An abstract framework for parabolic PDEs on evolving spaces

Abstract

We present an abstract framework for treating the theory of wellposedness of solutions to abstract parabolic partial differential equations on evolving Hilbert spaces. This theory is applicable to variational formulations of PDEs on evolving spatial domains including moving hypersurfaces. We formulate an appropriate time derivative on evolving spaces called the material derivative and define a weak material derivative in analogy with the usual time derivative in fixed domain problems; our setting is abstract and not restricted to evolving domains or surfaces. Then we show well-posedness to a certain class of parabolic PDEs under some assumptions on the parabolic operator and the data.

Error analysis for an ALE evolving surface finite element method

“…See [15,16,17] for further discussion on the well-posedness of the weak formulation of the continuous problem.…”

A Stefan problem on an evolving surface

“…That (2.2) defines a norm is easy to see once one checks that the integrals are well defined (the case p = ∞ is easy), which can be shown by a straightforward adaptation of the proof of theorem 2.8 in [6] for the case when each X(t) is separable (see also electronic supplementary material, S1) and the proof of lemma 3.5 in [14] for the non-separable case. The fact that L p X is a Banach space follows from lemma 2.3 below.…”

“…We show the case p = ∞ here; an adaptation of the p = 2 case done in [6] easily proves the lemma for p ∈ [1, ∞) (see also electronic supplementary material, S2). Let u ∈ L ∞ X .…”

“…The novelty of this work is that the Stefan problem itself is formulated on a moving hypersurface and our chosen method to treat this problem, which we believe is naturally suited to equations on moving domains, requires the use of some new function spaces and results that we shall introduce, building upon the spaces and concepts presented in [6,7]. There is, as alluded to above, a rich literature associated to Stefan-type problems [8][9][10][11][12][13].…”

“…In [6], we generalized some concepts from [14] and defined the Hilbert space L 2 H given a sufficiently smooth parametrized family of Hilbert spaces {H(t)} t∈ [0,T] . We need a generalization of this theory to Banach spaces.…”