Sub-unit initial-data index for Kawashima-type systems

Establish, for almost all Kawashima-type systems, global existence for initial perturbations of size \(\varepsilon^{\theta}\) with an index \(\theta<1\), including compressible elasticity with small viscosity, incompressible magnetohydrodynamics with small viscosity and zero magnetic diffusivity under a nonzero magnetic background, compressible magnetohydrodynamics with viscosity and magnetic diffusivity of size \(\varepsilon\), Boltzmann equations near global Maxwellians with small mean free path, compressible Euler equations with damping \(\varepsilon\rho u\), and one-dimensional quasilinear hyperbolic systems with small damping on the inhomogeneous term.

Background

The paper proves a global-existence result for the three-dimensional compressible Navier–Stokes system with a viscosity-dependent initial-data index close to one-half, up to logarithmic loss, rather than the classical index one. It then formulates a broader conjecture that the same improvement below one should hold for a range of Kawashima-type systems arising in fluid mechanics, elasticity, kinetic theory, and damped hyperbolic equations. The listed systems specify the intended scope of this unresolved generalization.