An existence result for evolution equations in non-cylindrical domains

Abstract

Abstract.We give an existence and uniqueness result for an evolution equation ut + Au = f , with suitable boundary data and where A is a strictly monotone operator, in a non-cylindrical domain.Mathematics Subject Classification. 35.01, 35K22, 35K55, 47H05.

An abstract framework for parabolic PDEs on evolving spaces

“…Partial differential equations on evolving or moving domains are an active area of research [9], [13], [25], [26], partly because their study leads to interesting analysis but also because models describing applications such as biological and physical phenomena can be better formulated on evolving domains (including hypersurfaces) rather than on stationary domains. For example, see [3], [20] for studies of pattern formation on evolving surfaces, [21] for the modelling of surfactants in two-phase flows, [14] for the modelling and numerical simulation of dealloying by surface dissolution of a binary alloy (involving a forced mean curavture flow coupled to a Cahn-Hilliard equation), [15] (and the references therein for applications) for the analysis of a diffuse interface model for a linear surface PDE, and [16] for the modelling and simulation of cell motility.…”

“…One aspect to consider in the study of such equations is how to formulate the space of functions that have domains which evolve in time. Taking a disjoint union of the domains in time to form a non-cylindrical set is standard: see [6], [33], [26] for example. Of particular interest is [22] where the problem of a semilinear heat equation on a time-varying domain is considered; the setup of the evolution of the domains is comparable to ours and similar function space results are shown (in the setting of Sobolev spaces).…”

Relative entropy and a weak–strong uniqueness principle for the compressible Navier–Stokes equations on moving domains

“…These types of results are extended in [1], [2], where so-called evolving spaces are defined via a pushforward/pullback map to associate function spaces at any time t > 0 with reference spaces at the initial time t = 0. For similar results on PDE with moving spatial domains, see [5], [16], and the references therein.…”

Global in Space Regularity Results for the Heat Equation with Robin-Neumann Type Boundary Conditions in Time-varying Domains

“…It is possible to consider similar questions with some other operators (see, for example, [4] for a 2m-th order operator in bounded non-rectangular domains). Whereas second-order parabolic equations in bounded non-cylindrical domains are well studied (see for instance [1,6,9,12,14,15,18,19,20,23] and the references therein), the literature concerning unbounded non-cylindrical domains does not seem to be very rich. The regularity of the heat equation solution in a non-smooth and unbounded domain (in the t direction) is obtained in [21] and [22] by using two different approaches.…”