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.

我们研究了退化的拟线性抛物线非牛顿薄膜方程强解的存在性和唯一性问题。该方程源自非牛顿的Navier–Stokes系统，是根据润滑理论推导出的，并假设毛细作用是唯一驱动力。流体的剪切稀化流变学由所谓的Ellis本构定律描述。对于流动行为指数\（\ alpha \ ge 2 \），相应的初始边界值问题适合Amann的抽象设置（函数空间，微分算子和非线性分析，Vieweg Teubner Verlag，斯图加特，1993）。由于缺乏规律性，因此对于流动行为指数\（\ alpha \ in（1,2）\）而言并非如此。因此，我们证明了具有Hölder连续依赖的抽象拟线性抛物线演化问题的存在性定理。该结果为分数Sobolev空间和（小）Hölder空间中的非牛顿薄膜问题提供了强有力的解决方案。强解决方案的唯一性是通过能量方法和使用方程的特定结构得出的。