A generalization of the Aubin–Lions–Simon compactness lemma for problems on moving domains

Abstract

This work addresses an extension of the Aubin-Lions-Simon compactness result to generalized Bochner spaces L 2 (0, T ; H(t)), where H(t) is a family of Hilbert spaces, parameterized by t. A compactness result of this type is needed in the study of the existence of weak solutions to nonlinear evolution problems governed by partial differential equations defined on moving domains. We identify the conditions on the regularity of the domain motion in time under which our extension of the Aubin-Lions-Simon compactness result holds. Concrete examples of the application of the compactness theorem are presented, including a classical problem for the incompressible, Navier-Stokes equations defined on a given non-cylindrical domain, and a class of fluid-structure interaction problems for the incompressible, Navier-Stokes equations, coupled to the elastodynamics of a Koiter shell. The compactness result presented in this manuscript is crucial in obtaining constructive existence proofs to nonlinear, moving boundary problems, using Rothe's method. *

Moving boundary problems

“…The work presented in [107] concerns a generalization of the Aubin-Lions-Simon result which involves Hilbert spaces that are solution dependent and not necessarily known a priori. This is a significant step forward, since the result can be applied to a large class of moving boundary problems, including numerical solvers.…”

“…More precisely, the compactness result in [107] is designed for problems which can be described in general as evolution problems, (5.1)…”

“…In both cases, certain conditions describing the regularity in time of the domain motion need to be satisfied, in order for a compactness argument to hold. In [107] those conditions are identified, and a generalization of the Aubin-Lions-Simon compactness result was obtained, which can be used in both approaches, mentioned above.…”

“…The compactness result from [107] was applied to study existence of solutions to FSI with Koiter shell [102,105], to FSI involving multi-layered structures [104], and to FSI with Koiter shell and Navier-slip coupling [106]. Since the result is based on the backwards Euler time discretization approaches to the coupled FSI problem, the compactness result from [107] is a promising tool for proving convergence of numerical schemes that use the backwards Euler scheme to discretize the problem in time; see [19][20][21][22].…”

Existence of a weak solution to the fluid-structure interaction problem in 3D

“…Next, To prove the assumption C given in [22,Theorem 3.1], one can use almost the same construction as in [22,Example 4.2] with the following changes. First, since η ∆t,k is only uniformly bounded in C 0,α (0, T ; C 0,1−2α (Γ)) for 0 < α < 1/2, the functions constructed in [22,Lemma 4.5] would satisfy a weaker inequality -on the right-hand side of the third inequality, we would have C(l∆t) α , instead of C √ l∆t. Consequently, the corresponding operator I i n,l,∆t would satisfy a weaker assumption C1 given in [22,Theorem 3.1], where the right-hand side of the inequality (3.3) is replaced by C(l∆t) α .…”