Abstract
We study a phase field model proposed recently in the context of tumour growth. The model couples a Cahn–Hilliard–Brinkman (CHB) system with an elliptic reaction-diffusion equation for a nutrient. The fluid velocity, governed by the Brinkman law, is not solenoidal, as its divergence is a function of the nutrient and the phase field variable, i.e., solution-dependent, and frictionless boundary conditions are prescribed for the velocity to avoid imposing unrealistic constraints on the divergence relation. In this paper we give a first result on the existence of weak and stationary solutions to the CHB model for tumour growth with singular potentials, specifically the double obstacle potential and the logarithmic potential, which ensures that the phase field variable stays in the physically relevant interval. New difficulties arise from the interplay between the singular potentials and the solution-dependent source terms, but can be overcome with several key estimates for the approximations of the singular potentials, which maybe of independent interest. As a consequence, included in our analysis is an existence result for a Darcy variant, and our work serves to generalise recent results on weak and stationary solutions to the Cahn–Hilliard inpainting model with singular potentials.
1 Introduction
Phase field models [17, 45] have recently emerged as a new mathematical tool for tumour growth, which offer new advantages over classical models [11, 24] based on a free boundary description, such as the ability to capture metastasis and other morphological instabilities like fingering in a natural way. In conjunction with clinical data, statistical methodologies and image analysis, they have begun to impact the development of personalised cancer treatments [5, 6, 37, 39, 41, 42].
To support this continuing effort, in this paper we expand the recent analysis performed in [19, 20] for a class of phase field tumour models based on the Cahn–Hilliard–Brinkman (CHB) system. For a bounded domain Ω ⊂ ℝd for d ∈ {2, 3} with boundary Σ : = ∂ Ω and outer unit normal n, and an arbitrary but fixed terminal time T, we consider for ΩT := Ω × (0, T) the following system of equations
where the viscous stress tensor T and the symmetrised velocity gradient Dv are given by
The model (1.1) is a description of the evolution of a two-phase cell mixture, containing tumour cells and healthy host cells, surrounded by a chemical species acting as nutrients only for the tumour cells, and is transported by a fluid velocity field. The variable σ denotes the concentration of the nutrient and is assumed to evolve quasi-statically, while φ denotes the difference in the volume fractions of the cells, with the region {φ = 1} representing the tumour cells and {φ = −1} representing the host cells. The fluid velocity v is taken as the volume averaged velocity, with pressure p, while μ denotes the chemical potential associated to φ.
Equations (1.1c)-(1.1d) comprise a convective Cahn–Hilliard system for (φ, μ) with source term Γφ(φ, σ), that couples the nutrient equation (1.1e) also through the consumption term h(φ) σ. The constant parameter χ appearing in (1.1d) captures the chemotaxis effect, see [32]. On the other hand, equations (1.1a)-(1.1b) form a Brinkman system for (v, p) with the term (μ + χ σ) ∇ φ modelling capillary forces, and in the definition of the stress tensor, the functions η and λ represent the shear and bulk viscosities, respectively. An interesting contrast with previous phase field models in two-phase flows that employ a volume averaged velocity, such as [4, 10, 18], is that the velocity in (1.1) is not solenoidal. This can be attributed to the fact that in the case of unmatched densities, the gain σ bφ(φ) and loss fφ(φ) of cellular volume leads to sources σ bv(φ) and sinks fv(φ) in the mass balance, see for example [32, 40] for more details in the model derivation.
As a consequence of (1.1a), we find the relation
and the typical no-penetration boundary condition v ⋅ n = 0 on Σ × (0, T) will lead to a mean-zero compatibility condition for Γv(φ, σ), which can not be fulfilled in general since Γv depends on φ and σ. For models with Darcy’s law instead of (1.1b), there are some works in the literature [25, 30] prescribing alternative boundary conditions, such as zero transport flux ∇ μ ⋅ n = φ v ⋅ n and zero pressure p = 0 on ΣT := Σ × (0, T) to circumvent the compatibility condition on Γv. For the Brinkman model, the compatibility issue can be avoided by prescribing the frictionless boundary condition T n = 0 on ΣT, as considered in [19, 20, 21, 22].
Hence, we furnish (1.1) with the following initial and boundary conditions
where ∂n f = ∇ f ⋅ n is the normal derivative on Σ, φ0 is a prescribed initial datum and K is a positive boundary permeability constant.
The resulting system (1.1)-(1.2) has been studied previously by the first author in a series of works [19, 20, 21, 22] concerning well-posedness, asymptotic limits and optimal control, in the setting where the double well potential ψ, whose first derivative appears in (1.1d), is continuously differentiable. The common example is the quartic potential ψ(s) = (s2 − 1)2. The purpose of the present paper is to provide a first result for the weak existence to the CHB model when ψ is a singular potential, where either the classical derivative of ψ does not exist, or the derivative ψ′ blows up outside a certain interval. For the former, the prime example is the double obstacle potential [8, 9]
and for the latter the logarithmic potential
with positive constants 0 < θ < θc is the standard example.
Let us briefly motivate the need to consider singular potentials such as ψdo and ψlog. The variable φ has a physical interpretation as the difference between the volume fractions of the tumour cells and the host cells. As volume fractions are non-negative and bounded above by 1, φ should lie in the physical range [−1, 1]. This natural boundedness of φ becomes particularly important for physical models in fluid flow and biological sciences, where the mass density of the mixture expressed as an affine linear function of φ may become negative if φ strays out of [−1, 1]. A counterexample in [14] shows that for polynomial ψ, the phase field variable φ can take values outside [−1, 1], and hence a remedy to enforce the natural bounds for φ is to introduce non-smooth potentials into the model.
For the original Cahn–Hilliard equation and its variants in solenoidal two-phase flow with the boundary condition (1.2a), there are numerous contributions in the literature involving singular potentials, of which we cite [1, 3, 8, 16, 27, 35, 36] and refer to the references cited therein. The most challenging aspect with singular potentials in the global existence of weak solutions is to show that the chemical potential μ is controlled in the Bochner space L2(0, T; H1(Ω)). This is equivalent to controlling the mean value μΩ :=
When the Cahn–Hilliard component is affixed with a source term, like (1.1c), then in general the mass conservation property is lost. Therefore, simply using previous techniques would only give a local-in-time weak existence result holding as long as φΩ(t) ∈ (−1, 1), see for instance the work of [13] on the so-called Cahn–Hilliard inpainting model [7] with logarithmic potential obtained from (1.1c)-(1.1d) by setting ψ = ψlog, v = 0, σ = 0 and fφ(φ) = λ (I − φ) for given non-negative λ ∈ L∞(Ω) and ∣I∣ ≤ 1 a.e. in Ω. Recently, the second author has established a global existence result to the inpainting model with the double obstacle potential ψdo in [33]. The key ingredients involved are a careful analysis of the interplay between the source terms and approximations of the singular part 𝕀[−1,1] of ψdo, as well as showing that the source terms lead to the conclusion that φΩ(t) ∈ (−1, 1) for all t > 0. Then, previous methods for controlling μΩ can be applied to achieve global weak existence.
To generalise the methodology of [33] for the CHB model (1.1) with ψdo, we found that it suffices to prescribe suitable conditions on bv, bφ, fv and fφ at φ = ± 1. Namely, we ask that bv and bφ are non-negative and vanish at ± 1, while fφ − fv is negative (resp. positive) at 1 (resp. −1). In Remark 2.1 below we give a biologically relevant example of bv, bφ, fv and fφ satisfying the above conditions. With the help of a modified version of a key estimate (see Proposition 3.3), the same methodology can be used to show global weak existence for the CHB model (1.1) with ψlog, thereby improving also the results of [13, 43] concerning the inpainting model with the logarithmic potential ψlog. Uniqueness for the CHB tumour model remains an open problem, as the frictionless boundary condition (1.2c) and the presence of source terms Γv and Γφ introduce new difficulties that do not permit us to mimic the arguments developed in [16, 34, 35].
By formally sending the bulk and shear viscosities in the stress tensor T to zero, the CHB model (1.1) reduces to a Cahn–Hilliard–Darcy (CHD) tumour model with
replacing (1.1b). Without source terms and the nutrient, the CHD model is also called the Hele-Shaw–Cahn–Hilliard system, where recent progress on strong well-posedness with the logarithmic potential can be found in [34, 35]. Meanwhile, for the CHD tumour model with polynomial ψ, it appears that the regularity of the weak solutions in [29, 30] is insufficient to replicate the usual procedure of approximating the singular potential with a sequence of polynomial potentials, deriving uniform estimates and passing to the limit. To the authors’ best knowledge the global weak existence to the CHD tumour model with singular potentials remains an open problem, and inspired by the relationship between the Brinkman and Darcy laws, we employ an idea of [20] to deduce the global weak existence to the CHD tumour model with singular potentials by scaling the shear and bulk viscosities in an appropriate way.
The second contribution of the present paper is to provide a first result for stationary solutions to the CHB model with singular potentials, where the time derivative in (1.1c) vanishes but the convection term remains, i.e., we allow for the possibility of a non-zero velocity field. This is interesting as the fluid velocity can be used to inhibit the growth of the tumour cells, and in an optimal control framework, these stationary solutions are ideal candidates in a tracking-type objective functional, e.g. see [12]. Unfortunately, the particular form of the source terms Γv(φ, σ) and Γφ(φ, σ) indicates that the CHB model does not admit an obvious Lyapunov structure, and so it is unlikely that the a priori estimates used to prove weak existence are uniform in time. This is in contrast to the phase field tumour model of [38] studied in [12, 15, 26, 48, 49] where tools such as global attractors and the Łojasiewicz–Simon inequality can be used to quantify the long-time behaviour of global solutions. We also mention the recent work [44] establishing a global attractor for a simplified model similar to the subsystem (1.1c)-(1.1e) with v = 0 and χ = 0, where despite the lack of an obvious Lyapunov structure, the authors can derive dissipative estimates under some constraints on the model parameters.
However, we opt to prove directly the existence of stationary solutions using the methodology of [33], as opposed to investigate the long-time behaviour of time-dependent solutions. Thanks to the well-posedness of the Brinkman subsystem (1.1a), (1.1b), (1.2c) and the nutrient subsystem (1.1e), (1.2b), we can express σ = σφ, v = vφ and p = pφ as functions of φ, since μ can also be interpreted as a function of φ by (1.1d). Then, the stationary CHB system is formally equivalent to the fourth order elliptic problem
The novelty of our work lies in controlling the convection term div(φ vφ) with the help of the unique solvability of the Brinkman system, and the existence result (Theorem 2) is a non-trivial extension of [33]. Furthermore, this also shows the existence of stationary solutions to the inpainting model with logarithmic potential, thereby completing the analysis of weak and stationary solutions to the Cahn–Hilliard inpainting model with both singular potentials. We mention that a similar strategy does not work for the stationary Cahn–Hilliard–Darcy system with non-zero velocity field, as regularity of the corresponding velocity field is simply insufficient.
The remainder of this paper is organised as follows. In section 2 we cover several useful preliminary results and state our main results on the existence of global weak solutions and stationary solutions to the CHB model (1.1)-(1.2) with singular potentials, as well as the global weak existence to the CHD tumour model. The proofs of these results are then contained in sections 3, 4 and 5, respectively. Proofs of several useful properties of approximations to both singular potentials are contained in appendix A, while in appendix B we give a proof of a well-posedness result for the Brinkman system that plays a significant role in our work.
2 Main results
2.1 Notation and preliminaries
For a real Banach space X we denote by ∥⋅∥X its norm, by X* its dual space and by 〈⋅, ⋅〉X the duality pairing between X* and X. If X is an inner product space the associated inner product is denoted by (⋅, ⋅)X. For two matrices A = (ajk)1≤j,k≤d, B = (bjk)1≤j,k≤d ∈ ℝd×d, the scalar product A : B is defined as the sum
For 1 ≤ p ≤ ∞, r ∈ (1, ∞), β ∈ (0, 1) and k ∈ ℕ, the standard Lebesgue and Sobolev spaces on Ω and on Σ are denoted by Lp := Lp(Ω), Lp(Σ), Wk,p := Wk,p(Ω), and Wβ,r(Σ) with norms ∥⋅∥Lp, ∥⋅∥Lp(Σ), ∥⋅∥Wk,p and ∥⋅∥Wβ,r(Σ), respectively. In the case p = 2 we use the notation Hk := Wk,2 with the norm ∥⋅∥Hk. The space
for two or more Bochner spaces A and B. For the dual space X* of a Banach space X, we introduce the (generalised) mean value by
and also the subspace of L2-functions with zero mean value:
The corresponding function spaces for vector-valued or tensor-valued functions are denoted in boldface, i.e., Lp, Wk,p, Hk, Lp(Σ) and Wβ,r(Σ). Furthermore, we introduce the space
equipped with the norm
where div is the weak divergence. We now state three auxiliary lemmas that are used for the study of our model. The first lemma concerns the solvability of the divergence problem, which will be useful for the mathematical treatment of equation (1.1a).
Lemma 2.1
([28, Sec. III.3]). Let Ω ⊂ ℝd, d ≥ 2, be a bounded domain with Lipschitz-boundary and let 1 < q < ∞. Then, for every f ∈ Lq and a ∈ W1−1/q,q(Σ) satisfying
there exists at least one solution u ∈ W1,q to the problem
satisfying for some positive constant C = C(Ω, q) the estimate
Since we invoke Lemma 2.1 frequently with
where by the smoothness of the normal n, see (A1) below, (2.3) yields
The second lemma concerns the existence of weak solutions to the model (1.1) which forms the basis of our approximation procedure.
Lemma 2.2
([20, Thm. 2.5]). Suppose the following assumptions are satisfied:
For d ∈ {2, 3}, Ω ⊂ ℝd is a bounded domain with C3-boundary.
The positive constants ν, K and the non-negative constant χ are fixed.
The viscosities η and λ belong to C2(ℝ) ∩ W1,∞(ℝ) and satisfy
for positive constants η0, η1 and a non-negative constant λ0. The function h ∈ C0(ℝ) ∩ L∞(ℝ) is non-negative
The source terms Γv and Γφ are of the form
where bv, fv ∈ C0,1(ℝ) ∩ W1,∞(ℝ) and bφ, fφ ∈ C0(ℝ) ∩ L∞(ℝ).
The initial condition φ0 belongs to H1.
The function ψ ∈ C2(ℝ) is non-negative and satisfies
where R0, R1, R2, R3 are positive constants.
Then, there exists a quintuple (φ, μ, σ, v, p) satisfying
and is a weak solution to (1.1)-(1.2) in the sense that (1.1a) holds a.e. in ΩT and
for a.e. t ∈ (0, T) and for all Φ ∈ H1 and ζ ∈ H1. Furthermore, there exists a positive constant C not depending on (φ, μ, σ, v, p) such that
Remark 2.1
We point out that the assumptions on the potential in [20] are more restrictive in the case of potentials with quadratic growth. However, it can be checked easily that the proof of [20, Thm. 2.5] follows through when using potentials that satisfy (2.6). Furthermore, in checking the proof of [20, Thm. 2.5], it turns out that Lipschitz continuity of bv and fv are sufficient for the assertions of Lemma 2.2, as opposed to the C1-regularity stated in [20, (A4)].
The third lemma concerns the solvability of the Brinkman system, which we will use to study stationary solutions. A proof is contained in appendix, and we mention that similar results can be found in [2] in the context of resolvent operators and H∞-calculus.
Lemma 2.3
Let Ω ⊂ ℝd, d = 2, 3, be a bounded domain with C3-boundary Σ and outer unit normal n. Let c ∈ W1,r with r > d be given and fix exponent q ≤ r such that q > 1 for d = 2 and q ≥
satisfying the following estimate
with a constant C depending only on η0, η1, λ0, ν, q, ∥c∥W1,r and Ω.
2.2 The time-dependent problem
We begin with a suitable notion of weak solutions for the model with the double obstacle potential ψdo and the logarithmic potential ψlog.
Definition 2.1
A quintuple (φ, μ, σ, v, p) is a weak solution to the CHB system (1.1)-(1.2) with the double obstacle potential ψdo if the following properties hold:
the functions satisfy
with φ(0) = φ0 a.e. in Ω.
for a.e. t ∈ (0, T), φ(t) ∈ 𝓚 := { f ∈ H1 : ∣f∣ ≤ 1 a.e. in Ω} and
We say that (φ, μ, σ, v, p) is a weak solution to (1.1)-(1.2) with the logarithmic potential ψlog if properties (a) and (b) hold along with
∣φ(x, t)∣ < 1 for a.e. (x, t) ∈ ΩT and for a.e. t ∈ (0, T),
Our first result concerns the existence of weak solutions to the CHB system (1.1)-(1.2) with singular potentials.
Theorem 1
Suppose (A1)-(A3) hold along with
The source terms Γv and Γφ are of the form (2.5) with fv ∈ C0,1([−1, 1]), fφ ∈ C0([−1, 1]), bv ∈ C0,1([−1, 1]), bφ ∈ C0([−1, 1]) satisfying
The initial condition φ0 belongs to 𝓚 with ∣(φ0)Ω∣ < 1.
Then, there exists a weak solution (φ, μ, σ, v, p) to (1.1)-(1.2) with the double obstacle potential ψdo in the sense of Definition 2.1, and additionally satisfies
for any p ∈ [2, ∞) if d = 2 and p = 6 if d = 3.
In addition, if the following assumptions are satisfied:
There exists a constant c such that for any 0 < δ ≪ 1,
bφ(s)
The initial condition φ0 ∈ H1(Ω) satisfies ∣φ0(x)∣ < 1 for a.e. x ∈ Ω and β̂log(φ0) ∈ L1(Ω).
Then, there exists a weak solution (φ, μ, σ, v, p) to (1.1)-(1.2) with the logarithmic potential ψlog in the sense of Definition 2.1 and satisfies (2.14). Furthermore, for a.e. t ∈ (0, T), it holds that
where u = 𝓓(Γv(φ, σ)).
Example 2.1
We now give a biologically relevant example of the source terms Γv and Γφ that satisfy (2.13). Following [32, Sec. 2.4.1], in a domain Ω occupied by both tumour cells and healthy cells, we denote by ρ1 the actual mass density of the healthy cells per unit volume in Ω and by ρ̄1 the (constant) mass density of the healthy cells occupying the whole of Ω. Then, it follows that ρ1 ∈ [0, ρ̄1] and the volume fraction of the healthy cells can be defined as the ratio
where P, A > 0 are the constant proliferation and apoptosis rates, so that proliferation occurs only at the interface region {−1 < φ < 1}. From the modelling, we have
and so we can set
where bv(± 1) = bφ(± 1) = 0 and
It is also clear that bφ(s) and bv(s) satisfy (C1) and (C2). Furthermore, we point out that it is possible to have α < 0 in the case ρ̄2 > ρ̄1.
2.3 The stationary problem
The stationary problem comprises of the equations (1.1a), (1.1b), (1.1d), (1.1e) posed in Ω and
together with the boundary conditions (1.2a)-(1.2c) posed on Σ.
Definition 2.2
A quintuple (φ, μ, σ, v, p) is a weak solution to the stationary CHB system with the double obstacle potential ψdo if the following properties hold:
the functions satisfy
(2.11) holds along with φ ∈ 𝓚 = {f ∈ H1 : ∣f∣ ≤ 1 a.e. in Ω}.
We say that (φ, μ, σ, v, p) is a weak solution to the stationary CHB system with the logarithmic potential ψlog if properties (d) and (e) are satisfied and
(2.12) holds along with ∣φ(x)∣ < 1 for a.e. x ∈ Ω.
Theorem 2
Under (A1)-(A3) and (B1), there exists a weak solution (φ, μ, σ, v, p) to the stationary CHB model with double obstacle potential ψdo in the sense of Definition 2.2. If, in addition, (C1) and (C2) hold, then there exists a weak solution (φ, μ, σ, v, p) to the stationary CHB model with logarithmic potential ψlog in the sense of Definition 2.2. Furthermore, in both cases, 0 ≤ σ ≤ 1 a.e. in ΩT and the following regularity properties hold
for q ∈ [2, ∞) if d = 2 and q = 6 if d = 3. In particular, the quintuple (φ, μ, σ, v, p) is a strong stationary solution.
2.4 Darcy’s law
By setting the viscosities η(⋅) and λ(⋅) to zero, the CHB model (1.1)-(1.2) reduces to a Cahn–Hilliard–Darcy (CHD) model consisting of (1.1a), (1.1c)-(1.1e) and the Darcy law
furnished with the initial-boundary conditions (1.2a)-(1.2b), (1.2d) together with the Dirichlet boundary condition
We begin with a notion of weak solutions for the CHD model with singular potentials.
Definition 2.3
A quintuple (φ, μ, σ, v, p) is a weak solution to the CHD system (1.1a), (1.1c)-(1.1e), (1.2a)-(1.2b), (1.2d), (2.18), (2.19) with the double obstacle potential ψdo if property (c1) from Definition 2.1 holds along with:
the functions satisfy
with φ(0) = φ0 a.e. in Ω.
for a.e. t ∈ (0, T) and for all Φ ∈ H1, χ ∈
We say that (φ, μ, σ, v, p) is a weak solution to the CHD system with the logarithmic potential ψlog if property (c2) in Definition 2.1 holds along with properties (g) and (h).
Remark 2.2
The variational equality (2.20a) comes naturally from (2.7a) when we neglect the viscosities η(φ) and λ(φ). Meanwhile, the variational equality (2.20b) arises from the weak formulation of the elliptic problem obtained from taking the divergence of Darcy’s law (2.18) in conjunction with the equation (1.1a) and the boundary condition (2.19).
Theorem 3
Under (A1)-(A3), (B1) and (B2), there exists a weak solution (φ, μ, σ, v, p) to the CHD model with double obstacle potential ψdo in the sense of Definition 2.3. If, in addition, (C1)-(C3) hold, then there exists a weak solution (φ, μ, σ, v, p) to the CHD model with logarithmic potential ψlog in the sense of Definition 2.3, which satisfies, for a.e. t ∈ (0, T), the inequality (2.15) with η(φ) ≡ 0. Furthermore, in both cases, 0 ≤ σ ≤ 1 a.e. in ΩT.
3 Proof of Theorem 1 – Time dependent solutions
The standard procedure is to approximate the singular potentials with a sequence of regular potentials, employ Lemma 2.2 to obtain approximate solutions, derive uniform estimates and pass to the limit.
3.1 Approximation potentials and their properties
3.1.1 Double obstacle potential
We point out that in order to use Lemma 2.2 the approximate potential should at least belong to C2(ℝ). Fix δ > 0, which serves as the regularisation parameter, and we define
Formally, it is easy to see that β̂do,δ(r) → 𝕀[−1,1](r) pointwise as δ → 0, and so
will serve as our approximation for the double obstacle potential. Let βdo,δ(r) = β̂′do,δ(r) = (r + ψ′do,δ(r)) ∈ C1(ℝ) denote the derivative of the convex part β̂do,δ, which has the form
Then, it is clear that βdo,δ is Lipschitz continuous with
Proposition 3.1
Let β̂do,δ and ψdo,δ be defined as above. Then, there exist positive constants C0 and C1 such that for all r ∈ ℝ and for all δ ∈ (0, 1/4),
Aside from approximating the singular potential, it would be necessary to extend the source functions bv, bφ, fv and fφ from [−1, 1] to the whole real line. Since the solution variable φ is supported in [−1, 1] (see (c1) of Definition (2.1)), the particular form of extensions outside [−1, 1] does not play a significant role and we have the flexibility to choose extensions that would easily lead to uniform estimates. Hence, unless stated otherwise we assume that bv, bφ, fv and fφ can be extended to ℝ such that fφ ∈ C0(ℝ) ∩ L∞(ℝ), fv ∈ C0,1(ℝ) ∩ W1,∞(ℝ), bφ ∈ C0(ℝ) ∩ L∞(ℝ), bv ∈ C0,1(ℝ) ∩ W1,∞(ℝ), and fulfil
The latter implies fφ(r) − fv(r)r is strictly negative (resp. positive) for r > 1 (resp. r < −1). For the functions stated in Example 2.1, we can consider following extensions: For r ∈ ℝ, we set
and
It is clear that (3.5) is fulfilled. For α = 0, we see that fv(r) = 0 and so fφ(r) − fv(r)r = fφ(r) satisfies (3.6). For α > 0, if r ≤ −1 we see that fφ(r) − fv(r)r = A(ρS − α r) > 0 and if r ≥ 1 we see that fφ(r) − fv(r)r attains its maximum at r = 1 with value A(α − ρS) < 0. Similarly, for α < 0, if r ≥ 1 we see that fφ(r) − fv(r)r = A(α r − ρS) < 0 and if r ≤ −1 we see that fφ(r) − fv(r)r attains its minimum at r = −1 with value A(ρS − α) > 0. Hence, the extensions fulfil (3.6). Then, we can employ Lemma 2.2 to deduce the existence of a quintuple (φδ, μδ, σδ, vδ, pδ) to (1.1)-(1.2) with ψ′do,δ replacing ψ′ and source terms bv, bφ, fv and fφ modified as above. Uniform estimates will be derived in the next section and then we can pass to the limit δ → 0 to infer the existence of a weak solution to (1.1) with the double obstacle potential in the sense of Definition 2.1.
3.1.2 Logarithmic potential
We define the convex part of ψlog and its corresponding derivative as
For fixed δ > 0, the approximation of ψlog is the standard one:
with convex part and corresponding derivative
Analogous to Proposition 3.1, we state several useful properties for β̂log,δ and ψlog,δ, whilst deferring the proof to the appendix.
Proposition 3.2
Let β̂log,δ and ψlog,δ be defined as above. Then, there exist positive constants C0, …, C3, such that for all r ∈ ℝ and for all 0 < δ ≤ min(1,θ/(4 θc)),
Once again, we extend the source functions bv and bφ from [−1, 1] to the whole real line so that (3.5) is satisfied, and instead of (3.6) we consider extensions of fv and fφ from [−1, 1] to the whole real line that satisfy
with continuous interpolation in [r0, 2r0] and [−2r0, −r0] for some fixed constant r0 ∈ (1, 2). For instance, with the functions fv and fφ introduced in Example 2.1, we can take as extensions
with
3.2 Existence of approximate solutions
To unify our analysis, we use the notation
and denote by β̂δ the convex part of ψδ. Furthermore, we emphasise that the property (3.6) is assumed when we work with the double obstacle potential ψdo, and (3.10) is assumed when we work with the logarithmic potential ψlog. Due to Proposition 3.1 and 3.2 and by the definition of ψδ(⋅), we can employ Lemma 2.2, and infer, for every δ ∈ (0, 1), the existence of a weak solution quintuple (φδ, μδ, σδ, vδ, pδ) to (1.1)-(1.2) with ψ′δ in (1.1d). More precisely,
and
for a.e. t ∈ (0, T) and for all Φ ∈ H1 and ζ ∈ H1.
3.3 Uniform estimates
We begin with an auxiliary result for the product of βlog,δ and the source terms, whose proof can be found in the appendix.
Lemma 3.1
Let βlog,δ denote the derivative of (3.8), and let bv ∈ C0,1(ℝ) ∩ W1,∞(ℝ), bφ ∈ C0(ℝ) ∩ L∞(ℝ), fv ∈ C0,1(ℝ) ∩ W1,∞(ℝ), and fφ ∈ C0(ℝ) ∩ L∞(ℝ) be given such that (C1), (C2), (3.5) and (3.10) are satisfied. Then, there exists δ0 > 0 and a positive constant C independent of δ ∈ (0, δ0) such that for all δ ∈ (0, δ0), s ∈ ℝ and r ∈ ℝ,
In the following, we derive uniform estimates in δ, and denote by C a generic constant independent of δ which may change its value even within one line.
3.3.1 Nutrient estimates
Choosing ζ = σδ in (3.12c) and using the non-negativity of h(⋅) leads to
from which we deduce that σδ is uniformly bounded in L∞(0, T; H1). Elliptic regularity additionally yields
By a standard argument with the comparison principle, testing with ζ = − (σδ)− and ζ = (σδ - 1)+ will yield the pointwise a.e. boundedness of σδ:
Then, the continuous embedding H2 ↪ L∞ and the assumptions on bv, bφ, fv and fφ lead to
3.3.2 Estimates from energy identity
Thanks to (3.16), the function u : = 𝓓(Γv) ∈ H1 satisfies the estimate
Technically, we should stress the dependence of u on δ, but in light of the uniform estimate (3.17) we infer that uδ is bounded in L∞(0, T; W1,p) for any p ∈ (1, ∞). Henceforth, we drop the index δ and reuse the variable u.
Choosing Φ = vδ − u in (3.12a), ζ = μδ + χσδ in (3.12b), using (3.11b) and summing the resulting identities, we obtain
where we used the fact that ψδ is a quadratic perturbation of a convex function β̂δ, in conjunction with [46, Lem. 4.1] to obtain the time derivative of the energy. By Young’s inequality and the estimates (3.15), (3.16) and (3.17), we find that
for a positive constant ε yet to be determined. It remains to control the two terms with βδ(φδ). Integrating by parts and employing the definition of u = 𝓓(Γv) leads to
Using (3.17) and the relation
Meanwhile, using (3.4b), (3.4c), (3.9c), (3.9d) and the trace theorem, it follows that
for positive constants C and C* independent of δ. To deal with the remaining term we use the definition (3.3) of βdo,δ and (3.5)-(3.6) to deduce that, for any s ≥ 0 and r ∈ ℝ,
which implies that
Meanwhile, for the other part with βlog,δ, we use (3.13) and (3.15) to obtain
Combining (3.23) with (3.24), we find that
and so, when substituting (3.19)-(3.22) and (3.24) into (3.18), we arrive at
Testing (3.11b) with − AΔφδ for some positive constant A, integrating by parts and using ∂n φδ = 0 on Σ and (3.15) yields
Then, summing up the last two inequalities and choosing A > C* and ε <
Before applying a Gronwall argument, we note that for the double obstacle potential, the assumption (B2) implies β̂do,δ(φ0) = 0, and for the logarithmic potential, using the property β̂log,δ(s) ≤ β̂log(s) for s ∈ [−1, 1] together with (C3) we infer β̂log,δ(φ0) is uniformly bounded. Hence, for 0 < δ < min (1, θ/(4 θc), δ0) = : δ*, we see that
Integrating (3.26) in time from 0 to s ∈ (0, T], using (3.4a), (3.9a), Korn’s inequality and elliptic regularity theory, we deduce the uniform estimate
where the L4(0, T; H2)-regularity comes from revisiting the equality in (3.25) and employing the boundedness of ∇ μδ in L2(QT) and of φδ in L∞(0, T; H1(Ω)), the non-negativity of
and also elliptic regularity theory. By the Sobolev embedding H1 ⊂ L6 and (3.27), it follows that
A similar argument together with (3.16) shows that φδ Γv is bounded in L2(0, T;
Furthermore, we find that the mean value (φδ)Ω satisfies
and so
In particular, by the fundamental theorem of calculus, it holds that
3.4 Estimates for the mean value of the chemical potential
In order to pass to the limit δ → 0 rigorously, it remains to derive uniform estimates for μδ, βδ(φδ) and pδ in L2(0, T; L2). To do so we appeal to the method introduced in [33], which involves first deducing that the limit φ of φδ has mean value in the open interval (−1, 1) for all times. We first state a useful auxiliary result, whose proof is deferred to the appendix.
Proposition 3.3
For δ ∈ (0, 1), let βlog,δ denote the derivative of (3.8). Then, there exist positive constants c1 and c2 independent of δ such that
Remark 3.1
This is a different type of estimate to the more commonly used inequality (see e.g. [13, (2.12)]):
with positive constant c̃0 and non-negative constant c̃1 that are independent of δ, provided δ is sufficiently small. We found that (3.32) is insufficient for our analysis as the constant c̃0 is not quantified.
Now, using reflexive weak compactness arguments (Aubin–Lions theorem) and [50, Sec. 8, Cor. 4], for δ → 0 along a non-relabelled subsequence, we infer that
for some limit functions ξ ∈ L2(0, T; L2), θ ∈ L2(0, T;
Let λ ∈ L4(0, T; L3) be an arbitrary test function, then (3.34) implies λ φδ → λ φ strongly in L2(0, T; L2) and λ ∇ φδ → λ ∇ φ strongly in L2(0, T;
This implies div(φδvδ) → div(φv) weakly in
for a.e. t ∈ (0, T) and for all ζ ∈ H1. Technically, one would multiply (2.7b) with a function κ ∈
Now, for the obstacle potential, the uniform boundedness of ψδ(φδ) in L1(0, T; L1) and (3.4b) imply
which implies that (c.f. [8, Proof of Thm. 2.2])
For the logarithmic potential, we use (3.9b) and the uniform boundedness of β̂log,δ(φδ) in L1(0, T; L1) to obtain that
Since φδ → φ a.e. in ΩT and strongly in L2(0, T; L2), passing to the limit δ → 0 in the inequality above also implies (3.38). From this we claim that φΩ(t) ∈ (−1, 1) for all t ∈ [0, T]. Indeed, choosing ζ = 1 in (3.36) leads to
Suppose to the contrary there exists a time t* ∈ (0, T] such that φΩ(t*) = 1 and (3.39) holds. Due to (3.38), this implies φ(t*, x) ≡ 1 a.e. in Ω and thus ∇ φ(t*, x) ≡ 0 a.e. in Ω. Using (3.39) and (3.5)-(3.6), we obtain
Hence, by continuity of t ↦ (φ(t))Ω, the mean value (φ(t))Ω must be strictly decreasing in a neighbourhood of t*, i.e., (φ(t))Ω > 1 for t < t*, which contradicts (3.38). Using a similar argument and the assumption fφ(−1) + fv(−1) > 0 leads to the conclusion that (φ(t))Ω > − 1 for all t ∈ (0, T]. In particular, together with (B2), we have (φ(t))Ω ∈ (−1, 1) for all t ∈ [0, T].
This allows us to derive uniform estimates on the mean value of μδ. Integrating (3.11b) and taking the modulus on both sides gives
for a.e. t ∈ (0, T). Using (3.31) and the fact
(which unfortunately does not hold for βlog,δ, hence the necessity of Proposition 3.3), we deduce that
and so, we arrive at
Together with the following equality obtained from testing (3.11b) with ζ = φδ,
we see that
Now, let
where by Poincaré’s inequality, we have
Testing (3.12b) with fδ, integrating by parts and rearranging yields
Plugging in this identity into (3.43) and rearranging again, we deduce that
for a.e. t ∈ (0, T). Recalling (3.29)-(3.30), we have the equiboundedness and equicontinuity of {(φδ)Ω}δ∈(0,1) so that by the Arzelá–Ascoli theorem,
along a non-relabelled subsequence. Then, one can find an index δ1 ∈ (0, 1) such that for all δ < min(δ1, δ*) = : δ2, where δ* is defined after (3.26),
where we know that the minimum is positive thanks to the property |(φ(t))Ω| < 1 for all t ∈ [0, T]. Then, we see that the prefactor on the left-hand side of (3.46) is bounded below by
Let us mention that if instead of (3.31) we employ the estimate (3.32), then we arrive at
and ultimately
Since c̃0 is not quantified, we may not be able to rule out the situation where c̃0 < 1, which may imply that the prefactor
The uniform estimate (3.47) for μδ allows us to infer further estimates for βδ(φδ) and pδ. Indeed, testing (3.11b) with βδ(φδ) yields
Integrating this identity in time from 0 to T, using the non-negativity of
For the pressure pδ, we invoke Lemma 2.1 to deduce the existence of qδ := 𝓓(pδ) ∈ H1 such that for a positive constant C depending only on Ω,
Then, testing (3.12a) with Φ = qδ yields
Applying Young’s inequality and using the uniform estimates (3.15), (3.27), (3.47) and (3.49) leads to
Following similar arguments in [16], with the uniform boundedness of μδ + χ σδ + Θc φδ in L2(0, T; Lp) for p = 6 if d = 3 and for any p ∈ [2, ∞) if d = 2, we can deduce from (3.11b) further uniform estimates
3.5 Passing to the limit
Let us first consider the double obstacle case. In addition to the convergence statements in (3.33), we further obtain
for some limit function τ ∈ L2(0, T; L2). Moreover, due to (3.47) we have ξ = ∇μ, which allows us to fully recover (2.7b) in the limit. To obtain (2.7a) and (2.7d) in the limit we refer the reader to similar arguments as outlined in [19], while the divergence equation (1.1a) can be obtained by similar arguments in [20]. It remains to show (2.11) is recovered in the limit δ → 0 from (3.11b). By arguing as in [31, Sec. 5.2], using the weak convergence βdo,δ(φδ) ⇀ τ in L2(0, T; L2), the strong convergence φδ → φ in L2(0, T; L2), and the maximal monotonicity of the subdifferential ∂ 𝕀[−1,1] we can infer that τ is an element of the set ∂ 𝕀[−1,1](φ), which implies that for any ζ ∈ 𝓚 and a.e. t ∈ (0, T),
Hence, testing (3.11b) (where ψδ = ψdo,δ) with ζ − φ and passing to the limit δ → 0 allows us to recover (2.11). This completes the proof of Theorem 1 for the double obstacle potential.
For the logarithmic case, the uniform estimate for βlog,δ(φδ) in L2(0, T; L2) allow us to infer, by the arguments in [13, Sec. 4] or [35, Sec. 3.3], that the limit φ satisfies the tighter bounds
Furthermore, by the a.e. convergence of φδ to φ we have βlog,δ(φδ) → βlog(φ) a.e. in ΩT. Hence, passing to the limit δ → 0 in (3.11b) we can recover (2.12). Meanwhile, the inequality (2.15) is obtained from integrating (3.18) over (0, t) for t ∈ (0, T) and then passing to the limit with the compactness assertions (3.33), weak lower semicontinuity, and Fatou’s lemma. This completes the proof of Theorem 1 for the logarithmic potential.
4 Proof of Theorem 2 – Stationary solutions
As with the time-dependent case, we extend bv, bφ, fv and fφ from [− 1, 1] to ℝ such that fφ ∈ C0(ℝ) ∩ L∞(ℝ), fv ∈ C0,1(ℝ) ∩ W1,∞(ℝ), bφ ∈ C0(ℝ) ∩ L∞(ℝ), bv ∈ C0,1(ℝ) ∩ W1,∞(ℝ), and fulfil (3.5), (3.6) (if ψ = ψdo) or (3.10) (if ψ = ψlog).
4.1 Approximation scheme
We consider a smooth function ĝ :ℝ → [0, 1] such that ĝ(r) = 1 for r ≥ 3 and ĝ(r) = 0 for r ≤ 2, and define F : L2(Ω) → ℝ as
where CF is a positive constant to be specified later. Furthermore, we denote by y(r, s) the function
and introduce, for δ ∈ (0, 1), the cut-off operator 𝓓δ ∈ C0,1(ℝ) defined as
It is clear that 𝓓δ and its derivative
where σδ ∈ H2 is the unique non-negative and bounded solution to the nutrient subsystem
and vδ is the first component of the unique solution (vδ, pδ) to the Brinkman subsystem
We aim to use pseudomonotone operator theory, akin to the methodology used in [33], to deduce the existence of at least one solution φδ ∈ W to (4.2) for each δ ∈ (0, 1). Then, we derive enough uniform estimates to pass to the limit δ → 0 in order to prove Theorem 2. The new element in the analysis is how we treat the convection term.
4.2 Preparatory results
For fixed u ∈ W :=
for any pair u1, u2 ∈ W with corresponding solutions σu1, σu2. For the Brinkman subsystem (4.4), invoking Lemma 2.3 with c = u ∈ W, g = Γv(u, σu) ∈ H1, f = (ψδ(u) − Δ u) ∇ 𝓓δ(u) ∈
Although this is sufficient for the upcoming analysis, we mention that an analogous continuous dependence result to (4.5) for the Brinkman subsystem (4.4) is not available from Lemma 2.3, and thus for completeness we provide the following result under a modification of the cut-off operator 𝓓δ.
Proposition 4.1
Suppose 𝓓δ ∈ C1,1(ℝ), and let (v1, p1), (v2, p2) ∈
One example of the modified cut-off operator 𝓓δ ∈ C1,1(ℝ) is the function
Proof
For convenience we write
Then, testing the difference of (4.4)1 with v̂ − ŵ yields
The left-hand side can be bounded below by
while the right-hand side can be bounded above by
In light of the regularities u1, u2 ∈ W and v1, v2 ∈
the Lipschitz continuity and boundedness of
for some small constant ε > 0. Invoking Korn’s inequality and (4.6) in (4.7) and combining with the above estimate gives
For the pressure, we test the difference of (4.4)1 with q̂ := 𝓓(p̂) ∈ H1 which satisfies
and in turn we obtain
This completes the proof. □
The unique solvability of the nutrient and Brinkman subsystems in turn yields the following result.
Lemma 4.1
For each δ ∈ (0, 1), the operator 𝓐 : W → W* defined as
is strongly continuous, i.e., if un ⇀ u in W, then 𝓐un → 𝓐u in W*.
Proof
Let {un}n∈ℕ ⊂ W be a sequence of functions such that un ⇀ u in W. Denoting by σn and (vn, pn) the corresponding unique solutions to the nutrient subsystem (4.3) and Brinkman subsystem (4.4) where φδ = un. Then, we easily infer that
for a positive constant C independent of n. Furthermore, using the assumptions on Γv by Lemma 2.1, there exists a function un := 𝓓(Γv(un, σn)) ∈ H1 satisfying
with C independent of δ. Testing (4.4)1 (for φδ = un) with vn − un, using (4.4)2-(4.4)3 and Korn’s inequality, we obtain
Similarly, let qn := 𝓓(pn) ∈ H1 which satisfies
Testing (4.4)1 (for φδ = un) with qn and using (4.4)2-(4.4)3, we obtain
In particular, this shows that
for a positive constant Cδ depending on δ but independent of n, thanks to the boundedness of {un}n∈ℕ in W. Hence, for fixed δ ∈ (0, 1), there exist functions σu ∈ H2, vu ∈ H1 and pu ∈ L2, such that along a non-relabelled subsequence,
It is clear that σu is the unique solution to (4.3) corresponding to u, while (vu, pu) is the unique weak solution to (4.4) corresponding to u. Indeed, for the highest order term in (4.4), we employ the dominated convergence theorem to deduce that
while the other terms in the Brinkman system can be handled using similar compactness assertions.
By Rellich’s theorem, it is easy to see that
for all ζ ∈ W. Meanwhile, integrating by parts and using div (vn) = Γv(un, σn) = : Γvn yields
on account of 𝓓δ(un) → 𝓓δ(u) in L2 and L2(Σ), vn → vu in L3 and L2(Σ) (see, e.g., [19, Lem. 1.3]), and Γvn → Γvu in L2 . This shows that 𝓐 is strongly continuous. □
4.3 Existence of approximate solutions
In this section we fix δ ∈ (0, 1), and define operators A1, A2 : W → W* by
Then, φδ ∈ W is a weak solution to (4.2) if 〈(A1 + A2)φδ, ζ〉W = 0 for all ζ ∈ W.
Since
We now claim that A is additionally coercive over W, i.e.,
Let us first treat the velocity term in 〈Au, u〉W. By the definition of 𝓓δ in (4.1), we see that u
Let uu := 𝓓(Γvu(u, σu)) ∈ H1 which satisfies
where the boundedness of Γv(u, σu) can be deduced using similar arguments in Section 3.3. Testing (4.4) for (vu, pu) with vu − uu then yields
By Korn’s inequality, Hölder’s inequality and Young’s inequality we obtain
where C* is the constant in (4.9). Let us note that as a consequence of elliptic regularity and integration by parts, for u ∈ W we obtain the following useful inequalities:
Employing the Gagliardo–Nirenberg inequality and the elliptic estimate (4.10), we have
Meanwhile, we observe that
since βdo,δ(s) = 0, βlog,δ(s) = βlog(s) for s ∈ [− 1 + δ, 1 − δ]. Hence, as 0 ≤
where for the second last inequality we used L’Hopital’s rule to confirm that
for a positive constant C1 independent of u and δ. Now, computing 〈Au, u〉W gives
for a positive constant C independent of u and δ. Recalling that F(u) = CF for
for
Invoking [52, Thm. 27.A], we deduce for every δ ∈ (0, 1) the existence of a weak solution φδ ∈ W to (4.2). Setting
we see that the equation A φδ = 0 in W* implies
with right-hand side
Thanks to the regularity φδ ∈ W, vδ ∈ H1, σδ ∈ H2, and the sublinear growth of βδ, we easily infer that fδ ∈ L2. On the other hand, choosing ζ = 1 in (4.15) implies that f ∈
4.4 Uniform estimates
From (4.2) and (4.15), the pair (φδ, μδ) ∈ W × W satisfies
for all ζ ∈ L2. Returning to the proof of the coercivity of the operator A, replacing u with φδ in (4.13) gives
If
If
where σφ is the unique solution to the nutrient subsystem (4.3) with data φ.
Convexity of β̂δ and β̂δ(0) = 0 imply the inequality
and together with (3.4b), (3.9b) and (4.18) we deduce that
Using (3.37) for the double obstacle potential, we deduce that for both cases the limit φ satisfies
In particular, we have
The latter can be deduced from the a.e. convergence 𝓓δ(φ) → φ, the Lipschitz continuity of 𝓓δ and hence a.e. convergence of 𝓓δ(φδ) − 𝓓δ(φ) → 0, and the dominating convergence theorem. Using the norm convergence
Choosing ζ = − Δ μδ in (4.16b), and choosing ζ = βδ(φδ) and also ζ = - Δ φδ in (4.16a) yields after summation and integrating by parts
on account of the boundedness of φδ and σδ in H2, and the estimates (3.23) and (3.24) for the term y(φδ, σδ) βδ(φδ). Let uδ := 𝓓(Γvδ) ∈ H1 which satisfies
Then, testing vδ − uδ with the Brinkman subsystem (4.4) yields
Recalling (4.9) and (4.11), when substituting (4.22) to (4.21), we have
where ε > 0 is a constant yet to be determined. Invoking Korn’s inequality and choosing ε sufficiently small, we arrive at
In particular, we deduce that vδ ⇀ v in H1 to some limit velocity v. Moreover,
and so
where the convergence of the convection term follows from an analogous integration by parts as in the proof of Lemma 4.1, and the convergence (4.20). Then, we infer from (2.13) and (4.24) that the limit φ has mean value φΩ ∈ (−1, 1). Indeed, substituting φ = 1 or − 1 in (4.24) leads to a contradiction on account of (2.13), and as |φ| ≤ 1 a.e. in Ω, we have that φΩ ∈ (−1, 1).
Arguing as in the time-dependent case we can derive a uniform estimate on the mean value of μδ, and consequently
where the boundedness of βlog,δ(φδ) in L2 implies the tighter bounds
in the case of the logarithmic potential.
4.5 Passing to the limit
In (4.16a) we take ζ ∈ H1 and apply integration by parts to get
Passing to the limit δ → 0 then yields (2.17). Meanwhile, (2.11) or (2.12) can be recovered in the limit δ → 0 from (4.16b) in a fashion similar to the time-dependence case, as with the recovery of (1.1a) and (2.7d).
It remains to show that the weak limit p of (pδ)δ∈(0,δ3) together with v constitutes a solution to the corresponding Brinkman system in the sense of (2.7a). From the definition (4.14) of μδ we observe from (4.4) that (vδ, pδ) satisfies
for Φ ∈ H1. For the last term, after integrating by parts and using 𝓣δ(φδ) Φ → φ Φ in L2(Ω) and in L2(Σ), μδ → μ in L4(Ω) and in L2(Σ), we see that
for all Φ ∈ H1. Hence, passing to the limit δ → 0 in (4.25) allows us to recover (2.7a) and thus the quintuple (φ, μ, σ, v, p) is a stationary solution in the sense of Definition 2.2.
Moreover, from the above estimates and weak lower semicontinuity of norms, we know that
Then, from (2.17) and elliptic regularity, we deduce that
In light of this improved regularity and the Sobolev embedding H2 ⊂ L∞, it is easy to see that (μ + χ σ) ∇ φ ∈ Lq where q < ∞ for d = 2 and q = 6 for d = 3. Invoking Lemma 2.3, we then infer
which completes the proof.
5 Proof of Theorem 3 – Darcy’s law
We can adapt most of the arguments and estimates from the proof of Theorem 1. The main idea is to consider a weak solution quintuple (φδ, μδ, σδ, vδ, pδ) to the CHB model (1.1)-(1.2) with stress tensor
where we have set η(⋅) = λ(⋅) = δ. Proceeding as in the proof of Theorem 1 we obtain the uniform estimates (3.15), (3.16) and
where in the above ψδ and βδ denote the approximations to either singular potentials and the derivatives of the corresponding convex part. From the first equality of (3.25) with A = 1, it holds that
so that by elliptic regularity we can infer
Moreover, by the Gagliardo–Nirenberg inequality, we find that
so that from (2.7b) and previous uniform estimates we arrive at
Let us mention that the sum ∂t φδ + div(φδ vδ) has better temporal integrability than either of its constituents, a fact which will play an important role for deriving uniform estimates for (μδ)Ω below.
By reflexive weak compactness arguments and [50, Sec. 8, Cor. 4], for δ → 0 along a non-relabelled subsequence, it holds that for any r ∈ [1, 6),
The identification θ = div(φ v) follows analogously as in Section 3.4, where the assertion (3.35) now holds for arbitrary λ ∈ L4(0, T; L6) by the strong convergence ∇ φδ → ∇ φ in L4(0, T; L3) and the weak convergence vδ harpoonup v in L2(0, T; L2).
In order to obtain uniform estimates for the chemical potential μδ in L2(0, T; L2), we again follow the argument in Section 3.4. Namely, we pass to the limit δ → 0 in (2.7b) to obtain (3.36), and use the uniform boundedness of ψδ(φδ) in L1(0, T; L1) from (5.1) to obtain that the limit φ satisfies the pointwise bound (3.38). Choosing ζ = 1 in (3.36) leads to (3.39) and obtain by contradiction argument that (φ(t))Ω ∈ (−1, 1) for all t ∈ [0, T].
Defining fδ ∈
which is bounded in L2(0, T) by (5.4). This modification allows us to infer that (μδ)Ω is uniformly bounded in L2(0, T), whereas simply using (5.3) would only give the uniform boundedness of (μδ)Ω in
Then, proceeding as in the proof of Theorem 1, we can recover (2.7b), (2.7d) and (2.11) (resp. (2.12)) for the double obstacle (resp. logarithmic) case in the limit δ → 0, whereas recovery of (2.20a), (2.20b), the improved regularity p ∈
A Properties of approximation potentials
A. 1 Proof of Proposition 3.1
Proof
As ψdo,δ is bounded for ∣r∣ ≤ 1 + δ, it suffices to show (3.4a) holds for ∣r∣ > 1 + δ. By Young’s inequality it is clear that for r > 1 + δ with δ ∈ (0, 1/4),
and a similar assertion holds also for r < −1 − δ. This establishes (3.4a).
From the definitions of β̂do,δ and βdo,δ we see that
for r ∈ (1, 1 + δ), and
for r > 1 + δ. Similar assertions also hold for the cases r ∈ (−1 − δ, −1) and r < −1 − δ, which then yield (3.4b).
A straightforward computation shows
and so (3.4c) is established.□
A.2 Proof of Proposition 3.2
Proof
For r ≥ 1 − δ with
where the inequality β′log(1-δ) ≥ 2 θc comes from the facts
This completes the proof of (3.9a).
For ∣r∣ ≤ 1, we see that
and similar arguments yield
For (3.9c) we first observe that δ (βlog,δ(0))2 = 0 = β̂log,δ(0) and for δ ∈ (0, 1] we have
which imply
Integrating the first inequality from 0 to r ∈ (0, 1 − δ] and the second inequality from r ∈ [−1 + δ, 0) to 0 yields
Taking note that over [−1 + δ, 1 − δ], β̂log,δ(r) is bounded uniformly in δ ∈ (0, 1], and so we easily infer the upper bound 2 θ β̂log,δ(r) ≤ C2 (δ (βlog,δ(r))2 + 1) for some positive constant C2 > 0 holding for all r ∈ [−1 + δ, 1 − δ]. Meanwhile, a direct calculation shows that for r ≥ 1 − δ and δ ∈ (0, 1] we have
on account of
For (3.9d) a straightforward calculation using
This completes the proof.□
A. 3 Proof of Lemma 3.1
Proof
We define
Then, due to (3.5), G(r, s) = 0 for ∣r∣ ≥ 1. Using (C1), we have for δ ∈ (0, 1), s ∈ ℝ and r ∈ [1 − δ, 1] that
where we used that ∣δlogδ∣ ≤ C for δ ∈ (0, 1). Consequently, we have that G(r, s) ≥ −C∣s∣ for δ ∈ (0, 1), s ∈ ℝ and r ∈ [1 − δ, 1]. A similar assertion holds for r ∈ [−1,−1 + δ]. Lastly, for ∣r∣ ≤ 1 − δ, we use (C2) to deduce that
and consequently G(r, s) ≥ −C∣s∣ for all ∣r∣ ≤ 1 − δ and all s ∈ ℝ. Therefore, for all δ > 0, s ∈ ℝ and r ∈ ℝ, it holds that G(r, s) ≥ − C∣s∣. Next, we define
By continuity of H(r) and (3.10), we can find a constant δ0 ∈ (0, r0 − 1) such that
Then, it is clear that for ∣r∣ ≥ 2 r0, H(r) = 0 thanks to (3.10) satisfied by the extensions of fv and fφ. Meanwhile, for any δ ∈ (0, δ0), we see that if ∣r∣ ≤ 1 − δ0 < 1 − δ, then
which implies that
On the other hand, as r0 > 1, for r ∈ [−2r0, −1 + δ0] ∪ [1 − δ0, 2r0], we use that βlog,δ(r) and H(r) have the same sign, so that their product H(r) βlog,δ(r) is non-negative. Hence, combining with the above analysis for the function G, we obtain the assertion (3.13).□
A. 4 Proof of Proposition 3.3
Proof
From the definition of β̂log,δ in (3.8), we infer that for r ≥ 1 − δ, δ ∈ (0, 1),
for some positive constant c independent of δ ∈ (0, 1). In a similar fashion, using βlog(δ − 1) < 0 and β′log(δ − 1) > 0, we have for r ≤ −1 + δ,
and when combined this yields
For the remaining case ∣r∣ ≤ 1 − δ, we employ the fact that βlog,δ(r) = βlog(r) and
to infer the existence of a constant c > 0 independent of δ ∈ (0, 1) such that
Hence, for δ ∈ (0, 1),
This completes the proof.□
B Well-posedness of the Brinkman system (2.9)
B. 1 Weak Solvability
Theorem 4
Let Ω ⊂ ℝd, d = 2, 3, be a bounded domain with C2,1-boundary Σ. Let c ∈ W1,r with r > d be given, fix exponent q such that q ∈ (1, 2] for d = 2 and q ∈
in the sense
for all Φ ∈ H1. Furthermore, it holds that
for a constant C > 0 depending only on Ω, q, η0, η1, λ0 and ν.
Proof
For (x1, …, xd)⊤ ∈ Ω, we define
Then, a short calculation shows that div(v0) = gΩ and Tc(v0, p0) = 0. Next, we will show there exist unique weak solutions (w, π) and (y, θ) to the systems
Then, one can check that the pair v := w + y + v0 and p := θ + π + p0 is the unique weak solution to (B.1).
B.1.1 Solvability of (P1)
By Lemma 2.1, there exists u := 𝓓(g − gΩ) ∈
Let f̃ := −ν u + div(2 η(c) Du + λ(c) div(u)I) ∈
and satisfies
Then, the function w := ŵ + u ∈
i.e., w is the first component of the solution to (P1). The recovery of the unique pressure variable π ∈
It is also clear that (w, π) constructed above is the unique solution to (P1), and for any Φ ∈ H1 it holds that
and so
B.1.2 Solvability of (P2)
We define the function space
By the Lax–Milgram theorem, there exists a unique solution y ∈
holding for all Φ ∈
By [51, p. 75, Lem. 2.2.2], there exists a unique pressure θ̂ ∈
It remains to adjust this pressure by a uniquely defined constant c0 so that y and θ := θ̂ + c0 satisfy the boundary condition Tc(y, θ) n = h − Tc(w, π)n. Let q′ =
and so 2 η(c) Dy − θ̂ I ∈
In turn, as W1,q′ ⊂ H1, this gives
From testing (B.9) with arbitrary Φ ∈
Since q′ ≥ 2, we have (H1/2(Σ))* ⊂ (W1/q,q′(Σ))* which implies that h − Tc(w, π) n ∈ (W1/q,q′(Σ))*. Comparing (B.6) with Φ ∈
The regularity of the boundary Σ implies the normal vector n ∈ W1/q,q′(Σ), and for arbitrary ψ ∈ W1/q,q′(Σ) we define
Then, substituting Φ = u͡ in (B.12) leads to
Setting θ := θ̂ + c0 in turn shows that (y, θ) satisfies the boundary condition Tc(y, θ) = h − Tc(w, π) on Σ in the following sense:
Lastly, combining (B.3), (B.4), (B.5), (B.7) and (B.8) and (B.10) it is easy to infer the estimate (B.2) for v = w + y + v0 and p = θ̂ + c0 + p0 + π0.□
B.2 Strong solvability
Let us first state the following auxiliary result for the Brinkman system with constant viscosities.
Lemma B.1
Let Ω ⊂ ℝd, d = 2, 3, be a bounded domain with C3-boundary Σ and outer unit normal n. Let η, λ and ν be positive constants, and fix an exponent q such that q > 1 for d = 2 and q ≥
satisfying the following estimate
with a constant C depending only on η, λ, ν, q and Ω.
Proof
We consider the functions analogous to v0, p0, w, π, y and θ defined in the proof of Theorem B.1 for the case η(c) = η and λ(c) = λ, whilst reusing the notation. It is clear that
By [23, Thm. 1.2], there exists a unique solution
satisfying
By the assumption on the exponent q, it holds that W2,q ⊂ H1 and W1,q ⊂ L2. Hence, the difference w͠ := w – z and π͠ := π – ϕ constitutes a weak solution in
Unique solvability implies (w͠, π͠) = (0, 0) and hence the unique weak solution (w, π) to (P1) with constant viscosities is in fact a strong solution satisfying
Furthermore, by the trace theorem, T(w, π)n := (2ηDw + λ div (w)I – πI)n ∈ W1/q′,q(Σ) where
Hence, v := v0 + w + y ∈ W2,q and p := p0 + π + θ ∈ W1,q constitute the unique strong solution to (B.13) satisfying (B.14).□
B.3 Proof of Lemma 2.3
Fix c ∈ W1,r with exponent r > d and data f ∈ Lq, g ∈ W1,q and h ∈ W1–1/q,q(Σ), where the exponent q ≤ r satisfies q > 1 for d = 2 and q ≥
and for d = 2 we have W1/q′,q(Σ) ⊂ Lk(Σ) and H1/2(Σ) ⊂ Lk/(k–1)(Σ) for any k > 1. Hence, h ∈ (H1/2(Σ))*, and we invoke Theorem 4 to deduce the existence of a unique weak solution (v, p̂) ∈ H1 × L2 to
in the sense that
for all Φ ∈ H1. Introducing ψ := η(c)Φ, we see that
for all ψ ∈ H1. Since q > 1 and 1 –
At this point the analysis is divided into two cases:
B.3.1 Case 1 (q ≤ s =
2 r 2 + r
)
We have q < 2 and
which implies k ∈ Lq. From (B.16), we see that (v,
where k ∈ Lq, g ∈ W1,q and
By the assumption on η, we easily infer that ∥p̂∥Lq ≤ C ∥
where we used
where we used r > d to deduce that
B.3.1.1 Case 2 (q > s =
2 r 2 + r
)
In this case, using s < 2 and r =
Using the embedding g ∈ W1,q ⊂ W1,s and
Then, the Sobolev embedding W1,q ⊂ W1,s ⊂
Let t be the exponent defined by
then by Sobolev embedding it holds that (v, p̂) ∈ W1,t × Lt. Since r > d, it follows that t > 2. The idea is to use this improved regularity as a starting point of a bootstrapping argument to show k ∈ Lq. Once we have k ∈ Lq, following the argument in Case 1 then gives the desired assertion.
Let s1 < t be the exponent defined by
Since r > d, we have s1 > s, and from (B.16),
If q ≤ s1 then k ∈ Lq and the proof is complete, otherwise if s1 < q, we deduce instead that (v, p̂) ∈ W2,s1 × W1,s1 ⊂ W1,t1 × Lt1 where
Then, k ∈ Lmin(s2, q) where the exponent t1 > s2 > s1 is defined as
For the bootstrapping argument, given sk–1 we set
then it holds that sk > sk–1 and tk > tk–1. After a finite number of iterations, we get min(sk, q) = q and the proof is complete.
Acknowledgments
M. Ebenbeck is supported by the RTG 2339 ”Interfaces, Complex Structures, and Singular Limits” of the German Science Foundation (DFG). The work of K.F. Lam is partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China [Project No.: CUHK 14302319].
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