Global classical solutions of 3D compressible viscoelastic system near equilibrium

TL;DR

This paper establishes the global existence of small solutions to the 3D compressible viscoelastic system without restrictive initial conditions or structural assumptions, broadening the class of initial states for which solutions exist.

Contribution

It removes previous assumptions on initial density and structural conditions, introducing a new effective flux to prove global solutions in a more general setting.

Findings

Proves global existence of solutions without initial state restrictions.

Introduces a new effective flux to handle nonlinear terms.

Shows solutions can exist even with increasing norms.

Abstract

In this paper, we prove the global existence of general small solutions to compressible viscoelastic system. We remove the "initial state" assumption ($\tilde \rho_0 \det F_0 =1$) and the "div-curl" structure assumption compared with previous works. It then broadens the class of solutions to a great extent, more precisely the initial density state would not be constant necessarily, and no more structure is need for global well-posedness. It's quite different from the elasticity system in which structure plays an important role. Since we can not obtain any dissipation information for density and deformation tensor, we introduce a new effective flux in the thought of regarding the wildest "nonlinear term" as "linear term". Although the norms of solution may increase now, we can still derive the global existence for it.