On quasilinear parabolic equations and continuous maximal regularity

Abstract

We consider a class of abstract quasilinear parabolic problems with lower-order terms exhibiting a prescribed singular structure. We prove well-posedness and Lipschitz continuity of associated semiflows. Moreover, we investigate global existence of solutions and we extend the generalized principle of linearized stability to settings with initial values in critical spaces. These general results are applied to the surface diffusion flow in various settings.

The surface diffusion and the Willmore flow for uniformly regular hypersurfaces

“…The results in Theorem 4.3 and Theorem 5.1 are new. However, we note that in case Σ is an infinitely long cylinder embedded in R 3 , an analogous result to Theorem 4.3 was obtained in [22] for the surface diffusion flow.…”

“…To the best of our knowledge, the current literature on the surface diffusion and Willmore flows for non-compact manifolds all concern surfaces defined over an infinite cylinder or entire graphs over R m , or the Willmore flow with small initial energy, cf. [8,16,17,21,22]. Our work generalizes the study of these two flows to a larger class of manifolds.…”

“…We use the convention that a sphere has negative mean curvature. We note that this convention is in agreement with [29,32,33], but differs from [13,20,22].…”

“…Following the convention in [30] and [22], we call the index j subcritical if (2.5) is a strict inequality and critical in case equality holds in (2.5).…”

“…In order to obtain our results, we show that the pertinent underlying evolution equations can be formulated as parabolic quasilinear equations of fourth order over the reference surface Σ. Our analysis relies on the theory of continuous maximal regularity and the results and techniques developed in [22,33,34].…”