Tomasz D Ębiec
email: t.debiec@mimuw.edu.pl.

Benoît Perthame
email: benoit.perthame@upmc.fr.

Markus Schmidtchen
email: markus.schmidtchen@upmc.fr.

AND Nicolas Vauchelet
email: vauchelet@math.univ-paris13.fr

INCOMPRESSIBLE LIMIT FOR A TWO-SPECIES MODEL WITH COUPLING THROUGH BRINKMAN'S LAW IN ANY DIMENSION

Keywords: 35B45, 35K57, 35K55, 35K65, 35Q92, 76N10, 76S99, TUMOUR GROWTH, INCOMPRESSIBLE LIMIT, FREE-BOUNDARY PROBLEMS, VISCO-ELASTIC MEDIA

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Incompressible limit for a two-species model with coupling through Brinkman's law in any dimension

Tomasz Dębiec, Benoît Perthame, Markus Schmidtchen, Nicolas Vauchelet

INTRODUCTION

In recent years there has been an increasing interest in multi-phase models applied to tumour growth. Traditionally, tumour growth was modelled using a single equation describing the evolution of the density of tumorous cells. This paper is dedicated to studying the two-species model

     ∂n (i) k ∂t -∇ • n (i) k ∇W k = n (i) k G (i) (p k ), -ν∆W k + W k = p k ,
where n (i) represents the density of the healthy (resp. cancerous) cells, for i = 1, 2, and p k , denotes the joint population pressure generated by the two species, i.e.,

p k := k k -1 n k-1 k
, where

n k := n (1) k + n (2)
k , denotes the total population. The model parameters k ∈ N and ν > 0 model the stiffness of the pressure and the level of viscosity, respectively. Note that the velocity field, W k , is generated by the joint population pressure through so-called Brinkman's law, for instance cf. [START_REF] Allaire | Homogenization of the Navier-Stokes equations and derivation of Brinkman's law[END_REF]. Unlike Darcy's law, the Brinkman flow incorporates viscosity effects into the model. In addition to the advection, the two cell densities are assumed to proliferate. This effect is encoded in the two functions G (i) , for i = 1, 2, which are assumed to be decreasing in their variable, p k , similar to [START_REF] Byrne | Individual-based and continuum models of growing cell populations: a comparison[END_REF][START_REF] Ranft | Fluidization of tissues by cell division and apoptosis[END_REF]. This accounts for an inhibited growth whenever the joint pressure gets too large. The total population plays an important role in the analysis of these types of models that was exploited in many related results, cf. [START_REF] Carrillo | Splitting schemes and segregation in reaction cross-diffusion systems[END_REF][START_REF] Dębiec | Incompressible limit for a two-species tumour model with coupling through Brinkman's law in one dimension[END_REF]. It is easily verified that the total population density, n k , satisfies

∂n k ∂t -∇ • (n k ∇W k ) = n k r k G (1) (p k ) + (1 -r k )G (2) (p k ) , (1)
where r k denotes the concentration

r k := n (1) k /n k , (2)
also referred to as the population fraction. Related models have been extensively studied in the past. We refer to [START_REF] Perthame | Derivation of a hele-shaw type system from a cell model with active motion[END_REF][START_REF] Perthame | Incompressible limit of a mechanical model of tumour growth with viscosity[END_REF], and references therein, for a study of the stiff limit for a single-species visco-elastic tumour model. As above, the velocity field is given by Brinkman's law. Another approach was taken in [START_REF] Kim | Uniform convergence for the incompressible limit of a tumor growth model[END_REF] where the authors construct viscosity solutions to the same viscoelastic single-species tissue growth model. Their result extends the result of [START_REF] Perthame | Incompressible limit of a mechanical model of tumour growth with viscosity[END_REF] in that they obtain pointwise convergence and, what is more, uniform convergence away from the boundary of the support. Similarly to [START_REF] Perthame | Incompressible limit of a mechanical model of tumour growth with viscosity[END_REF], a part of the paper is dedicated to studying the evolution of the support as a geometric flow. However, they observe that a regularisation of the velocity field is sufficient to apply the method of characteristics, to pass to the limit later, and to obtain the same complementarity relation of the limiting pressure. In their case, the pressure is directly linked to the density itself, rather than the sum of the two densities. As a matter of fact, coupling the two equations for the individual species through Brinkman's law, this time generated by the joint population pressure, changes the behaviour dramatically and the strategy of a kinetic reformulation used in [START_REF] Perthame | Incompressible limit of a mechanical model of tumour growth with viscosity[END_REF] cannot be applied directly. Two of the authors were able to establish the incompressible limit in the one dimensional case, cf. [START_REF] Dębiec | Incompressible limit for a two-species tumour model with coupling through Brinkman's law in one dimension[END_REF]. Their proof mainly relies on establishing uniform BV -bounds for the two species as well as the total population. In conjunction with the compactness criterion [18, Lemma A] they infer the strong compactness of the pressure which suffices to pass to the limit. In their paper, the authors already note that the BV -strategy fails in higher dimensions and they posit new techniques be indispensable in the quest of extending the result to higher dimensions. Even in the inviscid case, i.e., ν = 0, the system nature of the problem causes serious analytical difficulties, cf. [START_REF] Bubba | Hele-shaw limit for a system of two reaction-(cross-)diffusion equations for living tissues[END_REF][START_REF] Carrillo | Splitting schemes and segregation in reaction cross-diffusion systems[END_REF][START_REF] Gwiazda | A two-species hyperbolicparabolic model of tissue growth[END_REF]. In this case, the pressure does gain a bit in regularity. Yet, this gain in regularity is just about sufficient to obtain strong compactness of the pressure gradient. We highlight that similar difficulties arise when the pressure is not given in form of a power law but blows up at a finite threshold, cf. [START_REF] Chertock | Incompressible limit of a continuum model of tissue growth with segregation for two cell populations[END_REF][START_REF] Degond | Incompressible limit of a continuum model of tissue growth for two cell populations[END_REF][START_REF] Hecht | Incompressible limit of a mechanical model for tissue growth with non-overlapping constraint[END_REF]. All these results have one thing in commontheir minute study of the equation satisfied by the population pressure, cf. [9, 11-13, 16, 17, 20-22, 24, 25], which allows for proving the existence of solutions and obtaining estimates that are uniform in the stiffness parameter, k. The innovation of this work is to combine techniques for the one-species case in any space dimension and the two-species case in one dimension. In conjunction with a nonlocal compactness criterion, originally introduced and devised by BRESCH & JABIN, cf. [START_REF] Jabin | Differential equations with singular fields[END_REF], the required compactness is obtained and the passage to the incompressible limit is accomplished. This technique, just like the aforementioned results, rely on a clever choice of auxiliary variables: it is readily checked that the joint population pressure satisfies

∂p k ∂t -∇p k • ∇W k = k -1 ν p k W k -p k + νr k G (1) (p k ) + ν(1 -r k )G (2) (p k ) ,
where the population fraction r k , satisfies

∂r k ∂t -∇r k • ∇W k = r k (1 -r k ) G (1) (p k ) -G (2) (p k ) .
This change of variables was first introduced in [START_REF] Bertsch | On a degenerate diffusion equation of the form c(z) t = ϕ(z x ) x with application to population dynamics[END_REF][START_REF] Bertsch | On interacting populations that disperse to avoid crowding: the case of equal dispersal velocities[END_REF][START_REF] Bertsch | On interacting populations that disperse to avoid crowding: preservation of segregation[END_REF] in the context of a two-species system where the two species avoid overcrowding, paving the way for more modern approaches to tumour models linked through Darcy's law, cf. [START_REF] Bertsch | A nonlinear parabolic-hyperbolic system for contact inhibition of cell-growth[END_REF][START_REF] Bubba | Hele-shaw limit for a system of two reaction-(cross-)diffusion equations for living tissues[END_REF][START_REF] Carrillo | Splitting schemes and segregation in reaction cross-diffusion systems[END_REF][START_REF] Gwiazda | A two-species hyperbolicparabolic model of tissue growth[END_REF].

The rest of this paper is organised as follows. In the subsequent section, Section 2, we give a precise statement of the problem. In addition, we introduce the key assumptions on the growth terms and the initial data. Finally, we state the main result of this work, the incompressible limit and the complementarity relation. Section 3 is dedicated to establishing certain a priori estimates on the individual species, the pressure, and the velocity field given through Brinkman's law. Section 4 is devoted to establishing the strong compactness of the two individual species as well as the total population. Note that this is where we deviate from the strategy of [START_REF] Perthame | Incompressible limit of a mechanical model of tumour growth with viscosity[END_REF] and, instead, follow [START_REF] Vauchelet | Incompressible limit of the Navier-Stokes model with a growth term[END_REF]. A key step in the argument is to study not only the individual species but to include the total population in the estimates. At first glance, this approach may seem rather absurd and the reader might wonder what could possibly be gained by incorporating the total population into the estimates that has not already been obtained from the two individual species. However, this strategy already proved most useful in [START_REF] Dębiec | Incompressible limit for a two-species tumour model with coupling through Brinkman's law in one dimension[END_REF] as certain cancellations can be obtained in doing so. Having established the strong compactness of the individual species, we address the compactness of the pressure in Section 5. Here, we borrow the technique of the kinetic reformulation from [START_REF] Perthame | Incompressible limit of a mechanical model of tumour growth with viscosity[END_REF] to lift the strong compactness of the individual species to the joint population pressure. In Section 6 we gather all preceding results and establish the proof of the main result of this work, i.e., the incompressible limit and the complementarity relation.

PRELIMINARIES AND STATEMENT OF THE MAIN RESULTS

We study the system

         ∂n (1) k ∂t -∇ • n (1)
k ∇W k = n (1) k G (1) (p k ), ∂n (2)
k ∂t -∇ • n (2)
k ∇W k = n (2) k G (2) (p k ), (3a)
posed on (0, T ) × R d . It is coupled through an elliptic equation, the so-called Brinkman's law

-ν∆W k + W k = p k . ( 3b
)
The system is equipped with non-negative initial data

n (i) 0,k ∈ L 1 (R d ) ∩ L ∞ (R d ), (4a) for any k ≥ 2, such that p 0,k ∈ L 1 (R d ) ∩ L ∞ (R d ), as well as p 0,k ≤ P H ,
for some positive constant P H > 0. In addition, we assume that the initial data are compact, i.e., for i = 1, 2, there exist non-negative functions n

(i) 0,∞ ∈ L 1 (R d ) ∩ L ∞ (R d ), such that n (i) 0,k -→ n (i) 0,∞ , (4b) strongly in L 1 ((0, T ) × R d ), as k → ∞.
As before, the pressure is given in form of a power of the joint population, i.e.,

p k := k k -1 n (1) k + n (2) k k-1 = k k -1 n k-1 k . (5)
Recall that the pressure satisfies

∂p k ∂t -∇p k • ∇W k = k -1 ν p k W k -p k + νr k G (1) (p k ) + ν(1 -r k )G (2) (p k ) , (6)
with the population fraction,

r k := n (1) k /n k , given by ∂r k ∂t -∇r k • ∇W k = r k (1 -r k ) G (1) (p k ) -G (2) (p k ) . (7)
Throughout the paper we assume the following regularity and properties of the growth functions, G (i) ,

G (i) ∈ C 1 (R), G (i)
p ≤ -α < 0, and max i=1,2

G (i) (P H ) = 0, (8)
for i = 1, 2, and some α > 0, where G (i) p denotes the derivative of the function G (i) with respect to the pressure. We remark that the constant P H is usually referred to as the homeostatic pressure.

Remark 2.1 (Existence of solutions). Let us point out that existence of bounded weak solutions to System (3) has been recently established by D ĘBIEC & SCHMIDTCHEN, cf. [START_REF] Dębiec | Incompressible limit for a two-species tumour model with coupling through Brinkman's law in one dimension[END_REF], using a fixed point argument. For the reader's convenience we recall here the precise statement. Theorem 2.2. Let T > 0. For any initial data satisfying (4), System (3) admits a unique distributional solution (n

(1) k , n (2) k ), with n (i) k ∈ L ∞ 0, T ; L ∞ (R d ) .
In fact, in this paper we establish compactness of the sequences of individual densities, which would allow for another proof of existence, via a viscosity approximation approach, see also [START_REF] Belgacem | Compactness for nonlinear continuity equations[END_REF].

Below we formulate the main result of the current work.

Theorem 2.3 (Incompressible limit and complementarity relation). We may pass to the limit k → ∞ in the pressure equation, Eq. (6), using strong compactness of the pressure and the two species. This yields the so-called complementarity relation

p ∞ W ∞ -p ∞ + νn (1) ∞ G (1) (p ∞ ) + νn (2) ∞ G (2) (p ∞ ) = 0, (9)
almost everywhere, where n

(i) ∞ , satisfies      ∂n (i) ∞ ∂t -∇ • n (i) ∞ ∇W ∞ = n (i) ∞ G (i) (p ∞ ), -ν∆W ∞ + W ∞ = p ∞ ,
in the distributional sense, for i = 1, 2. Moreover, the following holds true almost everywhere

p ∞ (n ∞ -1) = 0.
In other words, Theorem 2.3 provides a rigorous link between the description of the evolution of the two populations, cf. Eq. (3a), and a geometric free-boundary model of Hele-Shaw flavour.

Regions with positive pressure correspond to fully saturated areas, since p ∞ (n ∞ -1) = 0, and on such domains, the pressure is given by the so-called complementarity relation, cf. Eq. ( 9). The rigorous derivation is already known in one spatial dimension, cf. [START_REF] Dębiec | Incompressible limit for a two-species tumour model with coupling through Brinkman's law in one dimension[END_REF] and, for one species, in any dimension, cf. [START_REF] Perthame | Incompressible limit of a mechanical model of tumour growth with viscosity[END_REF]. In both works the authors emphasised the possibility of jump discontinuities in the pressure which renders the problem of obtaining compactness rather challenging. An extension of the strategy of [START_REF] Dębiec | Incompressible limit for a two-species tumour model with coupling through Brinkman's law in one dimension[END_REF] to higher dimensions appears futile, as does the extension of [START_REF] Perthame | Incompressible limit of a mechanical model of tumour growth with viscosity[END_REF] to two species due to the contribution of the individual species and their role in the identification of weak-⋆ limits in the kinetic reformulation. The subsequent sections are concerned with the proof of the main theorem.

A PRIORI ESTIMATES

The proof of Theorem 2.3 relies on certain uniform bounds for the main quantities of interests, i.e., the population densities, the pressure, and the velocity field. These will be vital when passing to the limit k → ∞. The proofs of most of these assertions are rather straightforward, and have been carefully written in previous works of the authors, cf. [START_REF] Dębiec | Incompressible limit for a two-species tumour model with coupling through Brinkman's law in one dimension[END_REF][START_REF] Perthame | Incompressible limit of a mechanical model of tumour growth with viscosity[END_REF]. We therefore skip them here for brevity. The first lemma establishes uniform bounds for the densities and the pressure.

Lemma 3.1 (A priori estimates I).

The following hold uniformly in k for any T > 0.

(i) n (i) k ≥ 0, for i = 1, 2, (ii) n k ∈ L ∞ (0, T ; L 1 (R d )), (iii) p k ∈ L ∞ (0, T ; L ∞ (R d )), with 0 ≤ p k ≤ P H , (iv) n k ∈ L ∞ (0, T ; L ∞ (R d )), (v) p k ∈ L ∞ (0, T ; L 1 (R d ))
, and

(vi) n (i) k ∈ L ∞ (0, T ; L ∞ (R d )), for i = 1, 2.
The second lemma states some useful observations regarding regularity of the sequence (W k ) k .

Recall that a solution W k to Brinkman's equation, Eq. (3b), may be written as W k = K⋆p k , where K is the fundamental solution to the equation -ν∆K + K = δ 0 , i.e.,

K(x) = 1 4π ∞ 0 exp -(π|x| 2 /4sν + s/4π) s -d/2 ds.
Then K ≥ 0, K(x) dx = 1 and

K ∈ L q (R d ), for 1 ≤ q < d/(d -2), as well as, ∇K ∈ L q (R d ), for 1 ≤ q < d/(d -1)
. By the elliptic regularity theory we have

W k ∈ L ∞ 0, T ; W 2,q (R d ) , for 1 ≤ q ≤ ∞.

Lemma 3.2 (A priori estimates II).

The following hold uniformly in k for any T > 0.

(i

) W k ∈ L ∞ (0, T ; L 1 ∩ L ∞ (R d )), (ii) W k ∈ L ∞ (0, T ; W 1,q (R d )), for 1 ≤ q ≤ ∞, (iii) ∇ 2 W k ∈ L ∞ (0, T ; L q (R d )), for 1 < q < ∞, (iv) ∂ t W k ∈ L 1 (0, T ; L q (R d )), for 1 ≤ q ≤ ∞, and (v) ∂ t ∇W k ∈ L 1 (0, T ; L q (R d )), for 1 < q < d/(d -1).
Using the above lemma and the boundedness of W k , we have the following result.

Lemma 3.3 (Integrability and Segregation

). If both species are segregated initially, i.e., r 0,k (1 -r 0,k ) = 0, almost everywhere, then there holds

r k (t, x) (1 -r k (t, x)) = 0,
almost everywhere, for all times 0 ≤ t ≤ T . Moreover,

r 0,k (1 -r 0,k ) ∈ L 1 (R d ) implies r k (1 -r k ) ∈ L ∞ (0, T ; L 1 (R d )).
It is easy to infer from the preceding lemma that the population fraction itself is integrable.

Remark 3.4 (The population fraction is locally integrable). The population fraction r

k is bounded in L ∞ (0, T ; L 1 loc (R d )).
Finally, the following lemma establishes an L 1 -bound on the right-hand side of the pressure equation.

Lemma 3.5 (A priori estimates III).

The following estimate holds for any T > 0

k T 0 R d p k |W k -p k + νr k G (1) (p k ) + ν(1 -r k )G (2) (p k )| dx dt ≤ C(T ),
for a constant C(T ), independent of k.

This estimate follows directly from studying the dissipation of the quantity

R d W k -p k + νn (1) k G (1) (p k ) + νn (2) k G (2) (p k ) dx,
along solutions of System (3). Since the proof is parallel to the one dimensional case, we refer the reader to [START_REF] Dębiec | Incompressible limit for a two-species tumour model with coupling through Brinkman's law in one dimension[END_REF]Lemma 4.3] for a detailed derivation of this bound.

STRONG COMPACTNESS OF THE DENSITIES

The aim of this section is to establish local strong compactness of the two individual species, i.e.,

(n

(i) k ) k , for i = 1, 2.
To this end, we follow the strategy originally proposed by BRESCH & JABIN, cf. [START_REF] Bresch | Global existence of weak solutions for compressible Navier-Stokes equations: thermodynamically unstable pressure and anisotropic viscous stress tensor[END_REF], (see also [START_REF] Bresch | Global weak solutions of PDEs for compressible media: a compactness criterion to cover new physical situations[END_REF]) in the context of compressible Navier-Stokes equations with non-monotone pressure law and anisotropic stress tensor. Their approach has also been adapted to the case of the whole space, R d , with a growth term on the right-hand side of the continuity equation in [START_REF] Vauchelet | Incompressible limit of the Navier-Stokes model with a growth term[END_REF].

Here we adapt their strategy to the case of a system of two interacting species, where the interaction is given by the elliptic Brinkman's law and the individual growth terms. The main result of this section reads Theorem 4.1 (Compactness of the species). Suppose the initial data n (i) 0,k is compact, in the sense of Eq. (4b), and suppose the growth terms satisfy Eq. (8). Then both the individual species and the total population density (n k ) k , (n

(1) k ) k , (n (2) k ) k are compact in L 1 loc (0, T ) × R d .
In particular, this will be used to conclude the following.

Corollary 4.2 (Strong compactness of r k ). The sequence (r k ) k , of population fractions, is compact in L p 0, T, L q loc (R d ) , for any 1 ≤ p, q < ∞. Remark 4.3.
It is worth pointing out that even though our ultimate objective is to obtain strong compactness of the pressure, we cannot apply the method presented below directly to the pressure in order to accomplish this task. This is due to the fact that the right-hand side of the equation satisfied by the pressure, cf. Eq. ( 6), is merely in L 1 , cf. Lemma 3.5, which turns out to be borderline for obtaining estimates independent of the parameter k. As we will see, to establish compactness of the pressure, we need compactness of the densities, n (i) k , and the remarkable aspect of Theorem 4.1 is to prove compactness of the densities when the pressure is merely bounded (and not necessarily compact).

For clarity of exposition, throughout this section we will omit time dependence in the quantities of interest, writing n k (x), rather than n k (t, x).

The Compactness Criterion.

We begin the proof of Proposition 4.1 by setting out the compactness criterion of Jabin et al. We omit the proof, referring the reader to [2, Lemma 3.1], or [8, Proposition 4.1], for a detailed motivation and explanation thereof.

Lemma 4.4. There exists a family

(K h ) 0<h<1 of smooth, non-negative, symmetric functions on R d such that K h L 1 (R d ) ∼ |log h| as h → 0, and satisfying (10) |x||∇K h (x)| ≤ CK h (x),
such that the following holds. Let (n k ) k be a uniformly bounded sequence in L q ((0,

T ) × R d ), for some 1 ≤ q < ∞. If (∂ t n k ) k is uniformly bounded in L s (0, T ; W -1,s (R d )) with s ≥ 1 and lim sup k 1 K h L 1 (R d ) R 2d K h (x -y)|n k (x) -n k (y)| q dx dy -→ 0, as h → 0, then (n k ) k is compact in L q loc ((0, T ) × R d ). Conversely, if (n k ) k is compact in L q loc ((0, T ) × R d ),
then the above quantity vanishes as h → 0.

We refer the reader to the Appendix, where we provide further details of a possible choice for the kernels K h . The above lemma is the cornerstone of our compactness argument for the individual species and the total population. However, rather than directly investigating the quantity introduced in the above lemma, let us first consider a weighted version. The weights shall be chosen specifically to fit the problem at hand, as described below.

4.2 Definition of the Weights. We define the weights v k as solutions of the transport equation

   ∂v k ∂t -∇v k • ∇W k = -λB k v k , v k (0, x) = 1, (11)
where

B k = M|∇ 2 W k |.
Here λ is some non-negative constant which will be fixed later on. By M we denote the maximal operator, defined by

Mf (x) = sup 0<ǫ≤1 1 |B 0 (ǫ)| B 0 (ǫ) f (x + z) dz,
where B 0 (ǫ) denotes the Euclidean ball of radius ǫ centred at zero. Recall that for any p > 1, the maximal function is a bounded operator on L p , and we have the following inequality (see, e.g., [START_REF] Stein | Maximal functions: Poisson integrals on symmetric spaces[END_REF])

|Φ(x) -Φ(y)| ≤ C|x -y|(M|∇Φ|(x) + M|∇Φ|(y)), for any Φ ∈ W 1,1 (R d ).
Note that, due to the estimates in Lemma 3.2 and Eq. (3b), we have that B k , defined in Eq. ( 11), is uniformly bounded in L 2 ((0, T ) × R d ). This allows us to deduce the following properties of the weight v k . Proposition 4.5. Let us assume that ∇W k is given and that it is bounded in

L 2 loc ((0, T ) × R d ) ∩ L ∞ (0, T ; H 1 (R d ))
uniformly in k. Thus, there exists a unique solution to Eq. [START_REF] Carrillo | Splitting schemes and segregation in reaction cross-diffusion systems[END_REF]. Moreover, we have

(i) 0 ≤ v k (t, x) ≤ 1, for almost every (t, x) ∈ (0, T ) × R d .
(ii) If we assume moreover that (n

(i) k , W k ) satisfies System (3) and n (i) k is uniformly bounded in L 2 ((0, T ) × R d ), there exists C > 0, such that (12) R d n (i) k | log v k | dx ≤ Cλ. Proof. (i) Since ∇ 2 W k ∈ L 2 ((0, T ) × R d ), we have that B k ∈ L 2 ((0, T ) × R d ). Since also ∇W k ∈ L 2 loc ((0, T ) × R d ) ∩ L ∞ (0, T ; H 1 (R d ))
, by standard theory of renormalised solutions to the transport equations, cf. [START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF], we may construct a non-negative solution to Eq. [START_REF] Carrillo | Splitting schemes and segregation in reaction cross-diffusion systems[END_REF]. Moreover, since

B k is non-negative, we clearly have that v k ≤ 1, since it is satisfied initially. (ii) From part (i), we have | log v k | = -log v k .
By renormalisation of Eq. ( 11), we have

∂ ∂t | log v k | -∇W k • ∇| log v k | = λB k .
Therefore, using also the continuity equations, Eqs. (3), we get

∂ ∂t n (i) k | log v k | -∇ • n (i) k | log v k |∇W k = n (i) k | log v k |G (i) (p k ) + λn (i) k B k .
Upon integrating in space and using the assumptions on G (i) , cf. Eq. ( 8), we deduce

d dt R d n (i) k | log v k | dx ≤ G (i) (0) R d n (i) k | log v k | dx + λ R d n (i) k B k dx.
Using Gronwall's lemma, we obtain

R d n (i) k | log v k |(T, x) dx ≤ λe G (i) (0)T T 0 R d n (i) k B k dx dt.
Finally, since B k and n

(i)
k are uniformly bounded in L 2 ((0, T ) × R d ), we conclude using the Cauchy-Schwarz inequality. Note that this part of the proposition is also true for the total population n k , replacing "G (i) (0)" with "max i=1,2 G (i) (0)" in the proof.

Propagation of Regularity for the Transport Equation.

Here we carry out a preliminary calculation, which will be used in the following subsection. We first compute the equation satisfied by the difference n

(i) k (x) -n (i) k (y) . Using the equation for n (i) k , we get ∂ ∂t n (i) k (x) -n (i) k (y) + ∇ x • -∇W k (x) n (i) k (x) -n (i) k (y) + ∇ y • -∇W k (y) n (i) k (x) -n (i) k (y) = - 1 2 (∆W k (x) + ∆W k (y)) n (i) k (x) -n (i) k (y) + 1 2 (∆W k (x) -∆W k (y)) n (i) k (x) + n (i) k (y) + n (i) k (x)G (i) (p k (x)) -n (i) k (y)G (i) (p k (y)) .
Upon multiplying by σ (i) := sign n

(i) k (x) -n (i) k (y) , we deduce ∂ ∂t n (i) k (x) -n (i) k (y) + ∇ x • -∇W k (x) n (i) k (x) -n (i) k (y) + ∇ y • -∇W k (y) n (i) k (x) -n (i) k (y) = - 1 2 (∆W k (x) + ∆W k (y)) n (i) k (x) -n (i) k (y) + 1 2 (∆W k (x) -∆W k (y)) n (i) k (x) + n (i) k (y) σ (i) + n (i) k (x)G (i) (p k (x)) -n (i) k (y)G (i) (p k (y)) σ (i) . (13)
Similarly, with σ := sign(n k (x) -n k (y)), we obtain

∂ ∂t |n k (x) -n k (y)| + ∇ x • (-∇W k (x)|n k (x) -n k (y)|) + ∇ y • (-∇W k (y)|n k (x) -n k (y)|) = 1 2 (∆W k (x) + ∆W k (y))|n k (x) -n k (y)| + 1 2 (∆W k (x) -∆W k (y))(n k (x) + n k (y))σ + n (1) k (x)G (1) (p k (x)) -n (1)
k (y)G (1) (p k (y)) + n (2) k (x)G (2) (p k (x)) -n (2) k (y)G (2) (p k (y)) σ,
for the total population. The above computations can be made rigorous, for a fixed k, using the renormalisation technique of DIPERNA & LIONS, cf. [START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF].

Calculation towards Gronwall's Lemma

For i = 1, 2, we now introduce

R (i) h (t) := R 2d K h (x -y) n (i) k (x) -n (i) k (y) (v k (x) + v k (y)) dx dy,
where the weights v k satisfy Eq. [START_REF] Carrillo | Splitting schemes and segregation in reaction cross-diffusion systems[END_REF]. Similarly, for the total population n k = n

(1)

k + n (2)
k , we define

R (tot) h (t) := R 2d K h (x -y)|n k (x) -n k (y)|(v k (x) + v k (y)) dx dy.
Using Eq. ( 13) we have, for i = 1, 2,

d dt R (i) (t) = 2 R 2d K h n (i) k (x) -n (i) k (y) ∂v k ∂t (x) dx dy -2 R 2d K h (x -y)∆W k (x) n (i) k (x) -n (i) k (y) (v k (x) + v k (y)) dx dy + R 2d K h (x -y)(∆W k (x) -∆W k (y)) n (i) k (x) + n (i) k (y) σ (i) v k (x) dx dy + R 2d K h (x -y) n (i) k (x)G (i) (p k (x)) -n (i) k (y)G (i) (p k (y)) σ (i) (v k (x) + v k (y)) dx dy - R 2d ∇ x K h (x -y) • (∇W k (x) -∇W k (y)) n (i) k (x) -n (i) k (y) (v k (x) + v k (y)) dx dy - R 2d K h (x -y)(∇W k (x) • ∇v k (x) + ∇W k (y) • ∇v k (y)) n (i) k (x) -n (i) k (y) dx dy.
Upon integrating by parts and using symmetry of the kernel K h , we obtain

d dt R (i) (t) = - R 2d ∇ x K h (x -y) • (∇W k (x) -∇W k (y)) n (i) k (x) -n (i) k (y) (v k (x) + v k (y)) dx dy + 2 R 2d K h (x -y) n (i) k (x) -n (i) k (y) ∂v k ∂t (x) -∇W k (x) • ∇v k (x) dx dy + 2 R 2d K h (x -y)(∆W k (x) -∆W k (y))n (i) k (x)σ (i) v k (x) dx dy -2 R 2d K h (x -y)∆W k (x) n (i) k (x) -n (i) k (y) v k (x) dx dy + R 2d K h (x -y) n (i) k (x)G (i) (p k (x)) -n (i) k (y)G (i) (p k (y)) σ (i) (v k (x) + v k (y)) dx dy.
Using Brinkman's law and the a priori bounds on p k and W k , we therefore obtain

(14) d dt R (i) (t) ≤ A (i) 1 + A (i) 2 + A (i) 3 + ν -1 CR (i) (t) + React (i) ,
where

A (i) 1 = - R 2d ∇K h (x -y) • (∇W k (x) -∇W k (y)) n (i) k (x) -n (i) k (y) (v k (x) + v k (y)) dx dy, A (i) 2 = 2 R 2d K h (x -y) n (i) k (x) -n (i) k (y) ∂v k ∂t (y) -∇W k (y) • ∇v k (y) dx dy, A (i) 3 = 2 R 2d K h (x -y)(∆W k (x) -∆W k (y))n (i) k (x)σ (i) v k (x) dx dy,
as well as

React (i) = R 2d K h (x -y) n (i) k (x)G (i) (p k (x)) -n (i) k (y)G (i) (p k (y)) σ (i) (v k (x) + v k (y)) dx dy.
Similarly, we obtain for the total population, (

) d dt R (tot) (t) ≤ A (tot) 1 + A (tot) 2 + A (tot) 3 + ν -1 CR (tot) (t) + React (tot) , 15
where

A (tot) 1 = - R 2d ∇K h (x -y) • (∇W k (x) -∇W k (y))|n k (x) -n k (y)|(v k (x) + v k (y)) dx dy, A (tot) 2 = 2 R 2d K h (x -y)|n k (x) -n k (y)| ∂v k ∂t (y) -∇W k (y) • ∇v k (y) dx dy, A (tot) 3 = 2 R 2d K h (x -y)(∆W k (x) -∆W k (y))n k (x)σv k (x) dx dy, and
React (tot) = R 2d K h (x -y) 2 i=1 n (i) k (x)G (i) (p k (x)) -n (i) k (y)G (i) (p k (y)) σ(v k (x) + v k (y)) dx dy.
In what follows, we consider the quantity

R(t) := R (1) (t) + R (2) (t) + R (tot) (t).
As it turns out, this quantity allows for convenient cancellations in the reaction terms, and in the terms "A 3 ". This strategy is reminiscent of the BV-norm propagation argument employed in the one-dimensional case, cf. [START_REF] Dębiec | Incompressible limit for a two-species tumour model with coupling through Brinkman's law in one dimension[END_REF]. Accordingly, we define the following notation

A j = A (1)
j + A (2)
j + A (tot) j
, for j = 1, 2, 3, and React = React (1) + React (2) + React (tot) .

Combining Eq. ( 14) and Eq. ( 15) yields

(16) d dt R(t) ≤ A 1 + A 2 + A 3 + ν -1 CR(t) + React.
The terms A 1 and A 2 are (almost) the same as in previous works, cf. [START_REF] Bresch | Global weak solutions of PDEs for compressible media: a compactness criterion to cover new physical situations[END_REF][START_REF] Bresch | Global existence of weak solutions for compressible Navier-Stokes equations: thermodynamically unstable pressure and anisotropic viscous stress tensor[END_REF][START_REF] Vauchelet | Incompressible limit of the Navier-Stokes model with a growth term[END_REF], and can be handled using only the uniform bound on the velocity field -∇W k . We therefore begin by estimating the reaction terms and the term A 3 .

Treating the Reaction Terms.

First let us address the contribution coming from the reaction terms, React (tot) . Recalling the notation σ (i) = sign(n

(i) k (x) -n (i)
k (y)), we obtain for the two individual species

react (i) := n (i) k (x)G (i) (p k (x)) -n (i) k (y)G (i) (p k (y) σ (i) = G (i) (p k (x)) n (i) k (x) -n (i) k (y) + G (i) (p k (x)) -G (i) (p k (y)) n (i) k (y)σ (i) ≤ G (i) ∞ n (i) k (x) -n (i) k (y) + G (i) (p k (x)) -G (i) (p k (y)) n (i) k (y),
having used the fact that |σ (i) | ≤ 1.

Similarly, an expression for the sum, n k = n

(1)

k + n (2)
k , may be obtained, i.e., react (tot) := n

(1)

k (x)G (1) (p k (x)) -n (1)
k (y)G (1) (p k (y) + n

(2)

k (x)G (2) (p k (x)) -n (2) k (y)G (2) (p k (y) σ = G (1) (p k (x)) n (1) k (x) -n (1)
k (y) σ + G (1) (p k (x)) -G (1) (p k (y)) n (1) k (y)σ + G (2) (p k (x)) n (2) k (x) -n (2)
k (y) σ + G (2) (p k (x)) -G (2) (p k (y)) n (2) k (y)σ ≤ G (1) ∞ n (1) k (x) -n (1)
k (y) + G (2) ∞ n (2) k (x) -n (2)
k (y) + G (1) (p k (x)) -G (1) (p k (y)) n (1)
k (y)σ + G (2) (p k (x)) -G (2) (p k (y)) n (2)
k (y)σ. Using the fact that σ := sign(n k (x) -n k (y)) = sign(p k (x) -p k (y)), and that the functions G (i) are decreasing, it is readily verified that adding up all three contributions yields react (1) + react (2)

+ react (tot) ≤ C n (1) k (x) -n (1) k (y) + n (2) k (x) -n (2)
k (y) , for some constant C > 0, independent of k. It follows that the whole term can be estimated as React ≤ CR(t).

Treatment of the Highest Order Terms.

We now address the contribution of the term A 3 . Recall that

A 3 = R 2d K h (x -y)(∆W k (x) -∆W k (y)) 2 i=1 σ (i) n (i) k (x)v k (x) dx dy + R 2d K h (x -y)(∆W k (x) -∆W k (y))σn k (x)v k (x) dx dy.
Next, notice that

a (i) 3 := (∆W k (x) -∆W k (y))σ (i) n (i) k (x) = ν -1 (W k (x) -p k (x) -W k (y) + p k (y))σ (i) n (i) k (x) ≤ ν -1 |W k (x) -W k (y)| n (i) k L ∞ + ν -1 |p k (x) -p k (y)|n (i) k (x),
having used the fact that σ (i) ≤ 1, as before. Finally, the term stemming from the joint population reads a (tot) 3

:= (∆W k (x) -∆W k (y))σn k (x) = ν -1 (W k (x) -p k (x) -W k (y) + p k (y))σn k (x) = ν -1 σ(W k (x) -W k (y))n k (x) -ν -1 |p k (x) -p k (y)|n k (x) ≤ ν -1 |W k (x) -W k (y)| n k L ∞ -ν -1 |p k (x) -p k (y)|n k (x).
Again, using the a-priori bounds, it is immediate that adding up all three terms yields a

(1) 3 + a (2) 3 + a (tot) 3 ≤ ν -1 C|W k (x) -W k (y)|, due to the cancellation n (1) k + n (2)
k -n k = 0. Therefore we obtain the following bound

A 3 ≤ ν -1 C R 2d K h (x -y)|W k (x) -W k (y)|v k (x) dx dy. ( 17
)
This concludes the estimate of the highest order term for now. Let us just remark at this stage that the quantity on the right-hand side vanishes, as h → 0, due to local compactness of W k and the "converse" statement of Lemma 4.4.

4.7

Estimates of A 1 and A 2 . For the readers' convenience we recall how to estimate the remaining terms in Eq. ( 16). First, we make use of the following inequality

|∇W k (x) -∇W k (y)| ≤ C|x -y| D |x-y| ∇W k (x) + D |x-y| ∇W k (y) ,
where

D δ ∇W k (x) = 1 δ |z|≤δ ∇ 2 W k (x + z) |z| d-1
dz, see, for example, [START_REF] Jabin | Differential equations with singular fields[END_REF]Lemma 3.1]. We recall also that D δ ∇W k ≤ M ∇ 2 W k , Then, using the inequality in Eq. ( 10) and the symmetry of K h we get

A 1 ≤ C R 2d |x -y||∇K h (x -y)| D |x-y| ∇W k (x) + D |x-y| ∇W k (y) 2 i=1 n (i) k (x) -n (i) k (y) + |n k (x) -n k (y)| (v k (x) + v k (y)) dx dy ≤ C R 2d K h (x -y) D |x-y| ∇W k (x) + D |x-y| ∇W k (y) 2 i=1 n (i) k (x) -n (i) k (y) + |n k (x) -n k (y)| v k (y) dx dy.
Using that

D |x-y| ∇W k (x) + D |x-y| ∇W k (y) = D |x-y| ∇W k (x) -D |x-y| ∇W k (y) + 2D |x-y| ∇W k (y),
and changing the variables z = x -y, we may apply the Cauchy-Schwarz inequality and use the uniform L 2 -bounds on n

(i)
k and n k to deduce

A 1 ≤ C R d K h (z) D |z| ∇W k (•) -D |z| ∇W k (• + z) L 2 dz + C R 2d K h (x -y)D |x-y| ∇W k (y) 2 i=1 n (i) k (x) -n (i) k (y) v k (y) dx dy + C R 2d K h (x -y)D |x-y| ∇W k (y)|n k (x) -n k (y)|v k (y) dx dy.
We may bound D |x-y| ∇W k by the maximal operator M ∇ 2 W k , thus

A 1 ≤ C R d K h (z) D |z| ∇W k (•) -D |z| ∇W k (• + z) L 2 dz + C R 2d K h (x -y)M ∇ 2 W k (y) 2 i=1 n (i) k (x) -n (i) k (y) v k (y) dx dy + C R 2d K h (x -y)M ∇ 2 W k (y) |n k (x) -n k (y)|v k (y) dx dy. ( 18
)
The last two terms on the right-hand side of inequality (18) will be controlled by the term A 2 and the equation for the weight v k .

From Eq. ( 11), we have

A 2 = R 2d K h (x -y) 2 i=1 n (i) k (x) -n (i) k (y) (-2λB k (y))v k (y) dx dy + R 2d K h (x -y)|n k (x) -n k (y)|(-2λB k (y))v k (y) dx dy.
Therefore, combining the latter equality with Eq. ( 18), we deduce

A 1 + A 2 ≤ C R d K h (z) D |z| ∇W k (•) -D |z| ∇W k (• + z) L 2 dz + R 2d K h (x -y) 2 i=1 n (i) k (x) -n (i) k (y) v k (y) CM|∇ 2 [0.5em]W k (y)| -2λB k (y) dx dy + R 2d K h (x -y)|n k (x) -n k (y)|v k (y) CM|∇ 2 W k (y)| -2λB k (y) dx dy.
From the choice of B k in Eq. ( 11), we can find λ large enough such that

A 1 + A 2 ≤ C R d K h (z) D |z| ∇W k (•) -D |z| ∇W k (• + z) L 2 dz.
We shall now employ the following lemma, which we state without proof, referring the reader to [START_REF] Bresch | Global existence of weak solutions for compressible Navier-Stokes equations: thermodynamically unstable pressure and anisotropic viscous stress tensor[END_REF].

Lemma 4.6 (Lemma 6.3 in [START_REF] Bresch | Global existence of weak solutions for compressible Navier-Stokes equations: thermodynamically unstable pressure and anisotropic viscous stress tensor[END_REF]). There exists a constant, C > 0, such that for any u ∈ H 1 (R d ),

R d K h (z) D |z| u(•) -D |z| u(• + z) L 2 (R d ) dz ≤ C| log h| 1/2 u H 1 (R d ) .
In our case we take u = ∇W k , which indeed belongs to H 1 , uniformly. Thus, we have the estimate

A 1 + A 2 ≤ C| log h| 1/2 W k W 2,2 (R d ) ≤ C| log h| 1/2 . ( 19
)
4.8 Conclusions from the Estimates. Departing from Eq. ( 16), we obtain, in conjunction with Estimate (17) (coming from the term A 2 ) and Estimate [START_REF] Jabin | Differential equations with singular fields[END_REF] (coming from the terms A 1 and A 3 ),

d dt R h (t) ≤ C R h (t) + C| log h| 1/2 + ν -1 C R 2d K h (x -y)|W k (x) -W k (y)|v k (x) dx dy.
Then, integrating in time, we obtain, for all t ∈ [0, T ],

e -Ct R h (t) ≤ R h (0) + C T | log h| 1/2 + ν -1 C T T 0 R 2d K h (x -y)|W k (x) -W k (y)|v k (x) dx dy dt. ( 20
)
Regarding the last term we observe that

ν -1 C T T 0 R 2d K h (x -y)|W k (x) -W k (y)|v k (x) dx dy dt ≤ ν -1 C T T 0 R 2d K h (x -y)|W k (x) -W k (y)| dx dy dt. Since (W k ) k is locally compact in L 1 ((0, T ) × R d
), the right-most integral vanishes in the limit h → 0, due to the "converse" statement of Lemma 4.4.

Removing the Weights and Application of the Compactness

Argument. Let 0 < η < 1.
We define

ω η = {x ∈ R d : v k (x) ≤ η},
and denote by ω c η its complement in R d . We then split integration over the domain R 2d as follows

R 2d K h (x -y)|n k (x) -n k (y)| dx dy = {x∈ω c η }∪{y∈ω c η } K h (x -y)|n k (x) -n k (y)| dx dy + {x∈ωη}∩{y∈ωη} K h (x -y)|n k (x) -n k (y)| dx dy. ( 21
)
These integrals can now be estimated separately, in turn we have

{x∈ω c η }∪{y∈ω c η } K h (x -y)|n k (x) -n k (y)| dx dy ≤ 2 η R h (t),
and {x∈ωη}∩{y∈ωη} K h (x -y)|n k (x) -n k (y)| dx dy ≤ 2 {x∈ωη}∩{y∈ωη } K h (x -y)n k (x) dx dy ≤ 2 R d K h (z) dz {x∈ωη} n k (x) dx ≤ C|log h| {x∈ωη} n k (x) dx,
where we used the symmetry of K h and the fact that K h L 1 ∼ |log h|. In order to treat the last integral we observe that

{x∈ωη} n k (x) dx ≤ {x∈ωη} n k (x) | log v k (x)| | log η| dx ≤ C 1 | log η| ,
where the last inequality holds due to the uniform bound on n k | log v k |, cf. Eq. ( 12), Proposition 4.5, and the fact that, for η < 1, | log v k (x)| ≥ | log η|, whenever x ∈ ω η . Substituting these estimates into Eq. ( 21), we arrive at

R 2d K h (x -y)|n k (x) -n k (y)| dx dy ≤ 2 η R h (t) + C|log h| | log η| .
Therefore, from Eq. ( 20), we deduce

R 2d K h (x -y)|n k (x) -n k (y)| dx dy ≤ C T η R h (0) + | log h| 1/2 + C T νη T 0 R 2d K h (x -y)|W k (x) -W k (y)| dt dx dy + C|log h| | log η| .
Since we have K h L 1 ∼ | log h|, we obtain

1 K h L 1 R 2d K h (x -y)|n k (x) -n k (y)| dx dy ≤ C T η R h (0)| log h| -1 + | log h| -1/2 + C T νη| log h| T 0 R 2d K h (x -y)|W k (x) -W k (y)| dt dx dy + C | log η| .
Choosing η = | log h| -1/4 , we have η → 0 as h → 0. Then

1 K h L 1 R 2d K h (x -y)|n k (x) -n k (y)| dx dy ≤ C T R h (0)| log h| -3/4 + | log h| -1/4 + C T | log h| 3/4 T 0 R 2d K h (x -y)|W k (x) -W k (y)| dt dx dy + C | log | log h|| .
Of course, the above calculations are oblivious to replacing n k with n (i)

k . Hence, finally, we obtain the desired compactness of the sequences (n k ) k and (n

(i) k ) k , i = 1, 2,
as stated in Theorem 4.1, by applying the compactness criterion in Lemma 4.4. Indeed the estimate on the time derivative is a direct consequence of the a priori estimates.

Proof of Corollary 4.2. From Remark 3.4 in conjunction with the fact that 0 ≤ r k ≤ 1, we infer the boundedness of the sequence

(r k ) k in L ∞ (0, T ; L 1 loc ∩ L ∞ loc (R d )). Since both n (i)
k and n k converge not only strongly but also pointwise, we deduce the strong compactness of the sequence (r k ) k in any L p (0, T ; L q loc (R d )), for 1 ≤ p, q < ∞.

STRONG COMPACTNESS OF THE PRESSURE

In order to be able to pass to the incompressible limit, stated in Theorem 2.3, we require strong convergence of the pressure sequence (p k ) k . As pointed out before, cf. Remark 4.3, we cannot hope to propagate initial compactness for the pressure and therefore the strategy of Section 4 cannot be employed. Accordingly, a different approach has to be found. The main challenge in this endeavour is the absence of any estimates on the derivative of p k , unlike in the inviscid case, cf. [START_REF] Bubba | Hele-shaw limit for a system of two reaction-(cross-)diffusion equations for living tissues[END_REF][START_REF] Kim | Porous medium equation to Hele-Shaw flow with general initial density[END_REF][START_REF] Mellet | A Hele-Shaw problem for tumor growth[END_REF][START_REF] Perthame | Derivation of a hele-shaw type system from a cell model with active motion[END_REF][START_REF] Perthame | The Hele-Shaw asymptotics for mechanical models of tumor growth[END_REF]. To remedy this, we follow the idea of kinetic reformulations developed in [START_REF] Perthame | Incompressible limit of a mechanical model of tumour growth with viscosity[END_REF], and thus investigate possible oscillations of the sequence of pressures. Let us recall the equation satisfied by the pressure, p k , i.e.,

∂p k ∂t -∇p k • ∇W k = k -1 ν p k W k -p k + νr k G (1) (p k ) + ν(1 -r k )G (2) (p k ) = k -1 ν p k Q k . (22)
We now introduce a family of functions {H r } r as follows. Let H r (W ) := H(r, W ) := Id -νrG (1)

(•) -ν(1 -r)G (2) (•) -1 (W ),
where the inverse is to be understood in the p-variable. For a fixed r, we then have that H r is increasing and has a bounded derivative, i.e., H ′ r < C, for some C > 0. Moreover, notice that for any fixed r ∈ [0, 1] we have H r (0) > 0. We point out that, for the subsequent analysis, it will prove useful to introduce the number

p m := min r∈[0,1] H(r, 0) > 0. (23)
The remainder of this section is dedicated to proving the following result. Lemma 5.1 (Strong convergence of p k ). Up to a subsequence, p k converges strongly in L 1 loc (0, T ) × R d towards a function p ∞ . Moreover, the limit function is characterised by p ∞ = H(r ∞ , W ∞ )½ {p∞>0} a.e.. Furthermore, we have that

Ω(t) = {p ∞ (•, t) = H(r ∞ , W ∞ (•, t))} = R d \{p ∞ (•, t) = 0}, is the image of Ω 0 := {p ∞ (t = 0) > 0} by the limiting flow Y (x) (t), defined by    d dt Y (x) (t) = -∇W ∞ (t, Y (x) (t)), Y (x) (t = 0) = Id .
Finally, we have for all T > 0,

T 0 R d p k (t, x)|Q k (t, x)| dx dt -→ k→+∞ 0. (24) k

Proof of the Lemma.

Step 1. Representation of Nonlinear Weak Limits. We follow closely the proof in [START_REF] Perthame | Incompressible limit of a mechanical model of tumour growth with viscosity[END_REF], making appropriate adaptations to our current case. To this end, we need a representation of weak limits of p k which we can obtain thanks to a kinetic representation.

Our first result is to establish the existence of a measurable function 0 ≤ f (t, x) ≤ 1 such that, for all smooth functions S : [0, ∞) → R, we have, up to a subsequence,

S(p k ) ⇀ k→+∞ S(0)(1 -f ) + S(H(r ∞ , W ∞ ))f, in the weak-⋆ sense in L ∞ ((0, T ) × R d ) and (26a) S(0)(1 -f ) + S(H(r ∞ , W ∞ ))f = S(0) + ∞ 0 S ′ (ξ)χ(ξ) dξ, where (26b) χ(t, x; ξ) = f (t, x)½ {0<ξ<H(r∞(t,x),W∞(t,x))} . (25)
In particular, notice that for S(p) = p, we find

(27) p ∞ = f H(r ∞ , W ∞ ).
In order to prove these results, we define

χ k (t, x; ξ) = ½ {0<ξ<p k (t,x)} ,
and we write

(28) S(p k ) -S(0) = ∞ 0 S ′ (ξ)χ k (t, x; ξ) dξ.
We can extract a subsequence, still denoted

(p k ) k , such that {χ k } k converges in the weak-⋆ sense in L ∞ ((0, ∞) × R d ) towards a function χ(t, x; ξ) which satisfies 0 ≤ χ(t, x; ξ) ≤ 1.
We define f to be the weak-⋆ limit of ½ {p k (t,x)≥pm/2} , i.e.,

f (t, x) := weak-⋆ -lim ½ {p k (t,x)≥pm/2} ,
where we recall that p m is defined in Eq. [START_REF] Perthame | Derivation of a hele-shaw type system from a cell model with active motion[END_REF].

To prove the convergence in Eq. ( 25) we first split S(p k ) as follows

S(p k ) = S(p k )½ {p k <pm/2} + S(p k )½ {p k ≥pm/2} ,
and pass to the weak-⋆ limit in each term separately. Let now φ ∈ C ∞ c ((0, T ) × R d ) be a test function. We then have

T 0 R d φS(p k )½ {p k <pm/2} dx dt - T 0 R d φS(0)(1 -f ) dx dt ≤ T 0 R d φ(S(p k ) -S(0))½ {p k <pm/2} dx dt + T 0 R d φS(0) ½ {p k <pm/2} -(1 -f ) dx dt , (29)
and the last term converges to zero by definition of f . To deal with the other term on the righthand side of Eq. ( 29) we proceed as follows.

T 0 R d φ(S(p k ) -S(0))½ {p k <pm/2} dx dt ≤ φ L ∞ S ′ L ∞ T 0 R d p k ½ {p k ≤pm/2} dx dt.
Notice that H(r ∞ , W ∞ ) > p m . Therefore on the set {p k ≤ p m /2} we have

p k ≤ H(r ∞ , W ∞ ) -p m /2,
and thus

H -1 r∞ (p k ) ≤ H -1 r∞ (H(r ∞ , W ∞ ) -p m /2) ≤ W ∞ -ω(p m ), for some ω(p m ) > 0.
In particular, on the set {p k ≤ p m /2} we have

1 ≤ 1 ω(p m ) W ∞ -H -1 r∞ (p k ) . Therefore T 0 R d p k ½ {p k ≤pm/2} dx dt ≤ C T 0 R d p k W ∞ -H -1 r∞ (p k ) dx dt ≤ C T 0 R d p k |W ∞ -W k | dx dt + C T 0 R d p k W k -H -1 r k (p k ) dx dt + C T 0 R d p k H -1 r k (p k ) -H -1 r∞ (p k ) dx dt. (30)
Recalling that p k ≤ P M , and using the definition of the function H r (•), we arrive at

T 0 R d p k ½ {p k ≤pm/2} dx dt ≤ C T 0 R d P H |W ∞ -W k | dx dt + C T 0 R d p k |Q k | dx dt + C T 0 R d P H νG (1) (0) + νG (2) (0) |r k -r ∞ | dx dt.
Note that the last three integrals vanish in the limit k → ∞. In fact, the first integral converges to zero due to strong convergence of W k , the second one due to Lemma 3.5, and third integral vanishes due to strong convergence of r k , cf. Corollary 4.2.

Similarly, we have

T 0 R d φS(p k )½ {p k ≥pm/2} dx dt - T 0 R d φS(H(r ∞ , W ∞ ))f dx dt ≤ T 0 R d |φ||S(p k ) -S(H(r ∞ , W ∞ ))|½ {p k ≥pm/2} dx dt + T 0 R d φS(H(r ∞ , W ∞ )) f -½ {p k ≥pm/2} dx dt ,
and again we need only work on the first term on the right-hand side. We have

T 0 R d |φ||S(p k ) -S(H(r ∞ , W ∞ ))|½ {p k ≥pm/2} dx dt ≤ C T 0 R d |S(p k ) -S(H(r k , W k ))|½ {p k ≥pm/2} dx dt + C T 0 R d |S(H(r k , W k )) -S(H(r k , W ∞ ))|½ {p k ≥pm/2} dx dt + C T 0 R d |S(H(r k , W ∞ )) -S(H(r ∞ , W ∞ ))|½ {p k ≥pm/2} dx dt. ( 31
)
For the first term on the right-hand side of the last inequality we write

T 0 R d |S(p k ) -S(H(r k , W k ))|½ {p k ≥pm/2} dx dt ≤ S ′ L ∞ T 0 R d |p k -H(r k , W k )|½ {p k ≥pm/2} dx dt ≤ S ′ L ∞ H ′ r k L ∞ T 0 R d H -1 r k (p k ) -W k ½ {p k ≥pm/2} dx dt ≤ S ′ L ∞ H ′ r k L ∞ 2 p m T 0 R d p k H -1 r k (p k ) -W k dx dt = S ′ L ∞ H ′ r k L ∞ 2 p m T 0 R d p k |Q k | dx dt ≤ C(T ) k 0, (32)
as k → ∞, having applied Lemma 3.5 and used the fact that H ′ r k is uniformly bounded due to 0 ≤ r k ≤ 1. In a similar fashion the last two terms in Eq. (31) vanish in the limit k → ∞, due to strong convergence of W k and r k , respectively. This concludes the proof of the representation of weak limits, cf. Eq. [START_REF] Perthame | Incompressible limit of a mechanical model of tumour growth with viscosity[END_REF]. The final statement, Eq. (26b), follows from passing to the limit k → ∞ in the identity

S(0) + ∞ 0 S ′ (ξ)χ k (t, x; ξ) dξ = S(p k ) = S(p k )½ {p k (t,x)≥pm/2} + S(p k )½ {p k ≤pm/2} .
Using the fact that p k does not oscillate we deduce

S(0) + ∞ 0 S ′ (ξ)χ(t, x; ξ) dξ = S(H(r ∞ , W ∞ ))f + S(0)(1 -f ).

Rearranging this expression yields

f (t, x) ∞ 0 S ′ (ξ)½ {0≤ξ≤H(r∞,W∞)} (ξ) dξ = ∞ 0 S ′ (ξ) dξ,
whence χ(t, x; ξ) = f (t, x)½ {0≤ξ≤H(r∞,W∞)} , as claimed.

Proof of Lemma.

Step 2. Equation satisfied by χ k . Let S ∈ C 2 ([0, ∞)). Upon rewritting Eq. ( 22) in divergence form and multiplying by S ′ (p k ) we obtain

∂ ∂t S(p k ) -∇ • (S(p k )∇W k ) + S ′ (p k ) W k -p k ν = k -1 ν p k Q k S ′ (p k ) = ∞ 0 S ′ (ξ)µ k (dξ),
where the measure µ k is defined via

µ k (t, x; ξ) := k -1 ν p k Q k δ {ξ=p k } = k -1 ν p k W k -p k + νr k G (1) (p k ) + ν(1 -r k )G (2) (p k ) δ {ξ=p k } . (33)
Then, using Eq. ( 28) and the fact that

S(p k )p k = ∞ 0 S(ξ) + ξS ′ (ξ) χ k dξ, we have (34) ∞ 0 S ′ (ξ) ∂χ k ∂t -∇ • (χ k ∇W k ) + χ k W k -ξ ν dξ - ∞ 0 S(ξ) -S(0) ν χ k dξ = ∞ 0 S ′ (ξ)µ k (dξ). Since χ k (ξ) = -∂ ξ ∞ ξ χ k (t, x; η) dη, integrating by parts yields ∞ 0 S(ξ) -S(0) ν χ k dξ = ∞ 0 S ′ (ξ) ν ∞ ξ χ k (t, x; η) dη dξ.
Therefore, Eq. ( 34) is equivalent to

(35) ∂χ k ∂t -∇ • (χ k ∇W k ) + χ k W k -ξ ν - 1 ν ∞ ξ χ k (t, x; η) dη = µ k .
One can simplify this relation and write

∂χ k ∂t -∇ • (χ k ∇W k ) + χ k W k -ξ ν - (p k -ξ) + ν = µ k .
Finally, Eq. ( 35) is equivalent to

∂χ k ∂t -∇ • (χ k ∇W k ) + χ k W k -p k ν = µ k .
In particular, integrating in ξ we recover the expected formula

∂p k ∂t -∇ • (p k ∇W k ) + p k ν [W k -p k ] = ∞ 0 µ k dξ.

Proof of Lemma.

Step 3. Equation satisfied by f . We may pass to the limit in Eq. ( 35). For all T > 0, the sequence (µ k ) k is uniformly bounded in L 1 (0, T ) × R d × R thanks to estimate from Lemma 3.5. Thus we may extract a subsequence, converging, in the weak sense of measures, towards a measure µ ∈ M b (0, T ) × R d × R . Due to the fact that (2) (ξ), which is positive for ξ ≤ p m , we have µ(t, x; ξ) ≥ 0, for ξ ≤ p m . Therefore passing to the limit k → +∞ in Eq. (35), we obtain, in the sense of distributions,

Q k (t, x; ξ) = W k -ξ + νr k G (1) (ξ) + ν(1 -r k )G
∂χ ∂t -∇ • (χ∇W ∞ ) + χ W ∞ -ξ ν - 1 ν ∞ ξ χ(t, x; η) dη = µ.
Using Eq. ( 26), this equation can also be written as

∂χ ∂t -∇ • (χ∇W ∞ ) + χ W ∞ -ξ ν -f (t, x) (H(r ∞ , W ∞ ) -ξ) + ν = µ,
and thus

(36) ∂χ ∂t -∇ • (χ∇W ∞ ) + χ W ∞ -H(r ∞ , W ∞ ) ν = µ.
Using the assumption of compactness of initial data, cf. Eq. (4b), this equation is complemented with the initial condition

χ(t = 0, x; ξ) = ½ Ω 0 (x) ½ {0<ξ<H(r∞(t=0,x),W∞(t=0,x))} (ξ), and
f (t = 0, x) = ½ Ω 0 (x) =: f 0 (x).
It is useful to keep in mind the equivalent form of this equation,

∂χ ∂t -∇χ • ∇W ∞ + χ p ∞ -H(r ∞ , W ∞ ) ν = µ ≥ 0,
and thus, using Eq. ( 27),

(37) ∂χ ∂t -∇χ • ∇W ∞ + χ H(r ∞ , W ∞ ) f -1 ν = µ ≥ 0.
We can also integrate Eq. ( 36) and recover

∂p ∞ ∂t -∇ • (p ∞ ∇W ∞ ) + p ∞ ν [W ∞ -H(r ∞ , W ∞ )] = ∞ 0 µ dξ.
It is useful to consider the function

g(t, x) = f 0 X (t,x) (s = 0) ,
with the characteristics defined by

   d ds X (t,x) (s) = -∇W ∞ (s, X (t,x) (s)), X (t,x) (t) = Id .
This function g is a solution, in the distributional sense, of the transport equation

∂g ∂t -∇g • ∇W ∞ = 0,
equipped with g 0 = f 0 . Using Eq. (37) and 0 ≤ f ≤ 1, we find

(38) ∂f ∂t -∇f • ∇W ∞ = µ(t, x; ξ) + χ H(r ∞ , W ∞ ) 1 -f ν ≥ 0.
From the comparison principle, we conclude not only that f (t, x) ≥ g(t, x), but also that on the set {g(t, x) = 1} there holds f (t, x) = 1, and, moreover, µ(t, x; ξ) = 0, whenever ξ < p m .

Proof of Lemma.

Step 4. Identification of the function f . Another wording for the conclusion of the previous step is that

Ω(t) = Y (x) (t)[Ω 0 ] = {p ∞ (•, t) > 0},
with Y (x) (t) being the limiting flow of the forward flow Y

(k) (x) (t) corresponding to the velocity field -∇W k , i.e.,      d dt Y (k) (x) (t) = -∇W k t, Y (k)
(x) (t) , Y (k)
(x) (t = 0) = Id . We point out that ∇W k is slightly less regular than uniformly Lipschitz. Nevertheless, the flow Y (k) (x) (t) is well-defined almost everywhere thanks to the DIPERNA-LIONS theory, cf. [START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF]. Observe that we have 39) and the strong convergence of the flow, we infer that

p k (t, x) = 0, (39) for x ∈ R d \ Ω k (t) where, Ω k (t) := Y (k) (x) (t) Ω 0 . From Eq. (
p ∞ (t, •) = 0, on the set Y (x) (t)[R d \ Ω 0 ]. Then we have f (t, x) = ½ Ω(t) = ½ {p∞(t,x)>0}
. We recall that, by definition, f = weak-⋆ -lim k→+∞ ½ {p k ≥pm/2} .

p k -H(r ∞ , W ∞ )½ {p∞>0} dx ≤ I k + II k + III k , with I k = T 0 U ½ {p k ≥pm/2} |p k -H(r ∞ , W ∞ )| dx, II k = T 0 U ½ {p k <pm/2} p k dx, III k = T 0 U H(r ∞ , W ∞ ) ½ {p k ≥pm/2} 1 -½ {p∞>0} + ½ {p k <pm/2} ½ {p∞>0} dx.
We have already shown that I k and II k converge to zero as k → ∞, cf. Eq. (32) and Eq. (30), respectively. Using the fact that W ∞ is bounded in L ∞ , the last term may be estimated

III k ≤ C U ½ {p k ≥pm/2} 1 -½ {p∞>0} + 1 -½ {p k ≥pm/2} ½ {p∞>0} dx,
for some non-negative constant C. We have shown above that ½ {p k ≥pm/2} converges weakly towards ½ {p∞>0} . Then passing to the limit k → +∞ in the latter inequality, we deduce that lim k→+∞ III k = 0. We conclude from the inequality in Eq. (40) that

T 0 U |p k -H(r ∞ , W ∞ )½ {p∞>0} | dx -→ k→+∞ 0,
for any open and bounded set U ⊂ R d . By uniqueness of the weak limit, we deduce that

p ∞ = H(r ∞ , W ∞ )½ {p∞>0} almost everywhere in (0, T ) × R d .
Finally, from definition of the measures µ k , cf. Eq. (33), the limit in Eq. ( 24) is now a consequence of

k T 0 R d p k (t, x)|Q k (t, x)| dx dt = k k -1 T 0 R d ∞ 0 |µ k (t, x; ξ)| dξ dx dt,
and the fact that µ k vanishes for k → ∞. Indeed, recall that from Eq. (38) we infer that µ = 0 both when f = 1 and f = 0. Therefore, we recover Eq. ( 24) which concludes the proof of Lemma 5.1.

INCOMPRESSIBLE LIMIT AND COMPLEMENTARITY RELATION

Now, the proof of the Theorem 2.3 is easily deduced from Lemma 5.1. First, up to a subsequence, we have that p k converges almost everywhere towards p ∞ . On the one hand, recalling that the sequence (p k ) k is uniformly bounded in L ∞ , we use the Lebesgue dominated convergence Theorem to show that, for any open and bounded set U ,

T 0 U p k W k -p k + νr k G (1) (p k ) + ν(1 -r k )G (2) (p k ) dx dt -→ k→+∞ T 0 U p ∞ W ∞ -p ∞ + νr ∞ G (1) (p ∞ ) + ν(1 -r ∞ )G (2) (p ∞ ) dx dt.
On the other hand, from estimate Lemma 3.5 we have that T 0 U p k W k -p k + νr k G (1) (p k ) + ν(1 -r k )G (2) (p k ) dx dt -→ k→+∞ 0.

We deduce that p ∞ p ∞ -W ∞ -νn (1) ∞ G (1) (p ∞ ) -νn (2) ∞ G (2) (p ∞ ) = 0, almost everywhere in (0, T ) × R d , which is the required complementarity relation. Since the sequences (n (i) k ) k , (G(p k )) k and (W k ) k converge strongly, we readily pass to the limit in Eq. (3a) and Eq. (3b), to obtain the claimed limiting equations. Finally, passing to the limit in the relation,

n k p k = k -1 k 1/(k-1)
p k/(k-1) k , we deduce that (1 -n ∞ )p ∞ = 0.

Keeping Lemma 3.3 in mind, we know that phase segregation is preserved by the equation, i.e., initially segregated data remains segregated for all times. This allows us to characterise the free boundary motion more precisely.

Remark 6.1 (Segregation Patches). In the case of segregated densities the limiting equation may be simplified on each populated patch. Indeed, we can define three disjoint subsets

Ω 0 = {p ∞ = 0}, Ω 1 = {p ∞ > 0, n (1)
∞ = 1}, and Ω 2 = {p ∞ > 0, n (2) ∞ = 1}. On Ω 0 , the limiting system reduces to -ν∆W ∞ + W ∞ = 0, coupled with the equations for n (i) ∞ . This, in turn, implies that ∇W ∞ = 0, whence

∂n (i) ∞ ∂t = n (i)
∞ G (i) (0), i.e., the two species grow exponentially, yet at different rates, depending on the growth term, in the absence of pressure.

On the sets Ω i , for i = 1, 2, the complementarity relation takes on the form W ∞ + νG (i) (p ∞ ) = 0, i.e., p ∞ = G (i) -1 (-W ∞ /ν), and it remains to solve the elliptic equation

-ν∆W ∞ + W ∞ = (G (i) ) -1 (-W ∞ /ν), on each patch Ω i , i = 1, 2.
In summary, we only have to solve the elliptic equation -ν∆W ∞ + F (W ∞ ) = 0, where

F (W ∞ ) = W ∞ , on Ω 0 , W ∞ -(G (i) ) -1 (-W ∞ /ν), on Ω i , i = 1, 2.
The speed of each interface is given by v = -∇W ∞ .

CONCLUSIONS

In this work we establish the incompressible limit for a viscous two-species tissue model in arbitrary dimension. The two species are linked through growth and death processes and an elliptic equation, the so-called Brinkman law. In addition, we were able to show the phase segregation property, i.e., the fact that the two species remain segregated if they were segregated initially. Our results extend those in the literature to higher dimensions and to two species. Albeit technical, the proof hinges on three key observations: the transport of regularity à la BRESCH & JABIN, the observation of incorporating the total population as in [START_REF] Dębiec | Incompressible limit for a two-species tumour model with coupling through Brinkman's law in one dimension[END_REF] as an additional or "auxiliary" variable, and the kinetic formulation of [START_REF] Perthame | Incompressible limit of a mechanical model of tumour growth with viscosity[END_REF]. We stress that no additional information is gained from the estimates on the total population, on the contrary, its purpose is to make up for a change in regularity in the individual species. The same strategy may also be employed in order to perform the rigorous limit for crossreacting species, for instance, to account for a fraction of normal cells that become abnormal upon cell division. As a matter of fact, the a priori estimates are obtained in the same way, and estimating the two individual species and the total population simultaneously leads to the same beneficial cancellations. Finally, the kinetic reformulation requires a re-definition of the function H, however its core properties remain unchanged. Therefore no additional conceptual difficulties are to be encountered. While the result of this paper answers an important question it gives rise to a range of new and interesting problems. One cannot help but wonder as to whether the compactness of the initial data is, indeed, necessary. In the one species case, this is not a requirement, and the equation has, in some sense, an intrinsic regularising effect. Since the method of BRESCH & JABIN has a flavour of propagating compactness the initial compactness is a strong requirement. It would therefore be interesting to understand if this requirement is only necessary for this type of argument or if it can be removed. estimate the divergence of the flux after an integration by parts. Finally, we introduce the family of functions

K h (x) = 1 h K s (x) ds s ,
which has the following properties

K h L 1 (R d ) ∼ |log h|,
and, for each η > 0, sup

h {|x|≥η} K h dx < ∞.

5. 5

 5 Proof of Lemma. Step 5. Strong Convergence of the Pressure. We show that this implies the strong convergence of p k in L 1 loc ((0, T ) × R d ) towards the function H(r ∞ , W ∞ )½ {p∞>0} . Let U be an open and bounded subset of R d , we have

ACKNOWLEDGEMENTS

This work was completed while T.D. was a visitor at Laboratoire Jacques-Louis Lions, whose kind hospitality he appreciates. T.D. recognises the support of the Polish National Agency for Academic Exchange and National Science Center (Poland), grant no 2018/30/M/ST1/00423. B.P. has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 740623). M.S. fondly acknowledges the support of the Fondation Sciences Mathématiques de Paris (FSMP) for the postdoctoral fellowship.

APPENDIX

For the readers' convenience we provide a short and formal recapitulation of the compactness criterion employed in this work to propagated the compactness of the two individual species as well as the total population. To the best of the authors' knowledge, the method was first introduced in [START_REF] Belgacem | Compactness for nonlinear continuity equations[END_REF] in the context of nonlinear transport equations. It hinges on the observation that (local) compactness in Lebesgue spaces is equivalent to controlling quantities like

as h → 0, for certain families of kernels K h . In particular, compactness in the sense of the well-known Frechet-Kolmogorov-Riesz shift property is implied as it can be shown that

For more details and a rigorous argument we refer the reader to [2, Lemma 3.1]. The key observation of their paper is that this quantity is propagated along the flow induced by the transport equations under consideration. A subsequent Gronwall-type estimate allows them to infer the compactness of solutions at any time solely from the compactness of the initial data. With this formal motivation at hand, we shall now present a possible construction of a suitable family of kernels as in the statement of Lemma 4.4, while refering the reader to [8, Section 4] for further guidance. As in the main body, we denote by B ǫ 0 (x 0 ) the Euclidean ball centred at the point x 0 ∈ R d with radius ǫ 0 > 0. Next, we define an auxiliary family, (K h ) h>0 , of smooth, non-negative, symmetric functions given by

. The constants c h are chosen in such a way that each K h is normalised, i.e., K h L 1 (R d ) = 1. One can then obtain the inequality

which holds for some non-negative constant C, independent of h, due to the specific choice of K h . We remark that this property is not essential as far as the compactness criterion is concerned. However, applied to conservation laws, the property is indispensable for this allows to