Well-Posedness of Surface Wave Equations Above a Viscoelastic Fluid

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 in a domain with a free surface. Managing more general constitutive laws is also briefly depicted. The 2D 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 (Allain in Appl Math Optim 16:37–50, 1987) for the Navier–Stokes part. Then we solve the whole Lagrangian problem on [0, T0] for T0 small enough through a contraction mapping. Since the Lagrangian solution is regular enough and the change of coordinates invertible, we can come back to an Eulerian one.