not satisfy (H6).

References

Unlike in , where we only required the pair $$\bigl((\overline{S}0_{11}){-1} S_{11}(\omega), ( \overline{S}0_{11}){-1} S_{12}(\omega)Z(\omega){-1}S_{21}(\omega)\bigr)$$ to satisfy the (SK) condition (equivalently that $(N_\omega, M_\omega)$ satisfies the (SK) condition), it is unclear how to obtain global estimates on $V1$ in the $Lp$ framework without (H6).

— Global existence for viscous systems of conservation laws in $L^p$ framework  (2610.03011 - Adogbo, 2 Oct 2026) in Section 2, Subsection “Transfert of integrability from the parabolic mode to the hyperbolic mode,” immediately after Assumption (H6)