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

• Published in: Nonlinear differential equations and applications Vol. 27; no. 2
• Main Authors: Lienstromberg, Christina, Müller, Stefan
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 2020; Springer Nature B.V
• Subjects: Analysis; Boundary value problems; Capillarity; Differential equations; Energy methods; Existence theorems; Exponents; Fluid dynamics; Function space; Lubrication; Mathematics; Mathematics and Statistics; Nonlinear analysis; Operators (mathematics); Rheological properties; Rheology; Shear thinning (liquids); Sobolev space; Thin films; Uniqueness; 76A20; Classical solution; 76A05; 76D03; 35K59; 35K25; 35K35; 35K55; Non-Newtonian fluid; 35K65; Quasilinear parabolic equation; 35Q35; Thin-film equation
• ISSN: 1021-9722; 1420-9004
• DOI: 10.1007/s00030-020-0619-x

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 thin-film 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.