The current study aimed to analyze bacterial communities' diversity and abundance in three different deserted areas (Merzouga, Mhamid Elghizlane, and Erg lihoud) located in Moroccan Sahara, as well as to investigate osmotolerant microorganisms producing hydrolytic enzymes. The isolates were taxonomically affiliated using 16S rRNA gene sequencing. Four different hydrolase activities (amylase, lipase, cellulase, and protease) and osmotic stress tolerance were evaluated. The phylogenetic analysis of 364 screened isolates belonged to three phyla (Firmicutes 73%, Proteobacteria 26% and Actinobacteria 1%) and 18 different genera, from Bacillus, Ornithinibacillus, Paenibacillus, Geobacillus, Pseudomonas, Acinetobacter, Agrobacterium, Arthrobacter, Paenarthrobacter, Enterobacter, Staphylococcus, Erwinia, Herbasprillum, Ocuria, Massilia, Planomicrobium, Hodococcus, and Stenotrophomonas. The results detected a high proportion of osmotolerant and enzymes producing bacteria, many isolates can tolerate up to 55 °C (40%, 28%, and 30% in Merzouga, Mhamid Elghizlane, and Erg lihoudi, respectively). Meanwhile, the salinity tolerance reached 12% in some isolates with different proportions in each site, 29% in Merzouga, 24% in Mhamid Elghizlane, and 9% in Erg lihoudi. Furthermore, the enzymatic tests showed the presence of an amylolytic, lipolytic, cellulolytic, proteolytic activities in 20%, 31%, 63% and 72% of total strains, respectively.
As a result, the present study is thus a preliminary yet critical step towards identifying the best bacterial candidates for further biotechnological applications.
Citation: Amina Manni, Abdelkarim Filali-Maltouf. Diversity and bioprospecting for industrial hydrolytic enzymes of microbial communities isolated from deserted areas of south-east Morocco[J]. AIMS Microbiology, 2022, 8(1): 5-25. doi: 10.3934/microbiol.2022002
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[8] | Mingliang Liao, Danxia Wang, Chenhui Zhang, Hongen Jia . The error analysis for the Cahn-Hilliard phase field model of two-phase incompressible flows with variable density. AIMS Mathematics, 2023, 8(12): 31158-31185. doi: 10.3934/math.20231595 |
[9] | Alain Miranville . The Cahn–Hilliard equation and some of its variants. AIMS Mathematics, 2017, 2(3): 479-544. doi: 10.3934/Math.2017.2.479 |
[10] | Scala Riccardo, Schimperna Giulio . On the viscous Cahn-Hilliard equation with singular potential and inertial term. AIMS Mathematics, 2016, 1(1): 64-76. doi: 10.3934/Math.2016.1.64 |
The current study aimed to analyze bacterial communities' diversity and abundance in three different deserted areas (Merzouga, Mhamid Elghizlane, and Erg lihoud) located in Moroccan Sahara, as well as to investigate osmotolerant microorganisms producing hydrolytic enzymes. The isolates were taxonomically affiliated using 16S rRNA gene sequencing. Four different hydrolase activities (amylase, lipase, cellulase, and protease) and osmotic stress tolerance were evaluated. The phylogenetic analysis of 364 screened isolates belonged to three phyla (Firmicutes 73%, Proteobacteria 26% and Actinobacteria 1%) and 18 different genera, from Bacillus, Ornithinibacillus, Paenibacillus, Geobacillus, Pseudomonas, Acinetobacter, Agrobacterium, Arthrobacter, Paenarthrobacter, Enterobacter, Staphylococcus, Erwinia, Herbasprillum, Ocuria, Massilia, Planomicrobium, Hodococcus, and Stenotrophomonas. The results detected a high proportion of osmotolerant and enzymes producing bacteria, many isolates can tolerate up to 55 °C (40%, 28%, and 30% in Merzouga, Mhamid Elghizlane, and Erg lihoudi, respectively). Meanwhile, the salinity tolerance reached 12% in some isolates with different proportions in each site, 29% in Merzouga, 24% in Mhamid Elghizlane, and 9% in Erg lihoudi. Furthermore, the enzymatic tests showed the presence of an amylolytic, lipolytic, cellulolytic, proteolytic activities in 20%, 31%, 63% and 72% of total strains, respectively.
As a result, the present study is thus a preliminary yet critical step towards identifying the best bacterial candidates for further biotechnological applications.
In recent years there has been an increased focus on the mathematical modelling and analysis of tumour growth. Many new models have been proposed and numerical simulations have been carried out to provide new and important insights on cancer research, see for instance [8] and [13, Chap. 3]. In this work we analyse a diffuse interface model proposed in [20], which models a mixture of tumour cells and healthy cells in the presence of an unspecified chemical species acting as a nutrient. More precisely, for a bounded domain Ω⊂Rd where the cells reside and T > 0, we consider the following set of equations,
divv=ΓvinΩ×(0,T)=:Q, | (1.1a) |
v=−K(∇p−(μ+χσ)∇φ) in Q, | (1.1b) |
∂tφ+div(φv)=div(m(φ)∇μ)+Γφ in Q, | (1.1c) |
μ=AΨ′(φ)−BΔφ−χσ in Q, | (1.1d) |
∂tσ+div(σv)=div(n(φ)(D∇σ−χ∇φ))−S in Q. | (1.1e) |
Here, v denotes the volume-averaged velocity of the mixture, p denotes the pressure, σ denotes the concentration of the nutrient, φ∈[−1,1] denotes the difference in volume fractions, with {φ=1} representing the unmixed tumour tissue, and {φ=−1} representing the surrounding healthy tissue, and μ denotes the chemical potential for φ.
The model treats the tumour and healthy cells as inertia-less fluids, leading to the appearance of a Darcy-type subsystem with a source term Γv. The order parameter φ satisfies a convective Cahn--Hilliard type equation with additional source term Γφ, and similarly, the nutrient concentration σ satisfies a convection-reaction-diffusion equation with a non-standard flux and a source term S. We refer the reader to [20, x2] for the derivation from thermodynamic principles, and to [20, x2.5] for a discussion regarding the choices for the source terms Γφ,Γv and S.
The positive constants K and D denote the permeability of the mixture and the diffusivity of the nutrient, m(φ) and n(φ) are positive mobilities for φ and σ, respectively. The parameter χ≥0 regulates the chemotaxis effect (see [20] for more details), Ψ(⋅) is a potential with two equal minima at ±1, A and B denote two positive constants related to the thickness of the diffuse interface and the surface tension.
We supplement the above with the following boundary and initial conditions
∂nφ=∂nμ=0 on ∂Ω×(0,T)=:Σ, | (1.2a) |
v⋅n=∂np=0 on Σ, | (1.2b) |
n(φ)D∂nσ=b(σ∞−σ) on Σ, | (1.2c) |
φ(0)=φ0,σ(0)=σ0 on Ω. | (1.2d) |
Here φ0, σ0 and σ∞ are given functions and b>0 is a constant. We denote ∂nf:=∇f⋅n as the normal derivative of f at the boundary ∂Ω, where n is the outer unit normal. Associated to (1.1) is the free energy density N(φ,σ) for the nutrient, which is defined as
N(φ,σ):=D2|σ|2+χσ(1−φ). | (1.3) |
Note that
N,σ:=∂N∂σ=Dσ+χ(1−φ),N,φ:=∂N∂φ=−χσ, |
so that (1.1) may also be written as
divv=Γv, | (1.4a) |
v=−K(∇p−μ∇φ+N,φ∇φ), | (1.4b) |
∂tφ+div(φv)=div(m(φ)∇μ)+Γφ, | (1.4c) |
μ=AΨ′(φ)−BΔφ+N,φ, | (1.4d) |
∂tσ+div(σv)=div(n(φ)∇N,σ)−S, | (1.4e) |
which is the general phase field model proposed in [20]. In this work we do not aim to analyse such a model with a general free energy density N(φ,σ), but we will focus solely on the choice (1.3) and the corresponding model (1.1)-(1.2).
Our goal in this work is to prove the existence of weak solutions (see Definition 2.1 below) of (1.1)-(1.2) in two and three dimensions. Moreover, one might expect that by setting Γv=0 and then sending b→0 and K→0, the weak solutions to (1.1)-(1.2) will converge (in some appropriate sense) to the weak solutions of
∂tφ=div(m(φ)∇μ)+Γφ in Q, | (1.5a) |
μ=AΨ′(φ)−BΔφ−χσ in Q, | (1.5b) |
∂tσ=div(n(φ)(D∇σ−χ∇φ))−S in Q, | (1.5c) |
0=∂nφ=∂nμ=∂nσ on Σ. | (1.5d) |
We denote (1.5) as the limit system of vanishing permeability, where the effects of the volume-averaged velocity are neglected. By substituting
Γφ=S=f(φ)(Dσ+χ(1−φ)−μ) | (1.6) |
for some non-negative function f(φ) leads to the model derived in [21]. The specific choices for Γφ and S in (1.6) are motivated by linear phenomenological laws for chemical reactions. The analysis of (1.5) with the parameters
D=1,χ=0,n(φ)=m(φ)=1 |
has been the subject of study in [5, 6, 7, 16], where well-posedness and long-time behaviour have been established for a large class of functions Ψ(φ) and f(φ). Alternatively, one may consider the following choice of source terms
Γφ=h(φ)(λpσ−λa),S=λch(φ)σ, | (1.7) |
where λp, λa, λc are non-negative constants representing the tumour proliferation rate, the apoptosis rate, and the nutrient consumption rate, respectively, and h(φ) is a non-negative interpolation function such that h(−1)=0 and h(1)=1. The above choices for Γφ and S are motivated from the modelling of processes experienced by a young tumour.
The well-posedness of model (1.5) with the choice (1.7) has been studied by the authors in [17] and [18] with the boundary conditions (1.2) (neglecting (1.2b)) in the former and for non-zero Dirichlet boundary conditions in the latter. It has been noted in [17] that the well-posedness result with the boundary conditions (1.2) requires Ψ to have at most quadratic growth, which is attributed to the presence of the source term Γφμ=h(φ)μ(λpσ−λa) when deriving useful a priori estimates. Meanwhile in [18] the Dirichlet boundary conditions and the application of the Poincaré inequality allows us to overcome this restriction and allow for Ψ to be a regular potential with polynomial growth of order less than 6, and by a Yosida approximation, the case where Ψ is a singular potential is also covered.
We also mention the work of [19] that utilises a Schauder's fixed point argument to show existence of weak solutions for Ψ with quartic growth and Γφ,S as in (1.7). This is based on first deducing that σ is bounded by a comparison principle, leading to Γφ∈L∞(Ω). Then, the standard a priori estimates are derived for a Cahn--Hilliard equation with bounded source terms. The difference between [19] and [17, 18] is the absence of the chemotaxis and active transport mechanisms, i.e., χ=0, so that the comparison principle can be applied to the nutrient equation. We refer to [9] for the application of a similar procedure to a multi-species tumour model with logarithmic potentials.
On the other hand, by sending b→0 and χ→0 in (1.1), we should obtain weak solutions of
divv=Γv in Q, | (1.8a) |
v=−K(∇p−μ∇φ) in Q, | (1.8b) |
∂tφ+div(φv)=div(m(φ)∇μ)+Γφ in Q, | (1.8c) |
μ=AΨ′(φ)−BΔφ in Q, | (1.8d) |
∂tσ+div(σv)=div(n(φ)D∇σ)−S in Q, | (1.8e) |
0=∂nφ=∂nμ=∂nσ=v⋅n on Σ. | (1.8f) |
We denote (1.8) as the limit system of vanishing chemotaxis. If the source terms Γv and Γφ are independent of σ, then (1.8) consists of an independent Cahn--Hilliard--Darcy system and an equation for σ which is advected by the volume-averaged velocity field v. In the case where there is no nutrient and source terms, i.e., σ=Γv=Γφ=0, global existence of weak solutions in two and three dimensions has been established in [14] via the convergence of a fully discrete and energy stable implicit finite element scheme. For the well-posedness and long-time behaviour of strong solutions, we refer to [25]. Meanwhile, in the case where Γv=Γφ is prescribed, global weak existence and local strong well-posedness for (1.8) without nutrient is shown in [22].
We also mention the work of [3] on the well-posedness and long-time behaviour of a related system also used in tumour growth, known as the Cahn--Hilliard--Brinkman system, where in (1.8) without nutrient an additional viscosity term is added to the left-hand side of the velocity equation (1.8b) and the mass exchange terms Γv and Γφ are set to zero. The well-posedness of a nonlocal variant of the Cahn--Hilliard--Brinkman system has been investigated in [10]. Furthermore, when K is a function depending on φ, the model (1.8) with σ=Γv=Γφ=0 is also referred to as the Hele--Shaw--Cahn--Hilliard model (see [23, 24]). In this setting, K(φ) represents the reciprocal of the viscosity of the fluid mixture. We refer to [30] concerning the strong well-posedness globally in time for two dimensions and locally in time for three dimensions when Ω is the d-dimensional torus. Global well-posedness in three dimensions under additional assumptions and long-time behaviour of solutions to the Hele--Shaw--Cahn--Hilliard model are investigated in [29].
We point out that from the derivation of (1.1) in [20], the source terms Γv and Γφ are connected in the sense that Γv is related to sum of the mass exchange terms for the tumour and healthy cells, and Γφ is related to the difference between the mass exchange terms. Thus, if Γφ would depend on the primary variables φ, σ or μ, then one expects that Γv will also depend on the primary variables. Here, we are able to prove existence of weak solutions for Γφ of the form (2.1), which generalises the choices (1.6) and (1.7), but in exchange Γv has to be considered as a prescribed function. This is attributed to the presence of the source term Γv(φμ+D2|σ|2) when deriving useful a priori estimates. We see that if Γv depends on the primary variables, we obtain triplet products which cannot be controlled by the usual regularity of φ, μ and σ in the absence of a priori estimates.
In this work we attempt to generalise the weak existence results for the models studied in [5, 16, 17, 18, 22, 25] by proving that the weak solutions of (1.1) with Γv=0 converge (in some appropriate sense) to the weak solutions of (1.5) as b→0 and K→0, and the weak solutions of (1.1) converge to the weak solutions of (1.8) as b→0 and χ→0.
This paper is organised as follows. In Section 2 we state the main assumptions and the main results. In Section 3 we introduce a Galerkin procedure and derive some a priori estimates for the Galerkin ansatz in Section 4 for the case of three dimensions. We then pass to the limit in Section 5 to deduce the existence result for three dimensions, while in Section 6 we investigate the asymptotic behaviour of solutions to (1.1) as K→0 and χ→0. In Section 7, we outline the a priori estimates for two dimensions and show that the weak solutions for two dimensions yields better temporal regularity than the weak solutions for three dimensions. In Section 8 we discuss some of the issues present in the analysis of (1.1) using different formulations of Darcy's law and the pressure, and with different boundary conditions for the velocity and the pressure.
Notation. For convenience, we will often use the notation Lp:=Lp(Ω) and Wk,p:=Wk,p(Ω) for any p∈[1,∞], k>0 to denote the standard Lebesgue spaces and Sobolev spaces equipped with the norms ‖⋅‖Lp and ‖⋅‖Wk,p. In the case p=2 we use Hk:=Wk,2 and the norm ‖⋅‖Hk. For the norms of Bochner spaces, we will use the notation Lp(X):=Lp(0,T;X) for Banach space X and p∈[1,∞]. Moreover, the dual space of a Banach space X will be denoted by X∗, and the duality pairing between X and X∗ is denoted by ⟨⋅,⋅⟩X,X∗. For d=2 or 3, let Hd−1 denote the (d−1) dimensional Hausdorff measure on ∂Ω, and we denote Rd-valued functions and any function spaces consisting of vector-valued/tensor-valued functions in boldface. We will use the notation Df to denote the weak derivative of the vector function f.
Useful preliminaries. For convenience, we recall the Poincaré inequality: There exists a positive constant Cp depending only on Ω such that, for all f∈H1,
‖f−ˉf‖L2≤Cp‖∇f‖L2, | (1.9) |
where ˉf:=1|Ω|∫Ωf dx denotes the mean of f. The Gagliardo--Nirenberg interpolation inequality in dimension d is also useful (see [15, Thm. 10.1, p. 27], [11, Thm. 2.1] and [1, Thm. 5.8]): Let Ω be a bounded domain with Lipschitz boundary, and f∈Wm,r∩Lq, 1≤q,r≤∞. For any integer j, 0≤j<m, suppose there is α∈R such that
1p=jd+(1r−md)α+1−αq,jm≤α≤1. |
Then, there exists a positive constant C depending only on Ω, m, j, q, r, and α such that
‖Djf‖Lp≤C‖f‖αWm,r‖f‖1−αLq. | (1.10) |
We will also use the following Gronwall inequality in integral form (see [17, Lem. 3.1] for a proof): Let α,β,u and v be real-valued functions defined on [0,T]. Assume that α is integrable, β is non-negative and continuous, u is continuous, v is non-negative and integrable. If u and v satisfy the integral inequality
u(s)+∫s0v(t) dt≤α(s)+∫s0β(t)u(t) dt for s∈(0,T], |
then it holds that
u(s)+∫s0v(t) dt≤α(s)+∫s0β(t)α(t)exp(∫t0β(r) dr) dt. | (1.11) |
To analyse the Darcy system, we introduce the spaces
L20:={f∈L2:ˉf=0},H2N:={f∈H2:∂nf=0 on ∂Ω},(H1)∗0:={f∈(H1)∗:⟨f,1⟩H1=0}. |
Then, the Neumann-Laplacian operator −ΔN:H1∩L20→(H1)∗0 is positively defined and self-adjoint. In particular, by the Lax--Milgram theorem and the Poincaré inequality (1.9) with zero mean, the inverse operator (−ΔN)−1:(H1)∗0→H1∩L20 is well-defined, and we set u:=(−ΔN)−1f for f∈(H1)∗0 if ˉu=0 and
−Δu=f in Ω, ∂nu=0 on ∂Ω. |
We make the following assumptions.
Assumption 2.1.
(A1) The constants A, B, K, D, χ and b are positive and fixed.
(A2) The mobilities m, n are continuous on R and satisfy
m0≤m(t)≤m1, n0≤n(t)≤n1∀t∈R, |
for positive constants m0, m1, n0 and n1.
(A3) Γφ and S are of the form
Γφ(φ,μ,σ)=Λφ(φ,σ)−Θφ(φ,σ)μ,S(φ,μ,σ)=ΛS(φ,σ)−ΘS(φ,σ)μ, | (2.1) |
where Θφ,ΘS:R2→R are continuous bounded functions with Θφ non-negative, and Λφ,ΛS:R2→R are continuous with linear growth
|Θi(φ,σ)|≤R0, |Λi(φ,σ)|≤R0(1+|φ|+|σ|) for i∈{φ,S}, | (2.2) |
so that
|Γφ|+|S|≤R0(1+|φ|+|μ|+|σ|), | (2.3) |
for some positive constant R0.
(A4) Γv is a prescribed function belonging to L4(0,T;L20).
(A5) Ψ∈C2(R) is a non-negative function satisfying
Ψ(t)≥R1 |t|2−R2∀t∈R | (2.4) |
and either one of the following,
1: if Θφ is non-negative and bounded, then
Ψ(t)≤R3(1+|t|2), |Ψ′(t)|≤R4(1+|t|), |Ψ″(t)|≤R4; | (2.5) |
2: if Θφ is positive and bounded, that is,
R0≥Θφ(t,s)≥R5>0∀t,s∈R, | (2.6) |
then
|Ψ″(t)|≤R6(1+|t|q),q∈[0,4), | (2.7) |
for some positive constants R1, R2, R3, R4, R5, R6. Furthermore we assume that
A>2χ2DR1. | (2.8) |
(A6) The initial and boundary data satisfy
σ∞∈L2(0,T;L2(∂Ω)),σ0∈L2,φ0∈H1. |
We point out that some of the above assumptions are based on previous works on the well-posedness of Cahn--Hilliard systems for tumour growth. For instance, (2.5) and (2.8) reflect the situation encountered in [17], where if Θφ=0, i.e., Γφ is independent of μ, then the derivation of the a priori estimate requires a quadratic potential. But in the case where (2.6) is satisfied, we can allow Ψ to be a regular potential with polynomial growth of order less than 6, and by a Yosida approximation, we can extend our existence results to the situation where Ψ is a singular potential, see for instance [18]. Moreover, the condition (2.8) is a technical assumption based on the fact that the second term of the nutrient free energy χσ(1−φ) does not have a positive sign.
Meanwhile, the linearity of the source terms Γφ and S with respect to the chemical potential μ assumed in (2.1) is a technical assumption based on the expectation that, at best, we have weak convergence for Galerkin approximation to μ, which is in contrast with φ and σ where we might expect a.e convergence and strong convergence for the Galerkin approximations. Moreover, if we consider
Θφ(φ,σ)=ΘS(φ,σ)=f(φ),Λφ(φ,σ)=ΛS(φ,σ)=f(φ)(Dσ+χ(1−φ)), |
for a non-negative function f(φ), then we obtain the source terms in [5, 16, 21].
Compared to the set-up in [22], in (A4) we prescribe a higher temporal regularity for the prescribed source term Γv. This is needed when we estimate the source term ΓvD2|σ|2 in the absence of a priori estimates, see Section 4.1.2 for more details. The mean zero condition is a consequence of the no-flux boundary condition v⋅n=0 on ∂Ω and the divergence equation (1.1a). In particular, we can express the Darcy subsystem (1.1a)-(1.1b) as an elliptic equation for the pressure p:
−Δp=1KΓv−div((μ+χσ)∇φ) in Ω, | (2.9a) |
∂np=0 on ∂Ω. | (2.9b) |
Solutions to (2.9) are uniquely determined up to an arbitrary additive function that may only depend on time, and thus without loss of generality, we impose the condition ˉp=1|Ω|∫Ωp dx=0 to (2.9). We may then define p as
p=(−ΔN)−1(1KΓv−div((μ+χσ)∇φ)), | (2.10) |
if 1KΓv−div((μ+χσ)∇φ)∈(H1)∗0.
Remark 2.1. In the case Γv=0, one can also consider the assumption
SN,σ−Γφμ=S(Dσ+χ(1−φ))−Γφμ≥0 | (2.11) |
instead of (2.6), which holds automatically if Γφ and S are chosen to be of the form (1.6). In fact this property is used in [5, 16].
We make the following definition.
Definition 2.1 (Weak solutions for 3D). We call a quintuple (φ,μ,σ,v,p) a weak solution to (1.1)-(1.2) if
φ∈L∞(0,T;H1)∩L2(0,T;H3)∩W1,85(0,T;(H1)∗),σ∈L∞(0,T;L2)∩L2(0,T;H1)∩W1,54(0,T;(W1,5)∗),μ∈L2(0,T;H1),p∈L85(0,T;H1∩L20),v∈L2(0,T;L2), |
such that φ(0)=φ0,
⟨σ0,ζ⟩H1,(H1)∗=⟨σ(0),ζ⟩H1,(H1)∗∀ζ∈H1, |
and
⟨∂tφ,ζ⟩H1,(H1)∗=∫Ω−m(φ)∇μ⋅∇ζ+Γφζ+φv⋅∇ζ dx, | (2.12a) |
∫Ωμζ dx=∫ΩAΨ′(φ)ζ+B∇φ⋅∇ζ−χσζ dx, | (2.12b) |
⟨∂tσ,ϕ⟩W1,5,(W1,5)∗=∫Ω−n(φ)(D∇σ−χ∇φ)⋅∇ϕ−Sϕ+σv⋅∇ϕdx+∫∂Ωb(σ∞−σ)ϕ dHd−1, | (2.12c) |
∫Ω∇p⋅∇ζ dx=∫Ω1KΓvζ+(μ+χσ)∇φ⋅∇ζ dx, | (2.12d) |
∫Ωv⋅ ζ dx=∫Ω−K(∇p−(μ+χσ)∇φ)⋅ ζ dx, | (2.12e) |
for a.e. t∈(0,T) and for all ζ∈H1, ϕ∈W1,5, and ζ∈L2.
Neglecting the nutrient σ, we observe that our choice of function spaces for (φ,μ,p,v) coincide with those in [22, Defn. 2.1(i)]. In contrast to the usual L2(0,T;(H1)∗)-regularity (see [5, 16]) we obtain a less regular time derivative ∂tφ. The drop in the time regularity from 2 to 85 is attributed to the convection term div(φv) belonging to L85(0,T;(H1)∗). The same is true for the regularity for the time derivative ∂tσ in L54(0,T;(W1,5)∗) as the convection term div(σv) lies in the same space. We refer the reader to the end of Section 4.3 for a calculation motivating the choice of function spaces for div(σv) and ∂tσ. Furthermore, the embedding of L∞(0,T;H1)∩W1,85(0,T;(H1)∗) into C0([0,T];L2) from [28, x8, Cor. 4] guarantees that the initial condition for φ is meaningful. However, for σ we have the embedding L∞(0,T;L2)∩W1,54(0,T;(W1,5)∗)⊂⊂C0([0,T];(H1)∗), and so σ(0) makes sense as a function in (H1)∗. Thus, the initial condition σ0 is attained as an equality in (H1)∗. We now state the existence result for (1.1)-(1.2).
Theorem 2.1 (Existence of weak solutions in 3D and energy inequality). Let Ω⊂R3 be a bounded domain with C3-boundary ∂Ω. Suppose Assumption 2.1 is satisfied. Then, there exists a weak solution quintuple (φ,μ,σ,v,p) to (1.1)-(1.2) in the sense of Definition 2.1 with
p∈L87(0,T;H2),v∈L87(0,T;H1), | (2.13) |
and in addition satisfies
‖φ‖L∞(H1)∩L2(H3)∩W1,85((H1)∗)+‖σ‖L∞(L2)∩W1,54((W1,5)∗))∩L2(H1)+‖μ‖L2(H1)+b12‖σ‖L2(L2(∂Ω))+‖p‖L85(H1)∩L87(H2)+K−12(‖v‖L2(L2)∩L87(H1)+‖div(φv)‖L85((H1)∗)+‖div(σv)‖L54((W1,5)∗))≤C, | (2.14) |
where the constant C does not depend on (φ,μ,σ,v,p) and is uniformly bounded for b,χ∈(0,1] and is also uniformly bounded for K∈(0,1] when Γv=0.
The regularity result (2.13) is new compared to estimates for weak solutions in [22], which arises from a deeper study of the Darcy subsystem, and can be obtained even in the absence of the nutrient. We mention that higher regularity estimates for the pressure p in L2(0,T;H2) and the velocity v in L2(0,T;H1) are also established in [22, but these are for strong solutions local in time in three dimensions and global in time for two dimensions.
We now investigate the situation in two dimensions, where the Sobolev embeddings in two dimensions yields better integrability exponents.
Theorem 2.2 (Existence of weak solutions in 2D). Let Ω⊂R2 be a bounded domain with C3-boundary ∂Ω. Suppose Assumption 2.1is satisfied. Then, there exists a quintuple (φ,μ,σ,v,p) to (1.1)-(1.2) with the following regularity
φ∈L∞(0,T;H1)∩L2(0,T;H3)∩W1,w(0,T;(H1)∗),μ∈L2(0,T;H1),σ∈L2(0,T;H1)∩L∞(0,T;L2)∩W1,r(0,T;(H1)∗),p∈Lk(0,T;H1∩L20)∩Lq(0,T;H2),v∈L2(0,T;L2)∩Lq(0,T;H1), |
for
1≤k<2,1≤q<43,1<r<87,43≤w<2, |
such that (2.12a), (2.12b), (2.12d), (2.12e) and
⟨∂tσ,ζ⟩H1,(H1)∗=∫Ω−n(φ)(D∇σ−χ∇φ)⋅∇ζ−Sζ+σv⋅∇ζ dx+∫∂Ωb(σ∞−σ)ζ dHd−1 |
are satisfied for a.e. t∈(0,T), for all ζ∈H1, and all ζ∈L2. Furthermore, the initial conditions φ(0)=φ0 and σ(0)=σ0 are attained as in Definition 2.1, and an analogous inequality to (2.14) also holds.
The proof of Theorem 2.2 is similar to that of Theorem 2.1, and hence the details are omitted. In Section 7 we will only present the derivation of a priori estimates. It is due to the better exponents for embeddings in two dimensions and the regularity result for the velocity that we obtain better regularities for the time derivatives ∂tφ and ∂tσ, namely ∂tσ(t) belongs to the dual space (H1)∗ for a.e. t∈(0,T). Furthermore, as mentioned in Remark 7.1 below, if we only have v∈L2(0,T;L2), then the convection term div(σv) and the time derivative ∂tσ would only belong to the dual space L43(0,T;(W1,4)∗). However, even with the improved temporal regularity, as ∂tσ∉L2(0,T;(H1)∗), we do not have a continuous embedding into the space C0([0,T];L2) and so σ(0) may not be well-defined as an element of L2.
We now state the two asymptotic limits of (1.1) for three dimensions, and note that analogous asymptotic limits also hold for two dimensions.
Theorem 2.3 (Limit of vanishing permeability). For b,K∈(0,1], we denote a weak solution to (1.1)-(1.2) with Γv=0 and initial conditions (φ0,σ0) by (φK,μK,σK,vK,pK). Then, as b→0 and K→0, it holds that
φK→φ weakly - ∗ in L∞(0,T;H1)∩L2(0,T;H3)∩W1,85(0,T;(H1)∗),σK→σ weakly - ∗ in L2(0,T;H1)∩L∞(0,T;L2)∩W1,54(0,T;(W1,5)∗),μK→μ weakly in L2(0,T;H1),pK→p weakly in L85(0,T;H1)∩L87(0,T;H2),vK→0 strongly in L2(0,T;L2)∩L87(0,T;H1), |
where (φ,μ,σ,p) satisfies
⟨∂tφ,ζ⟩H1,(H1)∗=∫Ω−m(φ)∇μ⋅∇ζ+Γφ(φ,μ,σ)ζ dx, | (2.16a) |
∫Ωμζ dx=∫ΩAΨ′(φ)ζ+B∇φ⋅∇ζ−χσζ dx, | (2.16b) |
⟨∂tσ,ϕ⟩W1,5,(W1,5)∗=∫Ω−n(φ)(D∇σ−χ∇φ)⋅∇ϕ−S(φ,μ,σ)ϕ dx, | (2.16c) |
∫Ω∇p⋅∇ζ dx=∫Ω(μ+χσ)∇φ⋅∇ζ dx, | (2.16d) |
for all ζ∈H1, ϕ∈W1,5 and a.e. t∈(0,T). A posteriori, it holds that
∂tφ, ∂tσ∈L2(0,T;(H1)∗), |
and thus φ(0)=φ0 and σ(0)=σ0.
Theorem 2.4 (Limit of vanishing chemotaxis). For b,χ∈(0,1], we denote a weak solution to (1.1)-(1.2) with corresponding initial conditions (φ0,σ0) by (φχ,μχ,σχ,vχ,pχ). Then, as b→0 and χ→0, it holds that
φχ→φ weakly - ∗ in L∞(0,T;H1)∩L2(0,T;H3)∩W1,85(0,T;(H1)∗),σχ→σ weakly - ∗ in L2(0,T;H1)∩L∞(0,T;L2)∩W1,54(0,T;(W1,5)∗),μχ→μ weakly in L2(0,T;H1),pχ→p weakly in L85(0,T;H1)∩L87(0,T;H2),vχ→v weakly in L2(0,T;L2)∩L87(0,T;H1), |
and
div(φχvχ)→div(φv) weakly in L85(0,T;(H1)∗),div(σχvχ)→div(σv) weakly in L54(0,T;(W1,5)∗), |
where (φ,μ,σ,v,p) satisfies
⟨∂tφ,ζ⟩H1,(H1)∗=∫Ω−m(φ)∇μ⋅∇ζ+Γφ(φ,μ,σ)ζ+φv⋅∇ζ dx, | (2.19a) |
∫Ωμζ dx=∫ΩAΨ′(φ)ζ+B∇φ⋅∇ζ dx, | (2.19b) |
⟨∂tσ,ϕ⟩W1,5,(W1,5)∗=∫Ω−n(φ)D∇σ⋅∇ϕ−S(φ,μ,σ)ϕ+σ v⋅∇ϕ dx, | (2.19c) |
∫Ω∇p⋅∇ζ dx=∫Ω1KΓvζ+μ∇φ⋅∇ζ dx, | (2.19d) |
∫Ωv⋅ ζ dx=∫Ω−K(∇p−μ∇φ)⋅ζ dx, | (2.19e) |
for all ζ∈H1, ϕ∈W1,5, ζ∈L2 and a.e. t∈(0,T), with the attainment of initial conditions as in Definition 2.1.
We will employ a Galerkin approximation similar to the one used in [22]. For the approximation, we use the eigenfunctions of the Neumann--Laplacian operator {wi}i∈N. Recall that the inverse Neumann--Laplacian operator L:=(−ΔN)−1|L20:L20→L20 is compact, positive and symmetric. Indeed, let f,g∈L20 with z=Lf, y=Lg. Then,
(Lf,f)L2=∫Ωzf dx=∫Ω|∇z|2 dx≥0,(Lf,g)L2=∫Ω∇z⋅∇y dx=(f,Lg)L2. |
Furthermore, let {fn}n∈N⊂L20 denote a sequence with corresponding solution sequence {zn=Lfn}n∈N⊂H1∩L20. By elliptic regularity theory, we have that zn∈H2N for all n∈N. Then, by reflexive weak compactness theorem and Rellich--Kondrachov theorem, there exists a subsequence such that znj→z∈H1∩L20 as j→∞.
Thus, by the spectral theorem, the operator L admits a countable set of eigenfunctions {vn}n∈N that forms a complete orthonormal system in L20. The eigenfunctions of the Neumann--Laplacian operator is then given by w1=1, wi=vi−1 for i≥2, and {wi}i∈N is a basis of L2.
Elliptic regularity theory gives that wi∈H2N and for every g∈H2N, we obtain for gk:=∑ki=1(g,wi)L2wi that
Δgk=k∑i=1(g,wi)L2Δwi=k∑i=1(g,λiwi)L2wi=k∑i=1(g, Δwi)L2wi=k∑i=1(Δg,wi)L2wi, |
where λi is the corresponding eigenvalue to wi. This shows that Δgk converges strongly to Δg in L2. Making use of elliptic regularity theory again gives that gk converges strongly to g in H2N. Thus the eigenfunction {wi}i∈N of the Neumann--Laplace operator forms an orthonormal basis of L2 and is also a basis of H2N.
Later in Section 5, we will need to use the property that H2N is dense in H1 and W1,5. We now sketch the argument for the denseness of H2N in W1,5 and the argument for H1 follows in a similar fashion.
Lemma 3.1. H2N is dense in W1,5.
Proof. Take g∈W1,5, as Ω has a C3-boundary, by standard results [12, Thm. 3, x5.3.3] there exists a sequence gn∈C∞(¯Ω) such that gn→g strongly in W1,5. Let ε>0 be fixed, and define Dε:={x∈Ω: dist(x,∂Ω)≤ε}. Let ζε∈C∞c(Ω) be a smooth cut-off function such that ζε=1 in Ω∖¯Dε and ζε=0 in ¯Dε2.
As gn∈C∞(¯Ω), its trace on ∂Ω is well-defined. Choosing ε sufficiently small allows us to use a classical result from differential geometry about tubular neighbourhoods, i.e., for any z∈Tubε(∂Ω):={x∈Rd: |dist(z,∂Ω)|≤ε} there exists a unique y∈∂Ω such that
z=y+dist(z,∂Ω)n(y), |
where n is the outer unit normal of ∂Ω. We consider a bounded smooth function fn,ε:Rd→R such that
fn,ε(z)=gn(y) for all z∈Tubε(∂Ω) satisfying z=y+ dist(z,∂Ω)n(y). |
We now define the smooth function Gn,ε as
Gn,ε(x):=ζε(x)gn(x)+(1−ζε(x))fn,ε(x). |
By construction, the values of the function fn,ε in Dε⊂Tubε(∂Ω) are constant in the normal direction, so ∇Gn,ε⋅n=0 on ∂Ω and thus Gn,ε∈H2N. Furthermore, we compute that
‖Gn,ε−gn‖L5=‖(1−ζε)(fn,ε−gn)‖L5(Dε),‖∇(Gn,ε−gn)‖L5=‖(gn−fn,ε)∇ζε+(1−ζε)∇gn+(1−ζε)∇fn,ε‖L5. |
Using that gn,fn,ε are smooth functions on ¯Ω and that the Lebesgue measure of Dε tends to zero as ε→0 we have the strong convergence of Gn,ε to gn in L5. For the difference in the gradients, we use that ζε→1 a.e. in Ω, Lebesgue's dominated convergence theorem and the boundedness of ∇gn and ∇fn,ε to deduce that
‖(1−ζε)∇gn‖L5+‖(1−ζε)∇fn,ε‖L5→0 as ε→0. |
For the remaining term ‖(gn−fn,ε)∇ζε‖L5 we use that the support of ∇ζε lies in Dε∖¯Dε2 and for any z∈Dε∖¯Dε2,
|fn,ε(z)−gn(z)|=|gn(y)−gn(y+dist(z,∂Ω)n(y))|≤∫dist(z,∂Ω)0|∇gn(y+ξn(y))|dξ≤‖∇gn‖L∞dist(z,∂Ω)≤Cε. |
That is, fn,ε converges uniformly to gn in Dε∖¯Dε2. Furthermore, using ‖∇ζε‖L∞≤Cε in Dε∖¯Dε2 and |Dε∖¯Dε2|≤Cε we obtain ‖(gn−fε,n)∇ζε‖L5≤Cε15→0 as ε→0. This shows that Gn,ε converges strongly to gn in W1,5.
We denote
Wk:=span{w1,…,wk} |
as the finite dimensional space spanned by the first k basis functions and consider
φk(t,x)=∑ki=1αki(t)wi(x),μk(t,x)=∑ki=1βki(t)wi(x),σk(t,x)=∑ki=1γki(t)wi(x), | (3.1a) |
and the following Galerkin ansatz: For 1≤j≤k,
∫Ω∂tφkwj dx=∫Ω−m(φk)∇μk⋅∇wj+Γφ,kwj+φkvk⋅∇wjdx, | (3.2a) |
∫Ωμkwj dx=∫ΩAΨ′(φk)wj+B∇φk⋅∇wj−χσkwj dx, | (3.2b) |
∫Ω∂tσkwjdx=∫Ω−n(φk)(D∇σk−χ∇φk)⋅∇wj−Skwj+σkvk⋅∇wjdx+∫∂Ωb(σ∞−σk)wj dHd−1, | (3.2c) |
where we define the Galerkin ansatz for the pressure pk and the velocity field vk by
pk=(−ΔN)−1(1KΓv−div((μk+χσk)∇φk)), | (3.3) |
vk=−K(∇pk−(μk+χσk)∇φk), | (3.4) |
and we set
Γφ,k:=Γφ(φk,μk,σk),Sk:=S(φk,μk,σk). |
Note that in (3.3), the properties Γv∈L20 and ∇φk⋅n=0 on ∂Ω show that the term inside the bracket belongs to L20 and hence pk is well-defined. Let M and S denote the following mass and stiffness matrices, respectively: For 1≤i,j≤k,
Mij=∫Ωwiwj dx, Sij:=∫Ω∇wi⋅∇wj dx. |
Thanks to the orthonormality of {wi}i∈N in L2, we see that M is the identity matrix. It is convenient to define the following matrices with components
(Ck)ji:=∫Ωwivk⋅∇wjdx,(M∂Ω)ji:=∫∂ΩwiwjdH,(Skm)ji:=∫Ωm(φk)∇wi⋅∇wjdx,(Skn)ji:=∫Ωn(φk)∇wi⋅∇wjdx, |
for 1≤i,j≤k. Furthermore, we introduce the notation
Rkφ,j:=∫ΩΓφ,kwj dx,RkS,j:=∫ΩSkwj dx,ψkj:=∫ΩΨ′(φk)wj dx,Σkj:=∫∂Ωσ∞wj dH, |
for 1≤i,j≤k, and denote
Rkφ:=(Rkφ,1,…,Rkφ,k)⊤, RkS:=(RkS,1,…,RkS,k)⊤, ψk:=(ψk1,…,ψkk)⊤, Σk:=(Σk1,…,Σkk)⊤, |
as the corresponding vectors, so that we obtain an initial value problem for a system of equations for αk:=(αk1,…αkk)⊤,βk:=(βk1,…,βkk)⊤, and γk:=(γk1,…,γkk)⊤ as follows,
ddtαk=−Skmβk+Rkφ+Ckαk, | (3.5a) |
βk=Aψk+BSαk−χγk, | (3.5b) |
ddtγk=−Skn(Dγk−χαk)−RkS+Ckγk−bM∂Ωγk+bΣk, | (3.5c) |
pk=(−ΔN)−1(1KΓv−div((μk+χσk)∇φk)) | (3.5d) |
vk=−K(∇pk−(μk+χσk)∇φk), | (3.5e) |
and we supplement (3.5) with the initial conditions
(αk)j(0)=∫Ωφ0wj dx,(γk)j(0)=∫Ωσ0wj dx, | (3.6) |
for 1≤j≤k, which satisfy
‖k∑i=1(αk)i(0)wi‖H1≤C‖φ0‖H1, ‖k∑j=1(γk)i(0)wi‖L2≤‖σ0‖L2∀k∈N, | (3.7) |
for some constant C not depending on k.
We can substitute (3.5b), (3.5d) and (3.5e) into (3.5a) and (3.5c), and obtain a coupled system of ordinary differential equations for αk and γk, where Skm, Ck and Skn depend on the solutions αk and γk in a non-linear manner. Continuity of m(⋅), n(⋅), Ψ′(⋅) and the source terms, and the stability of (−ΔN)−1 under perturbations imply that the right-hand sides of (3.5) depend continuously on (αk,γk). Thus, we can appeal to the theory of ODEs (via the Cauchy--Peano theorem [4, Chap. 1, Thm. 1.2]) to infer that the initial value problem (3.5)-(3.6) has at least one local solution pair (αk,γk) defined on [0,tk] for each k∈N.
We may define βk via the relation (3.5b) and hence the Galerkin ansatz φk,μk and σk can be constructed from (3.1). Then, we can define pk and vk via (3.3) and (3.4), respectively. Furthermore, as the basis function wj belongs to H2 for each j∈N, by the Sobolev embedding H2⊂L∞, we obtain that div(wi∇wj)∈L2 for i,j∈N and hence the function div((μk+χσk)∇φk) belongs to L2. Then, by elliptic regularity theory, we find that pk(t)∈H2N∩L20 for all t∈[0,tk]. This in turn implies that
vk(t)∈{f∈ H1:divf=Γv, f⋅n=0 on ∂Ω} for all t∈[0,tk]. | (3.8) |
Next, we show that the Galerkin ansatz can be extended to the interval [0, T] using a priori estimates.
In this section, the positive constants C are independent of k, Γv, K, b and χ, and may change from line to line. We will denote positive constants that are uniformly bounded for b,χ∈(0,1] and are also uniformly bounded for K∈(0,1] when Γv=0 by the symbol E.
We first state the energy identity satisfied by the Galerkin ansatz. Let δij denote the Kronecker delta. Multiplying (3.2a) with βkj, (3.2b) with ddtαkj, (3.2c) with Nk,σ:=Dγkj+χ(δ1j−αkj), and then summing the product from j=1 to k lead to
∫Ω∂tφkμkdx=∫Ω−m(φk)|∇μk|2+Γφ,kμk+φkvk⋅∇μkdx,∫Ωμk∂tφkdx=ddt∫ΩAΨ′(φk)+B2|∇φk|2dx−∫Ωχσk∂tφkdx,∫Ω∂tσkNk,σdx=∫Ω−n(φk)|∇Nk,σ|2−SkNk,σ+σkvk⋅∇Nk,σdx +∫∂Ωb(σ∞−σk)Nk,σdHd−1. |
Here, we used that w1=1 and ∇w1=0. Then, summing the three equations leads to
ddt∫ΩAΨ(φk)+B2|∇φk|2+N(φk,σk)dx+∫Ωm(φk)|∇μk|2+n(φk)|∇Nk,σ|2dx+∫∂ΩDb|σk|2dHd−1=∫ΩΓφ,kμk−SkNk,σ+(φkvk⋅∇μk+σkvk⋅∇Nk,σ)dx+∫∂Ωb(σ∞Nk,σ−σkχ(1−φk))dHd−1. | (4.2) |
Next, multiplying (3.4) with 1Kvk, integrating over Ω and integrating by parts gives
∫Ω1K|vk|2dx=∫Ω−∇pk⋅vk+(μk+χσk)∇φk⋅vkdx=∫ΩΓvpk+(μk+χσk)∇φk⋅vk dx, |
where we used that divvk=Γv and vk⋅n=0 on ∂Ω. Similarly, we see that
∫Ω(φk∇μk+σk∇Nk,σ)⋅vkdx=∫Ωφkvk⋅∇μk+σkvk⋅∇(Dσk+χ(1−φk))dx=−∫ΩφkΓvμk+(μk+χσk)vk⋅∇φk−D2vk⋅∇|σk|2dx=−∫ΩΓv(φkμk+D2|σk|2)+(μk+χσk)∇φk⋅vk dx. |
In particular, we have
∫Ω1K|vk|2 dx=∫ΩΓv(pk−μkφk−D2|σk|2)−(φk∇μk+σk∇Nk,σ)⋅vk dx. | (4.4) |
Adding (4.3) to (4.2) leads to
ddt∫ΩAΨ(φk)+B2|∇φk|2+N(φk,σk)dx+∫Ωm(φk)|∇μk|2+n(φk)|∇Nk,σ|2+1K|vk|2dx+∫∂ΩDb|σk|2dHd−1=∫ΩΓφ,kμk−SkNk,σ+Γv(pk−μkφk−D2|σk|2)dx+∫∂Ωb(σ∞(Dσk+χ(1−φk))−σkχ(1−φk))dHd−1. | (4.3) |
To derive the first a priori estimate for the Galerkin ansatz, it suffices to bring (4.4) into a form where we can apply Gronwall's inequality. We start with estimating the boundary term on the right-hand side of (4.4). By Hölder's inequality and Young's inequality,
|∫∂Ωb(σ∞(Dσk+χ(1−φk))−σkχ(1−φk))dHd−1|≤b(‖σ∞‖L2(∂Ω)‖Dσk+χ(1−φk)‖L2(∂Ω)+χ‖σk‖L2(∂Ω)(|∂Ω|12+‖φk‖L2(∂Ω)))≤Db2‖σk‖2L2(∂Ω)+b(1+χ2D)‖φk‖2L2(∂Ω)+bC(χ+(1+χ2)‖σ∞‖2L2(∂Ω)). |
By the trace theorem and the growth condition (2.4), we have
‖φ‖2L2(∂Ω)≤C2tr(‖φ‖2L2+‖∇φ‖2L2)≤C2tr(1R1‖Ψ(φ)‖L1+‖∇φ‖2L2)+C(R2,|Ω|,Ctr), | (4.5) |
where the positive constant Ctr from the trace theorem only depends on Ω, and so
∫∂Ωb(σ∞(Dσk+χ(1−φk))−σkχ(1−φk))dHd−1≤Db2‖σk‖2L2(∂Ω)+Cb(1+χ2)(‖Ψ(φk)‖L1+‖∇φk‖2L2)+Cb(1+χ2)+bC(χ+(1+χ2)‖σ∞‖2L2(∂Ω)). | (4.6) |
For the source term
∫ΩΓφ,kμk−SkNk,σ+Γv(pk−μkφk−D2|σk|2) dx |
that appears on the right-hand side of (4.4) we will divide its analysis into two parts. We first analyse the part involving Γv, which will involve a closer look at the Darcy subsystem to deduce an estimate on ‖pk‖L2. For the remainder Γφ,kμk−SkNk,σ term we will estimate it differently based on the assumptions on Θφ.
Before we estimate the source terms involving Γv, we look at the Darcy subsystem, which can be expressed as an elliptic equation for the pressure (we will drop the subscript k for clarity)
−Δp=1KΓv−div((μ+χσ)∇φ) in Ω, with ˉp=0, | (4.7a) |
∂p=0 on ∂Ω. | (4.7b) |
The following lemma is similar to [22, Lem. 3.1], and the hypothesis is fulfilled by the Galerkin ansatz.
Lemma 4.1. Let Ω⊂R3 be a bounded domain with C3-boundary. Given φ∈H2N, μ,σ∈H1, the source term Γv∈L20, and the function p satisfying the above elliptic equation (4.7). Then, the following estimate hold
‖p‖L2≤CK‖Γv‖L2+C(‖∇μ‖L2+χ‖σ‖L6)‖∇φ‖L32+Cˉμ‖∇φ‖L2, | (4.8) |
for some positive constant C depending only on Ω.
Proof. We first recall some properties of the inverse Neumann-Laplacian operator. Suppose for g∈L20, f=(−ΔN)−1g∈H1∩L20 solves
−Δf=g in Ω, ∂f=0 on ∂Ω. | (4.9) |
Then, testing with f and integrating over Ω, applying integration by parts and the Poincaré inequality (1.9) leads to
‖(−ΔN)−1g‖H1=‖f‖H1≤c‖∇f‖L2≤C‖g‖L2, | (4.10) |
for positive constants c and C depending only on Cp. Elliptic regularity theory then gives that f∈H2N with
‖f‖H2≤C(‖f‖H1+‖g‖L2)≤C‖g‖L2, | (4.11) |
with a positive constant C depending only on Ω. Returning to the pressure system, we observe from (2.10) and the above that
‖p‖L2≤1K‖(−ΔN)−1Γv‖L2+‖(−ΔN)−1(div((μ+χσ)∇φ))‖L2≤CK‖Γv‖L2+‖(−ΔN)−1(div((μ−ˉμ+χσ)∇φ))‖L2+‖(−ΔN)−1(div(ˉμ∇φ))‖L2, | (4.12) |
for some positive constant C depending only on Cp. Note that the third term on the right-hand side can be estimated as
ˉμ‖(−ΔN)−1div∇(φ−ˉφ)‖L2=ˉμ‖φ−ˉφ‖L2≤Cpˉμ‖∇φ‖L2. | (4.13) |
We now consider estimating the second term on the right-hand side of (4.12). By assumption μ,σ∈H1 and φ∈H2N, we have that
‖(μ−ˉμ+χσ)∇φ‖L2≤‖μ−ˉμ+χσ‖L6‖∇φ‖L3, | (4.14) |
and so if we consider the function h:=(−ΔN)−1(div((μ−ˉμ+χσ)∇φ)), then we obtain that
∫Ω∇h⋅∇ζ dx=∫Ω−(μ−ˉμ+χσ)∇φ⋅∇ζ dx∀ζ∈H1 | (4.15) |
must hold, and by (4.14) and the Poincaré inequality (1.9) with zero mean it holds that h∈H1∩L20. We now define f:=(−ΔN)−1(h)∈H2N, and consider testing with ζ=f in (4.15), leading to
∫Ω|h|2 dx=∫Ω∇h⋅∇f dx=∫Ω−(μ−ˉμ+χσ)∇φ⋅∇f dx. |
Since f∈H2N, elliptic regularity theory and Hölder's inequality gives
‖h‖2L2≤‖(μ−ˉμ+χσ)∇φ‖L65‖∇f‖L6≤C‖(μ−ˉμ+χσ)∇φ‖L65‖f‖H2≤C‖(μ−ˉμ+χσ)∇φ‖L65‖h‖L2, |
where the constant C depends on Ω and the constant in (4.11). Thus we obtain
‖(−ΔN)−1(div((μ−ˉμ+χσ)∇φ))‖L2≤C‖(μ−ˉμ+χσ)∇φ‖L65≤C(‖μ−ˉμ‖L6+χ‖σ‖L6)‖∇φ‖L32 | (4.16) |
for some constant C depending only on Ω. By the Sobolev embedding H1⊂L6 (with constant CSob that depends only on Ω) and the Poincaré inequality, we find that
‖μ−ˉμ‖L6≤CSob‖μ−ˉμ‖H1≤c(CSob,Cp)‖∇μ‖L2. | (4.17) |
Substituting the above elements into (4.12) yields (4.8).
Remark 4.1. We choose not to use the estimate
c‖h‖L2≤‖∇h‖L2≤‖(μ−ˉμ+χσ)∇φ‖L2 | (4.18) |
obtained from substituting ζ=h in (4.15), where c is a positive constant depending only on Cp, since by (4.14) we require control of ∇φ in the L3(Ω)-norm and this is not available when deriving the first a priori estimate. Thus, we make use of the auxiliary problem f=(−ΔN)−1(h) to derive another estimate on ‖h‖L2 that involves controlling ∇φ in the weaker L32(Ω)-norm.
Next, we state regularity estimates for the pressure and the velocity field. The hypothesis will be fulfilled for the Galerkin ansatz once we derived the a priori estimates in Section 4. Note that in Lemma 4.2 below, we consider a source term Γv∈L2(0,T;L20), so that our new regularity results for the pressure and the velocity is also applicable to the setting considered in [22].
Lemma 4.2. Let φ∈L∞(0,T;H1)∩L2(0,T;H2N∩H3), σ∈L2(0,T;H1), μ∈L2(0,T;H1), the source term Γv∈L2(0,T;L20), and the function p satisfying (4.7). Then,
‖p‖85L85(H1)≤C1‖φ‖65L∞(H1)‖μ+χσ‖85L2(H1)‖φ‖25L2(H3)+C1KT15‖Γv‖85L2(L2), | (4.19) |
for some positive constant C1 depending only on Ω, and
‖p‖87L87(H2)≤C2T37K−87‖Γv‖87L2(L2)+C2T27‖p‖87L85(H1)+C2‖φ‖27L∞(H1)‖μ+χσ‖87L2(H1)‖φ‖67L2(H3), | (4.20) |
for some positive constant C2 depending only on Ω. Moreover, if we have the relation
v=−K(∇p−(μ+χσ)∇φ), |
then
‖Dv‖87L87(L2)≤C3K‖p‖87L87(H2)+C3K‖μ+χσ‖87L2(H1)‖φ‖67L2(H3)‖φ‖27L∞(H1), | (4.21) |
for some positive constant C3 depending only on Ω.
Proof. From (4.7) we see that p satisfies ˉp = 0 and
∫Ω∇p⋅∇ζ dx=∫Ω(μ+χσ)∇φ⋅∇ζ+1KΓvζ dx∀ζ∈H1(Ω). |
Testing with ζ=p and applying the Hölder's inequality and the Poincaré inequality (1.9) gives
‖∇p‖L2≤‖(μ+χσ)∇φ‖L2+CpK‖Γv‖L2. | (4.22) |
Applying Hölder's inequality and the Sobolev embedding H1⊂L6 yields that
‖(μ+χσ)∇φ‖L2≤‖μ+χσ‖L6‖∇φ‖L3≤CSob‖μ+χσ‖H1‖∇φ‖L3. | (4.23) |
By the Gagliardo--Nirenberg inequality (1.10) with parameters j=0, p=3, r=2, m=2, d=3 and q=2,
‖∇φ‖L3≤C‖∇φ‖14H2‖∇φ‖34L2≤C‖φ‖14H3‖φ‖34H1, | (4.24) |
where C>0 is a constant depending only on Ω. Then, the boundedness of μ,σ in L2(0,T;H1) and φ in L2(0,T;H3)∩L∞(0,T;H1) leads to
∫T0‖(μ+χσ)∇φ‖85L2dt≤C∫T0‖μ+χσ‖85H1‖φ‖25H3‖φ‖65H1dt≤C‖φ‖65L∞(H1)‖μ+χσ‖85L2(H1)‖φ‖25L2(H3). |
By (4.22) we find that
∫T0‖∇p‖85L2dt≤∫T0‖(μ+χσ)∇φ‖85L2+CK‖Γv‖85L2dt≤C[φ]65L∞(H1)‖μ+χσ‖85L2(H1)‖φ‖25L2(H3)+CKT15‖Γv‖85L2(L2), |
where the positive constant C depends only on Ω. As ˉp = 0, by the Poincarö inequality (1.9), we see that
‖p‖85L85(H1)≤C‖φ‖65L∞(H1)‖μ+χσ‖85L2(H1)‖φ‖25L2(H3)+CKT15‖Γv‖85L2(L2), |
for some positive constant C depending only on Ω. Next, we see that
‖div((μ+χσ)∇φ)‖L2≤‖(μ+χσ)Δφ‖L2+‖∇(μ+χσ)⋅∇φ‖L2≤‖μ+χσ‖L6‖Δφ‖L3+‖∇(μ+χσ)‖L2‖∇φ‖L∞. |
By the Gagliardo-Nirenberg inequality (1.10), we find that
‖D2φ‖L3≤C‖φ‖34H3‖φ‖14L6≤C‖φ‖34H3‖φ‖14H1,‖∇φ‖L∞≤C‖φ‖34H3‖φ‖14L6≤C‖φ‖34H3‖φ‖14H1, | (4.25) |
and so, we have
‖div((μ+χσ)∇φ)‖L2≤C‖μ+χσ‖H1‖φ‖34H3‖φ‖14H1. | (4.26) |
That is, div((μ+χσ)∇φ)∈L2. Since by assumption Γv∈L20, using elliptic regularity theory, we find that p(t,⋅)∈H2 for a.e. t and there exists a constant C depending only on Ω, such that
‖p‖H2≤C(‖p‖H1+‖div((μ+χσ)∇φ)‖L2+K−1‖Γv‖L2). | (4.27) |
Furthermore, from (4.26), we see that
∫T0div((μ+χσ)∇φ)87L2dt≤C‖φ‖27L∞(H1)∫T0‖μ+χσ‖87H1‖φ‖67H3dt≤C‖φ‖27L∞(H1)‖μ+χσ‖87L2(H1)‖φ‖67L2(H3), |
and so for some positive constant C depending only on Ω, it holds that
∫T0‖p‖87H2dt≤CT37K−87‖Γv‖87L2(L2)+CT27‖p‖87L85(H1)+C‖φ‖27L∞(H1)‖μ+χσ‖87L2(H1)‖φ‖67L2(H3). | (4.28) |
For the velocity field v we estimate as follows. Let 1≤i,j≤3 be fixed, we obtain from (4.25),
‖Divj‖L2=K‖DiDjp−Di(μ+χσ)Djφ−(μ+χσ)DiDjφ‖L2≤K(‖p‖H2+‖∇(μ+χσ)‖L2‖∇φ‖L∞+‖μ+χσ‖L6‖D2φ‖L3)≤K(‖p‖H2+C‖μ+χσ‖H1‖φ‖34H3‖φ‖14H1). | (4.29) |
Applying the same calculation as in (4.28) yields
∫T0‖Dv‖87L2dt≤CK∫T0‖p‖87H2+‖μ+χσ‖87H1‖φ‖67H3‖φ‖27H1dt≤CK(‖p‖87L87(H2)+‖μ+χσ‖87L2(H1)‖φ‖67L2(H3)‖φ‖27L∞(H1)), |
for some positive constant C depending only on Ω.
To estimate the third source term
∫ΩΓv(pk−μkφk−D2|σk|2) dx=∫ΩΓv(pk−ˉμkφk+(μk−ˉμk)φk−D2|σk|2) dx |
of the energy equality we use Hölder’s inequality to obtain
|∫ΩΓvD2|σk|2+Γvφk(μk−ˉμk)dx|≤D2‖Γv‖L2‖σk‖2L4+‖Γv‖L32‖μk−ˉμ‖kL6‖φ‖kL6. |
By the Gagliardo--Nirenberg inequality (1.10) with j=0, r=2, m=1, p=4, q=2 and α=34, we have
‖σk‖2L4≤C‖σk‖32H1‖σk‖12L2=C(‖σk‖2L2+‖σk‖12L2‖∇σ‖k32L2). |
By Young's inequality with Hölder exponents (i.e., ab≤εpap+ε−q/pqbq for 1p+1q=1 and ε>0), we find that
D2‖Γv‖L2‖σk‖2L4≤C(‖Γv‖L2‖σk‖2L2+‖Γv‖4L2‖σk‖2L2)+n0D24‖∇σk‖2L2, |
for some positive constant C depending only in n0, D and Ω. Then, by (4.17) we have
|∫ΩΓvD2|σk|2+Γvφk(μk−¯μk)dx|≤n0D24‖∇σk‖2L2+C(1+‖Γv‖4L2)‖σk‖2L2+‖Γv‖L32c(Cp,CSob)‖∇μk‖L2‖φk‖H1≤n0D24‖∇σk‖2L2+C(1+‖Γv‖4L2)‖σk‖2L2+m08‖∇μk‖2L2+C‖Γv‖2L2φk2H1, |
where the positive constant C depends only on Ω, m0, n0 and D. Here we point out that the assumption Γv∈L4(0,T;L20) is needed. For the remainder term Γv(pk−¯μkφk), we find that
pk−¯μkφk=((−ΔN)−1(1KΓv−div((μk−¯μk+χσk)∇φk)−¯μkdiv∇(φk−¯φk)))−¯μkφk=((−ΔN)−1(1KΓv−div((μk−¯μk+χσk)∇φk)))−¯μk¯φk, |
where we used
(−ΔN)−1(−¯μkdiv∇(φk−¯φk))=¯μk(φk−¯φk). |
Then, by
∫ΩΓv¯φk¯μk dx=¯μk¯φk∫ΩΓv dx=0, |
it holds that
∫ΩΓv(pk−¯μkφk) dx=∫ΩΓv((−ΔN)−1(1KΓv−div((μk−¯μk+χσk)∇φk))). |
Applying the calculations in the proof of Lemma 4.1 (specifically (4.10), (4.16) and (4.17)), H¨older’s inequality and Young’s inequality, we find that
|∫ΩΓv(pk−¯μkφk)dx|≤CK‖Γv‖2L2+C‖Γv‖L2(‖∇μk‖L2+χ‖σk‖H1)‖∇φk‖L32≤CK‖Γv‖2L2+m08‖∇μk‖2L2+n0D24‖∇σk‖2L2+‖σk‖2L2+C(1+χ2)‖Γv‖2L2‖∇φk‖2L2, |
where C is a positive constant depending only on |Ω|, Cp, CSob, D, n0 and m0. Here we point out that if we applied (4.18) instead of (4.8) then we obtain a term containing ‖∇φ‖L3 on the right-hand side and this cannot be controlled by the left-hand side of (4.4). Using (2.4) we have
‖φk‖2L2≤1R1‖Ψ(φk)‖L1+R2R1|Ω|. | (4.30) |
Then, we obtain the following estimate
|∫ΩΓv(pk−μkφk−D2|σk|2)dx|≤C(1K‖Γv‖2L2+1)+n0D22‖∇σk‖2L2+m04‖∇μk‖2L2+C(1+‖Γv‖4L2)‖σk‖2L2+C‖Ψ(φk)‖L1+C(1+χ2)‖Γv‖2L2‖∇φk‖2L2, | (4.31) |
for some positive constant C depending only on R1, R2, Ω, m0, n0 and D. Here we point out that it is crucial for the source term Γv to be prescribed and is not a function of φ, μ and σ, otherwise the product term ‖Γv‖4L2‖σk‖2L2 and ‖Γv‖2L2‖∇φk‖2L2 cannot be controlled in the absence of any a priori estimates. For the remaining source term
∫ΩΓφ,kμk−SkNk,σ dx |
we split the analysis into two cases and combine with (4.31) to derive an energy inequality.
Suppose Θφ is non-negative and bounded, and Ψ is a potential that satisfies (2.5). We will estimate the mean of μk by setting j=1 in (3.2b), and using the growth condition (2.5) to obtain
|∫Ωμkdx|2=|∫ΩAΨ′(φk)−χσkdx|2≤2A2‖Ψ′(φk)‖2L1+2χ2‖σk‖2L1≤2A2R24(|Ω|+|Ω|12‖φk‖L2)2+2χ2|Ω|‖σk‖2L2≤C(A,R4,|Ω|)+4A2R24|Ω|‖σk‖2L2+2χ2|Ω|‖σk‖2L2. |
Then, by the Poincarö inequality (1.9) and the growth condition (2.4), we find that
‖μk‖2L2≤2C2P‖∇μk‖2L2+2|Ω||¯μk|2≤2C2p‖∇μk‖2L2+8A2R24‖φk‖2L2+4χ2‖σk‖2L2+C(A,R4,|Ω|)≤2C2p‖∇μk‖2L2+8A2R24R1‖Ψ(φk)‖L1+4χ2‖σk‖2L2+C(A,R4,R1,R2,|Ω|). | (4.32) |
Note that by the specific form (2.1) for Γφ we have that
Γφ,kμk=Λφ(φk,σk)μk−Θφ(φk,σk) |μk|2. |
Moving the non-negative term Θφ(φk,σk)|μk|2 to the left-hand side of (4.4) and subsequently neglecting it, we estimate the remainder using the growth condition (2.3) and Hölder's inequality as follows (here we use the notation Λφ,k:=Λφ(φk,σk)),
|∫ΩΛφ,kμk−Sk(Dσk+χ(1−φk))dx|≤‖Λφ,k‖L2‖μk‖L2+(‖ΛS,k‖L2+R0‖μk‖L2)‖Dσk+χ(1−φk)‖L2≤C(1+χ+(1+χ)‖φk‖L2+(1+D)‖σk‖L2)‖μk‖L2+C(1+‖φk‖L2+‖σ‖L2)(χ|Ω|12+D‖σ‖L2+χ‖φk‖L2) | (4.33) |
where C is a positive constant depending only on R0 and |Ω|. By Young's inequality, (4.32) and (4.30), we have
|∫ΩΛφ,kμk−Sk(Dσk+χ(1−φk))dx|≤m08C2p‖μk‖2L2+C(1+χ+D+χ2)‖φk‖2L2+C(1+χ+D)2‖σk‖2L2+C(1+χ+χ2)≤m04‖∇μk‖2L2+C(1+χ2)‖σk‖2L2+C(1+χ2)‖Ψ(φk)‖L1+C(1+χ2), | (4.34) |
for some positive constant C depending only on |Ω|, R0, R1, R2, R4, A, D, Cp and m0. Using the fact that
‖D∇σ‖L2≤‖∇(Dσ+χ(1−φ))‖L2+‖χ∇φ‖L2, |
we now estimate the right-hand side of (4.4) using (4.6), (4.31) and (4.34), which leads to
ddt∫ΩAΨ(φk)+B2|∇φk|2+D2|σk|2+χσk(1−φk)dx+m02‖∇μk‖2L2+n0D22‖∇σk‖2L2+1K‖vk‖2L2+Db2‖σk‖2L2(∂Ω)≤C(1+b)(1+χ2)‖Ψ(φk)‖L1+C(‖Γv‖2L2+b)(1+χ2)‖∇φk‖2L2+C(1+χ2+‖Γv‖4L2)‖σk‖2L2+C(1+b)(1+χ2)+CK‖Γv‖2L2+bC(1+χ2)‖σ∞‖2L2(∂Ω), | (4.35) |
for some positive constant C not depending on Γv, K, b and χ. Integrating (4.35) with respect to t from 0 to s∈(0,T] leads to
A‖Ψ(φk(s))‖L1+B2‖∇φk(s)‖2L2+D2‖σk(s)‖2L2+∫Ωχσk(s)(1−φk(s))dx+∫s0m02‖∇μk‖2L2+n0D22‖∇σk‖2L2+1K‖vk‖2L2+Db2‖σk‖2L2(∂Ω)dt≤∫s0C(1+b)(1+χ2)(1+‖Γv‖4L2)(‖Ψ(φk)‖L1+‖∇φk‖2L2+‖σk‖2L2)dt+C(1+b)(1+χ2)T+CK‖Γv‖2L2(0,T;L2)+Cb(1+χ2)‖σ∞‖2L2(0,T;L2(∂Ω))+C(‖Ψ(φ0)‖L1+‖φ0‖2H1+‖σ0‖2L2), | (4.36) |
for some positive constant C independent of Γv, K, χ and b. Here we used σ0∈L2 and φ0∈H1, which implies by the growth condition (2.5) that Ψ(φ0)∈L1. Next, by Hölder's inequality and Young's inequality we have
|∫Ωχσk(x,s)(1−φk(x,s))dx|≤2D8‖σk(s)‖2L2+2χ2|Ω|D+2χ2D‖φk(s)‖2L2≤D4‖σk(s)‖2L2+2χ2DR1‖Ψ(φk(s))‖L1+2χ2|Ω|D(1+R2). | (4.37) |
Substituting (4.37) into (4.36) then yields
min(A−2χ2DR1,B2,D4)(‖Ψ(φk(s))‖L1(Ω)+‖∇φk(s)‖2L2(Ω)+‖σk(s)‖2L2(Ω))+∫s0‖∇μk‖2L2+‖∇σk‖2L2+1K‖vk‖2L2+b2‖σk‖2L2(∂Ω)dt≤∫s0C(1+b)(1+χ2)(1+‖Γv‖4L2)(‖Ψ(φk)‖L1+‖∇φk‖2L2+‖σk‖2L2)dt+C(1+b)(1+χ2)(1+T)+CK‖Γv‖2L2(0,T;L2), | (4.38) |
for some positive constant C independent of Γv, K, b and χ. Setting
α:=C(1+b)(1+χ2)(1+T)+CK‖Γv‖2L2(0,T;L2),β:=C(1+b)(1+χ2)(1+‖Γv‖4L2)∈L1(0,T), | (4.39) |
and noting that
α(1+∫s0β(t)exp(∫t0β(r) dr) dt)≤α(1+‖β‖L1(0,T)exp(‖β‖L1(0,T)))<∞, |
we find after applying the Gronwall inequality (1.11) to (4.38) leads to
sups∈(0,T](‖Ψ(φk(s))‖L1+‖∇φk(s)‖2L2+‖σk(s)‖2L2)+∫T0‖∇μk‖2L2+‖∇σk‖2L2+1K‖vk‖2L2+b2‖σk‖2L2(∂Ω)dt≤E, | (4.40) |
where we recall that E denotes a constant that is uniformly bounded for b,χ∈(0,1] and is also uniformly bounded for K∈(0,1] when Γv=0.
Suppose Θφ satisfies (2.6) and Ψ is a potential satisfying the growth condition (2.7). Similar to the previous case, we see that the specific form for Γφ leads to
Γφ,kμk=Λφ(φk,σk)μk−Θφ(φk,σk) |μk|2. |
We move the term Θφ(φk,σk)|μk|2 to the left-hand side of (4.4) and estimate the remainder as in (4.33). Using Young's inequality differently and also (4.30), we have
|∫ΩΛφ,kμk−Sk(Dσk+χ(1−φk))dx|≤R52‖μk‖2L2+C(1+χ+D+χ2)‖φk‖2L2+C(1+χ+D)2‖σk‖2L2+C(1+χ+χ2)≤R52‖μk‖2L2+C(1+χ2)‖σk‖2L2+C(1+χ2)‖Ψ(φk)‖L1+C(1+χ2), | (4.41) |
for some positive constant C depending only on |Ω|, R5, R1, R2, A, D, and Cp. Using (4.6), (4.31), (4.41) and the lower bound Θφ≥R5, instead of (4.35) we obtain from (4.4)
ddt∫ΩAΨ(φk)+B2|∇φk|2+D2|σ|k2+χσk(1−φk)dx+R52‖μk‖2L2+m02‖∇μk‖2L2(Ω)+n0D22‖∇σk‖2L2+1K‖vk‖2L2+Db2‖σk‖2L2(∂Ω)≤C(1+b)(1+χ2)‖Ψ(φk)‖L1+C(‖Γv‖2L2+b)(1+χ2)‖∇φk‖2L2+C(1+χ2+‖Γv‖4L2)‖σk‖2L2+C(1+b)(1+χ2)+CK‖Γv‖2L2+Cb(1+χ2)‖σ∞‖2L2(∂Ω), | (4.42) |
for some positive constant C independent of Γv, K, b and χ. We point out the main difference between (4.35) and the above is the appearance of the term R52‖μk‖2L2 on the left-hand side. The positivity of Θφ allows us to absorb the ‖μk‖2L2 term on the right-hand side of (4.41) and thus we do not need to use (4.32), which was the main reason why Ψ has to be a quadratic potential for a non-negative Θφ. Then, applying a similar argument as in Section 4.1.3, we arrive at an analogous energy inequality to (4.40),
sups∈(0,T](‖Ψ(φk(s))‖L1+‖∇φk(s)‖2L2+‖σk(s)‖2L2)+∫T0‖μk‖2H1+‖∇σk‖2L2+1K‖vk‖2L2+b2‖σk‖2L2(∂Ω)dt≤E. | (4.43) |
Using (4.32) and (4.30) applied to (4.40), and similarly using (4.30) applied to (4.43) we obtain
sups∈(0,T](‖Ψ(φk(s))‖L1+‖φk(s)‖2H1+‖σk(s)‖2L2)+∫T0‖μk‖2H1+‖∇σk‖2L2+1K‖vk‖2L2+b2‖σk‖2L2(∂Ω)dt≤E. | (4.44) |
This a priori estimate implies that the Galerkin ansatz φk, μk, σk and vk can be extended to the interval [0,T]. To determine if pk can also be extended to the interval [0,T] we require some higher order estimates for φk in order to use (4.19).
Let Πk denote the orthogonal projection onto the finite-dimensional subspace Wk. From (3.2b) we may view φk as the solution to the following elliptic equation
−BΔu+u=μk−AΠk(Ψ′(u))+χσk+u in Ω, | (4.45a) |
∂nu=0 on ∂Ω. | (4.45b) |
For the case where Ψ satisfies (2.5), as {φk}k∈N is bounded in L∞(0,T;H1), we have that {Ψ′(φk)}k∈N is also bounded in L∞(0,T;H1). Using the fact that our basis functions {wi}i∈N are the eigenfunctions of the inverse Neumann-Laplacian operator and is therefore orthogonal in H1, and the Sobolev embedding H1⊂Lr for r∈[1,6], there exists a positive constant C independent of φk such that
‖Πk(Ψ′(φk))‖X≤C‖Ψ′(φk)‖X for X=H1 or Lr,1≤r≤6. | (4.46) |
Then, this implies that {Πk(Ψ′(φk))}k∈N is also bounded in L∞(0,T;H1). As the right-hand side of (4.45a) belongs to H1 for a.e. t∈(0,T), and the boundary ∂Ω is C3, by elliptic regularity theory, we have
‖φk‖L2(H3)≤C(1+‖φk‖L2(H1)+‖μk+χσk‖L2(H1))≤E, | (4.47) |
for some positive constant C depending only on Ω and R4. For the case where Ψ satisfies (2.7), we employ a bootstrap argument from [18, x3.3]. The growth assumption (2.7) implies that
|Ψ′(y)|≤C(1+|y|m), |Ψ″(y)|≤C(1+|y|m−1) for m∈[1,5). | (4.48) |
For fixed m∈[1,5), we define a sequence of positive numbers {lj}j∈N by
l1m≤6,lj+1=6lj6−(5−m)lj. |
It can be shown that {lj}j∈N is a strictly increasing sequence such that lj→∞ as j→∞. The Gagliardo--Nirenberg inequality (1.10) then yields the following continuous embedding
L2(0,T;W2,lj)∩L∞(0,T;L6)⊂L2m(0,T;Lmlj+1). | (4.49) |
At the first step, the boundedness of {φk}k∈N in L∞(0,T;H1) yields
‖Πk(Ψ′(φk))‖2Ll1≤C(1+‖φk‖2mL6), |
which implies that {Πk(Ψ′(φk))}k∈N is bounded in L2(0,T;Ll1). As the other terms on the right-hand side of (4.45) are bounded in L2(0,T;H1), elliptic regularity then yields that {φk}k∈N is bounded in L2(0,T;W2,l1), and thus in L2m(0,T;Lml2) by (4.49).
At the j-th step, we have {φk}k∈N is bounded in L2(0,T,W2,lj)∩L2m(0,T;Lmlj+1). Then, it holds that
‖Πk(Ψ′(φk))‖2L2(Llj+1)≤C(1+‖φk‖2mL2m(Lmlj+1)), |
and so {Πk(Ψ′(φk))}k∈N is bounded in L2(0,T;Llj+1). Elliptic regularity then implies that {φk}k∈N is bounded in L2(0,T;W2,lj+1).
We terminate the bootstrapping procedure once lj≥6 for some j∈N. This occurs after a finite number of steps as limj→∞lj=∞. Altogether, we obtain that {φk}k∈N is bounded in L2(0,T;W2,6). From (4.48) it holds that
‖Ψ″(φk)∇φk‖2≤C(1+|φk|2m−2)|∇φk|2 for m∈[1,5), |
and by the following continuous embeddings obtain from the Gagliardo-Nirenberg inequality (1.10),
L2(0,T;W2,6)∩L∞(0,T;H1)⊂L2m(0,T;L2m)∩L2m−2(0,T;L∞) for m∈[1,5), |
we find that {Πk(Ψ′(φk))}k∈N is bounded in L2(0,T;H1). Applying elliptic regularity once more leads to the boundedness of {φk}k∈N in L2(0,T;H3). Consequently, the hypotheses of Lemma 4.2 are satisfied and we obtain that
‖pk‖L85(H1)≤E, |
which implies that the Galerkin ansatz pk can be extended to the interval [0,T].
By the Gagliardo-Nirenberg inequality (1.10) with j=0, p=∞, m=3, r=2, q=2 and d=3, we have
‖φk‖L∞≤C‖φk‖14H3‖φk‖34L6≤C‖φk‖14H3‖φk‖34H1. |
For any ζ∈L83(0,T;H1) with coefficients {ζkj}1≤j≤k⊂Rk such that Πkζ=∑kj=1ζkjwj, we can estimate
|∫T0∫Ωφkvk⋅∇Πkζdx dt|≤∫T0‖vk‖L2‖φk‖L∞‖∇Πkζ‖L2dt≤C‖φk‖34L∞(H1)‖vk‖L2(L2)‖φk‖14L2(H3)‖ζ‖L83(H1). | (4.50) |
Using (4.44) and (4.47), we find that
‖div(φkvk)‖L85((H1)∗)≤K12E. | (4.51) |
Next, multiplying (3.2a) by ζkj, summing from j=1 to k and then integrating in time from 0 to T leads to
|∫T0∫Ω∂tφkζdx dt|≤∫T0m1‖∇μk‖L2‖∇Πkζ‖L2dt+∫T0‖Γφ,k‖L2‖Πkζ‖L2+‖div(φkvk)‖(H1)∗‖Πkζ‖H1dt. |
By (2.1), (2.2) and (4.44), we find that
‖Γφ,k‖L2(L2)≤C(R0,|Ω|,T)(1+‖φk‖L2(L2)+‖μk‖L2(L2)+‖σk‖L2(L2))≤E, |
and so, by Hölder’s inequality, we find that
|∫T0∫Ω∂tφkζ dx dt|≤(ET18+‖div(φkvk)‖L85((H1)∗))‖ζ‖L83(H1). |
Taking the supremum over ζ∈L83(0,T;H1) and using (4.44) and (4.51) yields that
‖∂tφk‖L85((H1)∗)≤E(1+K12), | (4.52) |
Similarly, by Hölder's inequality and the following Gagliardo--Nirenberg inequality (1.10) with j=0, r=2, m=1, p=103, q=2 and d=3,
‖f‖L103≤C‖f‖35H1‖f‖25L2, |
which in turn implies that {σk}k∈N is bounded uniformly in L103(Q). Then, we find that for any ζ∈L5(0,T;W1,5),
|∫T0∫Ωσkvk⋅∇Πkζdx dt|≤∫T0‖σk‖L103‖vk‖L2‖∇ζ‖L5dt≤‖σk‖L103(Q)‖vk‖L2(L2)‖∇ζ‖L5(L5), | (4.53) |
and
‖div(σkvk)‖L54((W1,5)∗)≤K12E. | (4.54) |
A similar calculation to (4.52) yields that
‖∂tσk‖L54((W1,5)∗)≤E(1+K12). | (4.55) |
Remark 4.2. We may also use the Gagliardo-Nirenberg inequality to deduce that
‖f‖Lr≤C‖f‖3(r−2)2rH1‖f‖6−r2rL2 for any r∈(2,6). |
Then, the computation (4.53) becomes
|∫T0∫Ωσkvk⋅∇Πkζ dx dt|≤C‖vk‖L2(L2)‖σk‖6−r2rL∞(L2)‖σk‖3(r−2)2rL2(H1)‖∇ζ‖L4r6−r(L2rr−2), |
which implies that {div(σkvk)}k∈N and {∂tσk}k∈N are bounded uniformly in
L4r5r−6(0,T;(W1,2rr−2)∗) for r∈(2,6). |
Note that the temporal exponent decreases while the spatial exponent increases as r increases, and they intersect at the point r=103
Here we point out that even with the improved regularity vk∈L87(0,T;H1), we are unable to show div(σkvk) is bounded in the dual space (H1)∗. Indeed, let q, r>1 be constants yet to be determined such that 1q+1r=12. Then, from Hölder's inequality we have
|∫T0∫Ωσkvk⋅∇Πkζ dx dt|≤∫T0‖σk‖Lq‖vk‖Lr‖∇ζ‖L2 dt. |
By the Gagliardo--Nirenberg inequality we have for α=32−3q≤1, β=32−3r≤1,
|∫T0∫Ωσkvk⋅∇Πkζdx dt|≤C∫T0‖σk‖1−αL2‖σk‖αH1‖vk‖βH1‖vk‖1−βL2‖∇ζ‖L2dt≤C‖σk‖1−αL∞(L2)∫T0‖σk‖αH1‖vk‖βH1‖vk‖1−βL2‖ζ‖H1dt≤C‖σk‖1−αL∞(L2)‖σk‖αLαx1(H1)‖vk‖βLβx2(H1)‖vk‖1−βL(1−β)x3(L2)‖ζ‖Lx4(H1), |
where
1x1+1x2+1x3+1x4=1,αx1≤2,βx2≤87,(1−β)x3≤2. | (4.56) |
Note that α=32−3q=3r, and then substituting into the constraints (4.56) we find that
1x1+1x2+1x3≥α2+78β+1−β2=32r+2116−218r+32r−14=1716+38r>1. | (4.57) |
Hence, we cannot find x1, x2, x3 and x4 satisfying (4.56) and we are unable to deduce that div(σkvk) lies in the dual space (H1)∗ even with the improved regularity vk∈L87(0,T;H1).
From (4.44), (4.47), (4.19), (4.20), (4.21), (4.51), (4.52), (4.54), (4.55), we find that
{φk}k∈Nbounded in L∞(0,T;H1)∩L2(0,T;H3),{∂tφk}k∈N,{div(φkvk)}k∈N bounded in L85(0,T;(H1)∗),{σk}k∈N bounded in L∞(0,T;L2)∩L2(0,T;H1)∩L2(Σ),{∂tσk}k∈N,{div(σkvk)}k∈N bounded in L54(0,T;(W1,5)∗),{μk}k∈N bounded in L2(0,T;H1),{pk}k∈N bounded in L85(0,T;H1)∩L87(0,T;H2),{vk}k∈N bounded in L2(0,T;L2)∩L87(0,T;H1). |
By standard compactness results (Banach-Alaoglu theorem and reflexive weak compactness theorem), and [28, x8, Cor. 4], and the compact embeddings in dimension 3 (see [1, Thm. 6.3] and [15, Thm. 11.2, p. 31])
Hj+1(Ω)=Wj+1,2(Ω)⊂⊂Wj,q(Ω)∀j≥0,j∈Z,1≤q<6, |
and the compact embedding L2⊂⊂(H1)∗, we obtain, for a relabelled subsequence, the following weak/weak-* convergences:
φk→φ weakly - ∗ in L∞(0,T;H1)∩L2(0,T;H3)∩W1,85(0,T;(H1)∗),σk→σ weakly - ∗ in L2(0,T;H1)∩L∞(0,T;L2)∩L2(Σ),∂tσk→∂tσ weakly in L54(0,T;(W1,5)∗),μk→μ weakly in L2(0,T;H1),pk→p weakly in L85(0,T;H1)∩L87(0,T;H2),vk→v weakly in L2(0,T;L2)∩L87(0,T;H1),div(φkvk)→ξ weakly in L85(0,T;(H1)∗),div(σkvk)→θ weakly in L54(0,T;(W1,5)∗), |
and the following strong convergences:
φk→φ strongly in C0([0,T];Lr)∩L2(0,T;W2,r) and a.e. in Q,σk→σ strongly in C0([0,T];(H1)∗)∩L2(0,T;Lr) and a.e. in Q, |
for any r∈[1,6) and some functions ξ∈L85(0,T;(H1)∗), θ∈L54(0,T;(W1,5)∗).
For the rest of this section, we fix 1≤j≤k and δ∈C∞c(0,T). Then, we have δ(t)wj∈C∞(0,T;H2). By continuity of m(⋅), we see that m(φk)→m(φ) a.e. in Q. Thanks to the boundedness of m(⋅), applying Lebesgue's dominated convergence theorem to (m(φk)−m(φ))2|δ∇wj|2 yields
‖m(φk)δ∇wj−m(φ)δ∇wj‖L2(Q)→0 as k→∞. |
Together with the weak convergence μk⇀μ in L2(0,T;H1), we obtain by the product of weak-strong convergence
∫T0∫Ωm(φk)δ∇wj⋅∇μk dx dt→∫T0∫Ωm(φ)δ∇wj⋅∇μ dx dt as k→∞. |
Terms involving n(⋅) can be dealt with in a similar fashion. For the source term Γφ,k=Λφ(φk,σk)−Θφ(φk,σk)μk, by the continuity and boundedness of Θφ, the a.e. convergence of φk→φ and σk→σ in Q, we may apply Lebesgue's dominated convergence theorem to deduce that
∫T0∫Ω|δwj(Θφ(φk,σk)−Θφ(φ,σ))|2 dx dt→0 as k→∞, |
that is, we obtain the strong convergence δwjΘφ(φk,σk)→δwjΘφ(φ,σ) in L2(Q). Hence, the weak convergence μk⇀μ in L2(0,T;H1) yields
∫T0∫ΩδwjΘφ(φk,σk)μk dx dt→∫T0∫ΩδwjΘφ(φ,σ)μ dx dt as k→∞. |
Meanwhile, by the triangle inequality ||a|−|b||≤|a−b|, and Hölder's inequality, we obtain
∫T0∫Ω|(|φk|−|φ|)(δwj)| dx dt≤‖φk−φ‖L2(0,T;L2)‖δwj‖L2(0,T;L2)→0 |
and
∫T0∫Ω|(|σk|−|σ|)(δwj)| dx dt≤‖σk−σ‖L2(0,T;L2)‖δwj‖L2(0,T;L2)→0 |
as k→∞. In particular, we have
(1+|φk|+|σk|)|δwj|→(1+|φ|+|σ|)|δwj| strongly in L1(Q) as k→∞. |
By the continuity of Λφ we have
Λφ(φk,σk)→Λφ(φ,σ) a.e. as k→∞,|Λφ(φk,σk)δwj|≤R0(1+|φk|+|σk|)|δwj|. |
which leads to
Then, the generalised Lebesgue dominated convergence theorem (see [27, Thm. 1.9, p. 89], or [2, Thm. 3.25, p. 60]) yields
Λφ(φk,σk)δwj→Λφ(φ,σ)δwj strongly in L1(Q) as k→∞, |
which leads to
∫T0∫ΩΓφ(φk,μk,σk)δwj dx dt→∫T0∫ΩΓφ(φ,μ,σ)δwj dx dt as k→∞. | (5.1) |
The same arguments can be applied for the source term S and for the derivative Ψ′(φ) satisfying the linear growth condition (2.5). For potentials satisfying the growth condition (2.7), we refer to the argument in [18, x3.1.2].
To identify the limits ξ and θ of the convection terms div(φkvk) and div(σkvk), respectively, we argue as follows. Since δwj∈C∞(0,T;H2)⊂L83(0,T;H1), by the weak convergence div(φkvk)⇀ξ in L85(0,T;(H1)∗), we have
∫T0∫Ωdiv(φkvk)δwj dx dt→∫T0⟨ξ,wj⟩H1,(H1)∗δ dt as k→∞. |
Next, applying integrating by parts and by the boundary conditions vk⋅n=0 on ∂Ω (see (3.8)), we see that
∫T0∫Ωdiv(φkvk)δwj dx dt=−∫T0∫Ωδφkvk⋅∇wj dx dt. | (5.2) |
Moreover, we claim that δφk∇wj converges strongly to δφ∇wj in L2(0,T;L2). Indeed, we compute
∫T0∫Ω|δ|2|∇wj|2|φk−φ|2dx dt≤∫T0|δ|2‖∇wj‖2L6‖φk−φ‖2L3dt≤‖wj‖2H2‖δ‖2L∞(0,T)‖φk−φ‖2L2(L3)→0 |
as k→∞ by the strong convergence φk→φ in L2(0,T;Lr) for r∈[1,6). Together with the weak convergence vk⇀v in L2(0,T;L2), when passing to the limit k→∞ in (5.2) we find that
∫T0⟨ξ,wj⟩H1,(H1)∗δ dt=−∫T0∫Ωδφv⋅∇wj dx dt. |
Applying integration by parts on the right-hand side shows that ξ=div(φv) in the sense of distributions.
Now considering δ(t)wj as an element in L5(0,T;W1,5), a similar argument can be used to show θ=div(σv) in the sense of distributions using the strong convergence σk→σ in L2(0,T;Lr) for r∈[1,6), the weak convergence vk⇀v in L2(0,T;L2), and the weak convergence div(σkvk)⇀ϕ in L54(0,T;(W1,5)∗).
For the pressure and the velocity, we apply −ΔN to both sides of (3.3) and test with wj, then integrating by parts leads to
∫Ω∇pk⋅∇wj dx=∫Ω1KΓvwj+(μk+χσk)∇φk⋅∇wj dx. |
Multiplying by δ(t), integrating in time and passing to the limit k→∞, keeping in mind the weak convergences pk⇀p in L85(0,T;H1), μk⇀μ in L2(0,T;H1), σk⇀σ in L2(0,T;H1), and the strong convergence φk→φ in L2(0,T;W2,r) for r∈[1,6) leads to
∫T0∫Ωδ(t)∇p⋅∇wj dx dt=∫T0∫Ωδ(t)(1KΓvwj+(μ+χσ)∇φ⋅∇wj)dx dt. | (5.3) |
Here we used that wj∈H2, and
∫T0∫Ω|δ|2|∇φk−∇φ|2|∇wj|2dx dt≤‖δ‖2L∞(0,T)‖wj‖2W1,6‖φk−φ‖2L2(W1,3)→0 as k→∞, | (5.4) |
to deduce that δ∇φk⋅∇wj→δ∇φ⋅∇wj in L2(0,T;L2). Fix 1≤j1,j2,j3≤k, and define ζ=(wj1,wj2,wj3)⊤. Then, we can consider δ(t)ζ as an element in L83(0,T;L2)⊂L2(0,T;L2). Taking the scalar product of (3.4) with δζ, integrating over Ω and in time from 0 to T leads to
∫T0∫Ωδ(t)(vk+K∇pk)⋅ζ dx dt=∫T0∫ΩδK(μk+χσk)∇φk⋅ζ dx dt. | (5.5) |
By the weak convergences vk⇀v in L2(0,T;L2), μk⇀μ in L2(0,T;H1), σk⇀σ in L2(0,T;H1), ∇pk⇀∇p in L85(0,T;L2), and the strong convergence δ∇φk⋅ζ→δ∇φ⋅ζ in L2(0,T;L2) (which is proved in a similar manner as (5.4)), we find that passing to the limit in (5.5) yields
∫T0∫Ωδ(t)(v+K∇p)⋅ζ dx dt=∫T0∫Ωδ(t)K(μ+χσ)∇φ⋅ζ dx dt. | (5.6) |
Then, multiplying (3.2) with δ∈C∞c(0,T), integrating with respect to time from 0 to T, and passing to the limit k→∞, we obtain
∫T0δ(t)⟨∂tφ,wj⟩H1,(H1)∗dt=∫T0∫Ωδ(t)(−m(φ)∇μ⋅∇wj+Γφwj+φv⋅∇wj)dx dt,∫T0∫Ωδ(t)μwjdx dt=∫T0∫Ωδ(t)(AΨ′(φ)wj+B∇φ⋅∇wj−χσwj)dx dt,∫T0δ(t)⟨∂tσ,wj⟩W1,5,(W1,5)∗dt=∫T0∫Ωδ(t)(−n(φ)(D∇σ−χ∇φ)⋅∇wj−Swj)dx dt+∫T0δ(t)(∫Ωσv⋅∇wj dx+∫Γb(σ∞−σ)wj dHd−1)dt. |
Since the above, (5.3) and (5.6) hold for all δ∈C∞c(0,T), we infer that {φ,μ,σ,p,v} satisfies (2.12) with ζ=ϕ=wj for a.e. t∈(0,T) and for all j≥1. As {wj}j∈N is a basis for H2N, and H2N is dense in both H1 and W1,5 (see Section 3), we see that {φ,μ,σ,p,v} satisfy (2.12a), (2.12b), (2.12d) for all ζ∈H1, (2.12c) for all ϕ∈W1,5, and (2.12e) for all ζ∈L2.
Attainment of initial conditions. It remains to show that φ and σ attain their corresponding initial conditions. Strong convergence of φk to φ in C0([0,T];L2), and the fact that φk(0)→φ0 in L2 imply that φ(0)=φ0. Meanwhile, as the limit function σ belongs to the function space C0([0,T];(H1)∗), we see that σ(0):=σ(⋅,0) makes sense as an element of (H1)∗. Let ζ∈H1 be arbitrary, then by the strong convergence σk→σ in C0([0,T];(H1)∗) we see that
⟨σk(0),ζ⟩H1,(H1)∗→⟨σ(0),ζ⟩H1,(H1)∗ as k→∞. |
On the other hand, by (3.7), we have σk(0)→σ0 in L2. This yields
⟨σ0,ζ⟩H1,(H1)∗=limk→∞⟨σk(0),ζ⟩H1,(H1)∗=⟨σ(0),ζ⟩H1,(H1)∗. |
Energy inequality. For the energy inequality (2.14) we employ the weak/weak-* lower semicontinuity of the norms and dual norms to (4.44), (4.47), (4.19), (4.20), (4.21), (4.51), (4.52), (4.54), and (4.55).
For K,b∈(0,1] let (φK,μK,σK,vK,pK) denote a weak solution to (1.1)-(1.2) with Γv=0, obtain from Theorem 2.1. By we deduce that, for a relabelled subsequence as b→0 and K→0, the following weak/weak-* convergences:
φK→φ weakly - ∗ in L∞(0,T;H1)∩L2(0,T;H3)∩W1,85(0,T;(H1)∗),σK→σ weakly - ∗ in L2(0,T;H1)∩L∞(0,T;L2)∩W1,54(0,T;(W1,5)∗),μK→μ weakly in L2(0,T;H1),pK→p weakly in L85(0,T;H1)∩L87(0,T;H2), |
and the following strong convergences:
φK→φ strongly in C0([0,T];Lr)∩L2(0,T;W2,r) and a.e. in Q,σK→σ strongly in C0([0,T];(H1)∗)∩L2(0,T;Lr) and a.e. in Q,vK→0 strongly in L2(0,T;L2)∩L87(0,T;H1),div(φKvK)→0 strongly in L85(0,T;(H1)∗),div(σKvK)→0 strongly in L54(0,T;(W1,5)∗), |
for any r∈[1,6). The strong convergence of the velocity and the convection terms to zero follows from (2.14). Upon multiplying (2.12) by δ∈C∞c(0,T) and passing to the limit b,K→0, we obtain that the limit functions (φ,μ,σ,p) satisfy
⟨∂tφ,ζ⟩H1,(H1)∗=∫Ω−m(φ)∇μ⋅∇ζ+Γφ(φ,μ,σ)ζ dx, | (6.1a) |
∫Ωμζ dx=∫ΩAΨ′(φ)ζ+B∇φ⋅∇ζ−χσζ dx, | (6.1b) |
⟨∂tσ,ϕ⟩W1,5,(W1,5)∗=∫Ω−n(φ)(D∇σ−χ∇φ)⋅∇ϕ−S(φ,μ,σ)ϕ dx | (6.1c) |
∫Ω∇p⋅∇ζ dx=∫Ω(μ+χσ)∇φ⋅∇ζ dx, | (6.1d) |
for all ζ∈H1 and ϕ∈W1,5 and a.e. t∈(0,T).
Note that substituting any ζ∈L2(0,T;H1) into (6.1a), integrating in time from 0 to T, using Hölder's inequality and the linear growth condition for Γφ leads to the deduction that ∂tφ∈L2(0,T;(H1)∗). To show that ∂tσ∈L2(0,T;(H1)∗) we argue as follows. For any ξ∈L2(0,T;H1), we can define
F(ξ):=∫T0∫Ω−n(φ)(D∇σ−χ∇φ)⋅∇ξ−S(φ,μ,σ)ξ dx dt. |
By Hölder's inequality and the growth condition on S, we see that F∈L2(0,T;(H1)∗). It is known that the set of functions that are finite linear combinations of C1c(0,T)⋅H2N(Ω):={δ(t)ϕ(x):δ∈C1c(0,T),ϕ∈H2N(Ω)} is dense in C1c(0,T;H1) (see for instance [26, p. 384], and in fact this is what we use in Section 5). Let ζ∈C1c(0,T;H1) and let {ζn}n∈N denote a sequence of functions of the above form such that ζn→ζ in C1c(0,T;H1) as n→∞. Then, substituting ϕ=ζn in (6.1c), integrating over t from 0 to T, and passing to the limit n→∞ yields
limn→∞∫T0⟨∂tσ,ζn⟩W1,5,(W1,5)∗ dt=limn→∞F(ζn)=F(ζ). |
Moreover, by the definition of the weak time derivative, we have
∫T0⟨∂tσ,ζn⟩W1,5,(W1,5)∗ dt=−∫T0∫Ωσ∂tζn dx dt→−∫T0∫Ωσ∂tζ dx dt as n→∞. |
Hence, we obtain
−∫T0∫Ωσ∂tζ dx dt=F(ζ)=∫T0∫Ω−n(φ)(D∇σ−χ∇φ)⋅∇ζ−S(φ,μ,σ)ζ dx dt |
for all ζ∈C1c(0,T;H1). This implies that the weak time derivative ∂tσ satisfies
∫T0∂tσζH1,(H1)∗ dt=F(ζ)∀ζ∈C1c(0,T;H1), |
and as F belongs to L2(0,T;(H1)∗), we find that ∂tσ also belongs to L2(0,T;(H1)∗). Furthermore, due to the improved regularity ∂tσ∈L2(0,T;(H1)∗), we use the continuous embedding
L2(0,T;H1)∩H1(0,T;(H1)∗)⊂C0([0,T];L2) |
to deduce that σ(0)=σ0.
For χ,b∈(0,1], let (φχ,μχ,σχ,vχ,pχ) denote a weak solution to (1.1)-(1.2) obtain from Theorem 2.1. By (2.14) we deduce that, for a relabelled subsequence as b→0 and χ→0, the following weak/weak-* convergences:
φχ→φ weakly - ∗ in L∞(0,T;H1)∩L2(0,T;H3)∩W1,85(0,T;(H1)∗),σχ→σ weakly - ∗ in L2(0,T;H1)∩L∞(0,T;L2)∩W1,54(0,T;(W1,5)∗),μχ→μ weakly in L2(0,T;H1),pχ→p weakly in L85(0,T;H1)∩L87(0,T;H2),vχ→v weakly in L2(0,T;L2)∩L87(0,T;H1),div(φχvχ)→div(φv) weakly in L85(0,T;(H1)∗),div(σχvχ)→div(σv) weakly in L54(0,T;(W1,5)∗), |
and the following strong convergences:
φχ→φ strongly in C0([0,T];Lr)∩L2(0,T;W2,r) and a.e. in Q,σχ→σ strongly in C0([0,T];(H1)∗)∩L2(0,T;Lr) and a.e. in Q, |
for any r∈[1,6). For any δ∈C∞c(0,T) and ζ∈H1, we have
|∫T0∫Ωδχσχζdx dt|≤χ‖σχ‖L2(L2)‖ζ‖L2‖δ‖L2(0,T)→0,|∫T0∫Ωδn(φχ)χ∇φχ⋅∇ζdx dt|≤n1χ‖∇φχ‖L2(L2)‖∇ζ‖L2‖δ‖L2(0,T)→0,|∫T0∫Ωδχσχ∇φχ⋅∇ζ dx dt|≤χ‖∇ζ‖L2‖σχ‖L2(L4)‖∇φχ‖L2(L4)‖δ‖L∞(0,T)→0, |
as χ→0. Thus, multiplying (2.12) with δ∈C∞c(0,T), and then passing to the limit b,χ→0, we see that (φ,μ,σ,v,p) satisfies
⟨∂tφ,ζ⟩H1,(H1)∗=∫Ω−m(φ)∇μ⋅∇ζ+Γφ(φ,μ,σ)ζ+φv⋅∇ζ dx, | (6.2a) |
∫Ωμζ dx=∫ΩAΨ′(φ)ζ+B∇φ⋅∇ζ dx, | (6.2b) |
⟨∂tσ,ϕ⟩W1,5,(W1,5)∗=∫Ω−n(φ)D∇σ⋅∇ϕ−S(φ,μ,σ)ϕ+σv⋅∇ϕ dx, | (6.2c) |
∫Ω∇p⋅∇ζ dx=∫Ω1KΓvζ+μ∇φ⋅∇ζ dx, | (6.2d) |
∫Ωv⋅ζ dx=∫Ω−K(∇p−μ∇φ)⋅ζ dx, | (6.2e) |
for all ζ∈H1, ϕ∈W1,5, ζ∈L2 and a.e. t∈(0,T).
We first derive an analogous result to Lemma 4.2 for two dimensions.
Lemma 7.1. Let Ω⊂R2 be a bounded domain with C3-boundary. Let φ∈L∞(0,T;H1)∩L2(0,T;H2N∩H3), σ∈L2(0,T;H1), μ∈L2(0,T;H1), the source term Γv∈L2(0,T;L20), and the function p satisfying (4.7). Then,
p∈Lk(0,T;H1)∩Lq(0,T;H2), v∈Lq(0,T;H1), |
for any
1≤k<2,1≤q<43. |
Proof. We estimate (4.23) differently than in the proof of Lemma 4.2. By Hölder’s inequality for any 1≤s<∞ we have
‖(μ+χσ)∇φ‖L2≤‖μ+χσ‖L2s‖∇φ‖L2ss−1. |
Then, by the Gagliardo--Nirenberg inequality (1.10) with p=2ss−1, j=0, r=2, m=2, d=2, q=2, and α=12−s−12s=12s, we find that
‖∇φ‖L2ss−1≤C‖∇φ‖12sH2‖∇φ‖1−12sL2≤C‖φ‖12sH3‖φ‖1−12sH1. |
Then, by Hölder's inequality and the Sobolev embedding H1⊂Lr for 1≤r<∞ in two dimensions, we have for w,y≥1,
∫T0‖(μ+χσ)∇φ‖wL2dt≤C∫T0‖μ+χσ‖wL2s‖∇φ‖wL2ss−1dt≤C‖φ‖w2s−12sL∞(H1)‖μ+χσ‖wLwy(H1)‖φ‖w2sLw2syy−1(H3). |
As μ,σ belong to L2(0,T;H1) and φ belongs to L2(0,T;H3)∩L∞(0,T;H1), we need
wy=2,w2syy−1=2⇒y=2s+12s,w=4s1+2s. |
Since w=4s1+2s<2 for all s∈[1,∞), and Γv∈L2(0,T;L20), the computations in the proof of Lemma 4.2 yields that
p∈Lk(0,T;H1) for 1≤k<2 |
Next, we see that
‖div((μ+χσ)∇φ)‖L2≤‖(μ+χσ)Δφ‖L2+‖∇(μ+χσ)⋅∇φ‖L2≤‖μ+χσ‖L2s‖Δφ‖L2ss−1+‖∇(μ+χσ)‖L2‖∇φ‖L∞. |
By the Gagliardo--Nirenberg inequality (1.10) with p=∞, j=0, r=2, m=2, d=2, q=2 and α=12, we have
‖∇φ‖L∞≤C‖∇φ‖12H2‖∇φ‖12L2≤C‖φ‖12H3‖φ‖12H1, | (7.1) |
and with p=2ss−1, j=1, r=2, m=2, d=2, q=2 and α=s+12s∈(12,1] for s∈[1,∞), we have
‖Δφ‖L2ss−1≤C‖∇φ‖s+12sH2‖∇φ‖s−12sL2≤C‖φ‖s+12sH3‖φ‖s−12sH1. | (7.2) |
Hence, for w,y,z≥1, we find that
∫T0‖div((μ+χσ)∇φ)‖wL2dt≤C‖φ‖w2L∞(H1)‖μ+χσ‖wLwz(H1)‖φ‖w2Lw2zz−1(H3)+C‖φ‖ws−12sL∞(H1)‖μ+χσ‖wLwy(H1)‖φ‖ws+12sLws+12syy−1(H3). |
Since s+12s≤1 for all s∈[1,∞), we require
wy=2,wy(s+1)2s(y−1)=2⇒y=3s+12s,w=4s1+3s. |
We choose z=3s+12s∈(32,2] so that
wz=2,w2zz−1=2s1+3s3s+1s+1=2ss+1∈[1,2), |
and thus we obtain
∫T0‖div((μ+χσ)∇φ)‖4s1+3sL2dt≤C‖φ‖2s1+3sL∞(H1)‖μ+χσ‖4s1+3sL2(H1)‖φ‖2s1+3sL2ss+1(H3)+C‖φ‖2s−21+3sL∞(H1)‖μ+χσ‖4s1+3sL2(H1)‖φ‖2s+21+3sL2(H3). |
From (4.27) and using the fact that 4s1+3s<4s1+2s for all s∈[1,∞), we see that
p∈Lq(0,T;H2) for 1≤q<43. |
Similarly, from (4.29), (7.1) and (7.2), we obtain for fixed 1≤i,j≤2, and any s∈[1,∞),
‖Divj‖L2=K‖DiDjp−(Di(μ+χσ)Djφ−(μ+χσ)DiDjφ‖L2≤K(‖p‖H2+‖∇(μ+χσ)‖L2‖∇φ‖L∞+‖μ+χσ‖L2s‖D2φ‖L2ss−1)≤K(‖p‖H2+C‖μ+χσ‖H1(‖φ‖12H3‖φ‖12H1+‖φ‖s+12sH3‖φ‖s−12sH1)). | (7.3) |
Then, a similar calculation shows that the right-hand side is bounded in L4s1+3s(0,T), which in turn implies that
v∈Lq(0,T;H1) for 1≤q<43. |
By the above new estimates we can show that div(φv) and ∂tφ have improved temporal regularity, and that div(σv) and ∂tσ belong to the dual space (H1)∗.
Lemma 7.2. For dimension d=2, let (φk,μk,σk,pk,vk) denote the Galerkin ansatz from Section 3 satisfying (4.44). Then, it holds that for 43≤w<2 and 1<r<87,
‖div(φkvk)‖Lw((H1)∗)+‖div(σkvk)‖Lr((H1)∗)≤K12E,‖∂tφk‖Lw((H1)∗)+‖∂tσk‖Lr((H1)∗)≤E(1+K12), |
where E denotes positive constants that are uniformly bounded for b,χ∈(0,1] and are also uniformly bounded for K∈(0,1] when Γv=0.
Proof. The assertions for ∂tφk and ∂tσk will follow via similar arguments in Section 4.3 once we establish the assertion for the convection terms. In dimension d=2, we have the embedding L2(0,T;H1)∩L∞(0,T;L2)⊂L4(Q), and by the Gagliardo--Nirenberg inequality (1.10) with p=4, j=0, r=2, d=2, m=1, q=2 and α=12,
‖f‖L4≤C‖f‖12H1‖f‖12L2. |
Consider an arbitrary ζ∈Ls(0,T;H1) for some s≥1 yet to be determined. Then, we compute that
|∫T0∫Ωσkvk⋅∇Πkζdx dt|≤∫T0‖σk‖L4‖vk‖L4‖∇ζ‖L2dt≤C‖σk‖L4(Q)(∫T0‖vk‖23H1‖vk‖23L2‖ζ‖43H1dt)34≤C‖σk‖L4(Q)‖vk‖12L23x1(H1)‖vk‖12L23x2(L2)‖ζ‖L43x3(H1), |
where x1,x2,x3≥1 satisfy
1x1+1x2+1x3=1,23x1<43,23x2≤2⇒x3>6. |
Then, from (4.44) and (7.3), it holds that
|∫T0∫Ωσkvk⋅∇Πkζ dx dt|≤EK12‖ζ‖Ls(H1) for s=43x3>8, |
that is, {div(σkvk)}k∈N is uniformly bounded in the dual space of Ls(0,T;H1) for s>8. Similarly, by the Gagliardo--Nirenberg inequality (1.10) with p=∞, j=0, r=2, d=2, m=3, q∈[1,∞) and α=1q+1,
‖φk‖L∞≤C‖φk‖1q+1H3‖φk‖qq+1Lq≤C‖φk‖1q+1H3‖φk‖qq+1H1. |
Proceeding as in (4.50), we find that for an arbitrary ζ∈Ls(0,T;H1), where s≥1 is yet to be determined,
|∫T0∫Ωφkvk⋅∇Πkζdx dt|≤∫T0‖vk‖L2‖φk‖L∞‖∇Πkζ‖L2dt≤C‖φk‖qq+1L∞(H1)‖vk‖L2(L2)‖φk‖1q+1L2(H3)‖ζ‖L2(q+1)q(H1)≤EK12‖ζ‖L2(q+1)q(H1), |
and so {div(φkvk)}k∈N is uniformly bounded in the dual space of Ls(0,T;H1) for s=2+2q∈(2,4].
Remark 7.1. We point out that in the absence of the regularity result vk∈Lq(0,T;H1) from Lemma 7.1, and if we only have vk∈L2(0,T;L2), then we obtain
|∫T0∫Ωσkvk⋅∇Πkζ dx dt|≤‖σk‖L4(Q)‖vk‖L2(L2)‖∇ζ‖L4(L4), |
and this implies that both {div(σkvk)}k∈N and {∂tσk}k∈N are bounded uniformly only in L43(0,T;(W1,4)∗).
Reformulations of Darcy’s law and the pressure. Associated to Darcy's law (1.1b) is the term λv:=p−μφ−D2|σ|2 which will contribute the source term Γvλv in the energy identity (4.4). In [20, Rmk. 2.1] three other reformulations of Darcy's law (1.1b) and the pressure are considered:
(R1) Let q:=p−AΨ(φ)−B2|∇φ|2 so that
λv=q+AΨ(φ)+B2|∇φ|2−D2|σ|2+χσ(1−φ)−μφ, | (8.1a) |
v=K(∇(−q−B2|∇φ|2)−BΔφ∇φ)=−K(∇q+Bdiv(∇φ⊗∇φ)). | (8.1b) |
(R2) Let ˆp:=p+D2|σ|2+χσ(1−φ) so that
λv=ˆp−μφ−D|σ|2−χσ(1−φ), | (8.2a) |
v=−K(∇ˆp−μ∇φ−(Dσ+χ(1−φ))∇σ). | (8.2b) |
(R3) Let ˜p:=p−D2|σ|2−μφ so that
λv=˜p, | (8.3a) |
v=−K(∇˜p+φ∇μ+σ∇(Dσ+χ(1−φ))). | (8.3b) |
From the viewpoint of estimating the source term Γvλv, we see that (8.3a) has the advantage of being the simplest. Meanwhile, for (8.2a) the analysis for Γvλv is similar to that performed in Section 4.1.2, but for (8.1a) the main difficulty will be to estimate the terms (AΨ(φ)+B2|∇φ|2)Γv and (−D2|σ|2+χσφ)Γv, which at first glance would require the assumption that Γv∈L∞(Q), and obtaining an L2-estimate for the pressure q from the Darcy law (8.1b) would be difficult due to the term div(∇φ⊗∇φ).
Other boundary conditions for the pressure and velocity. In [20, x2.4.4] the authors have discussed possible boundary conditions for the velocity and for the pressure. As discussed in Section 2 following Assumption 2.1, we require the source term Γv to have zero mean due to the no-flux boundary condition v⋅n=0 on ∂Ω. The general energy identity (with homogeneous Neumann boundary conditions for φ and μ) from [20, Equ. (2.27)] reads as
ddt∫ΩAΨ(φ)+B2|∇φ|2+D2|σ|2+χσ(1−φ)dx+∫Ωm(φ)|∇μ|2+n(φ)|∇(Dσ+χ(1−φ))|2+1K|v|2dx=∫ΩΓφμ−S(Dσ+χ(1−φ))+Γvλvdx+∫∂Ω(Dσ+χ(1−φ))n(φ)(D∂nσ)−(v⋅n)(D2|σ|2+χσ(1−φ)+p) dHd−1, |
and we see the appearance of an extra boundary source term involving the normal component of the velocity and the pressure. Here it would be advantageous to use the rescaled pressure ˆp and the Darcy law (8.2b), as the extra boundary source term will become
∫∂Ω−(v⋅n)ˆp dHd−1, |
which motivates the consideration of a Robin-type boundary condition for ˆp
g=aˆp−v⋅n=aˆp+K∂nˆp−K(Dσ+χ(1−φ))∂nσ on ∂Ω, |
for some given datum g and positive constant a. On one hand, this would allow us to consider source terms Γv that need not have zero mean, but on the other hand, the analysis of the Darcy system becomes more complicated. In particular, the weak formulation of the pressure system now reads as
∫ΩK∇ˆp⋅∇ζdx+∫∂ΩaˆpζdHd−1=∫ΩΓvζ+K(μ∇φ+(Dσ+χ(1−φ))∇σ)⋅∇ζdx+∫∂ΩgζdHd−1, |
and we observe that the term Dσ∇σ on the right-hand side belongs to L1 as σ has at most H1-spatial regularity from the energy identity. Thus, it is not clear if the pressure system can be solved with the regularities stated in Lemma 4.1. A deeper study into the theory of linear elliptic equations with right-hand sides of the form divf where f∈L1 is required.
All authors declare no conflicts of interest in this paper.
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