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

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 α ≥ 2 the corresponding initial boundary value problem fits into the abstract setting of Amann (Function Spaces, Differential Operators and Nonlinear Analysis, Vieweg Teubner Verlag, Stuttgart, 1993). Due to a lack of regularity this is not true for flow behaviour exponents α ∈ (1, 2). For this reason we prove an existence theorem for abstract quasilinear parabolic evolution problems with Hölder continuous dependence. This result provides existence of strong solutions to the non-Newtonian thinfilm problem in the setting of fractional Sobolev spaces and (little) Hölder spaces. Uniqueness of strong solutions is derived by energy methods and by using the particular structure of the equation.

Analysis of a Two-Fluid Taylor–Couette Flow with One Non-Newtonian Fluid

“…(1.1), respectively (1.5), belongs to a class of non-Newtonian thin-film equations with strain-dependent viscosity. Similar equations have been studied in different settings, for instance, in Giacomelli (2002, 2004); King (2001a, b); Lienstromberg and Müller (2020). In Ansini and Giacomelli (2004), the authors consider a single thin film occupied by a power-law fluid.…”

Non-Newtonian two-phase thin-film problem: Local existence, uniqueness, and stability

“…In [28] the authors prove the existence of local strong solutions of (1.6) and uniqueness with respect to initial data. Moreover, the authors of [3] investigate a class of quasi-self-similar solutions describing the spreading of a droplet in the limit of a Newtonian rheology.…”

“…In view of a continuation argument, it is sufficient to show that there exists a time * ≤ such that ( , ) = ( , ) on [0, * ]. Following the approach in [28], which was applied in the context of a single non-Newtonian thin film, we show that…”

“…If ∈ (1, 2), then  and the derivative of  are merely ( − 1)-Hölder continuous and problem (P) does not fall into the frame of [2, Theorem 12.1]. However, on the a priori cost of uniqueness of solutions, we may apply a recently developed result on existence of classical solutions of quasilinear Cauchy problems of the form (CP), where the dependence of the functionals  and  on its coefficients is assumed to be only Hölder continuous [28,Theorem 4.2]: Theorem 3.1. Suppose that ( 0 , 1 ) is a densely and compactly injected Banach couple.…”

“…Suppose that̄ < ∞.for all ∈ ( , 1].The proof is based on the standard continuation argument. We follow the lines of the proof of[28, Thm. 7.1].Proof.…”