Establish the full conjectural pointwise endpoint estimate

Establish the full conjectural cone-preserving pointwise endpoint estimate for the physical remainder of the one-dimensional compressible isentropic Navier–Stokes equations after subtracting the Burgers diffusion waves, the complete convergent higher-order diffusion-wave hierarchy, and the cross-family viscous corrections, including every component of the pointwise weight conjectured by Koike.

Background

The paper studies the limiting pointwise decay of localized perturbations of the one-dimensional compressible isentropic Navier–Stokes equations. Earlier work by Koike established pointwise remainder estimates after any fixed finite number of higher-order diffusion-wave corrections, but the associated decay exponent approaches its limiting value only as the correction order tends to infinity. Because the constants may depend on the finite order, those estimates do not directly justify an infinite-order endpoint estimate.

The paper proves a weaker endpoint result for the exact physical remainder after subtracting the entire convergent hierarchy and the cross-family viscous corrections. Its cone-preserving weight reaches the limiting global algebraic exponent up to a single logarithmic loss, but the authors explicitly state that it is weaker than the full conjectural pointwise weight and that not every component of the conjecture is established. Thus, the complete conjectural pointwise endpoint estimate remains unresolved.