Effective interface conditions for a porous medium type problem

Motivated by biological applications on tumour invasion through thin membranes, we study a porous-medium type equation where the density of the cell population evolves under Darcy's law, assuming continuity of both the density and flux velocity on the thin membrane which separates two domains. The drastically different scales and mobility rates between the membrane and the adjacent tissues lead to consider the limit as the thickness of the membrane approaches zero. We are interested in recovering the effective interface problem and the transmission conditions on the limiting zero-thickness surface, formally derived by Chaplain et al. (2019), which are compatible with nonlinear generalized Kedem-Katchalsky ones. Our analysis relies on a priori estimates and compactness arguments as well as on the construction of a suitable extension operator which allows to deal with the degeneracy of the mobility rate in the membrane, as its thickness tends to zero.

where the subscript indicates that (•) is evaluated as the limit to a point of the interface coming from the subdomain Ω1 , Ω3 , respectively.
Motivations and previous works.Nowadays, a huge literature can be found on the mathematical modeling of tumour growth , see, for instance, [27,30,32,36], on a domain Ω ⊆ R d (with d = 2, 3 for in vitro experiments, d = 3 for in vivo tumours).Studying tumour's evolution, a crucial and challenging scenario is represented by cancer cells invasion through thin membranes.
In particular, one of the most difficult barriers for the cells to cross is the basement membrane.This kind of membrane separates the epithelial tissue from the connective one (mainly consisting in extracellular matrix, ECM), providing a barrier that isolates malignant cells from the surrounding environment.At the early stage, cancer cells proliferate locally in the epithelial tissue originating a carcinoma in situ.Unfortunately, cancer cells could mutate and acquire the ability to migrate by producing matrix metalloproteinases (MMPs), specific enzymes which degrade the basement membrane, allowing cancer cells to penetrate into it, invading the adjacent tissue.A specific study can be done on the relation between MMP and their inhibitors as in Bresch et al. [35].Instead, we are interested in modeling cancer transition from in situ stage to the invasive phase.This transition is described both by System (1) and (2).In fact, for the both of them, the left domain can be interpreted as the domain in which the primary tumor lives, whereas the one on the right is the connective tissue.Between them, the basal membrane is penetrated by cancer cells either with a mobility coefficient (in the case of a nonzero thickness membrane) or with particular membrane conditions, in the case of a zero-thickness interface.Since in biological systems the membrane is often much smaller than the size of the other components, it is then convenient and reasonable to approximate the membrane as a zero-thickness one, as done in [10,15], differently from [35].In particular, it is possible to mathematically describe cancer invasion through a zero thickness interface considering a limiting problem defined on two domains.The system is then closed by transmission conditions on the effective interface which generalise the classical Kedem-Katchalsky conditions.The latter were first formulated in [20] and are used to describe different diffusive phenomena, such as, for instance, the transport of molecules through the cell/nucleus membrane [9,12,38], solutes absorption processes through the arterial wall [34], the transfer of chemicals through thin biological membranes [8], or the transfer of ions through the interface between two different materials [2].In our description, the transmission conditions define continuity of cells density flux through the effective interface Γ1,3 and their proportionality to the jump of a term linked to cells pressure.The coefficient of proportionality is related to the permeability of the effective interface with respect to a specific population.
For these reasons, studying the convergence as the thickness of the membrane tends to zero represents a relevant and interesting problem both from a biological and mathematical point of view.In the literature, this limit has been studied in different fields of applications other than tumour invasion, such as, for instance, thermal, electric or magnetic conductivity, [24,37], or transport of drugs and ions through an heterogeneous layer, [29].Physical, cellular and ecological applications characterised the bulk-surface model and the dynamical boundary value problem, derived in [25] in the context of boundary adsorption-desorption of diffusive substances between a bulk (body) and a surface.Another class of limiting systems is offered by [23], in the case in which the diffusion in the thin membrane is not as small as its thickness.Again, this has a very large application field, from thermal barrier coatings (TBCs) for turbine engine blades to the spreading of animal species, from commercial pathways accelerating epidemics to cell membrane.
As it is now well-established, see for instance [7], living tissues behave like compressible fluids.Therefore, in the last decades, mathematical models have been more and more focusing on the fluid mechanical aspects of tissue and tumour development, see for instance [3,6,7,10,17,30].Tissue cells move through a porous embedding, such as the extra-cellular matrix (ECM).This nonlinear and degenerate diffusion process is well captured by filtration-type equations like the following, rather than the classical heat equation, Here F (u) represents a generic density-dependent reaction term and the model is closed with the velocity field equation and a density-dependent law of state for the pressure p := f (u).The function µ = µ(t, x) ≥ 0 represents the cell mobility coefficient and the velocity field equation corresponds to the Darcy law of fluid mechanics.This relation between the velocity of the cells and the pressure gradient reflects the tendency of the cells to move away from regions of high compression.
Our model is based on the one by Chaplain et al. [10], where the authors formally recover the effective interface problem, analogous to System (2), as the limit of a transmission problem, (or thin layer problem) cf.System (1), when the thickness of the membrane converges to zero.They also validate through simulations the numerical equivalence between the two models.When shrinking the membrane Ω 2,ε to an infinitesimal region, Γ1,3 , (i.e. when passing to the limit ε → 0, where ε is proportional to the thickness of the membrane), it is important to guarantee that the effect of the thin membrane on cell invasion remains preserved.To this end, it is essential to make the following assumption on the mobility coefficient in the subdomain Ω 2,ε , This condition implies that, when shrinking the pores of the membrane, the local permeability of the layer decreases to zero proportionally with respect to the local shrinkage.The function μ1,3 represents the effective permeability coefficient of the limiting interface Γ1,3 , i.e. the permeability of the zero-thickness membrane.We refer the reader to [10,Remark 2.4] for the derivation of the analogous assumption in the case of a fluid flowing through a porous medium.In [10], the authors derive the effective transmission conditions on the limiting interface, Γ1,3 , which relates the jump of the quantity Π := Π(u), defined by Π (u) = uf (u) and the normal flux across the interface, namely These conditions turns out to be the well-known Kedem-Katchalsky interface conditions when f (u) := ln(u), for which Π(u) = u + C, C ∈ R, i.e. the linear diffusion case.
In this paper, we provide a rigorous proof to the derivation of these limiting transmission conditions, for a particular choice of the pressure law.To the best of our knowledge, this question has not been addressed before in the literature for a non-linear and degenerate model such as System (1).Although our system falls into the class of models formulated by Chaplain et al., we consider a less general case, making some choices on the quantities of interest.First of all, for the sake of simplicity, we assume the mobility coefficients µ i,ε to be positive constants, hence they do not depend on time and space as in [10].We take a reaction term of the form uG(p), where G is a pressure-penalized growth rate.Moreover, we take a power-law as pressure law of state, i.e. p = u γ , with γ ≥ 1.Hence, our model turns out to be in fact a porous medium type model, since Equations (4, 5) read as follows The nonlinearity and the degeneracy of the porous medium equation (PME) bring several additional difficulties to its analysis compared to its linear and non-degenerate counterpart.In particular, the main challenge is represented by the emergence of a free boundary, which separates the region where u > 0 from the region of vacuum.On this interface the equation degenerates, affecting the control and the regularity of the main quantities.For example, it is well-known that the density can develop jumps singularities, therefore preventing any control of the gradient in L 2 , opposite to the case of linear diffusion.On the other hand, using the fundamental change of variables of the PME, p = u γ , and studying the equation on the pressure rather than the equation on the density, turns out to be very useful when searching for better regularity of the gradient.Nevertheless, since the pressure presents "corners" at the free boundary, it is not possible to bound its laplacian in L 2 (uniformly on the entire domain).For these reasons, we could not straightforwardly apply some of the methods previously used in the literature in the case of linear diffusion.For instance, the result in [5] is based on proving H 2 -a priori bounds, which do not hold in our case.The authors consider elliptic equations in a domain divided into three subdomains, each one contained into the interior of the other.The coefficients of the second-order terms are assumed to be piecewise continuous with jumps along the interior interfaces.Then, the authors study the limit as the thickness of the interior reinforcement tends to zero.In [37], Sanchez-Palencia studies the same problem in the particular case of a lense-shaped region, I ε , which shrinks to a smooth surface in the limit, facing also the parabolic case.The approach is based on H 1 -a priori estimates, namely the L 2 -boundedness of the gradient of the unknown.Considering the variational formulation of the problem, the author is able to pass to the limit upon applying an extension operator.In fact, if the mobility coefficient in I ε converges to zero proportionally with respect to ε, it is only possible to establish uniform bounds outside of I ε .The extension operator allows to "truncate" the solution and then "extend" it into I ε reflecting its profile from outside.Therefore, making use of the uniform control outside of the ε-thickness layer, the author is able to pass to the limit in the variational formulation.Let us also mention that, in the literature, one can find different methods and strategies for reaction-diffusion problems with a thin layer.For instance, in [28] the notion of two-scale convergence for thin domains is introduced which allows the rigorous derivation of lower dimensional models.Some other papers have deepened the case of heterogeneous membrane.We cite [29], where the authors develop a multiscale method which combines classical compactness results based on a priori estimates and weak-strong two-scale convergence results in order to be able to pass to the limit in a thin heterogeneous membrane.In [13], a transmission problem involving nonlinear diffusion in the thin layer is treated and an effective model was derived.Finally, in [14], the accuracy of the effective approximations for processes through thin layers is studied by proving estimates for the difference between the original and the effective quantities.The passage at the limit allows to infer the existence of weak solutions for the effective Problem (2), thanks to the existence result for the ε−problem provided in Appendix A. In the case of linear diffusion, the existence of global weak solutions for the effective problem with the Kedem-Katchalsky conditions is provided by [11].In particular, the authors prove it under weaker hypothesis such as L 1 initial data and reaction terms with sub-quadratic growth in an L 1 -setting.
Outline of the paper.The paper is organised as follows.In Section 2, we introduce the assumptions and notations, including the definition of weak solution of the original problem, System (1).In Section 3, a priori estimates that will be useful to pass to the limit are proven.
Section 4 is devoted to prove the convergence of Problem (1), following the method introduced in [37] for the (non-degenerate) elliptic and parabolic cases.The argument relies on recovering the L 2 -boundedness (uniform with respect to ε) of the velocity field, in our case, the pressure gradient.As one may expect, since the permeability of the membrane, µ 2,ε , tends to zero proportionally with respect to ε, it is only possible to establish a uniform bound outside of Ω 2,ε .For this reason, following [37], we introduce an extension operator (Subsection 4.1) and apply it to the pressure in order to extend the H 1 -uniform bounds in the whole space Ω \ Γ1,3 , hence proving compactness results.We remark that the main difference between the strategy in [37] and our adaptation, is given by the fact that due to the non-linearity of the equation, we have to infer strong compactness of the pressure (and consequently of the density) in order to pass to the limit in the variational formulation.For this reason, we also need the L 1 -boundedness of the time derivative, hence obtaining compactness with a standard Sobolev's embedding argument.Moreover, since solutions to the limit Problem (2) will present discontinuities at the effective interface, we need to build proper test functions which belong to H 1 (Ω \ Γ1,3 ) that are zero on ∂Ω and are discontinuous across Γ1,3 , (Subsection 4.2).
Finally, using the compactness obtained thanks to the extension operator, we are able to prove the convergence of solutions to Problem (1) to couples (ũ, p) which satisfy Problem (2) in a weak sense, therefore inferring the existence of solutions of the effective problem, as stated in the following theorem.
for all test functions w(t, x) with a proper regularity (defined in Theorem 4.3) and w(T, x) = 0 a.e. in Ω.We used the notation Section 5 concludes the paper and provides some research perspectives.

Assumptions and notations
Here, we detail the problem setting and assumptions.For the sake of simplicity, we consider as domain Ω ⊂ R 3 a cylinder with axis x 3 , see Figure 1.Let us notice that it is possible to take a more general domain Ω defining a proper diffeomorfism F : Ω → Ω.Therefore, the results of this work extend to more general domains as long as the existence of the map F can be proved (this implies that Ω is a connected open subset of R d and has a smooth boundary).Therefore, we assume that the domain Ω has a C 1 -piecewise boundary.We also want to emphasize the fact that our proofs hold in a 2D domain considering three rectangular subdomains.We introduce We define the interfaces between the domains Ω i,ε and Ω i+1,ε for i = 1, 2, as We denote with n i,i+1 the outward normal to Γ i,i+1,ε with respect to Ω i,ε , for i = 1, 2. Let us notice that n i,i+1 = −n i+1,i .We define two trace operators Therefore, for any z ∈ W k,p (Ω \ Γ1,3 ), we have the following decomposition Obviously, we have that z α ∈ W k,p ( Ωα ) (α = 1, 3).Thus, we denote and the following continuity property holds [4] We assume W k,p (Ω \ Γ1,3 ) is endowed with the norm We make the following assumptions on the initial data: there exists a positive constant p H , such that Moreover, we assume that there exists a function ũ0 ∈ L 1 + (Ω) (i.e.ũ0 ∈ L 1 (Ω) and non-negative) such that The growth rate G(•) satisfies The value p H , called homeostatic pressure, represents the lowest level of pressure that prevents cell multiplication due to contact-inhibition.We assume that the mobility coefficients satisfy µ i,ε > 0 for i = 1, 3 and Notations.For all T > 0, we denote Ω T := (0, T ) × Ω.We use the abbreviated form From now on, we use C to indicate a generic positive constant independent of ε that may change from line to line.Moreover, we denote and sign(w) = sign + (w) + sign − (w).

A priori estimates
We show that the main quantities satisfy some uniform a priori estimates which will later allow us to prove strong compactness and pass to the limit.
Lemma 3.1 (A priori estimates).Given the assumptions in Section 2, let (u ε , p ε ) be a solution of Problem (1).There exists a positive constant C independent of ε such that Remark 3.2.We remark that statement (i) implies that for all p ∈ [1, ∞], we have Remark 3.3.The following proof can be made rigorous by performing a parabolic regularization of the problem, namely by adding δ∆u i,ε , for δ > 0, to the left-hand side of the equation and in the flux continuity conditions.In fact, the following estimates can be obtained uniformly both in ε and δ.
Proof.Let us recall the equation satisfied by u ε on Ω i,ε , namely The L ∞ -bounds of the density and the pressure are a straight-forward consequence of the comparison principle applied to Equation ( 9), which can be rewritten as Indeed, summing up Equations ( 10) for i = 1, 2, 3, we obtain Then, we also have Let us recall Kato's inequality, [19], i.e.
If we multiply by sign − (u H − u i,ε ), thanks to Kato's inequality, we infer that where we have used the assumption (A-G).We integrate over the domain Ω.Thanks to the boundary conditions in System (1), i.e. the density and flux continuity across the interfaces, and the homogeneous Dirichlet conditions on ∂Ω, we gain Hence, from Equation ( 12), we find Finally, Gronwall's lemma and hypothesis (A-data1) on u 0 i,ε imply We then conclude the boundedness of u i,ε by u H for all i = 1, 2, 3. From the relation p ε = u γ ε , we conclude the boundedness of p ε .
By arguing in an analogous way, replacing u H by 0 and multiplying by sign + (u i,ε ), we obtain namely, u ε ≥ 0, and consequently, p ε ≥ 0.
We derive Equation (10) with respect to time to obtain Upon multiplying by sign(∂ t u i,ε ) and using Kato's inequality, we have since u i,ε and p i,ε are both nonnegative and ∂ t p i,ε = γu γ−1 i,ε ∂ t u i,ε .We integrate over Ω i,ε and we sum over i = 1, 2, 3, namely where we use that G ≤ 0. Now we show that the term J vanishes.Integration by parts yields For the sake of simplicity, we denote n := n i,i+1 .Let us recall that, by definition, n i+1,i = −n.
Combing Equation (15) and Equation ( 14) we get Moreover, Equation ( 14) also implies which, combined with Equation ( 15) gives also Now we may come back to the computation of the term J .By Equations (16), and (17) we directly infer that J 1 vanishes.
We rewrite the term , where we used Equation ( 16), which also implies The terms J 2,1 and J 2,2 vanish thanks to Equation (18) and Equation ( 19), respectively.Hence, from Equation ( 13), we finally have and, using Gronwall's inequality, we obtain Thanks to the assumptions on the initial data, cf.Equation (A-data2), we conclude.
As known, in the context of a filtration equation, we can recover the pressure equation upon multiplying the equation on u i,ε , cf.System (1), by p (u i,ε ) = γu γ−1 i,ε .Therefore, we obtain Studying the equation on p ε rather than the equation on u ε turns out to be very useful in order to prove compactness, since, as it is well-know for the porous medium equation (PME), the gradient of the pressure can be easily bounded in L 2 , while the density solution of the PME can develop jump singularities on the free boundary, [39].We integrate Equation ( 20) on each Ω i,ε , and we sum over all i to obtain Integration by parts yields we have homogeneous Dirichlet boundary conditions on ∂Ω and the flux continuity conditions (17).
Hence, from Equation ( 21), we have We integrate over time and we deduce that Finally, we conclude that Since we have already proved that p i,ε is bounded in L ∞ (Ω T ) and by assumption G is continuous, we finally find that where C denotes a constant independent of ε.Since both µ 1,ε and µ 3,ε are bounded from below away from zero, we conclude that the uniform bound holds in Ω \ Ω 2,ε .
Remark 3.4.Let us also notice that, differently from [37], where the author studies the linear and uniformly parabolic case, proving weak compactness is not enough.Indeed, due to the presence of the nonlinear term u∇p, it is necessary to infer strong compactness of u.For this reason, the L 1 -uniform estimate on the time derivative proven in Lemma 3.1 is fundamental.
4 Limit ε → 0 We have now the a priori tools to face the limit ε → 0. We need to construct an extension operator with the aim of controlling uniformly, with respect to ε, the pressure gradient in L 2 (Ω).Indeed, from (25), we see that one cannot find a uniform bound for ∇p 2,ε L 2 (Ω 2,ε ) .The blow-up of Estimate ( 25) for i = 2, is in fact the main challenge in order to find compactness on Ω.To this end, following [37], we introduce in Subsection 4.1 an extension operator which projects the points of Ω 2,ε inside Ω 1,ε ∪ Ω 3,ε .Then, introducing proper test functions such that the variational formulation for ε > 0 in ( 8) and ε → 0 in ( 6) are well-defined, we can pass to the limit (Subsection 4.2).

Extension operator and compactness
2: Representation of the spatial symmetry used in the definition of the extension operator, cf.Equation ( 26) and of the two subdomains of Ω 2,1,ε and Ω 2,3,ε .
As mentioned above, in order to be able to pass to the limit ε → 0, we first need to define the following extension operator as follows for a general function z ∈ L q (0, T ; W 1,p (Ω \ Ω 2,ε )), where x is the symmetric of x with respect to Γ 1,2,ε (or ), defined by the function g : x → x for x = (x 1 , x 2 , x 3 ) ∈ Ω 2,ε such that where d(Γ 1,2,ε , x) (respectively d(Γ 2,3,ε , x)) denotes the distance between x and the surface Γ 1,2,ε (respectively Γ 2,3,ε ).The point x is illustrated in Figure 2. It can be easily seen that the function g and its inverse have uniformly bounded first derivatives.Hence, we infer that P ε is linear and bounded, i.e.
Let us notice that the extension operator is well defined also from Hence, we can apply it also on u ε and ∂ t p ε .
Remark 4.1.Thanks to the properties of the extension operator, the estimates stated in Lemma 3.1 hold true also upon applying P ε (•) on p ε , u ε , and ∂ t p ε , namely The two bounds hold thanks to the following arguments and Lemma 4.2 (Compactness of the extension operator).Let (u ε , p ε ) be the solution of Problem (1).There exists a couple (ũ, p) with such that, up to a subsequence, it holds Proof.(i).Since both )) uniformly with respect to ε, we infer the strong compactness of Let us also notice that since both u ε and p ε are uniformly bounded in L ∞ (0, T ; L ∞ (Ω \ Γ1,3 )) then the strong convergence holds in any L p (0, T ; L p (Ω \ Γ1,3 )) with 1 ≤ p < ∞.
(ii).From (i), we can extract a subsequence of P ε (p ε ) which converges almost everywhere.Then, remembering that u ε = p 1/γ ε , with γ > 1 fixed, we have convergence of P ε (u ε ) almost everywhere.Thanks to the uniform L ∞ -bound of P ε (u ε ), Lebesgue's theorem implies the statement.Let us point out that, in particular, the L ∞ -uniform bound is also valid in the limit.
4.2 Test function space and passage to the limit ε → 0 Since in the limit we expect a discontinuity of the density on Γ1,3 , we need to define a suitable space of test functions.Therefore we construct the space E as follows.Let us consider a function ζ ∈ D(Ω) (i.e.C ∞ c (Ω)).For any ε > 0 small enough, we build the function v ε = P ε (ζ), using the extension operator previously defined.The space of all linear combinations of these We stress that the functions of E are discontinuous on Γ1,3 .
In the weak formulation of the limit problem (6), we will make use of piece-wise C ∞ -test functions (discontinuous on Γ1,3 ) of the type w(t, x) = ϕ(t)v(x), where ϕ ∈ C 1 ([0, T )) with ϕ(T ) = 0 and v ∈ E * .Therefore, w belongs to C 1 ([0, T ); E * ).On the other hand, in the variational formulation (8), i.e. for ε > 0, H 1 (0, T ; H 1 0 (Ω)) test functions are required.Thus, in order to study the limit ε → 0, we need to introduce a proper sequence of test functions depending on ε that converges w.To this end, we define the operator In this way, L ε (w) belongs to H 1 (0, T ; H 1 0 (Ω)), therefore, it can be used as test function in the formulation (8).
Following Sanchez-Palancia, [37], for all t ∈ [0, T ] and x = (x 1 , x 2 , x 3 ) ∈ Ω, we define It can be easily verified that L ε (w) is linear with respect to x 3 in Ω 2,ε and is continuous on ∂Ω 2,ε .Let us notice that it holds ∂L ε (w) Furthermore, thanks to the mean value theorem, the partial derivatives of L ε (w) with respect to x 1 and x 2 are bounded by a constant (independent of ε), and since the measure of Ω 2,ε is proportional to ε, we have Given w ∈ C 1 ([0, T ); E ), we take L ε (w) as a test function in the variational formulation of the problem, i.e.Equation ( 8), and we have Thanks to the a priori estimates already proven, cf.Lemma 3.1, Remark 4.1 and the convergence result on the extension operator, cf.Lemma 4.2, we are now able to pass to the limit ε → 0 and recover the effective interface problem.
Theorem 4.3.For all test functions of the form w(t, x) := ϕ(t)v(x) with ϕ ∈ C 1 ([0, T )) and v ∈ E * , the limit couple (ũ, p) of Lemma 4.2 satisfies the following equation where Proof.may pass to the limit in Equation ( 29), computing each term individually.
Step 1.Time derivative integral.We split the first integral into two parts Since outside of Ω 2,ε the extension operator coincides with the identity, and L ε (w) = w, we have Thanks to Remark 4.1, we know that the last integral converges to zero, since both P ε (u ε ) and ∂ t w are bounded in L 2 and the measure of Ω 2,ε tends to zero as ε → 0.Then, by Lemma 4.2, we have where we used the weak convergence of P ε (u ε ) to ũ in L 2 (0, T ; L 2 (Ω \ Γ1,3 )).The term I 2 vanishes in the limit, since both u ε and ∂ t L ε (w) are bounded in L 2 uniformly with respect to ε.
Hence, we finally have Step 2. Reaction integral.We use the same argument for the reaction term, namely Using again the convergence result on the extension operator, cf.Lemma 4.2, we obtain ũ G(p)w, as ε → 0, since both P ε (u ε ) and G(P ε (p ε )) converge strongly in L 2 (0, T ; L 2 (Ω \ Γ1,3 )).Arguing as before, it is immediate to see that K 2 vanishes in the limit.Hence Step 3. Initial data integral.From (A-data3), it is easy to see that Step 4. Divergence integral.Now it remains to treat the divergence term in Equation ( 29), from which we recover the effective interface conditions at the limit.
Since the extension operator P ε is in fact the identity operator on Ω \ Ω 2,ε , we can write We treat the two terms Since we want to use the weak convergence of ∇P ε (p ε ) in L 2 (0, T ; L 2 (Ω \ Γ1,3 )) (together with the strong convergence of P ε (u ε ) in L 2 (0, T ; L 2 (Ω \ Γ1,3 ))) we need to write the term H 1 as an integral over Ω.To this end, let µ ε := µ ε (x) be a function defined as follows Then, we can write Let us notice that as ε goes to 0, µ ε converges to μ1 in Ω1 and μ3 in Ω3 .Therefore, by Lemma 4.2, we infer Now we treat the term H 2 , which can be written as By the Cauchy-Schwarz inequality, the a priori estimate (25), and Equation ( 28), we have On other hand, by Fubini's theorem, the following equality holds Therefore, In order to conclude the proof, we state the following lemma, which is proven below.
We may finally find the limit of the term H 2 , using Assumption (7), and applying Lemma 4.4 to Equation ( 35) as ε → 0. Combining the above convergence to Equation (33) and Equation (34), we find the limit of the divergence term as ε goes to 0, which, together with Equations ( 29), (31), and (32), concludes the proof.
We now turn to the proof of Lemma 4.4 Proof of Lemma 4.4.Since by definition w(t, x) = ϕ(t)v(x), with ϕ ∈ C 1 ([0, T )) and v ∈ E * , the uniform convergence in Equation ( 36) comes from the piece-wise differentiability of w.
Recalling that L is the length of Ω, trivially, we find the following estimate Remark Although not relevant from a biological point of view, let us point out that, in the case of dimension greater than 3, the analysis goes through without major changes.It is clear that the a priori estimates are not affected by the shape or the dimension of the domain (although some uniform constants C may depend on the dimension, this does not change the result in Lemma 3.1).The following methods, and in particular the definition of the extension operator and the functional space of test functions, clearly depends on the dimension, but the strategy is analogous for a d-dimensional cylinder with axis {x 1 = • • • = x d−1 = 0}.
Remark 4.6.We did not consider the case of non-constant mobilities, i.e. µ i,ε := µ i,ε (x), but continuity and boundedness are the minimal hypothesis to succeed in the proof.

Conclusions and perspectives
We proved the convergence of a continuous model of cell invasion through a membrane when its thickness is converging to zero, hence giving a rigorous derivation of the effective transmission conditions already conjectured in Chaplain et al., [10].Our strategy relies on the methods developed in [37], although we had to handle the difficulties coming from the nonlinearity and degeneracy of the system.A very interesting direction both from the biological and mathematical point of view, could be coupling the system to an equation describing the evolution of the MMP concentration.In fact, as observed in [10], the permeability coefficient can depend on the local concentration of MMPs, since it indicates the level of "aggressiveness" at which the tumour is able to destroy the membrane and invade the tissue.In a recent work [16], a formal derivation of the multi-species effective problem has been proposed.However, its rigorous proof remains an interesting and challenging open question.Indeed, introducing multiple species of cells, hence dealing with a cross-(nonlinear)-diffusion system, adds several challenges to the problem.As it is well-known, proving the existence of solutions to cross-diffusion systems with different mobilities is one of the most challenging and still open questions in the field.Nevertheless, even when dealing with the same constant mobility coefficients, the nature of the multi-species system (at least for dimension greater than one) usually requires strong compactness on the pressure gradient.We refer the reader to [18,33] for existence results of the two-species model without membrane conditions.
Another direction of further investigation of the effective transmission problem (2) could be studying the so-called incompressible limit, namely the limit of the system as γ → ∞.The study of this limit has a long history of applications to tumour growth models, and has attracted a lot of interest since it links density-based models to a geometrical (or free boundary) representation, cf.[21,31].
Moreover, including the heterogeneity of the membrane in the model could not only be useful in order to improve the biological relevance of the model, but could bring interesting mathematical challenges, forcing to develop new methods or adapt already existent ones, [29], from the parabolic to the degenerate case.