Well-posedness of the equations of a viscoelastic fluid with a free boundary

TL;DR

This paper establishes the local well-posedness of equations governing a viscoelastic fluid with a free boundary, using a Lagrangian approach to handle the complex free surface and constitutive laws.

Contribution

It proves local existence and uniqueness of solutions for viscoelastic fluid equations with free boundary conditions, extending to more general constitutive laws.

Findings

Proves local well-posedness for viscoelastic fluid equations with free boundary.

Uses a Lagrangian formulation and fixed point arguments for the proof.

Ensures smoothness of solutions allows transition from Lagrangian to Eulerian framework.

Abstract

In this article, we prove the local well-posedness, for arbitrary initial data with certain regularity assumptions, of the equations of a Viscoelastic Fluid of Johnson-Segalman type with a free surface. More general constitutive laws can be easily managed in the same way. The geometry is defined by a solid fixed bottom and an upper free boundary submitted to surface tension. The proof relies on a Lagrangian formulation. First we solve two intermediate problems through a fixed point using mainly G. Allain Appl. Math. Optim. 16 (1987) 37-50 for the Navier-Stokes part. Then we solve the whole Lagrangian problem on $[0,T_0]$ for $T_0$ small enough through a contraction mapping. Since the Lagrangian solution is smooth, we can come back to an Eulerian one.