Global Solvability for the Compressible Hookean Viscoelastic Fluids with a Free Boundary in Some Classes of Large Data

TL;DR

This paper extends the understanding of global solutions for compressible Hookean viscoelastic fluids with free boundaries from 2D to certain 3D cases under restrictive conditions, revealing stability and approximation properties for large elasticity.

Contribution

It establishes a 3D global solvability result for stratified compressible Hookean viscoelastic fluids under specific conditions, extending prior 2D results and analyzing nonlinear interaction phenomena.

Findings

Global existence of solutions in 3D under restrictive conditions

Solutions can be approximated by linear problems for large elasticity coefficient

Nonlinear interactions diminish with large elasticity coefficient

Abstract

Recently Jiang-Jiang established a global (in time) existence result for unique strong solutions of the two-dimensional (2D) free-boundary problem of an incompressible Hookean viscoelastic fluid, the rest state of which is defined in a slab, in some classes of large data [28]. In particular, Jiang-Jiang's mathematical result shows that, if the initial free boundary is flat, the way the elastic deformation under the large elasticity coefficient $\kappa$ acts on the free boundary prevents the natural tendency of the fluid to form singularities, even when the initial velocity is properly large. However it is not clear whether their result can be extended to the corresponding 3D case. In this paper, we further find a similar result in the 3D stratified (immiscible) compressible Hookean viscoelastic fluids in an infinite slab with two restrictive conditions: that the elasticity coefficients of two fluids are equal, and that the initial density functions satisfy the asymptotic stability condition in Lagrangian coordinates. These two restrictive conditions in the compressible case contribute us to avoid the essential obstacles that would be faced in the extension of Jiang-Jiang's result from two dimensions to our 3D case. In addition, we can further obtain a new result regarding the vanishing phenomena of the nonlinear interactions of solutions with the fixed initial velocity and the initial zero perturbation deformation. Such a new result roughly presents that the solutions of the problem considered by us can be approximated by the ones of a linear problem for sufficiently large $\kappa$.