Global Solvability for the Compressible Hookean Viscoelastic Fluids with a Free Boundary in Some Classes of Large Data
Fei Jiang, Youyi Zhao
TL;DR
The paper develops a global-in-time strong-solution theory for a 3D stratified compressible Hookean viscoelastic fluid with a free internal interface under two restrictive hypotheses: equal elastic coefficients across fluids and a density-stability condition in Lagrangian coordinates. By reformulating in Lagrangian coordinates and decomposing the system into an inhomogeneous form, the authors prove global solvability for certain large-data classes and derive exponential decay of the energy with κ-dependent rates. They further show a vanishing nonlinear-interaction phenomenon as κ becomes large, establishing that nonlinear solutions can be closely approximated by linear ones in this asymptotic regime. The results extend 2D incompressible free-boundary insights to a 3D compressible stratified setting and illuminate the stabilizing role of elasticity in preventing boundary singularities, with precise quantitative estimates and rigorous a priori energy bounds.
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 $κ$ 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 $κ$.