Noemi David
email: ndavid@

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

Mainak Mandal

AND Markus Schmidtchen
email: markus.schmidtchen@tu-dresden.de

A DEGENERATE CROSS-DIFFUSION SYSTEM AS THE INVISCID LIMIT OF A NONLOCAL TISSUE GROWTH MODEL

Keywords: 2010 Mathematics Subject Classification. 35K57, 47N60, 35B45, 35K55, 35K65, 35Q92; Keywords and phrases. Inviscid Limit, Brinkman-to-Darcy Limit, Tissue Growth, Parabolic-Hyperbolic Cross-Diffusion Systems;

In recent years, there has been a spike in interest in multi-phase tissue growth models. Depending on the type of tissue, the velocity is linked to the pressure through Stoke's law, Brinkman's law or Darcy's law. While each of these velocity-pressure relations has been studied in the literature, little emphasis has been placed on the fine relationship between them. In this paper, we want to address this dearth of results in the literature providing a rigorous argument that bridges the gap between a viscoelastic tumour model (of Brinkman type) and an inviscid tumour model (of Darcy type). Specifically, we prove the convergence of solutions of the Brinkman nonlocal transport system towards a weak solution of the Darcy nonlinear parabolic system in the limit of vanishing viscosity.

INTRODUCTION

A Beautiful Link between Local and Nonlocal Dispersal.

In recent years, a beautiful link between nonlocal dispersal and nonlinear, local dispersal has been experiencing a renaissance. By nonlocal dispersal, we refer to dynamics governed by the equation

∂n ∂t = ∇ • (n∇K ϵ ⋆ n), (1)
where the interaction kernel, K ϵ , is a nonnegative, radial function with unit mass. In the context of dispersal, it is important and commonly assumed that the kernel is, furthermore, strictly decreasing in the radial variable. Now, if the interaction kernel is scaled in such a way that the interactions become stronger and more localised, approaching a Dirac delta, i.e., K ϵ → δ 0 as ϵ → 0, solutions to the nonlocal dispersion equation should, in a sense, converge to solutions of

∂n ∂t = ∇ • (n∇δ 0 ⋆ n) = ∇ • (n∇n). (2)
Of course, it is immediately clear that this limit is singular in the sense that the very nature of the equation changes from transport to degenerate parabolic. Both equations as well as the meta-problem of connecting both viewpoints have received considerable attention from different communities.

Applications in Biomathematical

Modelling. Nonlinear dispersal plays an important role in the evolution of large systems of interacting particles, [START_REF] Bruna | Active crowds[END_REF][START_REF] Carrillo | Aggregation-diffusion equations: dynamics, asymptotics, and singular limits[END_REF][START_REF] Naldi | Giuseppe: Mathematical modeling of collective behavior in socio-economic and life sciences[END_REF]. The importance of nonlinear, local dispersal is particularly relevant in contexts of mathematical biology. While linear diffusion is used frequently in order to incorporate the random dispersal of particles, the tendency to use nonlinear diffusion for random dispersal represents a paradigm shift. It was ushered in and motivated by several independent biological experiments (see [START_REF] Busenberg | Epidemic models with spatial spread due to population migration[END_REF][START_REF] Carl | Population control in arctic ground squirrels[END_REF][START_REF] Morisita | Population density and dispersal of a water strider. Gerris lacustris: Observations and considerations on animal aggregations[END_REF][START_REF] Morisita | Dispersion and population pressure: experimental studies on the population density of an ant-lion, Glenuroides japonicus m'l (2)[END_REF]) that suggested that the diffusion coefficient depends on the local density, i.e., ∂n ∂t = ∇ • (D(n)∇n),

with D(n) = n, in contrast to linear diffusion, D(n) = D. Particularly in the context of tissue growth and multi-phase tumour modelling, a balance law reminiscent of a nonlinear Fisher-KPP equation, i.e., ∂n ∂t

+ ∇ • (nv) = nG(n),
is used, where the velocity is related to the pressure gradient, v = -∇p = -∇n and G represents growth-death dynamics; most notably in [START_REF] Perthame | The Hele-Shaw asymptotics for mechanical models of tumor growth[END_REF], which acts as a starting point for this work.

Aggregation Equation and Nonlocal Dispersal.

We return to Eq. ( 1) which has received a lot of attention in its own right. It is customarily referred to as the aggregation equation, cf. [START_REF] Carrillo | Adhesion and volume constraints via nonlocal interactions determine cell organisation and migration profiles[END_REF][START_REF] Carrillo | Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations[END_REF][START_REF] Mogilner | Leah: A non-local model for a swarm[END_REF][START_REF] Volkening | Modelling stripe formation in zebrafish: an agent-based approach[END_REF], and references therein. Using the method of matched asymptotic expansions, the authors of [START_REF] Bruna | Coarse graining of a Fokker-Planck equation with excluded volume effects preserving the gradient flow structure[END_REF][START_REF] Bruna | Diffusion of particles with short-range interactions[END_REF] were able to systematically establish a link between Brownian particles with finite size, where the size-exclusion effects are incorporated by a very short-range, strong interaction potential, W ϵ , similarly to Eq. ( 1) and nonlinear diffusion similar to Eq. [START_REF] Alasio | Trend to equilibrium for systems with small cross-diffusion[END_REF]. In a regime of low volume fraction and extremely localised repulsion, they showed that the probability density of a representative particle evolves according to a linear diffusion equation with a nonlinear correction of the form as in Eq. [START_REF] Alasio | Trend to equilibrium for systems with small cross-diffusion[END_REF]. Indeed, systems of interacting particles with size-exclusion effects share one important feature, the presence of degenerate diffusion, as observed in various contexts, cf. [START_REF] Bruna | Diffusion of multiple species with excludedvolume effects[END_REF][START_REF] Bruna | Excluded-volume effects in the diffusion of hard spheres[END_REF][START_REF] Burger | Bärbel: Nonlinear cross-diffusion with size exclusion[END_REF].

Analytical Nonlocal

Approximations in the Literature. Concomitantly, the link between nonlocal dispersal and local nonlinear diffusion has also been addressed rigorously. Indeed, to the best of our knowledge, it dates back to a deterministic particle method for nonlinear diffusion equations introduced in [START_REF] Degond | A deterministic approximation of diffusion equations using particles[END_REF]. An idea of the proof based on a double-mollification argument was given in [START_REF] Lions | Une méthode particulaire déterministe pour des équations diffusives non linéaires[END_REF]. Subsequently, fully rigorous proofs were given in [START_REF] Burger | Porous medium equation and cross-diffusion systems as limit of nonlocal interaction[END_REF][START_REF] Carrillo | A blob method for diffusion[END_REF][START_REF] Craig | A blob method for inhomogeneous diffusion with applications to multi-agent control and sampling[END_REF], for the passage from Eq. (1) to Eq. [START_REF] Alasio | Trend to equilibrium for systems with small cross-diffusion[END_REF], culminating in the very recent work by Carrillo, Esposito, and Wu [START_REF] Carrillo | Nonlocal approximation of nonlinear diffusion equations[END_REF] that provides a full answer to nonlocal approximations of general degenerate diffusion equations. A remarkable fact about all of the aforementioned results is their technique which is based on an astute mollification argument. In its essence, it allows for a chain rule in the nonlocal equation mimicking the entropy dissipation structure of the local limit equation. Since the methods are based on techniques from optimal transportation they are unable to incorporate growth dynamics, in general. While there are some variational results on studying these tissue growth models incorporating growth dynamics [START_REF] Marino | Lénaïc: A tumor growth model of Hele-Shaw type as a gradient flow[END_REF][START_REF] Jacobs | Jiajun: Darcy's law with a source term[END_REF], no nonlocal-to-local limits have been reported in this variational framework. Moreover, the extension to multiple species (i = 1, 2) with joint population pressure, i.e., ∂n ∂t

(i) = ∇ • (n (i) ∇p(n (1) + n (2) )), (3)
coupled through a constitutive pressure law is anything but straightforward. While, to the best of the authors' knowledge, there are currently only two results on nonlocal approximations of cross-diffusion systems without linear self-diffusion, [START_REF] Burger | Porous medium equation and cross-diffusion systems as limit of nonlocal interaction[END_REF][START_REF] Hecht | Multispecies cross-diffusions: from a nonlocal mean-field to a porous medium system without self-diffusion[END_REF], any attempts to show existence of solutions (even in one dimension, leave alone multiple dimensions) to System (3) using a nonlocal approximation have remained futile, so far. We will address this gap in the literature in this work.

Cross-Interaction Systems.

Unlike other multi-species cross-diffusion systems that incorporate size-exclusion effects that were recently introduced and considered, see [START_REF] Alasio | Existence and regularity for a system of porous medium equations with small cross-diffusion and nonlocal drifts[END_REF][START_REF] Alasio | Trend to equilibrium for systems with small cross-diffusion[END_REF][START_REF] Berendsen | On a crossdiffusion model for multiple species with nonlocal interaction and size exclusion[END_REF][START_REF] Bruna | Derivation of a macroscopic model for Brownian hard needles[END_REF][START_REF] Burger | Bärbel: Nonlinear cross-diffusion with size exclusion[END_REF][START_REF] Di | Nonlinear degenerate cross-diffusion systems with nonlocal interaction[END_REF][START_REF] Jüngel | The boundedness-by-entropy method for cross-diffusion systems[END_REF], System (3) assumes a rather prominent place. This is due to the high symmetry in the velocity-pressure relation which renders the system non-strictly parabolic and is only convex rather than strictly convex. If convexity is completely lost, convexification arguments can be employed [START_REF] Ducasse | A cross-diffusion system obtained via (convex) relaxation in the JKO scheme[END_REF], however, the approximation might not converge to solutions of the original system. Returning to System (3), from the very early works, it has been reported, shown numerically, and proven analytically, that solutions may exhibit a propagation of segregation property, [START_REF] Carrillo | Splitting Schemes and Segregation in Reaction Cross-Diffusion Systems[END_REF][START_REF] Gurtin | A note on interacting populations that disperse to avoid crowding[END_REF]. Thereupon, even arbitrarily smooth initial data may lose regularity at the onset of sharp interfaces between both densities. Formally speaking, this sharp interface is transported with the same velocity, v = -∇p, rendering the quest of obtaining regularity of the individual species, n (i) , a demanding challenge. Indeed, any regularity better than bounded variation cannot be expected, [START_REF] Bertsch | Travelling wave solutions of a 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]. A different approach to prove the existence of solutions relies on the strong compactness of the pressure gradient, either by variations of the Aronson-Bénilan estimate [START_REF] Bubba | Hele-Shaw Limit for a System of Two Reaction-(Cross-)Diffusion Equations for Living Tissues[END_REF][START_REF] Gwiazda | Agnieszka: A two-species hyperbolic-parabolic model of tissue growth[END_REF][START_REF] Jacobs | Non-mixing Lagrangian solutions to the multispecies Porous Media Equation[END_REF] or convergence of the norm [START_REF] David | Phenotypic heterogeneity in a model of tumour growth: existence of solutions and incompressible limit[END_REF][START_REF] Jacobs | Existence of solutions to reaction cross diffusion systems[END_REF][START_REF] Liu | Xiangsheng: Existence and incompressible limit of a tissue growth model with autophagy[END_REF][START_REF] Price | Xiangsheng: Global existence theorem for a model governing the motion of two cell populations[END_REF]. Historically, System (3) first appeared in [START_REF] Busenberg | Epidemic models with spatial spread due to population migration[END_REF] in the context of epidemiological modelling of polymorphic populations and in [START_REF] Gurtin | A note on interacting populations that disperse to avoid crowding[END_REF] with applications to population dynamics that avoid overcrowding. It was subsequently analysed in [START_REF] Bertsch | Danielle: 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]. These early works were ensued by [START_REF] Bertsch | A free boundary problem arising in a simplified tumour growth model of contact inhibition[END_REF][START_REF] Bertsch | A nonlinear parabolichyperbolic system for contact inhibition of cell-growth[END_REF] which study a variation of System (3) incorporating growth dynamics first in one and then in arbitrary spatial dimensions for tissues with cell-cell contact inhibitions. While the argument relied upon the introduction of transformed variables, a different regularisation approach was introduced in [START_REF] Galiano | On a cross-diffusion segregation problem arising from a model of interacting particles[END_REF]. The models with growth and joint population pressure were applied and further scrutinised in [START_REF] Carrillo | A population dynamics model of cell-cell adhesion incorporating population pressure and density saturation[END_REF][START_REF] Murakawa | Hideru: Continuous models for cell-cell adhesion[END_REF] for human embryonic kidney cells and they are able to explain the phase separation of normal and abnormal tissue. Following a diametrically different approach based on a splitting scheme, the authors of [START_REF] Carrillo | Splitting Schemes and Segregation in Reaction Cross-Diffusion Systems[END_REF] were able to remove restrictive assumptions on the initial data, namely those of being bounded away from zero or strictly ordered in one dimension. Successively, assumptions on initial data and space dimension were removed, see [START_REF] Bubba | Hele-Shaw Limit for a System of Two Reaction-(Cross-)Diffusion Equations for Living Tissues[END_REF][START_REF] David | Phenotypic heterogeneity in a model of tumour growth: existence of solutions and incompressible limit[END_REF][START_REF] Gwiazda | Agnieszka: A two-species hyperbolic-parabolic model of tissue growth[END_REF][START_REF] Jacobs | Existence of solutions to reaction cross diffusion systems[END_REF][START_REF] Liu | Xiangsheng: Existence and incompressible limit of a tissue growth model with autophagy[END_REF][START_REF] Price | Xiangsheng: Global existence theorem for a model governing the motion of two cell populations[END_REF]. Alongside these analytical advances and the growing understanding of the system, it is also worthwhile mentioning several results on travelling wave solutions, steady states, and pattern formation for variations of Eq. (3), see [START_REF] Bertsch | Travelling wave solutions of a parabolic-hyperbolic system for contact inhibition of cell-growth[END_REF][START_REF] Bertsch | A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic fisher kpp equation[END_REF][START_REF] Burger | Segregation effects and gap formation in cross-diffusion models[END_REF][START_REF] Carrillo | Zoology of a Nonlocal Cross-Diffusion Model for Two Species[END_REF][START_REF] Galiano | Víctor: Turing instability analysis of a singular cross-diffusion problem[END_REF]. To this day, the existence of solutions to Eq. (3) with different diffusivity, apart from [START_REF] Gurtin | A note on interacting populations that disperse to avoid crowding[END_REF], is widely open. Similarly, the inclusion of different drifts remains a challenging open question with two very partial results given in [START_REF] Carrillo | Zoology of a Nonlocal Cross-Diffusion Model for Two Species[END_REF][START_REF] Kim | On nonlinear cross-diffusion systems: an optimal transport approach[END_REF], in one dimension and only for specific drifts or strong assumptions on the drifts. The interest in System (3) and the wealth of new methods for proving existence leads us to believe that our work connects pleasingly with the advances in the literature in the last few years: not only do we prove the existence of solutions, but we propose yet a new approach with the hope it may interest the community.

Goal of this Paper. The starting point of our analysis is the model

     ∂n (i) ν ∂t -∇ • n (i) ν ∇W ν = n (i) ν G (i) (n ν ), -ν∆W ν + W ν = n ν , (4)
where n

(i) ν = n (i)
ν (x, t), i = 1, 2, represents the normal (resp. abnormal) cell density at location x ∈ R d at time t ∈ [0, T ), proposed in [START_REF] Ębiec | Nicolas: Incompressible limit for a two-species model with coupling through Brinkman's law in any dimension[END_REF][START_REF] Ębiec | Incompressible limit for a two-species tumour model with coupling through Brinkman's law in one dimension[END_REF] as a two-species version of [START_REF] Perthame | Incompressible limit of a mechanical model of tumour growth with viscosity[END_REF]. Here,

n ν = n (1) ν + n (2)
ν denotes the total population. The system is equipped with some nonnegative initial data n

(i),in ν ∈ L 1 (R d )∩L ∞ (R d ).
Here, ν > 0 is a fixed positive constant representing a measure of the tissue's viscosity. The elliptic equation linking the macroscopic velocity potential, W ν , with the density, n ν , is typically referred to as Brinkman's law, see for instance [START_REF] Allaire | Homogenization of the Navier-Stokes equations and derivation of Brinkman's law[END_REF][START_REF] Perthame | Incompressible limit of a mechanical model of tumour growth with viscosity[END_REF]. The growth of the two densities is assumed to be modulated by density-dependent growth rates, G (i) , [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]. Throughout, we shall make the following assumptions on the growth rate: (G1) regularity:

G (i) ∈ C 1 (R), for i = 1, 2, (G2) monotonicity: ∂ n G (i) ≤ -α < 0, for some α > 0, for i = 1, 2, (G3) critical densities: there exists n > 0 such that G (i) (n) ≤ 0, ∀n ≥ n, and i = 1, 2.
Formally, it is easy to observe that solutions to the Brinkman model, System (4), converge to solutions to the Darcy model [START_REF] Berendsen | On a crossdiffusion model for multiple species with nonlocal interaction and size exclusion[END_REF] ∂n

(i) 0 ∂t -∇ • n (i) 0 ∇n 0 = n (i) 0 G (i) (n 0 ), i = 1, 2,
where n

(i) 0 = lim ν→0 n (i)
ν , and n 0 = n

(1) 0 + n (2)
0 . Of course, this argument only holds on a formal level. The challenge is immediately clear when directing our focus on the nature of the two models -the Brinkman model is of transport type while the Darcy model is a degenerate parabolic equation. In a sense, solutions to the limit equation, System (5), have higher regularity than solutions for ν > 0, and this gain of regularity at the limit classifies this question as a singular limit problem. It is the goal of this paper to establish the limit process rigorously. Before we state the precise statement of the main result let us briefly introduce the notion of weak solutions to the two equations. Definition 1.1 (Weak Solutions -Brinkman). We say the pair (n

(1) ν , n (2)
ν ) ≥ 0 is a weak solution to System (4) with nonnegative initial data n

(i),in ν ∈ L 1 (R d ) ∩ L ∞ (R d ), if n (i) ν ∈ L ∞ (0, T ; L 1 (R d ) ∩ L ∞ (R d ))
, for i = 1, 2, and there holds

T 0 R d n (i) ν ∂φ ∂t dx dt - T 0 R d n (i) ν ∇W ν • ∇φ dx dt = - T 0 R d φn (i) ν G (i) (n ν ) dx dt - R d φ(x, 0)n (i),in (x) dx, for any test function φ ∈ C ∞ c (R d × [0, T )), as well as -ν∆W ν + W ν = n ν ,
almost everywhere in R d × (0, T ).

Similarly, let us introduce our notion of weak solutions to the limiting equation.

Definition 1.2 (Weak Solutions -Darcy

). We say a pair of nonnegative functions, (n

(1) 0 , n (2)
0 ) is a weak solution to System [START_REF] Berendsen | On a crossdiffusion model for multiple species with nonlocal interaction and size exclusion[END_REF] with nonnegative initial data n

(i),in 0 ∈ L 1 (R d ) ∩ L ∞ (R d ), if n (i) 0 ∈ L ∞ (0, T ; L 1 (R d ) ∩ L ∞ (R d ))
, for i = 1, 2, and n 0 ∈ L 2 (0, T ; H 1 (R d )) and there holds

T 0 R d n (i) 0 ∂φ ∂t dx dt - T 0 R d n (i) 0 ∇n 0 • ∇φ dx dt = - T 0 R d φn (i) 0 G (i) (n 0 ) dx dt - R d φ(x, 0)n (i),in 0 (x) dx, for any test function φ ∈ C ∞ c (R d × [0, T )).
In our analysis of the inviscid limit, we shall make use of the entropy structure of the primitive system and the limit system, respectively. Let us introduce the usual entropy functional

H[f ](t) := R d f (x, t)(log f (x, t) -1) dx.
Finally, let us recall that any solution of Brinkman's equation can be expressed as convolution with the fundamental solution, denoted by K ν , i.e., the solution of

-ν∆W ν + W ν = n ν ,
can be expressed as

W ν = K ν ⋆ n ν ,
where explicitly,

K ν (x) = 1 4π ∞ 0 e - π|x| 2 4sν + 4s π ds s d/2
. With these definitions at hand, we can now state the main result of this paper.

Theorem 1.3 (Inviscid Limit). Let n (i),in 0 ∈ L 1 (R d ) ∩ L ∞ (R d ) be given and suppose that n in 0 := n (1),in 0 + n (2),in 0 satisfies H[n in 0 ] < ∞, R d |x| 2 n in 0 dx < ∞, R d n in 0 K ⋆ n in 0 dx < ∞. Let (n (i)
ν ) ν>0 be a sequence of solutions to System (4) in the sense of Definition 1.1 with initial data n

(i),in ν = n (i),in 0
. Then, there exists a subsequence, still denoted (n

(i) ν ) ν ,

and functions n

(i) 0 ∈ L ∞ (0, T ; L 1 (R d ) ∩ L ∞ (R d )), such that n (i) ν ⇀ n (i) 0 , i = 1, 2, weakly in L 2 (0, T ; L 2 (R d )), as well as n ν → n 0 := n (1) 0 + n (2) 0 , strongly in L 2 (0, T ; L 2 (R d )), and
W ν → n 0 , strongly in L 2 (0, T ; H 1 (R d )). Furthermore, (n (1) 0 , n (2)
0 ) is a weak solution to System (5) in the sense of Definition 1.2, with initial data (n

(1),in 0 , n (2)
,in 0

).

Remark 1.4. The result remains true for more general sequences of "well-prepared" initial data n (i),in ν . Indeed, any family (n

(i),in ν ) ν>0 which is weakly compact in L 1 (R d ) and such that n in ν = n (1),in ν + n (2),in ν satisfies sup ν>0 H[n in ν ] < ∞, sup ν>0 R d |x| 2 n in ν dx < ∞, sup ν>0 R d n in ν K ⋆ n in ν dx < ∞, as well as lim inf ν→0 H[n in ν ] ≤ H[n in 0 ],
where n in 0 is the weak limit of n in ν , can serve as initial data for the Brinkman system (4). To indicate this generality, we keep the distinction between n in 0 and n in ν , even though in the statement of the main theorem they are the same.

Summary of the Strategy and Outline of this Paper.

It is routine to derive uniform integrability bounds on the solution (n

(1) ν , n (2)
ν ) of System (4), see bounds (B1)-(B3) in the subsequent section. The assumptions (G2) and (G3) on the reaction rates are crucial for the derivation of uniform L ∞ -bounds on n (i) ν , as they allow for a maximum principle for the total population density, and consequently for the individual densities. The resulting bound (B2) together with assumption (G1), guarantees integrability of the reaction term in the equation for n (i) , which is used several times in deriving further properties of the densities. The uniform L p -bounds on the densities readily imply their weak precompactness, which is, of course, insufficient to pass to the limit in the nonlinear terms of (4). Particularly for the transport term, we have to establish either strong compactness of the individual densities and weak compactness of the velocity, or weak compactness of the individual densities and strong compactness of the velocity. While proving that the velocity is weakly precompact is feasible with some effort, as explained below, improving the convergence of the individual densities appears to be a challenging problem. Therefore, we shall pursue the second strategy and prove the velocity's strong convergence. The proof of strong compactness of W ν in L 2 (0, T ; H 1 (R d )) without any uniform control on any second derivatives therefore constitutes the main challenge of this paper. We adapt the ideas used in [START_REF] Jacobs | Existence of solutions to reaction cross diffusion systems[END_REF][START_REF] Price | Xiangsheng: Global existence theorem for a model governing the motion of two cell populations[END_REF], which were applied to prove the existence of solutions and the so-called incompressible limit for hyperbolic-parabolic systems such as System [START_REF] Berendsen | On a crossdiffusion model for multiple species with nonlocal interaction and size exclusion[END_REF]. Our strategy relies on first establishing the limit equation on the joint density, n 0 , and exploiting its structure to infer the strong compactness of the velocity field, ∇W ν , by comparing the entropy dissipation inequality satisfied by n ν with the entropy dissipation equality satisfied by n 0 . More precisely, we will first show that the total population density n ν satisfies the inequality ( 6)

H[n ν ](T ) -H[n in ν ] - T 0 R d n ν ∆W ν dx dt ≤ T 0 R d log n ν n (1) ν G (1) (n ν ) + n (2) ν G (2) (n ν ) dx dt.
Using Brinkman's law to relate the quantity n ν ∆W ν to |∇W ν | 2 , we infer a uniform L 2bound on the velocity ∇W ν , and therefore its weak precompactness. Then, we identify the limit to be ∇n 0 . This information is enough to deduce that the limiting total density n 0 satisfies the equation

∂n 0 ∂t -∇ • (n 0 ∇n 0 ) = n (1) 0 G (1) (n 0 ) + n (2) 0 G (2) (n 0 ),
and consequently the entropy identity ( 7)

H[n 0 ](T ) -H[n in 0 ] + T 0 R d |∇n 0 | 2 dx dt = T 0 R d log n 0 n (1) 0 G (1) (n 0 ) + n (2) 0 G (2) (n 0 ) dx dt.
Under the assumption that H[n in 0 ] dominates H[n in ν ], we can compare ( 6) and ( 7) to deduce

T 0 R d |∇W ν | 2 dx dt ≤ T 0 R d |∇n 0 | 2 dx dt + E ν ,
the error term E ν comprising the difference between final-time entropies and the reaction terms. The former has the correct sign by convexity, while the latter can be shown to vanish in the limit by a simple measure-theoretic argument. Thus, we infer the upper semicontinuity property

lim sup ν→0 ∥∇W ν ∥ L 2 (R d ×(0,T )) ≤ ∥∇n 0 ∥ L 2 (R d ×(0,T )) .
Together with the lower semicontinuity furnished by the weak convergence, this implies convergence of norms, and consequently (in conjunction with the weak convergence), convergence in the strong topology of L 2 (R d ×(0, T )). We refer the reader to [START_REF] Jacobs | Existence of solutions to reaction cross diffusion systems[END_REF], where essentially the same argument is used. Finally, let us stress that the uniform L 2bound on ∇W ν cannot be used to derive any such bound on the gradient of the total density ∇n ν and no convergence ∇n ν ⇀ ∇n 0 can be shown.

The rest of the paper is organised as follows. In the subsequent section, we recall some regularity properties satisfied by the solution of System (4). Since the regularity of n ν is insufficient to rigorously compute the time evolution of the entropy functional, we first introduce a regularised system, prove its fundamental properties (Lemma 2.1 and 2.2) and show convergence to the solution of System (4) (in Proposition 2.4). We then proceed to deriving the entropy dissipation inequality [START_REF] Bertsch | A free boundary problem arising in a simplified tumour growth model of contact inhibition[END_REF], see Proposition 2.6. In Section 3, we use the entropy dissipation inequality to provide uniform bounds on ∇W ν and ∂ t W ν . Subsequently, by means of the Aubin-Lions lemma, we prove the strong compactness of W ν , and consequently of n ν , in L 2 (0, T ; L 2 (R d )). Having garnered all the ingredients necessary to show that the limit n 0 satisfies a porous medium equation with growth (and therefore an entropy dissipation equality), in Section 4, we conclude the proof of Theorem 1.3 by showing the convergence in norm of the velocity ∇W ν , hence enhancing the weak convergence to strong convergence in L 2 (0, T ; L 2 (R d )). Finally, in Section 5, we provide a remark on how our strategy could possibly be adapted to stratified models including phenotypic heterogeneity.

A PRIORI ESTIMATES

We begin by summarising some fundamental regularity results for System (4) that were obtained recently in a series of works, see [START_REF] Ębiec | Nicolas: Incompressible limit for a two-species model with coupling through Brinkman's law in any dimension[END_REF][START_REF] Ębiec | 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]. Most notably, for any ν > 0, there is a unique solution (n

(1) ν , n (2)
ν ) to System (4) satisfying:

(B1) n (i) ν ∈ L ∞ (0, T ; L 1 (R d )), for i = 1, 2, (B2) 0 ≤ n (i) ν (x, t) ≤ n, almost everywhere, with i = 1, 2, (B3) W ν ∈ L ∞ (0, T ; L 1 (R d ) ∩ L ∞ (R d )),
uniformly in ν > 0, provided that the initial data n (i),in ν are uniformly bounded in L 1 (R d ). In particular, from standard function-space interpolation, it follows that n (i) ν are bounded uniformly in L 2 (0, T ; L 2 (R d )). Therefore, passing to a subsequence, they converge to some functions n (i) 0 in the weak topology of L 2 (0, T ; L 2 (R d )). As a consequence, the total density, n ν , converges weakly to the function n 0 := n

(1) 0 + n (2)
0 . The potential, W ν , enjoys the following regularity properties:

(B4) W ν ∈ L ∞ (0, T ; W 1,q (R d )), for 1 ≤ q ≤ ∞, (B5) D 2 W ν ∈ L ∞ (0, T ; L q (R d )), for 1 < q < ∞,
for any given ν > 0. These properties may be obtained directly from the representation formula W ν = K ν ⋆n ν . However, let us stress that the last two statements are not claimed to hold uniformly in ν. Indeed, it is a substantial part of this work to establish (B4) uniformly in ν, at least in L 2 (0, T ; H 1 (R d )). One might hope to derive such an estimate directly from the representation formula; however, we shall not pursue this strategy here. Moreover, let us point out that (B5) cannot be expected to hold true uniformly as, in the limit, ∆n 0 is only a measure in general.

Entropy Dissipation

Inequality. This section is dedicated to establishing an entropy dissipation inequality for the joint population density. Indeed, for a solution, (n

(1) ν , n (2)
ν ), to System (4), we can show that n ν = n

(1) ν + n (2) ν satisfies H[n ν ](T ) -H[n in ν ] - T 0 R d n ν ∆W ν dx dt ≤ T 0 R d log n ν n (1)
ν G (1) (n ν ) + n (2) ν G (2) (n ν ) dx dt.

In order to establish this inequality rigorously, let us begin by regularising the equations satisfied by the two densities, n

ν . Specifically, for any ϵ > 0, we consider

     ∂n (i) ϵ,ν ∂t -ϵ∆n (i) ϵ,ν = ∇ • (n (i) ϵ,ν ∇W ν ) + n (i) ϵ,ν G (i) (n ν ) in R d × (0, T ), n (i) ϵ,ν (x, 0) = n (i),in ϵ,ν (x), (8)
for i = 1, 2, where n ν and W ν are the unique solutions to System (4), and n

(i),in ϵ,ν ∈ C ∞ c (R d ) denotes some nonnegative, regularised initial data such that sup ϵ>0 ∥n (i),in ϵ,ν ∥ L p (R d ) ≤ ∥n (i),in ν ∥ L p (R d ) , p = 1, ∞,
and satisfying n

(i),in ϵ,ν → n (i),in ν in L 1 (R d
) and almost everywhere in R d . The existence of a pair of unique nonnegative L 2 (0, T ;

H 1 (R d ))-solutions (n (1) ϵ,ν , n (2)
ϵ,ν ) is well known, see for instance [START_REF] Ladyženskaja | Linear and quasilinear equations of parabolic type[END_REF], and

(i) n (i) ϵ,ν ∈ L ∞ (0, T ; L 1 (R d ) ∩ L ∞ (R d )) ∩ L 2 (0, T ; H 1 (R d )), (ii) ∂ t n (i) ϵ,ν ∈ L 2 (0, T ; L 2 (R d )).
We omit details of the existence proof for brevity. Let us point out that the norm in L ∞ (0, T ; L 1 (R d )) is bounded uniformly in both ϵ and ν. This follows from integrating the equation, which leads to the inequality

∥n (i) ϵ,ν (t)∥ L 1 (R d ) ≤ e ∥G (i) ∥ L ∞ ([0,n]) T ∥n (i),in ϵ,ν ∥ L 1 (R d ) ≤ C(T )∥n (i),in ν ∥ L 1 (R d ) .
On the other hand, the norm in L ∞ (0, T ; L ∞ (R d )) is bounded uniformly in ϵ, but not in ν. This follows from a standard maximum principle argument using the bound (B2) and the regularity of G (i) , as well as controlling the L ∞ -norm of ∆W ν (which depends on ν).

We will now discuss some important properties of the total regularised density, n ϵ,ν := n

(1)

ϵ,ν + n (2)
ϵ,ν , which satisfies the equation

   ∂n ϵ,ν ∂t -ϵ∆n ϵ,ν = ∇ • (n ϵ,ν ∇W ν ) + n (1) ϵ,ν G (1) (n ν ) + n (2) ϵ,ν G (2) (n ν ), in R d × (0, T ), n ϵ,ν (x, 0) = n in ϵ,ν (x) := n (1),in ϵ,ν (x) + n (2),in ϵ,ν (x). (9)
Lemma 2.1. Let n ϵ,ν be the solution to Eq. (9). Assume that there is a constant, C > 0, such that

sup ϵ>0 R d n in ϵ,ν (x)|x| 2 dx ≤ C,
i.e., the initial data has finite second moment. Then, the second moment remains bounded and there holds

sup t∈[0,T ] R d n ϵ,ν (x, t)|x| 2 dx ≤ C + C T 0 R d n ϵ,ν |∇W ν | 2 dx dt =: C(ν),
for some constant C(ν) > 0, independent of ϵ.

Proof. We compute

1 2 d dt R d n ϵ,ν |x| 2 dx = ϵd R d n ϵ,ν dx + R d n ϵ,ν x • ∇W ν dx + 1 2 R d |x| 2 n (1) ϵ,ν G (1) (n ν ) + n (2) ϵ,ν G (2) (n ν ) dx ≤ ϵd∥n ϵ,ν ∥ L ∞ (0,T ;L 1 (R d )) + 1 2 + max i=1,2 ∥G (i) ∥ L ∞ (0,n) R d n ϵ,ν |x| 2 dx + 1 2 R d n ϵ,ν |∇W ν | 2 dx ≤ C 1 + 1 2 R d n ϵ,ν |∇W ν | 2 dx + C 2 2 R d n ϵ,ν |x| 2 dx.
Then, by Gronwall's lemma, we get

1 2 R d |x| 2 n ϵ,ν (x, t) dx ≤ 1 2 R d |x| 2 n in ϵ,ν (x) dx + C 1 T + 1 2 T 0 R d n ϵ,ν |∇W ν | 2 dx dt e C 2 T .
This completes the proof. □

Let us point out that the above lemma implies also uniform second moment control for the individual species, n

(i) ϵ,ν , i = 1, 2.
Lemma 2.2 (Entropy Bounds). Let n ϵ,ν be the solution to Eq. (9). Then, there holds

sup ϵ>0 sup t∈[0,T ] R d n ϵ,ν (t)| log n ϵ,ν (t)| dx ≤ C(ν),
for some constant, C(ν) > 0, independent of ϵ.

Proof. Let us consider

R d n ϵ,ν |log n ϵ,ν | dx = {nϵ,ν ≥1} n ϵ,ν log n ϵ,ν dx - {nϵ,ν <1} n ϵ,ν log n ϵ,ν dx ≤ ∥n ϵ,ν ∥ L ∞ (0,T ;L ∞ (R d )) ∥n ϵ,ν ∥ L ∞ (0,T ;L 1 (R d )) + J,
where

J := - {nϵ,ν <1} n ϵ,ν log n ϵ,ν dx.
In order to estimate J, let N (x) denote the standard normal Gaussian. Then, we have

J = - {nϵ,ν <1} n ϵ,ν log n ϵ,ν dx = - R d n ϵ,ν 1 {nϵ,ν <1} N log n ϵ,ν 1 {nϵ,ν <1} N N dx + R d n ϵ,ν 1 {nϵ,ν <1} |x| 2 dx ≤ - R d n ϵ,ν 1 {nϵ,ν <1} N log n ϵ,ν 1 {nϵ,ν <1} N N dx + C(ν),
having used the second-order moment bound from Lemma 2.1. Applying Jensen's inequality to the first term, we observe

- R d n ϵ,ν 1 {nϵ,ν <1} N log n ϵ,ν 1 {nϵ,ν <1} N N dx ≤ - R d n ϵ,ν 1 {nϵ,ν <1} N N dx log R d n ϵ,ν 1 {nϵ,ν <1} N N dx ≤ e -1 .
In conclusion, we obtain J ≤ C(ν) + e -1 , whence

R d n ϵ,ν | log n ϵ,ν | dx ≤ ∥n ϵ,ν ∥ L ∞ (0,T ;L ∞ (R d )) ∥n ϵ,ν ∥ L ∞ (0,T ;L 1 (R d )) + C(ν) + e -1 ,
which concludes the proof. □ Proposition 2.3. Let (n ϵ,ν ) ϵ>0 be a family of solutions to Eq. (9). Then, there holds

√ ϵ∥∇n ϵ,ν ∥ L 2 (0,T ;L 2 (R d )) ≤ C(ν),
for some constant C(ν) > 0 independent of ϵ.

Proof. Testing Eq. ( 9) by n ϵ,ν and integrating by parts, we obtain

1 2 d dt R d n 2 ϵ,ν dx + ϵ R d |∇n ϵ,ν | 2 dx = 1 2 R d n 2 ϵ,ν ∆W ν dx + R d n ϵ,ν n (1) ϵ,ν G (1) (n ν ) + n (2) ϵ,ν G (2) (n ν ) dx ≤ C(ν) R d n 2 ϵ,
ϵ,ν ) ϵ>0 , (n ϵ,ν ) ϵ>0 are compact in L 1 (0, T ; L 1 (R d )).

Proof. We will prove compactness for the individual densities, n

(i) ϵ,ν . It is easy to see that ∂ t n (i)
ϵ,ν is bounded uniformly (with respect to ϵ) in L 2 (0, T ; H -1 (R d )). For space compactness, we follow the strategy of Belgacem-Jabin [4, Proposition 3.1]. They guarantee local compactness by considering the evolution of the following quantity

Q ϵ (t) := R 2d K h (x -y)|n (i) ϵ,ν (x, t) -n (i) ϵ,ν (y, t)| dx dy,
where (K h ) 0<h<1 is a family of nonnegative, smooth functions such that

∥K h ∥ L 1 (R d ) ∼ |log h|, and |x||∇K h (x)| ≤ CK h (x).
Then, the sequence (n

(i) ϵ,ν ) ϵ>0 is compact in L 1 loc (R d ), provided that lim h→0 lim sup ϵ→0 | log h| -1 Q ϵ (t) = 0.
For the precise statement see [4, Lemma 3.1]. Our situation is of course simpler since Eq. ( 8) is linear and the velocity field is independent of ϵ. The only term that we need to be careful about is the reaction. The contribution of this term in d dt Q ϵ is

R 2d K h (x -y)sign n (i) ϵ,ν (x) -n (i) ϵ,ν (y) n (i) ϵ,ν (x)G (i) (n ν (x)) -n (i) ϵ,ν (y)G (i) (n ν (y)) dx dy ≤ ∥n (i) ϵ,ν ∥ L ∞ (0,T ;L ∞ (R d )) R 2d K h (x -y) G (i) (n ν (x)) -G (i) (n ν (y)) dx dy + ∥G (i) ∥ L ∞ (R) R 2d K h (x -y) n (i) ϵ,ν (x) -n (i) ϵ,ν (y) dx dy ≤ C R 2d K h (x -y)|n ν (x) -n ν (y)| dx dy + CQ ϵ (t),
having used the fact that G (i) is Lipschitz. Since n ν ∈ L ∞ (0, T ; L 1 (R d )), the first term on the right-hand side satisfies

lim h→0 | log h| -1 R 2d K h (x -y)|n ν (x, t) -n ν (y, t)| dx dy = 0.
We conclude that n

(i) ϵ,ν is compact in L 1 (0, T ; L 1 loc (R d ))
, for i = 1, 2. Finally, using the second moment control of Lemma 2.1, we can deduce global compactness. □

By the compactness of solutions to the regularised equations, we can always extract subsequences, still denoted (n

(i) ϵ,ν ) ϵ>0 , such that n (i) ϵ,ν → n (i)
ν almost everywhere, as ϵ → 0. Using Proposition 2.3, it is easy to identify the limit of n ϵ,ν as the unique solution satisfied by the sum of the n (i) ν , cf. System (4). Passing to the limit ϵ → 0, we can now ensure that the limit n ν satisfies the second moment bound, and hence the entropy bounds, uniformly in ν.

Proposition 2.5. The limit n ν of the sequence (n ϵ,ν ) ϵ>0 satisfies

sup ν>0 sup t∈[0,T ] R d n ν |x| 2 dx < ∞, and
sup ν>0 sup t∈[0,T ] R d n ν | log n ν | dx < ∞.
Proof. From the proofs of Lemmata 2.1 and 2.2 we see that the only obstruction to νuniform bounds is caused by the quantity n ϵ,ν |∇W ν | 2 . We will now show that in the vanishing viscosity limit, this quantity is independent of ν. Indeed, multiplying Eq. ( 9) by K ν ⋆ n ϵ,ν and integrating in space, we obtain (10)

1 2 d dt R d K ν ⋆ n ϵ,ν n ϵ,ν dx + ϵ R d ∇K ν ⋆ n ϵ,ν • ∇n ϵ,ν dx = - R d ∇K ν ⋆ n ϵ,ν • n ϵ,ν ∇W ν dx + R d K ν ⋆ n ϵ,ν n (1) ϵ,ν G (1) (n ν ) + n (2) ϵ,ν G (2) (n ν ) dx.
The bound follows after integrating in time and bounding all remaining terms. Beginning with the gradient term, we observe it vanishes in the ϵ-limit thanks to Proposition 2.3, since

ϵ T 0 R d ∇K ν ⋆ n ϵ,ν • ∇n ϵ,ν dx dt ≤ ϵ∥∇n ϵ,ν ∥ L 2 (R d ×(0,T )) ∥∇K ν ∥ L 1 (R d ) ∥n ϵ,ν ∥ L 2 (R d ×(0,T )) ≤ C(ν) √ ϵ.
Next, we claim that

T 0 R d ∇K ν ⋆ n ϵ,ν • n ϵ,ν ∇W ν dx dt → T 0 R d n ν |∇W ν | 2 dx dt.
To see this, we first observe that the strong L 1 -convergence and the uniform L ∞ -bound of n ϵ,ν imply that n ϵ,ν → n ν strongly in L p (R d × (0, T )), for any p ∈ [1, ∞). Adding and subtracting n ϵ,ν |∇W ν | 2 , we can now write

T 0 R d ∇K ν ⋆ n ϵ,ν • n ϵ,ν ∇W ν dx dt - T 0 R d n ν |∇W ν | 2 dx dt ≤ T 0 R d |n ϵ,ν -n ν ||∇W ν | 2 dx dt + T 0 R d n ϵ,ν |∇W ν ||∇K ν ⋆ (n ϵ,ν -n ν )| dx dt ≤ C(ν)∥n ϵ,ν -n ν ∥ L 1 (0,T ;L 1 (R d )) + C(ν)∥n ϵ,ν ∥ L 2 (0,T ;L 2 (R d )) ∥∇K ν ⋆ (n ϵ,ν -n ν )∥ L 2 (0,T ;L 2 (R d )) ≤ C(ν)∥n ϵ,ν -n ν ∥ L 1 (0,T ;L 1 (R d )) + C(ν)∥n ϵ,ν ∥ L 2 (0,T ;L 2 (R d )) ∥∇K ν ∥ L 1 (R d ) ∥n ϵ,ν -n ν ∥ L 2 (0,T ;L 2 (R d )) ,
where in the last line we used Young's inequality. The claim follows. Similarly, for the reaction term, we have

T 0 R d K ν ⋆ n ϵ,ν n (i) ϵ,ν G (i) (n ν ) dx dt → T 0 R d K ν ⋆ n ν n (i) ν G (i) (n ν ) dx dt;
and the last integral is bounded uniformly in ν

T 0 R d K ν ⋆ n ν n (i) ν G (i) (n ν ) dx dt ≤ C∥K ν ∥ L 1 (R d ) ∥n ν ∥ 2 L 2 (R d ×(0,T )) .
Finally, we observe that

1 2 R d K ν ⋆ n in ϵ,ν n in ϵ,ν dx - 1 2 R d K ν ⋆ n ϵ,ν (T )n ϵ,ν (T ) dx ≤ 1 2 R d K ν ⋆ n in ϵ,ν n in ϵ,ν dx → 1 2 R d K ν ⋆ n in ν n in ν dx.
Integrating [START_REF] Bertsch | On interacting populations that disperse to avoid crowding: the case of equal dispersal velocities[END_REF] in time and passing to the limit ϵ → 0, we thus obtain

T 0 R d n ν |∇W ν | 2 dx dt ≤ C + 1 2 R d n in ν K ⋆ n in ν dx ≤ C,
where the last constant is independent of ν. Now, using Lemma 2.1, we can write

R d |x| 2 n ν dx = R d |x| 2 (n ν -n ϵ,ν ) dx + R d |x| 2 n ϵ,ν dx ≤ R d |x| 2 (n ν -n ϵ,ν ) dx + C + C T 0 R d n ϵ,ν |∇W ν | 2 dx dt.
Passing to the limit ϵ → 0, we obtain

R d |x| 2 n ν dx ≤ C + C T 0 R d n ν |∇W ν | 2 dx dt ≤ C.
Having this ν-uniform bound, we can repeat the proof of Lemma 2.2 to obtain the required uniform entropy bound for the family n ν . □

We now state and prove the main result of this section, namely that the solutions to the equation for the total density, n ν , are entropy weak solutions.

Proposition 2.6 (Entropy Inequality). The limit n ν of the sequence (n ϵ,ν ) ϵ>0 satisfies the entropy inequality

H[n ν ](T ) -H[n in ν ] - T 0 R d n ν ∆W ν dx dt ≤ T 0 R d log n ν n (1) ν G (1) (n ν ) + n (2) ν G (2) (n ν ) dx dt. (11)
Proof. Let δ > 0 and ϕ ∈ C ∞ c (R d ), ϕ ≥ 0. We consider the following regularised form of the entropy functional

H ϕ δ [n ϵ,ν ](t) := R d (n ϵ,ν (x, t) + δ) (log(n ϵ,ν (x, t) + δ) -1)ϕ(x) dx,
and by H ϕ [f ] we denote the above functional with δ = 0. Given the regularity of n ϵ,ν provided by parabolic theory, the weak form of Eq. ( 9) can be written as

T 0 R d ∂n ϵ,ν ∂t φ(x, t) + n ϵ,ν ∇φ(x, t) • ∇W ν dx dt + ϵ T 0 R d ∇n ϵ,ν • ∇φ(x, t) dx dt = T 0 R d n (1) ϵ,ν G (1) (n ν ) + n (2) ϵ,ν G (2) (n ν ) φ(x, t) dx dt, for any φ ∈ L 2 (0, T ; H 1 (R d )). Choosing φ(x, t) := log(n ϵ,ν + δ)ϕ(x),
we obtain ( 12)

T 0 d dt H ϕ δ [n ϵ,ν ](t) dt + I ϵ,δ 1 + I ϵ,δ 2 + I ϵ,δ 3 + I ϵ,δ 4 = I ϵ,δ 5 ,
where

I ϵ,δ 1 = T 0 R d n ϵ,ν n ϵ,ν + δ ∇n ϵ,ν • ∇W ν ϕ dx dt, I ϵ,δ 2 = T 0 R d n ϵ,ν log(n ϵ,ν + δ)∇ϕ • ∇W ν dx dt, I ϵ,δ 3 = ϵ T 0 R d |∇n ϵ,ν | 2 n ϵ,ν + δ ϕ dx dt, I ϵ,δ 4 = ϵ T 0 R d log(n ϵ,ν + δ)∇ϕ • ∇n ϵ,ν dx dt, I ϵ,δ 5 = T 0 R d log(n ϵ,ν + δ) n (1) ϵ,ν G (1) (n ν ) + n (2) ϵ,ν G (2) (n ν ) ϕ dx dt.
Now we treat the terms individually. Starting with I ϵ,δ 1 we see

I ϵ,δ 1 = T 0 R d n ϵ,ν n ϵ,ν + δ ∇n ϵ,ν • ∇W ν ϕ dx dt, = T 0 R d ∇n ϵ,ν • ∇W ν ϕ dx dt -δ T 0 R d ∇n ϵ,ν • ∇W ν n ϵ,ν + δ ϕ dx dt, = - T 0 R d n ϵ,ν ∆W ν ϕ dx dt - T 0 R d n ϵ,ν ∇W ν • ∇ϕ dx dt + δ T 0 R d log(n ϵ,ν + δ)∆W ν ϕ dx dt + δ T 0 R d log(n ϵ,ν + δ)∇W ν • ∇ϕ dx dt.
Passing to the limit ϵ → 0, we readily obtain

I ϵ,δ 1 → I δ 1 = - T 0 R d n ν ∆W ν ϕ dx dt - T 0 R d n ν ∇W ν • ∇ϕ dx dt + δ T 0 R d log(n ν + δ)∆W ν ϕ dx dt + δ T 0 R d log(n ν + δ)∇W ν • ∇ϕ dx dt.
For the last two integrals, we observe that

δ T 0 R d | log(n ν + δ)||∆W ν |ϕ dx dt + δ T 0 R d |log(n ν + δ)||∇W ν • ∇ϕ| dx dt ≤ Cδ|log δ|.
Thus, when ϵ → 0 and δ → 0,

I ϵ,δ 1 → - T 0 R d n ν ∆W ν ϕ dx dt - T 0 R d n ν ∇W ν • ∇ϕ dx dt.
Now we look at the next term from Eq. ( 12), namely

I ϵ,δ 2 = T 0 R d n ϵ,ν log(n ϵ,ν + δ)∇ϕ • ∇W ν dx dt.
Using the Dominated Convergence Theorem, in the limit ϵ → 0 and δ → 0 we find that

I ϵ,δ 2 → T 0 R d n ν log n ν ∇ϕ • ∇W ν dx dt.
The next term from Eq. ( 12), I ϵ,δ 3 , is nonnegative and can be dropped in the limit. The fourth term from Eq. ( 12) is estimated by

I ϵ,δ 4 = ϵ T 0 R d log(n ϵ,ν + δ)∇ϕ • ∇n ϵ,ν dx dt ≤ ϵ ∥n ϵ,ν ∥ L 2 (0,T ;L 2 (R d )) ∥∇ϕ∥ L ∞ (R d ) + | log δ|∥∇ϕ∥ L 2 (R d ) ∥∇n ϵ,ν ∥ L 2 (0,T ;L 2 (R d )) ≤ C √ ϵ,
having used Lemma 2.3. Thus, when ϵ → 0 we have I ϵ,δ 4 → 0. The final term from Eq. ( 12) is given by

I ϵ,δ 5 = T 0 R d log(n ϵ,ν + δ) n (1) ϵ,ν G (1) (n ν ) + n (2) ϵ,ν G (2) (n ν ) ϕ dx dt,
which, again by Dominated Convergence, converges to

I ϵ,δ 5 → T 0 R d log n ν n (1) ν G (1) (n ν ) + n (2) ν G (2) (n ν ) ϕ dx dt. Finally, since n ϵ,ν ∈ C([0, T ]; L 2 (R d )), the mapping t → H ϕ δ [n ϵ,ν (t)] is continuous in [0, T ]. We therefore have T 0 d dt H ϕ δ [n ϵ,ν ] dt = H ϕ δ [n ϵ,ν ](T ) -lim s→0 H ϕ δ [n ϵ,ν ](s) = H ϕ δ [n ϵ,ν ](T ) -H ϕ δ [n in ϵ,ν ],
and, as ϵ → 0 and then δ → 0, we obtain

H ϕ δ [n ϵ,ν ](T ) -H ϕ δ [n in ϵ,ν ] → R d (n ν (T ) + δ)(log(n ν (T ) + δ) -1)ϕ dx - R d (n in ν + δ)(log(n in ν + δ) -1)ϕ dx → R d n ν (T )(log n ν (T ) -1)ϕ dx - R d n in ν (log n in ν -1)ϕ dx.
Combining all the pieces of the jigsaw, we can pass to the limits ϵ → 0 and δ → 0 in Eq. ( 12) to get

H ϕ [n ν ](T ) -H ϕ [n in ν ] - T 0 R d n ν ∆W ν ϕ dx dt - T 0 R d n ν ∇W ν • ∇ϕ dx dt + T 0 R d n ν log n ν ∇ϕ • ∇W ν dx dt ≤ T 0 R d log n ν n (1) ν G (1) (n ν ) + n (2) ν G (2) (n ν ) ϕ dx dt, for any ϕ ∈ C ∞ c (R d ), ϕ ≥ 0. Let us now choose ϕ = χ R , where χ R is a sequence of smooth cut-off functions such that |∇χ R | ≲ R -1 .
Then, using the L ∞ L 1 -control of n ν log n ν from Proposition 2.5, we can pass to the limit R → ∞ by the Monotone Convergence Theorem to obtain Eq. [START_REF] Bertsch | On interacting populations that disperse to avoid crowding: preservation of segregation[END_REF]. □

Let us conclude this section with a short remark that, by virtue of Proposition 2.5, the entropy as well as the right-hand side of Eq. ( 11) are bounded uniformly in ν > 0.

COMPACTNESS OF W ν

This section is dedicated to establishing the strong convergence of W ν in L 2 (0, T ; L 2 (R d )) using the celebrated Aubin-Lions Lemma.

Proposition 3.1 (Regularity of W ν ). Let (W ν ) ν>0 be a sequence of potentials associated to solutions of System (4). Then, there exists a constant C > 0 such that

∥∇W ν ∥ L 2 (0,T ;L 2 (R d )) ≤ C, and
∥∂ t W ν ∥ L 2 (0,T ;H -1 (R d )) ≤ C.
Remark 3.2 (Weak Convergence of Velocity). It is worthwhile to point out that the uniform bound on ∇W ν implies its weak convergence (up to a subsequence) which plays an important role in the subsequent analysis.

Proof. We begin by proving the spatial regularity. To this end, we observe that

T 0 R d |∇W ν | 2 dx dt = - T 0 R d W ν ∆W ν dx dt.
Using Brinkman's law, we get

T 0 R d |∇W ν | 2 dx dt = - T 0 R d n ν ∆W ν dx dt -ν T 0 R d |∆W ν | 2 dx dt ≤ - T 0 R d n ν ∆W ν dx dt.
Rearranging the entropy dissipation inequality, Eq. ( 11), we find (13)

- T 0 R d n ν ∆W ν dx dt ≤ T 0 R d log n ν n (1) ν G (1) (n ν ) + n (2) ν G (2) (n ν ) dx dt + H[n in ν ] -H[n ν ](T ) ≤ C ∥n ν ∥ L ∞ (0,T ;L 1 (R d )) + ∥n ν log n ν ∥ L ∞ (0,T ;L 1 (R d )) + 1 ,
which is bounded uniformly. Hence ∇W ν ∈ L 2 (0, T ; L 2 (R d )) uniformly in ν. As for the bound on the time derivative, let us recall that W ν = K ν ⋆ n ν , and

∥K ν ∥ L 1 (R d ) = 1. Then, for any φ ∈ L 2 (0, T ; H 1 (R d )), we have T 0 R d φ ∂W ν ∂t dx dt = T 0 R d ∂n ν ∂t K ν ⋆ φ dx dt = - T 0 R d n ν ∇W ν • ∇φ ⋆ K ν dx dt + T 0 R d n (1) ν G (1) (n ν ) + n (2) ν G (2) (n ν ) φ ⋆ K ν dx dt ≤ ∥n ν ∥ L ∞ (0,T ;L ∞ (R d )) ∥∇W ν ∥ L 2 (0,T ;L 2 (R d )) ∥∇φ∥ L 2 (0,T ;L 2 (R d )) + C∥φ∥ L 2 (0,T ;L 2 (R d )) .
It follows that ∂ t W ν ∈ L 2 (0, T ; H -1 (R d )), uniformly in ν > 0. □ From Proposition 3.1, we deduce the existence of a function χ ∈ L 2 (0, T ; H 1 (R d )) such that, after passing to a (non-relabelled) subsequence,

W ν → χ, strongly in L 2 (0, T ; L 2 (R d ))
, and ∇W ν ⇀ ∇χ, weakly in L 2 (0, T ; L 2 (R d )), as ν → 0. From the previous section and Bounds (B1) and (B2) we deduce that the sequence n ν has a weak limit in L 2 (0, T ; L 2 (R d )), denoted by n 0 . It is not difficult to identify the limit of W ν , χ, as n 0 -this is done in the next proposition.

Proposition 3.3 (Identification of limit). Let n 0 be the weak limit of n ν . Then, the strong limit of W ν in L 2 (0, T ; L 2 (R d )) is n 0 . Consequently, n 0 is well defined as an element of L 2 (0, T ; H 1 (R d )).

Proof. Let φ ∈ C ∞ c (R d × (0, T )) be a test function. Then T 0 R d W ν φ dx dt → T 0 R d χφ dx dt.
Concomitantly, we know,

T 0 R d W ν φ dx dt = T 0 R d K ν ⋆ n ν φ dx dt = T 0 R d n ν K ν ⋆ φ dx dt.
Combining the two equations, we get

T 0 R d n ν K ν ⋆ φ dx dt → T 0 R d χφ dx dt.
On the other hand, we know that

n ν ⇀ n 0 in L 2 (0, T ; L 2 (R d )) and K ν ⋆ φ → φ in L 2 (0, T ; L 2 (R d )), as ν → 0. Thus, T 0 R d n ν K ν ⋆ φ dx dt → T 0 R d n 0 φ dx dt.
This shows that χ = n 0 , almost everywhere in R×(0, T ). It follows that n 0 ∈ L 2 (0, T ; L 2 (R d )) with ∇n 0 ∈ L 2 (0, T ; L 2 (R d )). □

Using the entropy inequality, the strong compactness of W ν can be leveraged to get the compactness of the total densities n ν .

Proposition 3.4. We have

n ν → n 0 in L 2 (0, T ; L 2 (R d )).
Proof. Using Brinkman's law and the entropy inequality, cf. (13), we have

T 0 R d |n ν -W ν | 2 dx dt = T 0 R d (n ν -W ν )(n ν -W ν ) dx dt = -ν T 0 R d n ν ∆W ν dx dt + ν T 0 R d W ν ∆W ν dx dt = -ν T 0 R d n ν ∆W ν dx dt -ν T 0 R d |∇W ν | 2 dx dt ≤ Cν.
Consequently,

∥n ν -n 0 ∥ L 2 (0,T ;L 2 (R d )) ≤ ∥n ν -W ν ∥ L 2 (0,T ;L 2 (R d )) + ∥W ν -n 0 ∥ L 2 (0,T ;L 2 (R d )) ≤ C √ ν + ∥W ν -n 0 ∥ L 2 (0,T ;L 2 (R d )) . □ 4 PASSAGE TO THE LIMIT
In this section, we conclude the passage to the limit ν → 0. Let us recall that, thanks to the a priori estimates of Section 2, there are functions n

(i) 0 ∈ L ∞ (0, T ; L 1 (R d ) ∩ L ∞ (R d )
) such that, up to extracting subsequences

n (i) ν ⇀ n (i) 0 , i = 1, 2, and n ν ⇀ n 0 = n (1) 0 + n (2)
0 , weakly in L 2 (0, T ; L 2 (R d )). In Section 3 we proved that n ν → n 0 , as well as W ν → n 0 , strongly in L 2 (0, T ; L 2 (R d )); as well as

∇W ν ⇀ ∇n 0 , weakly in L 2 (0, T ; L 2 (R d ))
. The last remaining ingredient in the proof of Theorem 1.3 is to upgrade weak convergence in the last term to convergence in norm. This shall be the major focus of this section.

4.1 Limit equation for the total density. First, having established the strong compactness of the total density n ν , we can pass to the limit in the equation for n ν to obtain (in the distributional sense) ( 14)

∂n 0 ∂t -∇ • (n 0 ∇n 0 ) = n (1) 0 G (1) (n 0 ) + n (2)
0 G (2) (n 0 ). To pass to the limit in the equations for the individual densities we will use the entropy inequality [START_REF] Bertsch | On interacting populations that disperse to avoid crowding: preservation of segregation[END_REF] together with the corresponding entropy equality satisfied by the solution to Eq. ( 14) to upgrade the weak convergence of the velocity ∇W ν to strong convergence. From the previous estimates we clearly have

n 0 ∈ L ∞ (0, T ; L 1 (R d ) ∩ L ∞ (R d )),
and we recall from Proposition 3.3 that

n 0 ∈ L 2 (0, T ; H 1 (R d )).
It now follows from Eq. ( 14) that

∂n 0 ∂t ∈ L 2 (0, T ; H -1 (R d )).
Consequently, we deduce that

n 0 ∈ C([0, T ]; L 2 (R d )).
Finally, we observe that Proposition 2.5 implies that

sup t∈[0,T ] R d |x| 2 n 0 dx < ∞,
and therefore

sup t∈[0,T ] R d n 0 | log n 0 | dx < ∞.
This regularity is enough to guarantee the entropy equality ( 15)

H[n 0 ](T ) + T 0 R d |∇n 0 | 2 dx dt -R n (1) 0 , n (2)
0 , n 0 = H[n in 0 ], where R n (1) 0 , n (2)
0 , n 0 := T 0 R d log n 0 n (1) 0 G (1) (n 0 ) + n (2) 0 G (2) (n 0 ) dx dt.
The last term is well-defined, since R n

(1) 0 , n (2) 0 , n 0 ≤ C T 0 R d n 0 | log n 0 | dx dt ≤ CT.

Strong convergence of the velocity.

We now aim to prove that ( 16) ∇W ν → ∇n 0 strongly in L 2 (0, T ; L 2 (R d )). To do so, we will show that, in addition to the weak convergence ∇W ν ⇀ ∇n 0 , there holds convergence of L 2 norms:

∥∇W ν ∥ L 2 (0,T ;L 2 (R d )) → ∥∇n 0 ∥ L 2 (0,T ;L 2 (R d )) . Since L 2 (0, T ; L 2 (R d )) ≃ L 2 (R d × (0, T )
) is a uniformly convex Banach space, the conjunction of these two properties implies convergence in norm topology [START_REF] Bruna | Diffusion of multiple species with excludedvolume effects[END_REF].

To achieve this goal, we shall compare the two entropy inequalities [START_REF] Bertsch | On interacting populations that disperse to avoid crowding: preservation of segregation[END_REF] and [START_REF] Bruna | Derivation of a macroscopic model for Brownian hard needles[END_REF]. A connection is of course provided by the terms involving the initial data. Let us therefore recall the assumption:

H[n in ν ] ≤ H[n in 0 ]. Let R n (1) ν , n (2) ν , n ν := T 0 R d log n ν n (1) ν G (1) (n ν ) + n (2) ν G (2) (n ν ) dx dt.
Upon rearranging the inequalities ( 11) and ( 15) we get

- T 0 R d n ν ∆W ν dx dt ≤ H[n in ν ] -H[n ν ](T ) + R n (1) ν , n (2) ν , n ν ≤ H[n in 0 ] -H[n ν ](T ) + R n (1) ν , n (2) ν , n ν = T 0 R d |∇n 0 | 2 dx dt + E ν , (17)
where, for convenience, we use the notation

E ν := H[n 0 ](T ) -H[n ν ](T ) + R n (1) ν , n (2) ν , n ν -R n (1) 0 , n (2)
0 , n 0 .

Recall that

T 0 R d |∇W ν | 2 dx dt = - T 0 R d W ν ∆W ν dx dt = - T 0 R d n ν ∆W ν dx dt + T 0 R d (n ν -W ν )∆W ν dx dt = - T 0 R d n ν ∆W ν dx dt - 1 ν T 0 R d |n ν -W ν | 2 dx dt ≤ - T 0 R d n ν ∆W ν dx dt.
This, in conjunction with Eq. ( 17), gives

T 0 R d |∇W ν | 2 dx dt ≤ T 0 R d |∇n 0 | 2 dx dt + E ν . (18)
We will now investigate the term E ν and show that it vanishes in the limit ν → 0.

Given the uniform L 1 ∩L ∞ -bound on n ν (T ) and uniform control of the second moment, we infer that the sequence n ν (T ) converges to n 0 (T ) weakly in L 1 (R d ). Therefore, by the convexity of the entropy functional, we have

H[n 0 ](T ) ≤ lim inf ν→0 H[n ν ](T ), or equivalently lim sup ν→0 (H[n 0 ](T ) -H[n ν ](T )) ≤ 0.
Let us now focus on the reaction terms. We shall use the tightness of the sequence (n ν ) ν , as well as its pointwise convergence to n 0 , to split the domain of integration into a subdomain on which the terms corresponding to n ν converge to their counterparts for n 0 and remainders which can be made arbitrarily small. Let ε > 0 and 0 < β < e -1 . By tightness, there exists a set

K ⊂ R d × [0, T ] of finite measure such that sup ν>0 K c n ν ≤ ε.
Furthermore, by Egorov's Theorem there exists A ⊂ K with meas(A) < ε such that n ν converges uniformly to n 0 on K \ A. Then, for ν small enough, we have

n 0 ≥ β 2 =⇒ n ν ≥ β 4 , n 0 < β 2 =⇒ n ν < β. Writing T 0 R d n (i) ν log n ν G (i) (n ν ) dx dt = K c n (i) ν log n ν G (i) (n ν ) dx dt+ K n (i) ν log n ν G (i) (n ν ) dx dt,
we can guarantee that the first integral is small. Indeed, letting n(i) be the zero of G (i) , we have

K c ∩{nν <n (i) } n (i) ν log n ν G (i) (n ν ) dx dt ≤ K c ∩{nν <n (i) } n (i) ν (n ν -1)G (i) (n ν ) dx dt ≤ ∥G (i) ∥ L ∞ (R) + ∥n ν ∥ L ∞ (R d ) K c n ν dx dt ≤ Cε,
having used that log s ≤ s -1 for s > 0 and that G (i) (n ν ) ≥ 0 on {n ν < n(i) }. Next,

K c ∩{nν ≥n (i) } n (i) ν log n ν G (i) (n ν ) dx dt ≤ ∥G (i) ∥ L ∞ (R) K c ∩{nν ≥n (i) } n ν | log n ν | dx dt ≤ ∥G (i) ∥ L ∞ (R) K c ∩{nν ≥n (i) } ( √ n ν + n 2 ν ) dx dt ≤ ∥G (i) ∥ L ∞ (R) ∥n ν ∥ L ∞ (0,T ;L ∞ (R d )) + 1 √ n(i) K c n ν dx dt ≤ Cε.
It remains to treat the integral over K. This we split into three disjoint subdomains as follows

K = (K \ A ∩ {n 0 ≥ β/2}) ∪ (K \ A ∩ {n 0 < β/2}) ∪ A ≡ K 1 ∪ K 2 ∪ A.
For the integral over A, we notice that A n (i) ν log n ν G (i) (n ν ) dx dt ≤ ∥G (i) ∥ L ∞ (R) 1 + ∥n ν ∥ 2 L ∞ (0,T ;L ∞ (R d )) meas(A) ≤ Cε, while the integral over K 2 is bounded as follows

K 2
n (i) ν log n ν G (i) (n ν ) dx dt ≤ ∥G (i) ∥ L ∞ (R) meas(K)β| log β| ≤ Cβ| log β|.

Finally, on the set K 1 , the sequence log n ν converges strongly to log n 0 and therefore

K 1 n (i) ν log n ν G (i) (n ν ) dx dt → K 1 n (i) 0 log n 0 G (i) (n 0 ) dx dt.
It follows that R n (1) ν , n (2) ν , n ν -R n

(1)

0 , n (2)
0 , n 0

≤ K 1 n (1)
ν log n ν G (1) (n ν ) dx dt -

K 1 n (1)
0 log n 0 G (1) (n 0 ) dx dt

+ K 1 n (2) ν log n ν G (2) (n ν ) dx dt - K 1 n (2)
0 log n 0 G (2) (n 0 ) dx dt + C(ε + β| log β|).

Passing to the limit in Eq. ( 18), we therefore deduce lim sup It follows that ∇W ν → ∇n 0 , strongly in L 2 (0, T ; L 2 (R d )). Armed with the strong convergence of the velocities, we can now pass to the limit in System (4) to obtain

∂ t n (i) 0 + ∇ • n (i) 0 ∇n 0 = n (i) 0 G (i) (n 0 ), i = 1, 2.
This concludes the last step of the proof of Theorem 1.3.

A SHORT REMARK ON PHENOTYPICALLY STRATIFIED MODELS

Let us conclude our analysis with a short remark on an interesting extension of the model. In recent years, there has been a growing interest in structured models for tissue growth. As the aggressiveness of a tumour is intimately connected to the cells' phenotypic traits which, in turn, influence several characteristics, such as cell motility or cell division rate, a phenotypic variable, y ∈ [0, 1], may be introduced to account for the population's heterogeneity. In [START_REF] David | Phenotypic heterogeneity in a model of tumour growth: existence of solutions and incompressible limit[END_REF], the author shows the existence of weak solutions to ∂n ∂t (y, x, t) = ∇ • (n∇N ) + nG(y, N ), where all operators act on the x-variable, which can be understood as a structured counterpart of System [START_REF] Berendsen | On a crossdiffusion model for multiple species with nonlocal interaction and size exclusion[END_REF]. Here, the total population density is given by the integral of the phenotypic densities along the interval of traits, i.e., N (x, t) := 1 0 n(y, x, t) dy, acting as the infinite-dimensional counterpart of n = n (1) + n (2) . In the context of viscoelastic tissues, we propose a structured Brinkman model of the form

   ∂n ν ∂t (y, x, t) -∇ • (n ν ∇W ν ) = n ν G(y, N ν ), -ν∆W ν + W ν = N ν ,
where, again, all operators act in physical space. Formally, we obtain the entropy dissipation inequality of the total population, i.e., which gives strong reason to believe that our strategy is applicable in other contexts.

|∇W ν | 2

 2 |∇n 0 | 2 dx dt + C(ε + β| log β|).Since ε and β can be chosen arbitrarily small, we deduce the desired upper semicontinuity propertylim sup ν→0 T 0 R d |∇W ν | 2 dx dt ≤ T 0 R d |∇n 0 | 2 dx dt.On the other hand, from the weak convergence∇W ν ⇀ ∇n 0 in L 2 (0, T ; L 2 (R d )) we have dx dt = T 0 R d |∇n 0 | 2 dx dt.

HR d log N ν 1 0

 1 [N ν ](T ) -H[N in ν ] -T 0 R d N ν ∆W ν dx dt ≤ T 0 n ν (y)G(y, N ν ) dy dx dt,paralleling that of Lemma 2.6. Using a similar argument as in the main part of the paper, we expect to obtainlim ν→0 T 0 R d |∇W ν | 2 dx dt = T 0 R d |∇N 0 | 2 dx dt,

ACKNOWLEDGEMENTS T.D. would like to acknowledge the hospitality of Technische Universität Dresden during his research stay, where the foundation for this project was laid. M.S. is happy to acknowledge the fruitful discussions during his time as visiting professor at the University of Warsaw during the Thematic Research Programme 'PDEs in Mathematical Biology' (Excellence Initiative Research University). N.D. was supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program « Investissements d'Avenir » (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).