Chapter 2 Applications of the abstract framework to evolving sur-
2.3 The equations
We now state the equations we will study. Three of the problems are posed on evolving bounded open sets inRn. In this case, we shall denote by Ω(t) the evolving domain and Γ(t) will denote the evolving compact hypersurface∂Ω(t). In the equa- tions given below,w is a velocity field which has a normal component wν agreeing with the normal velocity of the evolving hypersurface or domain associated to the problem and an arbitrary tangential componentwa.
Surface heat equation Suppose we have an evolving compact hypersurface Γ(t) that evolves with normal velocity wν. Given a surface flux q, we consider the conservation law d dt Z M(t) u=− Z ∂M(t) q·µ
on an arbitrary portion M(t) ⊂ Γ(t), where µ denotes the conormal on ∂M(t). Without loss of generality we can assume that q is tangential. This conservation law implies the pointwise equationut+∇u·wν+u∇Γ·wν+∇Γ·q= 0.Assuming that the flux is a combination of a diffusive flux and an advective flux, so that
q = −∇Γu +ubτ where bτ is an advective tangential velocity field, we obtain ut+∇u·wν +u∇Γ·wν −∆Γu+∇Γu·bτ +u∇Γ·bτ = 0.Setting b =wν +bτ, and recalling (2.3), we end up with the surface heat equation
˙
u−∆Γu+u∇Γ·b+∇Γu·(b−w) = 0 u(0) =u0
(2.4)
supplemented with an initial condition u0 ∈ L2(Γ0). Clearly, this surface heat equation is the archetypal example of a parabolic equation on a moving domain so
fitting it into the framework is worthwhile. The heat equation on an evolving surface was first considered by Dziuk and Elliott in [48] where well-posedness (in slightly different function spaces) through a Galerkin method and finite element analysis was done. These types of reaction-diffusion equations have an extensive literature; let us name a few papers. See [9] for numerical analysis of such equations with a nonlinear source term, [59] for an ALE evolving surface finite element method, [125] for well-posedness and optimal control, and see also the introduction of Chapter 1 for many other references.
A bulk equation With f(t) : Ω(t) → R and u0: Ω0 → R given, consider the boundary value problem
˙
u(t) + (b(t)−w(t))· ∇u(t) +u(t)∇ ·b(t)−D∆u(t) =f(t) on Ω(t) u(t,·) = 0 on Γ(t) u(0,·) =u0(·) on Ω0
(2.5)
where D > 0 is a constant and the physical material velocity b(t) : Ω(t) → Rn is sufficiently smooth with kb(t)kL∞(Ω(t)) ≤ C1 and k∇ ·b(t)kL∞(Ω(t)) ≤ C2 for constants C1 and C2 uniform for all almost time. We refer the reader to [38] for a formulation of balance equations on moving time-dependent bulk domains. The problem (2.5) is a moving hypersurface version of a problem considered in [21] on a movingdomain. This type of equation is a first approximation of Navier–Stokes equations describing fluid-structure interactions [21].
A coupled bulk-surface system In [55], the authors consider the well-posedness of an elliptic coupled bulk-surface system on a (static) domain; we now extend this to the parabolic case in a moving framework. These types of models arise in cell biology and in particular in cellular signalling and metabolism which can be medi- ated by membrane receptors in the interior that can diffuse on the boundary (the cell membrane). There may also be diffusion processes on the boundary coupled to diffusion processes in the interior [36]. These types of coupled bulk-surface problems are abundant in the mathematical biology literature [88, 37, 103] and indeed these applications are a rich source of interesting PDE problems for analysts. Recently, a parabolic coupled bulk-surface system arising from modelling receptor-ligand dy- namics in cells was shown to be well posed in [56] (on a stationary domain). A number of free boundary problems were also derived as limits of certain parameters in the coupled system. There, the coupling between the interior and the boundary quantities is nonlinear. As a start, we will study a linear problem on an evolving
domain.
Given f(t) : Ω(t) → R, g(t) : Γ(t) → R, u0 ∈ H1(Ω0) and v0 ∈ H1(Γ0), we want to find solutionsu(t) : Ω(t)→Randv(t) : Γ(t)→Rof the coupled bulk-surface system ˙ u−∆Ωu+u∇Ω·w=f on Ω(t) (2.6) ˙ v−∆Γv+v∇Γ·w+∇Ωu·ν =g on Γ(t) (2.7) ∇Ωu·ν =βv−αu on Γ(t) (2.8) u(0) =u0 on Ω0 (2.9) v(0) =v0 on Γ0 (2.10) whereα,β >0 are constants. One can think of u and v as being chemical species interacting through the Robin boundary condition (2.8). Note that we reused the notationufor denoting the trace ofu. We use the physical material velocity to define the mapping F and assume there is just the one velocity field w which advects u within Ω andv on Γ.
A dynamic boundary problem for an elliptic equation Given v0 ∈L2(Γ0)
andf(t)∈H−12(Γ(t)), we consider the problem of finding a functionv(t) : Ω(t)→R
such that, withu(t) :=v(t)|Γ(t) denoting the trace,
∆v(t) = 0 on Ω(t) ˙ u(t) +∂v(t) ∂ν(t) +u(t) =f(t) on Γ(t) u(0) =v0 on Γ0 (2.11)
holds in a weak sense. Here we assume that Γ(t) evolves with the velocitywwhich we suppose is a normal velocity. This is a natural (linearised) extension to evolving domains of a problem considered by Lions in [84,§1.11.1]. One reason why problems with dynamic boundary conditions are interesting is because (by definition) the paraboic nature of the problem is found in the boundary of the domain and this leads to a more interesting functional setting. It will turn out that this problem can be formulated in the fractional Sobolev spaceH12(Γ(t)), meaning that we need
to check a number of technical assumptions on this kind of space in order to apply the abstract framework. Doing this work here becomes enormously useful when we study a fractional porous medium equation in Chapter 4.
boundary conditions is treated on a bounded domain, where well- and ill-posedness results are given. In [126], the problem studied is a Laplace equation with zero Dirichlet and nonlinear dynamical boundary conditions on two disjoint parts of the boundary of the domain, and the author proves existence through the use of Dirichlet-to-Neumann maps like we also will do. The latter paper also contains many references for the curious reader. Both papers are in the setting of stationary domains, so our work, though linear, is new.
In order to formulate these equations in an appropriate weak sense and carry out the analysis, we will need Bochner-type function spaces for evolving hypersur- faces and the associated theory. This is done in the next section.