Local strong solutions to a quasilinear degenerate fourth-order non-Newtonian thin-film equation
Tuesday, January 28, 2020, 10:30am
Christina Lienstromberg, Universität Bonn
This talk is motivated by questions for existence and uniqueness of strong solutions to a degenerate quasilinear fourth-order 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.
Regarding the existence and uniqueness of strong solutions, it turns out that there is a qualitative difference between flow behaviour exponents α ∈ (1, 2) and those larger than or equal 2. If we associate to the equation an abstract quasilinear Cauchy problem, it turns out that in the case α ≥ 2 the differential operator and the right-hand side are Lipschitz continuous in an appropriate sense and the classical Hölder theory of Eidel’man (1969) as well as the abstract theory of Amann (1993) are applicable.
The situation is more delicate for flow behaviour exponents α ∈ (1, 2). In this regime the operator and the right-hand side are only (α −1)-Hölder continuous, whence there is no hope to obtain existence and uniqueness by Banach’s fixed point theorem. For this reason we prove an abstract existence result for quasilinear parabolic problems of fourth order with Hölder continuous coefficients. 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.
Raum 008/SeMath, Pontdriesch 14-16, 52062 Aachen