Local strong solutions to a quasilinear degenerate fourth-order thin-film equation

TL;DR

This paper establishes existence and uniqueness of strong solutions for a degenerate quasilinear non-Newtonian thin-film equation, covering cases with different flow behavior exponents and employing advanced functional analysis techniques.

Contribution

It introduces a new existence theorem for quasilinear parabolic problems with fractional Sobolev and H"older spaces, addressing cases where regularity is limited.

Findings

Existence of strong solutions for flow exponents   2.

Uniqueness established via energy methods.

Solutions constructed in fractional Sobolev and Hf6lder spaces.

Abstract

We study the problem of existence and uniqueness of strong solutions to a degenerate quasilinear parabolic non-Newtonian thin-film equation. Originating from a non-Newtonian Navier--Stokes system the equation is derived by lubrication theory and under the assumption that capillarity is the only driving force. The fluid's shear-thinning rheology is described by the so-called Ellis constitutive law. For flow behaviour exponents $\alpha \geq 2$ the corresponding initial boundary value problem fits into the abstract setting of [4,Thm. 12.1]. Due to a lack of regularity this is not true for flow behaviour exponents $\alpha \in (1,2)$. For this reason we prove an existence theorem for abstract quasilinear parabolic evolution problems with H\"older continuous dependence. This result provides existence of strong solutions to the non-Newtonian thin-film problem in the setting of fractional Sobolev spaces and (little) H\"older spaces. Uniqueness of strong solutions is derived by energy methods and by using the particular structure of the equation.