Global well-posedness of the three-dimensional free boundary problem for viscoelastic fluids without surface tension

Abstract

In this paper, we consider the three-dimensional free boundary problem of incompressible and compressible neo-Hookean viscoelastic fluid equations in an infinite strip without surface tension, provided that the initial data is sufficiently close to the equilibrium state. By reformulating the problems in Lagrangian coordinates, we can get the stabilizing effect of elasticity. In both cases, we utilize the elliptic estimates to improve the estimates. Moreover, for the compressible case, we find there is an extra ODE structure that can improve the regularity of the free boundary, thus we can have the global well-posedness. To prove the global well-posedness for the incompressible case, we employ two-tier energy method introduced in [11][12][13] to compensate for the inferior structure.