On a non-isothermal Cahn-Hilliard model for tumor growth
aa r X i v : . [ m a t h . A P ] F e b On a non-isothermal Cahn-Hilliard modelfor tumor growth
Erica Ipocoana ∗ March 1, 2021
Abstract
We introduce here a new diffuse interface thermodynamically con-sistent non-isothermal model for tumor growth in presence of a nu-trient in a domain Ω ⊂ R . In particular our system describes thegrowth of a tumor surrounded by healthy tissues, taking into accountchanges of temperature, proliferation of cells, nutrient consumptionand apoptosis. Our aim consists in proving an existence result forweak solutions to our model. Keywords:
Cahn-Hilliard, non-isothermal, tumor growth, weak solutions,existence.
MSC 2020:
The study of tumor growth processes has become of great interest also formathematicians in recent years [1, 2, 6, 7, 21, 25, 26]. Indeed, mathematicalmodels might be able to give further insights in tumor growth behaviour. Inparticular, the framework of diffuse interface modeling with Cahn-Hilliardequations [5] has received increasing attention. In this context, the tumoris seen as an expanding mass surrounded by healthy tissues. Its evolutionis assumed to be governed by mechanisms such as proliferation of cells vianutrient consumption, apoptosis [12, 19, 23] and, in more complex models ∗ University of Modena and Reggio Emilia, Dipartimento di Scienze Fisiche, Infor-matiche e Matematiche, via Campi 213/b, I-41125 Modena (Italy).E-mail: [email protected] ϕ t = ∆ µ + ( P σ − A ) h ( ϕ ) (1.1) µ = − ε ∆ ϕ + 1 ε F ′ ( ϕ ) − θ − χ ϕ σ (1.2) θ t + θϕ t − div ( κ ( θ ) ∇ θ ) = |∇ µ | (1.3) σ t = ∆ σ − C σh ( ϕ ) + B ( σ B − σ ) . (1.4)We carry out our analysis in Ω × (0 , ∞ ), where Ω ⊂ R is a smooth domain.We complement the above PDE system with homogeneus Neumann bound-ary conditions for all the unknowns.The evolution of the tumor is described by the order parameter ϕ whichrepresents the local concentration of tumor cells, ϕ ∈ [ − , { ϕ = 1 } representing the tumor phase and { ϕ = − } the healthy one. Moreover µ denotes the chemical potential of phase transition from healthy to tumorcells, θ is the absolute temperature, κ ( θ ) represents the heat conductivityand ε is a small parameter related to the thickness of interfacial layers. Wedenote by σ the concentration of a nutrient consumed (only) by the tumorcells (e.g. oxygen and glucose). The parameter χ ϕ ≥ P , A , C B indicate respectively the tumor proliferation rate, apoptosis rate, nu-trient consuption rate and nutrient supply rate. The function h is chosen asmonotone increasing, nonnegative in [ − ,
1] and such that h ( −
1) = 0 and h (1) = 1. The tumor growth is thus described by the term P σh ( ϕ ), whichreasonably increases proportionally to the concentration of tumor cells, whilethe death of tumor cells is modelled by the term A h ( ϕ ). Therefore, accordingto (1.1), if P σ − A >
0, then the tumor expands and it happens faster whenthe concentration of tumor cells is already high. If otherwise P σ − A < C σh ( ϕ ) represents the consumptionof the nutrient by the tumor cells. The term B ( σ − σ B ) is due to the factthat we consider here the case where the tumor has its own vasculature(as in e.g. [4], [23]), where the threshold σ B ∈ (0 ,
1) is the constant nutri-ent concentration in the pre-existing vasculature. In particular, if σ B > σ , B ( σ B − σ ) models the supply of nutrient from the blood vessels, on the otherhand if σ B < σ , B ( σ B − σ ) represents the transport of nutrient away fromthe domain. Eventually, the function F ( s ) represents a polynomial potentialhaving at least cubic growth at infinity, whose assumptions will be specifiedin Section 3.2. A simple choice might be a double-well potential with equalminima at s = ± | ϕ | from its naturalvalue 1. This more general potential allows ϕ to take values also outside ofthe significance interval [ − , | ϕ | > h . We also remarkthat although among Cahn-Hilliard literature the singular potentials, suchas logarithm type (see e.g. [11]), are very common, the growth conditionsthat the problem requires make them unsuitable for our case, as it will beclear in Section 4.In this work we derive a new phase field model according to the laws of ther-modynamics describing the tumor growth. The novelty of this contributionis to include possible variations of temperature in the model. The presenceof nutrient concentration σ in the system implies that here the spatial meanof ϕ is not conserved in time (as we can see from equation (1.1)), thereforethe derivation of the model cannot follow the standard techniques proposede.g. in [9]. However we are able to gain enough regularity for the quadruple( ϕ, µ, θ, σ ) in order to prove the existence of weak solutions to the initial-boundary value problem associated to (1.1)–(1.4).The structure of this paper is the following. In Section 2 we derive sys-tem (1.1)–(1.4) according to the approach proposed by Gurtin in [16]. Thenwe proceed with the mathematical analysis of our problem in the case ε =1 , χ ϕ = 0. In particular, Section 3 is devoted to give the setting and topresent the main result of this work (which is Theorem 3.1) concerning the3xistence of weak solutions to our problem. The proof is carried out in twosteps. In Section 4 we gain a priori bounds for ( ϕ, µ, θ, σ ). In Section 5 weuse the weak sequential stability argument to prove the existence of weaksolutions. Namely, we exploit the a priori bounds obtained for a sequenceof weak solutions together with standard compactness results to pass to thelimit. We suppose that a two-component mixture consisting of healthy cells andtumor cells occupies an open spatial domain Ω ⊂ R . We denote by ϕ ( x, t )the tumor phase concentration, θ ( x, t ) is the absolute temperature and σ ( x, t )is the concentration of a nutrient for the tumor cells. According to theGinzburg-Landau theory for phase transitions, we postulate the free energydensity ψ in the form ψ = ε |∇ ϕ | + 1 ε F ( ϕ ) + f ( θ ) − θϕ + N ( ϕ, σ ) . (2.1)Here, ε is a positive constant depending on the interface thickness. Thefunction F represents a polynomial potential having at least cubic growthat infinity. The easiest choice is taking F ( ϕ ) = ( | ϕ | − , known in theliterature as the double-well potential .The term f in (2.1) describes the part of free energy which is purely caloricand is related to the specific heat c V ( θ ) = Q ′ ( θ ) through relation Q ( θ ) = f ( θ ) − θf ′ ( θ ). In the following we assume the specific heat c V ≡
1. Moreoverwe recall that it holds q = Qθ (2.2)where q denotes the heat flux. Eventually, the latter term N in equation (2.1)describes both the chemical energy of the nutrient and the energy contribu-tions given by the interactions between the tumor tissues and the nutrient.One of the main difficulties we have to afford in the derivation of our modelis that, differently from standard Cahn-Hilliard models (such as [9]), the spa-tial mean of the tumor phase concentration ϕ is not conserved. Indeed thetumor may grow or shrink according to the right hand side of (1.1). In orderto deal with this issue, we follow Gurtin’s approach (used e.g. in [22, 24])proposed in [16], namely we treat separately the balance laws and the con-stitutive relations, moreover we introduce the following new balance law forinternal microforces div ζ + π = 0 , (2.3)4here ζ is a vector representing the microstress and π is a scalar correspond-ing to the internal microforces. We remark that here we neglect the externalactions.The mass balance law reads ϕ t = − div h + m (2.4)where h is the mass flux and m is the external mass supply. Moreover theinternal energy density of the system is given by e = ψ + θs. (2.5)Here, s denotes the entropy of the system, which has the following expression,according to (2.1) s = − ∂ψ∂θ = − f ′ ( θ ) + ϕ. (2.6)Combining the two previous formulas we infer ∂e∂t = ∂ψ∂t + θ ∂s∂t + s ∂θ∂t = ∂ψ∂ϕ ∂ϕ∂t + ∂ψ∂ ∇ ϕ ∂ ∇ ϕ∂t + θ ∂s∂t (2.7)and consequently ∂ψ∂t + s ∂θ∂t = ∂ψ∂ϕ ∂ϕ∂t + ∂ψ∂ ∇ ϕ ∂ ∇ ϕ∂t . (2.8) Cahn-Hilliard system
The derivation of our system is based on the two fundamental laws of ther-modynamics. According to [16], we write the first law in the form ddt Z R e d x = − Z ∂ R q · ν d η + W ( R ) + M ( R ) (2.9)where R is the control volume, ν is the outward unit normal to ∂R and W ( R ) = Z ∂ R ( ζ · ν ) ∂ϕ∂t d η M ( R ) = − Z ∂ R µh · ν d η + Z R µm d x are the rate of working and the rate at which free energy is added to R (as-suming no heat supply) respectively. Using Green’s formula, we can rewrite(2.9) as ∂e∂t = − div q + ∂ϕ∂t div ζ + ζ · ∇ ∂ϕ∂t − h ∇ µ − µ div h + µm. (2.10)5ince the control volume R is arbitary, exploiting the mass balance (2.4) andthe microforce balance (2.3), we infer ∂e∂t = − div q + ( µ − π ) ∂ϕ∂t + ζ · ∇ ∂ϕ∂t − h ∇ µ. (2.11)We now impose the validity of the second law of thermodinamics in the formof the Clausius-Duhem inequality θ (cid:18) ∂s∂t + div Q (cid:19) ≥ . (2.12)We develop the left hand side of (2.12) as follows θ (cid:18) ∂s∂t + div Q (cid:19) (2.5) = ∂e∂t − ∂ψ∂t − s ∂θ∂t + θ div Q (2.2) = ∂e∂t − ∂ψ∂t − s ∂θ∂t + div q − Q · ∇ θ (2.11) = ( µ − π ) ∂ϕ∂t + ζ · ∇ ∂ϕ∂t − h ∇ µ − ∂ψ∂t − s ∂θ∂t − Q · ∇ θ (2.8) = (cid:18) µ − π − ∂ψ∂ϕ (cid:19) ∂ϕ∂t + (cid:18) ζ − ∂ψ∂ ∇ ϕ (cid:19) ∂ ∇ ϕ∂t − h ∇ µ − Q · ∇ θ In order to satisfy relation (2.12), we impose µ − π − ∂ψ∂ϕ = 0 , (2.13) ζ = ∂ψ∂ ∇ ϕ , (2.14) h ∇ µ + Q · ∇ θ ≤ , (2.15)where in particular in order for (2.15) to hold, we exploited Fourier’s law q = − κ ( θ ) ∇ θ, (2.16)with κ = κ ( θ ) > ζ = ε ∇ ϕ (2.17)which leads to, according to (2.1) and (2.13), µ = − ε ∆ ϕ + 1 ε F ′ ( ϕ ) − θ + ∂N∂ϕ . (2.18)6ventually, inequality (2.15) can be satisfied choosing h = −∇ µ , which is asuitable assumption according to [16]. Therefore equation (2.4) reads ϕ t = ∆ µ + m. (2.19) Temperature equation.
We start from the internal energy equation (2.11), taking advantage of (2.14)and of the expression for the chemical potential (2.13), therefore ∂e∂t = − div q + ∂ψ∂ϕ ∂ϕ∂t + ∂ψ∂ ∇ ϕ ∂ ∇ ϕ∂t − h ∇ µ. Now, exploiting the assumption h = −∇ µ and Fourier’s law (2.16), we infer ∂e∂t = div ( κ ( θ ) ∇ θ ) + ∂ψ∂ϕ ∂ϕ∂t + ∂ψ∂ ∇ ϕ ∂ ∇ ϕ∂t + |∇ µ | and by identity (2.7), θ ∂s∂t − div ( κ ( θ ) ∇ θ ) = |∇ µ | . Recalling the definition of Q and relation (2.6), we finally get the expectedequation θ t + θϕ t − div ( κ ( θ ) ∇ θ ) = |∇ µ | . (2.20) Nutrient equation.
We postulate the nutrient balance equation in the form σ t = − div J − S , (2.21)where J is the nutrient flux and S denotes a source/sink term for the nutrient.Motivated by [15], we choose J = −∇ σ , therefore equation (2.21) reads σ t = ∆ σ − S . (2.22) Owing to [4, 15, 23], we now make the following constitutive assumptions: m = ( P σ − A ) h ( ϕ ) , (2.23) ∂N∂ϕ = − χ ϕ σ, (2.24) S = C σh ( ϕ ) − B ( σ B − σ ) , (2.25)7here h ( ϕ ) is a monotone increasing, nonnegative function in [ − ,
1] andsuch that h ( −
1) = 0 and h (1) = 1. Hence equation (2.23) states that onone hand the tumor growth is proportional to the nutrient supply in thetumoral region. This assumption reflects the fact that it often happens thattumors bring mutations which switch off certain growth inhibiting proteins.Therefore the tumor cells increasing is limited only by the supply of nutrients,despite of healthy cells where the mitotic cycle regulates the growth. Onthe other hand, when we are in the healthy region, equation (2.23) showsthat the proliferation rate of the tumor is greater than the one of healthycells. Equation (2.24) is due to the mechanism of chemotaxis. According toequation (2.25), we here assume that the sink/source of nutrient is regulatedby consumption of nutrients and the term B ( σ B − σ ) which models the factthat we here consider the case in which the tumor has its own vasculature.In particular the threshold σ B indicates whether the nutrient is supplied tothe tumor or transported away. In this section we present the main result of this work, concerning the exis-tence of solutions for the tumor growth model (1.1)–(1.4). We carry out theanalysis for χ ϕ = 0 and ε = 1. In order to carry out a mathematical analysis of our problem, let us introducesome notation we will use in the sequel. We recall that Ω is a smooth domainof R and we denote by Γ its boundary. For sake of semplicity, let us assume | Ω | = 1. We denote by (0 , T ) an assigned but otherwise arbitrary timeinterval. We set H := L (Ω) and V := H (Ω) and we will use these symbolsalso referring to vector valued functions. The symbol ( · , · ) will indicate thestandard scalar product in H , while h· , ·i will stand for the duality between V ′ and V . We denote by k · k X the norm in the generic Banach space X .For brevity we will write k · k instead of k · k H . Still for brevity, we omitthe variables of integration. We will specify them when there could be amisinterpretation.For any function v ∈ V , we define v Ω := 1 | Ω | Z Ω v = Z Ω v, (3.1)8here the last equality holds since we assumed | Ω | = 1.We recall the Poincar´e-Wirtinger inequality k v − v Ω k ≤ c Ω k∇ v k ∀ v ∈ V (3.2)and the non-linear Poincar´e inequality k v p k V ≤ c p (cid:16) k v k pL (Ω) + k∇ v p k (cid:17) , (3.3)which holds ∀ v ∈ L (Ω) s.t. ∇ v p ∈ L (Ω) and ∀ p ∈ [2 , ∞ ). We assume the coefficients P , A , B and C to be strictly positive and σ B ∈ (0 , F ∈ C loc ( R , R )decomposes as a sum of a monotone part β and a linear perturbation, namely F ′ ( r ) = β ( r ) − λr λ ≥ , r ∈ R . (3.4)Moreover we normalize β s.t. β (0) = 0 and we require ∃ c β > | β ( r ) | ≤ c β (1 + F ( r )) ∀ r ∈ R , (3.5)lim inf | r |→∞ F ( r ) | r | > , (3.6)where (3.5) means that F has at most an exponential growth at infinity,while (3.6) states a minimal growth condition at infinity for the potential.Moreover, according to (3.5), we suppose min F = 0.Next, we assume h ∈ C ( R ) increasingly monotone s.t.i) h ( −
1) = 0 , h ( r ) ≡ ∀ r ≥ ∃ h ≥ ϕ ≤ − h ( r ) ≡ − h ∀ r ≤ ϕ .Therefore h is globally Lipschitz continuous and there exists a constant c > | h ( r ) | + | h ′ ( r ) | ≤ c ∀ r ∈ R . (3.7)Moreover we assume the thermal conductivity to depend on the absolutetemperature θ as follows κ ( θ ) = 1 + θ q , q ∈ [2 , ∞ ) , θ ≥ . (3.8)9ventually, we require the initial data to be such that ϕ | t =0 = ϕ , ϕ ∈ V, F ( ϕ ) ∈ L (Ω) θ | t =0 = θ , θ ∈ L (Ω) , θ > , log θ ∈ L (Ω) σ | t =0 = σ , σ ∈ L ∞ (Ω) , ≤ σ ≤ σ is due to the interpretation of σ as a nutrientconcentration. We also recall that we couple our system with homogeneusNeumann boundary conditions for all the unknowns. Theorem 3.1.
Suppose that the assumptions in Section 3.2 hold and let
T > . Then there exists at least one weak solution to our model, namely aquadruple ( ϕ, µ, θ, σ ) with regularity ϕ ∈ C ([0 , T ]; V ) ∩ H (0 , T ; V ′ ) ∩ L (0 , T ; H (Ω)) β ( ϕ ) ∈ L (0 , T ; H ) µ ∈ L (0 , T ; V ) θ ∈ L (0 , T ; V ) ∩ L ∞ (0 , T ; L (Ω)) ∩ L q (0 , T ; L q (Ω)) , q ≥ , θ > a.e. in Ω σ ∈ C ([0 , T ]; H ) ∩ H (0 , T ; V ′ ) ∩ L ∞ (0 , T ; L ∞ (Ω)) ∩ L (0 , T ; V ) satisfying system (1.1) – (1.4) in the following weak sense h ϕ t , ξ i = − Z Ω ∇ µ · ∇ ξ + Z Ω ( P σ − A ) h ( ϕ ) ξ a.e. in (0 , T ) and ∀ ξ ∈ V,µ = − ∆ ϕ + F ′ ( ϕ ) − θ a.e. in (0 , T ) and a.e. in Ω − Z T Z Ω θξ t − Z Ω θ ξ ( · ,
0) + Z Ω θ ( · , T ) ξ ( · , T ) + Z T Z Ω θϕ t ξ + Z T Z Ω ( κ ( θ ) ∇ θ ) · ∇ ξ = Z T Z Ω |∇ µ | ξ ∀ ξ ∈ C ∞ ([0 , T ] × Ω) , h σ t , ξ i = − Z Ω ∇ σ · ∇ ξ − Z Ω C σh ( ϕ ) ξ + Z Ω B ( σ B − σ ) ξ a.e. in (0 , T ) and ∀ ξ ∈ V and complying a.e. in Ω with the initial conditions (3.9) . This section is devoted to gain the suitable regularity for the quadruple( ϕ, µ, θ, σ ) to prove the existence of solutions in Section 5. These a priori10ounds are obtained formally, working on our system (1.1)-(1.4). We remarkthat the existence (of weak solutions) argument might be made rigorous bythe Faedo-Galerkin method that we decided not to detail here.
We first search for a priori bounds for the nutrient, following [23]. Testing(1.4) by − σ − , where σ − ≥ σ ,exploting the initial conditions on σ and applying the Gronwall lemma, wegain σ ( t, x ) ≥ t ≥ , x ∈ Ω . Now, testing (1.4) by ( σ − ¯ σ ) + (where ¯ σ ≥ h and σ B , it is possible to obtain k σ k L ∞ (0 ,T ; L ∞ (Ω)) ≤ c T , (4.1)where c T is a constant depending on time. We test (1.1) by µ , (1.2) by ϕ t and (1.3) by 1 and then sum up. This yields,taking into account the boundary conditions, ddt (cid:18) k∇ ϕ k + Z Ω F ( ϕ ) + Z Ω θ (cid:19) = Z Ω ( P σ − A ) h ( ϕ ) µ. (4.2)We take care of the right hand side, in particular Z Ω ( P σ − A ) h ( ϕ ) µ (1.2) = − Z Ω ( P σ − A ) h ( ϕ )∆ ϕ + Z Ω ( P σ − A ) h ( ϕ ) F ′ ( ϕ ) − Z Ω ( P σ − A ) h ( ϕ ) θ (3.4) = Z Ω ( P σ − A ) h ′ ( ϕ ) |∇ ϕ | + Z Ω P h ( ϕ ) ∇ σ · ∇ ϕ + Z Ω β ( ϕ )( P σ − A ) h ( ϕ )+ Z Ω λϕ ( A − P σ ) h ( ϕ ) + Z Ω ( A − P σ ) h ( ϕ ) θ. ddt (cid:18) k∇ ϕ k + Z Ω F ( ϕ ) + Z Ω θ (cid:19) = Z Ω ( P σ − A ) h ′ ( ϕ ) |∇ ϕ | + Z Ω ( P σ − A ) β ( ϕ ) h ( ϕ )+ Z Ω λ ( A − P σ ) h ( ϕ ) ϕ + P Z Ω h ( ϕ ) ∇ σ · ∇ ϕ + Z Ω ( A − P σ ) h ( ϕ ) θ := I + II + III + IV. (4.3)
We now estimate each term on the right hand side separately. The estimateon the nutrient (4.1) is a key point for all these bounds. Exploting theassumption (3.7), we infer I ≤ c T k∇ ϕ k . (4.4)According to (3.5) it is straightforward that II ≤ c T (cid:18) Z Ω F ( ϕ ) (cid:19) . (4.5)Moreover, using once again the assumption (3.7) on h and Young’s inequality,we get III ≤ k∇ σ k + c T (cid:0) k ϕ k L (Ω) + k∇ ϕ k (cid:1) . (4.6)Eventually, by the same tools used to estimate III , it holds IV ≤ c T k θ k L (Ω) . (4.7)Combining estimates (4.4)–(4.7), (4.3) reads ddt (cid:18) k∇ ϕ k + Z Ω F ( ϕ ) + Z Ω θ (cid:19) (4.8) ≤ k∇ σ k + c T (cid:18) k ϕ k L (Ω) + k∇ ϕ k + Z Ω F ( ϕ ) + k θ k L (Ω) (cid:19) Our aim is to apply Gronwall’s lemma in order to gain the energy estimate.Therefore we estimate and reabsorb the term k ϕ k L (Ω) according to (3.6).Moreover we test (1.4) by σ which yields12 ddt k σ k + k∇ σ k ≤ c (1 + k σ k ) . (4.9)12ence, summing this last estimate to (4.8) we finally get ddt (cid:18) k∇ ϕ k + Z Ω F ( ϕ ) + Z Ω θ + 12 k σ k (cid:19) + 12 k∇ σ k ≤ c T (cid:18) k∇ ϕ k + Z Ω F ( ϕ ) + k θ k L (Ω) + k σ k (cid:19) . (4.10)We are now able to apply Gronwall’s lemma to (4.8), therefore we obtain thefollowing a priori estimates k∇ ϕ k L ∞ (0 ,T ; H ) ≤ c T (4.11) k F ( ϕ ) k L ∞ (0 ,T ; L (Ω)) ≤ c T (4.12) k θ k L ∞ (0 ,T ; L (Ω)) ≤ c T (4.13) k σ k L ∞ (0 ,T ; H ) ∩ L (0 ,T ; V ) ≤ c T . (4.14)In particular, combining (3.5) and (3.6) with (4.12), we gain k ϕ k L ∞ (0 ,T ; L (Ω)) ≤ c T (4.15) We now derive the entropy estimate testing (1.3) by − θ . Therefore ddt Z Ω ( − log θ − ϕ ) + Z Ω θ |∇ µ | + Z Ω (cid:0) |∇ log θ | + k q |∇ θ q/ | (cid:1) = 0 , (4.16)where k q > q ≥ | r | ≤ r − log r ∀ r >
0, we infer k log θ k L ∞ (0 ,T ; L (Ω)) + k log θ k L (0 ,T ; V ) ≤ c T , (4.17) k∇ θ q/ k L (0 ,T ; H ) ≤ c T . (4.18)Then, combining (3.3) with (4.13) and (4.18), it holds k θ q k L (0 ,T ; V ) ≤ c T (4.19)which implies in particular, since q ≥ k θ k L (0 ,T ; V ) ≤ c T . (4.20)On the other hand, using Sobolev embedding theorems, (4.19) also implies k θ q k L (0 ,T ; L (Ω)) ≤ c T and hence k θ k L q (0 ,T ; L q (Ω)) ≤ c T . (4.21)13 .4 Chemical potential estimate Integrating (1.3) over Ω we infer k∇ µ k = ddt Z Ω θ + Z Ω θϕ t . (4.22)We now rewrite the latter term according to (1.1), then using (3.7) and (4.1),it follows that (4.22) reads12 k∇ µ k ≤ ddt Z Ω θ + 12 k∇ θ k + c T k θ k L (Ω) . (4.23)Thus from (4.13) and (4.20), we obtain k∇ µ k L (0 ,T ; H ) ≤ c T . (4.24)Now we integrate (1.2) over Ω, then | µ Ω | (3.1) , (3.4) = (cid:12)(cid:12)(cid:12) Z Ω ( β ( ϕ ) − λϕ ) − Z Ω θ (cid:12)(cid:12)(cid:12) (4.25) ≤ Z Ω | β ( ϕ ) | + Z Ω | λϕ | + k θ k L (Ω) (4.26) (3.5) , (4.13) ≤ c β (cid:18) Z Ω F ( ϕ ) (cid:19) + c T . (4.27)Using now the bound (4.12), we get k µ Ω k L ∞ (0 ,T ) ≤ c T . (4.28)Combining this last bound with the Poincar´e inequality (3.2) and the previ-ous estimate (4.24), we achieve k µ k L (0 ,T ; V ) ≤ c T . (4.29) ϕ -dependent estimates We start testing (1.1) by ϕ , which leads to ddt Z Ω | ϕ | = − Z Ω ∇ µ · ∇ ϕ + Z Ω ( P σ − A ) h ( ϕ ) ϕ. (4.30)Exploiting Young’s inequality, the uniform bounds on h and (4.1) we infer ddt k ϕ k ≤ k∇ µ k + 12 k∇ ϕ k + c T k ϕ k L (Ω) . k ϕ k L ∞ (0 ,T ; H ) ≤ c T , whence estimate (4.11) gives k ϕ k L ∞ (0 ,T ; V ) ≤ c T . (4.31)Next we test (1.2) by β ( ϕ ) and we obtain Z Ω | β ( ϕ ) | + Z Ω β ′ ( ϕ ) |∇ ϕ | = Z Ω µβ ( ϕ ) + Z Ω λϕβ ( ϕ ) + Z Ω θβ ( ϕ )Now, from (4.29), (4.31), (4.20) and the monotonicity of β , it follows k β ( ϕ ) k L (0 ,T ; H ) ≤ c T . (4.32)Taking advantage of this last estimate with (3.5) and again of (4.31) and(4.20), equation (1.2) yields k ϕ k L (0 ,T ; H ) ≤ c T (4.33) We start testing (1.1) by a nonzero test function v ∈ V and we infer h ϕ t , v i = − Z Ω ∇ µ · ∇ v + Z Ω ( P σ − A ) h ( ϕ ) v. Now, according to estimates (4.1), (4.24) and (4.31) it follows k ϕ t k L (0 ,T ; V ′ ) ≤ c T . (4.34)Taking advantage of this last estimate and exploting (4.33) together with(4.31), we infer (for example from [20]) ϕ ∈ C ([0 , T ]; V ) . (4.35)Similarly, multiplying equation (1.4) by a nonzero test function v ∈ V and exploiting the bound (4.14), it holds k σ t k L (0 ,T ; V ′ ) ≤ c T . (4.36)From standard embedding results (see e.g. [3]), combining (4.36) and (4.14),we gain the additional regularity for the nutrient σ ∈ C ([0 , T ]; H ) . (4.37)15 Weak sequential stability
We assume to have a sequence of weak solutions ( ϕ n , µ n , θ n , σ n ) which sat-isfies the a priori estimates obtained in Section 4 uniformly with respect to n . We then show that, up to the extraction of a subsequence, ( ϕ n , µ n , θ n , σ n )converges in a suitable way to a quadruple ( ϕ, µ, θ, σ ) solving (1.1)–(1.4) inthe sense of Theorem 3.1.Indeed, exploiting the above bounds (4.1), (4.12)–(4.20), (4.29), (4.31)–(4.36), together with standard weak compactness results, it is possible toextract a nonrelabelled subsequence of n such that ϕ n → ϕ weakly star in L ∞ (0 , T ; V ) ∩ L (0 , T ; H (Ω)) ∩ H (0 , T ; V ′ ) (5.1) µ n → µ weakly in L (0 , T ; V ) (5.2) θ n → θ weakly star in L (0 , T ; V ) ∩ L ∞ (0 , T ; L (Ω)) ∩ L q (0 , T ; L q (Ω)) (5.3) σ n → σ weakly star in L ∞ (0 , T ; L ∞ (Ω)) ∩ L (0 , T ; V ) ∩ H (0 , T ; V ′ ) (5.4) Moreover combining (4.34) and (4.36) with (5.1) and (5.4) respectively andapplying the Aubin-Lions lemma, we infer that ϕ n → ϕ and σ n → σ stronglyin L (0 , T ; H ). Moreover convergence (5.3) and Sobolev embeddings implythat θ converges strongly in L p (0 , T ; L p (Ω)) p ∈ [1 , q + ]. Therefore it ispossible to pass to the limit also in the nonlinear terms, according to thecontinuity of κ, β and h . Indeed, by a generalized version of Lebesgue’sdominated convergence theorem it holds κ ( θ n ) → κ ( θ ) strongly in L p (0 , T ; L p (Ω)) , p ∈ h , q + 23 q (cid:17) β ( ϕ n ) → β ( ϕ ) weakly in L (0 , T ; H ) . This concludes the procedure and so the proof of existence of weak solutions.
Remark . We notice that we have assumed throughout the proof that theabsolute temperature is a.e. positive. This is crucial in order for estimates inSection 4 to make sense. In particular it should be shown that the solution θ n of the discretized problem (for instance in a Faedo-Galerkin scheme, thatwe decided not to detail here) is positive. At least, according to (4.17) it isreasonable to think that the strict positivity of θ n should be preserved a.e.in (0 , T ) × Ω also in the limit.
Acknowledgements
The author is supported by GNAMPA (Gruppo Nazionale per l’Analisi Matem-atica, la Probabilit`a e le loro Applicazioni) of INdAM (Istituto Nazionale diAlta Matematica) and by MIUR through the project FFABR (M. Eleuteri).16 eferences [1] R.P. Araujo and D.L.S. McElwain, A History of the Study of SolidTumour Growth: The Contribution of Mathematical Modelling,
Bull.Math. Biol. (2004), 1039–1091.[2] N. Bellomo, N.K. Li, and P.K. Maini, On the foundations of cancermodelling: selected topics, speculations, and perspectives, Math. Mod-els Methods Appl. Sci. , (4) (2008), 593–646.[3] F. Brezzi and G. Gilardi, Functional Analysis, Functional Spaces, Par-tial Differential Equations in: H. Kardestuncer and Norrie eds., FiniteElement Handbook, McGraw-Hill Book Co., New York, (1987). Chap-ters 1-3, pp. 1-121 of Part 1.[4] H.M. Byrne and M.A.J. Chaplain, Growth of Nonnecrotic Tumors inthe Presence and Absence of Inhibitors, Mathematical Biosciences , (1995), 151–181.[5] J. Cahn and J. Hilliard, Free energy of a nonuniform system. I. Inter-facial free energy, J. Chem. Phys. , (1958), 258–267.[6] L. Cherfils, S. Gatti, A. Miranville and R. Guillevin, Analysis of amodel for tumor growth and lactate exchanges in a glioma, Discreteand Continuous Dynamical Systems - S , doi: 10.3934/dcdss.2020457.[7] V. Cristini and J. Lowengrub,
Multiscale modeling of cancer. An Inte-grated Experimental and Mathematical Modeling Approach,
CambridgeUniv. Press, Cambridge, (2010).[8] M. Eleuteri, S. Gatti and G. Schimperna, Regularity and long-timebehavior for a thermodynamically consistent model for complex fluidsin two space dimensions,
Indiana Univ. Math. J. (5) (2019), 1465–1518.[9] M. Eleuteri, E. Rocca and G. Schimperna, On a non-isothermal diffuseinterface model for two phase flows of incompressible fluids, DCDS (2015), 2497–2522.[10] M. Eleuteri, E. Rocca and G. Schimperna, Existence of solutions toa two-dimensional model for nonisothermal two-phase flows of incom-pressible fluids,
Ann. Inst. H. Poincare Anal. Non Lineaire (2016),1431–1454. 1711] S. Frigeri and M. Grasselli, Nonlocal Cahn-Hilliard-Navier-Stokes sys-tems with singular potentials, arXiv DOI: 10.4310/DPDE.2012.v9.n4.a1(2012).[12] S. Frigeri, M. Grasselli and E. Rocca, On a diffuse interface model oftumor growth, Eur. J. Appl. Math. , (2015) 215–243.[13] H. Garcke and K.F. Lam, Well-posedness of a Cahn-Hilliard systemmodelling tumour growth with chemotaxis and active transport, Eur.J. Appl. Math. (2017), 284–316.[14] H. Garke, K.F. Lam, R. N¨urnberg and E. Sitka, A multiphase Cahn-Hilliard-Darcy model for tumour growth with necrosis, Math. ModelsMethods Appl. Sci. (2018), 525–577.[15] H. Garke, K.F. Lam, E. Sitka and V. Styles, A Cahn-Hilliard-Darcymodel for tumour growth with chemotaxis and active transport, Math.Models Methods Appl. Sci. (6) (2016), 1095–1148.[16] M.E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equa-tions based on a microforce balance, Physica D (1996), 178–192.[17] A. Hawkins-Daarud, S. Prudhomme, K.G. van der Zee and J.T. Oden,Bayesian calibration, validation, and uncertainty quantification of dif-fuse interface models of tumor growth, Journal of Mathematical Biology (6) (2013), 1457–1485.[18] E. Ipocoana and A. Zafferi, Further regularity and uniqueness resultsfor a non-isothermal Cahn-Hilliard equation, Comm. Pure Appl. Anal.
DOI: 10.3934/cpaa.2020289, (2020).[19] J. Jiang, H. Wu and S. Zheng, Well-posedness and long-time behav-ior of a non-autonomous Cahn-Hilliard-Darcy system with mass sourcemodeling tumor growth,
J. Differ. Equ. (2015), 3032–3077.[20] J.L. Lions and E. Magenes,
Non-Homogeneous Boundary Value Prob-lems and Applications: Vol. 1 , Springer (1971).[21] J. Lowengrub, E. Titi and K. Zhao, Analysis of a mixture model oftumor growth,
European J. Appl. Math. (2013), 1–44.[22] A. Marveggio and G. Schimperna, On a non-isothermal Cahn-Hilliardmodel based on a microforce balance, arXiv:2004.02618 [math.AP] (2020). 1823] A. Miranville, E. Rocca and G. Schimperna, On the long time behaviorof a tumor growth model, J. Differential Equations (2019), 2616–2642.[24] A. Miranville and G. Schimperna, Nonisothermal phase separationbased on a microforce balance,
Discrete Contin. Dynam. Systems Ser.B , (2005), 753-768.[25] J.T. Oden, A. Hawkins and S. Prudhomme, General diffuse-interfacetheories and an approach to predictive tumor growth modeling, Math.Models Methods Appl. Sci. (2010), 477–517.[26] X. Wu, G.J. van Zwieten and K.G. van der Zee, Stabilized second-orderconvex splitting schemes for Cahn-Hilliard models with applications todiffuse-interface tumor-growth models, Int. J. Numer. Meth. Biomed.Engng.30