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 ge 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 alpha in (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.
Highlights
Thin-film equation, Non-Newtonian fluid, Classical solution, Quasilinear parabolic equation. This contribution is motivated by questions for existence and uniqueness of strong solutions to the degenerate quasilinear fourth-order evolution problem
In this paper we focus onpositive strong solutions, emanating from a positive initial film height
Cauchy problems, based on the theory for linear parabolic problems. This result will later be applied to the non-Newtonian thin-film equation in the setting of fractional Sobolev spaces and Holder spaces
Summary
NoDEA describing the evolution of the height u(t, x) of a non-Newtonian incompressible thin liquid film on a solid bottom. In [33] the authors establish the existence of weak solutions to a non-Newtonian Stokes equation with a viscosity that depends on the fluid’s shear rate and its pressure at the same time. Due to their weak regularity properties, weak solutions do usually not provide much control at the contact points, where the film height u tends to zero, i.e. where solid, liquid (and gas) meet These are probably the reasons why existence and uniqueness of non-negative strong solutions of the free-boundary value problem associated to (1.2) have recently attracted much attention. The proof exploits the a-priori estimates and the smoothing properties of the corresponding abstract linear equation to obtain a solution for the quasilinear problem by a fixed-point argument We apply this result to obtain existence of solutions to (1.1) in (fractional) Sobolev spaces as well as in (little) Holder spaces. We close this introduction by briefly outlining the organisation of this work
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