Nonlocal and local models for taxis in cell migration: a rigorous limit procedure

Maria Krasnianski,a Kevin J. Painter,b Christina Surulescu,c and Anna Zhigund

Abstract A rigorous limit procedure is presented which links nonlocal models involving adhesion or nonlocal chemotaxis to their local counterparts featuring haptotaxis and classical chemotaxis, respectively. It relies on a novel reformulation of the involved nonlocalities in terms of integral operators applied directly to the gradients of signal-dependent quantities. The proposed approach handles both model types in a unified way and extends the previous mathematical framework to settings that allow for general solution-dependent coefficient functions. The previous forms of nonlocal operators are com- pared with the new ones introduced in this paper and the advantages of the latter are highlighted by concrete examples. Numerical simulations in 1D provide an illustration of some of the theoretical findings.

Keywords: cell-cell and cell-tissue adhesion; nonlocal and local chemotaxis; haptotaxis; integro- differential equations; unified approach; global existence; rigorous limit behaviour; weak solutions. MSC 2010: 35Q92, 92C17, 35K55, 35R09, 47G20, 35B45, 35D30.

1 Introduction

Macroscopic equations and systems describing the evolution of populations in response to soluble and insoluble environmental cues have been intensively studied and the palette of such reaction-diffusion- taxis models is continuously expanding. Models of such form are motivated by problems arising in various contexts, a large part related to cell migration and proliferation connected to tumor invasion, embryonal development, wound healing, biofilm formation, insect behavior in response to chemical cues, etc. We refer, e.g. to [5] for a recent review also containing some deduction methods for taxis equations based on kinetic transport equations. Apart from such purely local PDE systems with taxis, several spatially nonlocal models have been in- troduced over the last two decades and are attracting ever increasing interest. They involve integro- differential operators in one or several terms of the featured reaction-diffusion-drift equations. Their aim is to characterize interactions between individuals or signal perception happening not only at a specific location, but over a whole set (usually a ball) containing (centered at) that location. In the context of cell populations, for instance, this seems to be a more realistic modeling assumption, as cells are able to extend various protrusions (such as lamellipodia, filopodia, cytonemes, etc.) into their surroundings, which can reach across long distances compared against cell size, see [26, 45] and references therein. Moreover, the cells are able to relay signals they perceive and thus transmit them to cells with which they are not in direct contact, thereby influencing their motility, see e.g., [20, 22]. Cell-cell and cell- tissue adhesion are essential for mutual communication, homeostasis, migration, proliferation, sorting, arXiv:1908.10287v2 [math.AP] 26 Nov 2019 and many other biological processes. A large variety of models for adhesive behavior at the cellular level have been developed to account for the dynamics of focal contacts, e.g. [3,4, 50] and to assess their influ- ence on cytoskeleton restructuring and cell migration, e.g. [13, 12, 32, 49]. Continuous, spatially nonlocal models involving adhesion were introduced more recently [2] and are attracting increasing interest from the modeling [6,8,9, 14, 24, 25, 38, 40], analytical [10, 16, 17, 46, 30], and numerical [23] viewpoints. Yet more recent models [15, 19] also take into account subcellular level dynamics, thus involving fur- ther nonlocalities (besides adhesion), with respect to some structure variable referring to individual cell

aFelix-Klein-Zentrum f¨urMathematik, Technische Universit¨atKaiserslautern, Paul-Ehrlich-Str. 31, 67663 Kaiser- slautern, Germany, [email protected] bDepartment of Mathematics & Maxwell Institute, Heriot-Watt University, Edinburgh, Scotland, EH14 4AS, [email protected] cFelix-Klein-Zentrum f¨urMathematik, Technische Universit¨atKaiserslautern, Paul-Ehrlich-Str. 31, 67663 Kaiser- slautern, Germany, [email protected] dSchool of Mathematics and Physics, Queen’s University Belfast, University Road, Belfast BT7 1NN, Northern Ireland, UK, [email protected]

1 state. Thereby, multiscale mathematical settings are obtained, which lead to challenging problems for analysis and numerics. Another essential aspect of cell migration is the directional bias in response to a diffusing signal, commonly termed chemotaxis. A model of cell migration with finite sensing radius, thus featuring nonlocal chemotaxis has been introduced in [39] and readdressed in [29] from the perspective of well-posedness, long time behaviour, and patterning. We also refer to [36] for further spatially nonlocal models and their formal deduction. For adhesion and nonlocal chemotaxis models, a gradient of some nondiffusing or diffusing signal is replaced by a nonlocal integral term. Here we are only interested in this type of model, and refer to [11, 18, 31] for reviews on settings involving other types of nonlocality. Specifically, following [2, 24, 29, 39], we consider the subsequent systems, whose precise mathematical formulations will be specified further below: 1. a prototypical nonlocal model for adhesion

Btcr “ ∇ ¨ pDcpcr, vrq∇cr ´ crχpcr, vrqArpgpcr, vrqqq ` fcpcr, vrq, (1.1a)

Btvr “ fvpcr, vrq, (1.1b)

where 1 ξ Arupxq :“ - upx ` ξq Frp|ξ|q dξ (1.2) r |ξ| ż Br

is referred to as the adhesion velocity, and the function Fr describes how the magnitude of the interaction force depends on the interaction range |ξ| within the sensing radius r. We require this function to satisfy

Assumptions 1.1 (Assumptions on Fr).

2 (i) pr, ρq ÞÑ Frpρq is continuous and positive in r0, r0s for some r0 ą 0; 1 (ii) F0p0q “ n ` 1.

The quantity Fpcr, vrq “ crχpcr, vrqArpgpcr, vrqq is often referred to as the total adhesion flux, possibly scaled by some constant involving the typical cell size or the sensing radius, see e.g., [2,8]. Here we also include a coefficient χpcr, vrq that depends on cell and tissue (extracellular matrix, ECM) densities, which can be seen as characterizing the sensitivity of cells towards their neighbours and the surrounding tissue. It will, moreover, help provide in a rather general framework a unified presentation of this and the subsequent local and nonlocal model classes for adhesion, haptotactic, and chemotactic behavior of moving cells. System (1.1) is a simplification of the integro-differential system (4) in [24]. The main difference between the two settings is that in our case we ignore the so-called matrix-degrading enzymes (MDEs). Instead, we assume cells directly degrade the tissue directly: this fairly standard simpli- fication (e.g., [40]) effectively assumes that proteolytic enzymes remain localised to the cells, and helps simplify the analysis. On the other hand, (1.1) can also be viewed as a nonlocal version of the haptotaxis model with nonlinear diffusion:

Btc “ ∇ ¨ pDcpc, vq∇c ´ cχpc, vq∇gpc, vqq ` fcpc, vq, (1.3a)

Btv “ fvpc, vq; (1.3b)

2. a prototypical nonlocal chemotaxis-growth model

Btcr “∇ ¨ Dcpcr, vrq∇cr ´ crχpcr, vrq∇˚rvr ` fcpcr, vrq, (1.4a)

Btvr “Dv∆´vr ` fvpcr, vrq ¯ (1.4b)

with the nonlocal gradient n ∇˚ upxq :“ - upx ` rξqξ dξ. r r ż Sr 1In Section3 we will see that this is, indeed, the ’right’ normalisation. If we assume, as in [2], that this function is a constant involving some viscosity related proportionality, then this choice provides the value of that constant.

2 System (1.4) can be seen as a nonlocal version of the chemotaxis-growth model

Btc “∇ ¨ pDcpc, vq∇c ´ cχpc, vq∇vq ` fcpc, vq, (1.5a)

Btv “Dv∆v ` fvpc, vq, (1.5b) where χpc, vq is the chemotactic sensitivity function. As mentioned above, in order to have a unified description of our systems (1.3) and (1.5) and of their respective nonlocal counterparts (1.1) and (1.4), we later introduce a more general version of the nonlocal chemotaxis flux, similar to the above adhesion velocity Ar. n Here and below Br and Sr denote the open r-ball and the r-sphere in R , both centred at the origin, and 1 - upξq dξ :“ upξq dξ, |B | ż Br r żBr 1 - upξq dξ :“ upξq dS pξq |S | r ż Sr r żSr are the usual mean values of a function u over Br and Sr, respectively. The nonlocal systems (1.3) and (1.5) are stated for n t ą 0, x P Ω Ă R . Unless the spatial domain Ω is the whole Rn, suitable boundary conditions are required. In the latter case, usually periodicity is assumed, which is not biologically realistic in general. Still, this offers the easiest way to properly define the output of the nonlocal operator in the boundary layer where the sensing region is not fully contained in Ω. Very recently various other boundary conditions have been derived and compared in the context of a single equation modeling cell-cell adhesion in 1D [7]. Few previous works focus on solvability for models with nonlocality in a taxis term. Some of them deal with single equations that only involve cell-cell adhesion [17, 16,7], others study nonlocal systems of the sort considered here for two [29] or more components [19]. The global solvability and boundedness study in [30] is obtained for the case of a nonlocal operator with integration over a set of sampling directions being an open, not necessarily strict subset of Rn. The systems studied there include settings with a third equation for the dynamics of diffusing MDEs. Conditions which secure uniform boundedness of solutions to such cell-cell and cell-tissue adhesion models in 1D were elaborated in [46]. Some heuristic analysis via local Taylor expansions was performed in [24] and [28] showing that as r Ñ 0 the outputs Aru and ∇˚ru, respectively, converge pointwise to ∇u for a fixed and sufficiently smooth u. In [29] it was observed that it would be interesting to study rigorously the limiting behaviour of solutions of the nonlocal problems involving ∇˚ru. The authors ask in which sense, if at all, do these solutions converge to solutions of the corresponding local problem as r Ñ 0. Numerical results appeared to confirm that, in certain cases, the answer is positive. Still, to the best of our knowledge, no rigorous analytical study of this issue has as yet been performed. Clearly, any approach based on representations using Taylor polynomials requires a rather high order regularity of solution components and a suitable control on the approximation errors, and that uniformly in r. This is difficult or even impossible to obtain in most cases, particularly when dealing with weak solutions. In this work we propose a different approach based on the representation of the input u in terms of an integral of ∇u over line segments. This leads to a new description of the nonlocal operators Ar and ∇˚r in terms of nonlocal operators applied to gradients (see Section3 below). Moreover, it turns out that redefining their outputs inside the vanishing boundary layer in a suitable way allows one to perform a rigorous proof of convergence: Under suitable assumptions on the system coefficients and other parameters, appropriately defined sequences of solutions to nonlocal problems involving the mentioned modified nonlocal operators converge for r Ñ 0 to those of the corresponding local models (1.3) and (1.5), respectively. Our convergence proof is based on estimates on cr and vr which are uniform in r and on a compactness argument. The two models (1.1) and (1.4) are chosen as illustrations, however our idea can be further applied to other integro-differential systems with similar properties. The rest of the paper is organised as follows. Section2 introduces some basic notations to be used throughout this paper. In Section3 we introduce the aforementioned adaptations of the nonlocal operators Ar and ∇˚r and study their limiting properties as r becomes infinitesimally small. This turns out to be useful for our convergence proof later. We also establish in Section4 the well-posedness for a certain class of equations including such operators. In the subsequent Section5 we introduce a couple of nonlocal models that involve the previously considered averaging operators, prove the global existence of solutions of the respective systems, and investigate their limit behaviour as r Ñ 0. Section6 provides some numerical simulations comparing various nonlocal and local models considered in this work in the 1D case. Finally, Section7 contains a discussion of the results and a short outlook on open issues.

3 2 Basic notations and function spaces

We denote the Lebesgue measure of a set A by |A|. Let Ω Ă Rn be a bounded domain with smooth enough boundary. For a function w :Ω Ñ Rn we assume, by convention, that

n w :“ 0 in R zΩ. For r ą 0 we introduce the following subdomain of Ω

Ωr :“ tx P Ω : distpx, BΩq ą ru.

Partial derivatives, in both classical and distributional sense, with respect to variables t and xi, will be denoted respectively by Bt and Bxi . Further, ∇, ∇¨ and ∆ stand for the spatial gradient, divergence and Laplace operators, respectively. Bν is the derivative with respect to the outward unit normal of BΩ. We assume the reader to be familiar with the definitions and the usual properties of such spaces as: the standard Lebesgue and Sobolev spaces, spaces of functions with values in these spaces, and with 2 anisotropic Sobolev spaces. In particular, we denote by Cwpr0,T s; L pΩqq the space of functions u : r0,T s Ñ L2pΩq which are continuous w.r.t. the weak topology of L2pΩq. ˚ Throughout the paper h¨, ¨iX˚,X denotes a duality paring between a space X and its dual X . Finally, we make the following useful convention: For all indices i, the quantity Ci denotes a positive constant or, alternatively, a positive function of its arguments. Moreover, unless explicitly stated, these constants do not depend upon r.

3 Operators Ar and ∇˚r and averages of ∇

In this section we study the applications of the non-local operators Ar and ∇˚r to fixed, i.e. independent of r, functions u. Our focus is on the limiting behaviour as r Ñ 0. Formal Taylor expansions performed in [24, 29] anticipate that the limit is the gradient operator in both cases. This we prove here rigorously under rather mild regularity assumptions on u. To be more precise, we replace Ar and ∇˚r by certain integral operators Tr and Sr (see (3.2) and (3.7) below) applied to ∇u and show that these operators are pointwise approximations of the identity operator in the Lp spaces. 1 We start with the operator Ar. For r P p0, r0s, u P C pΩq, and x P Ωr we compute that 1 ξ Arupxq “ - upx ` ξq Frp|ξ|q dξ r |ξ| ż Br 1 ξ “ - pupx ` ξq ´ upxqq Frp|ξ|q dξ r |ξ| ż Br 1 1 ξ “ - p∇upx ` sξq ¨ ξq ds Frp|ξ|q dξ r |ξ| ż Br ż0 1 1 ξ “ - p∇upx ` sξq ¨ ξq Frp|ξ|q dξ ds r |ξ| ż0 ż Br 1 y “ - p∇upx ` rsyq ¨ yq Frpr|y|q dyds. (3.1) |y| ż0 ż B1 Formula (3.1) extends to arbitrary u P W 1,1pΩq by means of a density argument. Motivated by (3.1) we introduce the averaging operator

1 y Trwpxq :“ - pwpx ` rsyq ¨ yq Frpr|y|q dyds. (3.2) |y| ż0 ż B1 1 n In Subsection 3.1 we check that Trwpxq is well-defined for all w P pL pΩqq and a.a. x P Ω. In this 1,1 notation, for all r P p0, r0s and u P W pΩq identity (3.1) takes the form

Aru “ Trp∇uq a.e. in Ωr.

In the limiting case r “ 0 we have that

1 y T0wpxq “ - pwpxq ¨ yq F0p0q dyds, |y| ż0 ż B1

4 n yiyj “F0p0q wipxqej - dy |y| i,j“1 B1 ÿ ż n 2 yi “F0p0q wipxqejδij - dy |y| i,j“1 B1 ÿ ż n 2 yi “F0p0q wipxqei - dy |y| i“1 B1 ÿ ż n 1 |y|2 “F0p0q wipxqei - dy n |y| i“1 ż B1 1ÿ “F p0q - |y| dy wpxq 0 n ż B1 “wpxq. (3.3)

In the final step we used Assumptions 1.1(ii) which says that F0p0q “ n ` 1 (this explains our choice) and the trivial identity n - |y| dy “ . (3.4) n ` 1 ż B1 Thus, we have just proved the following lemma:

1,1 Lemma 3.1 (Adhesion velocity vs. Tr). Let u P W pΩq. Then it holds that

Aru “ Trp∇uq a.e. in Ωr for r P p0, r0s. (3.5)

Moreover, if F0p0q “ n ` 1, then

∇u “ T0p∇uq in Ω. (3.6)

In a very similar manner one can establish a representation for ∇˚r. For this purpose we define the averaging operator

1

Srwpxq :“n - pwpx ` rsyq ¨ yqy dS1 pyqds for r P p0, r0s. (3.7) ż0 ż S1 The corresponding result then reads:

1,1 Lemma 3.2 (Non-local gradient vs. Sr). Let u P W pΩq. Then it holds that

∇˚ru “Srp∇uq a.e. in Ωr for r P p0, r0s, (3.8)

∇u “S0p∇uq a.e. in Ω. (3.9)

The proof of Lemma 3.2 is very similar to that of Lemma 3.1 and we omit it here. Next, we observe that identity (3.5) was established for Ωr. In the boundary layer ΩzΩr the definition (1.2) of the adhesion velocity allows various extensions. For example, one could keep (1.2) by assuming (as done, e.g., in [19]) that u :“ 0 in RnzΩ. An alternative would be to average over the part of the r-ball that lies inside the domain. Let us have a closer look at the first option (the second can be handled similarly). Consider the following example:

1 Example 3.3 (Ar vs. Trp∇¨q in 1D). Let Ω “ p´1, 1q, r0 “ 1, Fr ” 2, and u ” 1. In this case, u ” 0, hence 1 1 Trpu q ” 0 ” u .

For Ar one readily computes by assuming u “ 0 in Rzp´1, 1q that for x P p´1, 1q 2 1 A upxq “ signpξq dξ r r 2r żp´1´x,1´xqXp´r,rq 1 r2 p´1 ` r ´ xq in r´1, ´1 ` rs, “ $0 in p´1 ` r, 1 ´ rq “ Ωr, ’ 1 & r2 p1 ´ r ´ xq in r1 ´ r, 1s, %’ 5 so that 1 ´1`r 1 1 }A u} 1 “}A u} 1 “ |´1 ` r ´ x| dx ` |1 ´ r ´ x| dx “ 1, r L p´1,1q r L pΩzΩr q r2 r2 ż´1 ż1´r although

|ΩzΩr| “ 2r Ñ 0. rÑ0 Thus, 1 Aru Ñ 0 ” u rÑ0 in the measure but not in L1pΩq.

Example 3.3 supports our idea to average ∇u instead of u itself. The same applies to ∇˚ru vs. Srp∇uq. Averaging w.r.t. y P B1 and then also w.r.t. s P p0, 1q might appear superfluous in the definition of the operator Tr. The following example compares the effect of Tr with that of an operator which averages w.r.t. to y only.

n Example 3.4. Let Ω “ R , n ě 2, and r ą 0, Fr ” n ` 1. In this case 1 y Trwpxq :“ pn ` 1q - pwpx ` rsyq ¨ yq dyds. |y| ż0 ż B1 Consider also the operator y Trwpxq :“ pn ` 1q - pwpx ` ryq ¨ yq dy. |y| ż B1 It is easy to see that both operatorsr are well-defined, linear, continuous, and self-adjoint in the space 2 n n n L pR q. Moreover, they map the dense subspace C0pR ; R q into itself. This suggests the following n n ˚ natural extension to pC0pR ; R qq :

hTrµ, ϕi n n ˚ n n :“ hµ, Trϕi n n ˚ n n , pC0pR ;R qq ,C0pR ;R q pC0pR ;R qq ,C0pR ;R q D E D E Trµ, ϕ :“ µ, Trϕ . n n ˚ n n n n ˚ n n pC0pR ;R qq ,C0pR ;R q pC0pR ;R qq ,C0pR ;R q Let, for instance, r r w :“ δ0e1,

δ0 and e1 mean the usual Dirac delta and the vector p1, 0,..., 0q, respectively. One readily computes that

n ` 1 x1 x Trpδ0e1qpxq “ χBr pxq , |Br| r |x| whereas r

1 n ` 1 ´n´1 x1 x Trpδ0e1qpxq “ s χB pxq ds |B | rs r |x| r ż0 n ` 1 r n x x “ ´ 1 1 . n|B | |x| r |x| r ˆˆ ˙ ˙`

For n ě 2, the operator Tr retains the singularity at the origin, however making it less concentrated, while Tr eliminates that singularity entirely and produces instead jump discontinuities all over Sr.

r 3.1 Properties of the averaging operators Tr and Sr

In this section we collect some properties of the averaging operators Tr and Sr.

Lemma 3.5 (Properties of Tr). Let Fr satisfy Assumptions 1.1 and let r P p0, r0s. Then:

6 p n (i) Tr is a well-defined continuous linear operator in pL pΩqq for all p P r1, 8s. The corresponding operator norm satisfies

}Tr}LppLppΩqqnq ď C1pr, pq, (3.10)

where

1 1 p˚ n´1`p˚ p˚ 1 1 n ρ pFrprρqq dρ for p P p1, 8s, ` ˚ “ 1, C pr, pq :“ p p 1 $ˆ 0 ˙ ’ maxş ρFrprρq for p “ 1. &ρPr0,1s ’ %’ n ˚ 1 1 p n p˚ (ii) Let p, p P r1, 8s be such that p ` p˚ “ 1. For all w1 P pL pΩqq and w2 P L pΩq it holds: ´ ¯

pTrw1pxq ¨ w2pxqq dx “ pw1pxq ¨ Trw2pxqq dx. (3.11) żΩ żΩ (iii) Let p P r1, 8q. For all w P pLppΩqqn it holds that

p n Trw Ñ T0w “ w in pL pΩqq . (3.12) rÑ0

(iv) For p “ 2 it holds that

}Tr}LppL2pΩqqnq Ñ 1. (3.13) rÑ0

Remark 3.6. Due to the assumptions on Fr we have in the limit that

1 n p˚ 1 1 pn ` 1q n p˚ for p P p1, 8szt2u, p ` p˚ “ 1, C1pr, pq Ñ C2ppq :“ ` (3.14) rÑ0 $n 1´ ¯ for p 1. & ` “ Proof. (of Lemma 3.5) %

y (i) Since w is measurable and ρ ÞÑ Frpρq, px, s, yq ÞÑ x ` rsy, py, zq ÞÑ pz ¨ yq |y| are continuous, we have that y px, y, sq ÞÑ pwpx ` rsyq ¨ yq Frpr|y|q |y|

1 1 is well-defined a.e. in Ω ˆ B1 ˆ p0, 1q and is measurable. Let p P p1, 8q and p ` p˚ “ 1. Using H¨older’sinequality, Fubini’s theorem, and our convention that w vanishes outside Ω, we deduce for all w P pLppΩqqn that

1 p p y }Trw} p n “ - pwpx ` rsyq ¨ yq Frpr|y|q dyds dx pL pΩqq |y| żΩ ˇż0 ż B1 ˇ ˇ ˇ p 1 p˚ ˇ p ˇ p˚ ď ˇ - |wpx ` rsyq| dy - p|y|Frpr|yˇ|qq dy dsdx żΩ ż0 ż B1 ˆż B1 ˙ 1 p p “C1 pr, pq - |wpx ` rsyq| dxdyds, ż0 ż B1 żΩ 1 p p ďC1 pr, pq - |wpzq| dzdyds 0 B1 Ω p ż żp ż “C1 pr, pq}w}pLppΩqqn .

p n This implies that for all p P p1, 8q operator Tr is well-defined in pL pΩqq and satisfies (3.10). It is p n also clearly linear. Taken together we then have that Tr P LppL pΩqq q and (3.10) holds. The cases p “ 1 and p “ 8 can be treated similarly.

7 n p n p˚ (ii) Let w1 P pL pΩqq and w2 P L pΩq . We compute by using Fubini’s theorem, the symmetry of B1, and simple variable transformations´ ¯ that

pTrw1pxq ¨ w2pxqq dx żΩ 1 y “ - pw1px ` rsyq ¨ yq Frpr|y|q dyds ¨ w2pxq dx |y| żΩ ż0 ż B1 1 y y “ - |y|Frpr|y|q w1px ` rsyq ¨ w2pxq ¨ dx dy ds |y| |y| ż0 ż B1 żΩ ˆ ˙ ˆ ˙ 1 y y “ - |y|Frpr|y|q w1px ` rsyq ¨ w2pxq ¨ dx dy ds (3.15) |y| |y| ż0 ż B1 żΩXp´rsy`Ωq ˆ ˙ ˆ ˙ 1 y y “ - |y|Frpr|y|q w1pzq ¨ w2pz ´ rsyq ¨ dzdyds |y| |y| ż0 ż B1 żprsy`ΩqXΩ ˆ ˙ ˆ ˙ 1 y y “ - |y|Frpr|y|q w1pzq ¨ w2pz ` rsyq ¨ dzdyds. (3.16) |y| |y| ż0 ż B1 żp´rsy`ΩqXΩ ˆ ˙ ˆ ˙ Thereby we used our convention that each function defined in Ω is assumed to be prolonged by zero outside Ω. Comparing (3.15) and (3.16) we obtain (3.11).

(iii) We apply the Banach-Steinhaus theorem. Due to (i) and (3.14), tTrurPp0,r0s is a family of uniformly p n n p n bounded linear operators in the Banach space pL pΩqq . Thus, as CcpΩ; R q is dense in pL pΩqq n for p ă 8, we only need to check (3.12) for w P CcpΩ; R q. But for such w we can directly pass to the limit under the integral and thus obtain using (3.3) and the dominated convergence theorem that

p n Trw Ñ T0w “ w for all x P Ω and in pL pΩqq . rÑ0

(iv) Here we make use of the Fourier transform, which we denote by the hat symbol. A straightforward calculation shows that

Trw “ Φrw,

where y p 1 T yy irsy¨ξ Φrpξq :“ - Frpr|y|qe dyds. (3.17) |y| ż0 ż B1 Combining (3.17) with the Plancherel theorem and using our convention that w vanishes outside Ω, we can estimate as follows:

}Tr}LppL2pΩqqnq “ sup }Trw}pL2pΩqqn }w}pL2pΩqqn “1

2 n n ď sup }Trw}pL pR qq }w}pL2pΩqqn “1

8 n y 2 n n ď}|Φr|2}L pR q sup }w}pL pR qq }w}pL2pΩqqn “1

n 2 n “}|Φr|2}L8pR q sup }wp}pL pΩqq }w}pL2pΩqqn “1

n “}|Φr|2}L8pR q. (3.18)

nˆn Here |M|2 denotes the spectral norm of a matrix M P R . Further, observe that

T nˆn n ΦrpOξq “ OΦrpξqO for all orthogonal O P R and ξ P R . (3.19)

n Consequently, denoting by e1 the first canonical vector of R and appropriately constructing an orthogonal matrix O in order for Oξ “ |ξ|e1 to hold, we obtain that

n |Φrpξq|2 “ |Φrp|ξ|e1q|2 for all ξ P R . (3.20)

8 Since 1 yyT irs|ξ|y1 Φrp|ξ|e1q “ - Frpr|y|qe dyds (3.21) |y| ż0 ż B1 is a diagonal matrix, its spectral norm is given by the spectral radius. Estimating the right-hand side of (3.21) we then conclude that

1 n |Φrp|ξ|e1q|2 ď - |y|Frpr|y|q dy Ñ 1 for all ξ P R (3.22) n rÑ0 ż B1 due to F0p0q “ n ` 1 and (3.4). Combining (3.18), (3.20), and (3.22) we arrive at

lim sup }Tr}LppL2pΩqqnq ď 1. (3.23) rÑ0 Finally, the pointwise convergence (3.12) and the Banach-Steinhaus theorem imply that

lim inf }Tr}LppL2pΩqqnq ě 1, rÑ0 concluding the proof.

A similar result holds for Sr:

Lemma 3.7 (Operator Sr). Let r P r0, r0s. Then: p n (i) Sr is a well-defined continuous linear operator in pL pΩqq for all p P r1, 8s. The corresponding operator norm satisfies

}Sr}LppLppΩqqnq ď n. (3.24)

n ˚ 1 1 p n p˚ (ii) Let p, p P r1, 8s be such that p ` p˚ “ 1. For all w1 P pL pΩqq and w2 P L pΩq it holds: ´ ¯ pSrw1pxq ¨ w2pxqq dx “ pw1pxq ¨ Srw2pxqq dx. żΩ żΩ (iii) Let p P r1, 8q. For all w P pLppΩqqn it holds that p n Srw Ñ S0w “ w in pL pΩqq . rÑ0

(iv) For p “ 2 it holds that

}Sr}LppL2pΩqqnq Ñ 1. rÑ0 Proof. The proof almost repeats that of Lemma 3.5. Therefore, we only check (3.24) and omit further 1 1 details. Let p P r1, 8q and p ` p˚ “ 1. Using H¨older’s inequality, Fubini’s theorem, and our convention that w vanishes outside Ω we deduce for all w P pLppΩqqn that 1 p p p }Srw}pLppΩqqn “n - pwpx ` rsyq ¨ yqy dS1 pyqds dx żΩ ˇż0 ż S1 ˇ ˇ 1 ˇ p ˇ p ˇ ďn ˇ - |wpx ` rsyq| dS1 pyqdsdx ˇ żΩ ż0 ż S1 1 p p “n - |wpx ` rsyq| dxdS1 pyqds, ż0 ż S1 żΩ 1 p p ďn - |wpzq| dzdS1 pyqds ż0 ż S1 żΩ p p “n }w}pLppΩqqn , which means that

}Sr}LppLppΩqqnq ď n. (3.25) The proof in the case p “ 8 follows the same steps, or, alternatively, one passes to the limit as p Ñ 8 in (3.25).

9 Remark 3.8. The constants in (3.10) for any n ě 1 and in (3.24) for n ě 2 are not necessarily optimal. For p ‰ 2 it remains open whether or not

lim inf }Tr}L Lp Ω n “ 1, rÑ0 pp p qq q

lim inf }Sr}L Lp Ω n “ 1. rÑ0 pp p qq q The answer may depend upon Ω and p.

4 Well-posedness for a class of evolution equations involving Tr or Sr In this Section we establish the existence and uniqueness of solutions to a certain class of single evolution equations involving Tr or Sr. This result is an important ingredient for our analysis of nonlocal systems in Section5 . Thus, we consider the following initial boundary value problem:

Btcr “ ∇ ¨ pa1∇cr ´ a2GεpRrpa3∇crqqq ` f in p0,T q ˆ Ω, (4.1a)

pa1∇cr ´ a2Rrpa3∇crqq ¨ ν “ 0 in p0,T q ˆ BΩ, (4.1b)

crp0, ¨q “ c0 in Ω. (4.1c)

Here

Rr P tTr, Sru, and for ε ě 0 we set

n n x Gε : R Ñ R , x ÞÑ . (4.2) 1 ` ε|x|

A standard calculation shows that Gε is globally Lipschitz with a Lipschitz constant 1. Remark 4.1. Observe that for ε “ 0 equation (4.1a) is linear, whereas for ε ą 0 the nonlocal part of the flux is a priori bounded. The latter helps us to construct nonnegative solutions in Section5 . We make the following assumptions:

8 8 a1, a2, a3 P L p0,T ; L pΩqq, (4.3) ´1 8 8 a1 ą 0 and a1 P L p0,T ; L pΩqq, (4.4) 1 1 ´ 2 ´ 2 a1 a2 a1 a3 }Rr}LppL2pΩqqnq ă 1, (4.5) L8p0,T ;L8pΩqq L8p0,T ;L8pΩqq › › › › f› P L2p0›,T ; pH1pΩqq˚q›, › (4.6) › › › › 2 c0 P L pΩq. (4.7)

To shorten the notation, we introduce a pair of constants

1 1 ´1 ´1 ´ 2 ´ 2 αr :“}a1 }L8p0,T ;L8pΩqq 1 ´ a1 a2 a1 a3 }Rr}LppL2pΩqqnq , L8p0,T ;L8pΩqq L8p0,T ;L8pΩqq ˆ › › › › ˙ M :“}a } 8 8 ` }a } ›8 ›8 }a } 8 › 8 ›}R } 2 n . r 1 L p0,T ;L pΩqq 2 L› p0,T ;L› pΩqq 3 L p0,T› ;L pΩqq› r LppL pΩqq q Due to assumptions (4.3)–(4.5) it is clear that

0 ăαr,Mră 8.

We introduce a family of operators

hMpt, uq, ϕipH1pΩqq˚,H1pΩq :“ a1pt, ¨q∇u ¨ ∇ϕ dx ´ a2pt, ¨qGεpa3pt, ¨qRrp∇uqq ¨ ∇ϕ dx, Ω Ω ż T ż

hMpuq, ϕiL2p0,T ;pH1pΩqq˚q,L2p0,T ;H1pΩqq :“ hMpt, uq, ϕptqipH1pΩqq˚,H1pΩq dt. ż0 Lemma 4.2. Let (4.3)–(4.5) be satisfied. Then:

10 (i) For a.a. t P r0,T s the operator

Mpt, ¨q : H1pΩq Ñ pH1pΩqq˚

is well-defined, monotone, hemicontinuous, and satisfies for all u P H1pΩq the bounds

2 hMpt, uq, uipH1pΩqq˚,H1pΩq ě αr||∇u||pL2pΩqqn , (4.8)

||Mpt, uq||pH1pΩqq˚ ď Mr||∇u||pL2pΩqqn . (4.9) Moreover, for all u P H1pΩq the function Mp¨, uq is measurable. (ii) The operator

M : L2p0,T ; H1pΩqq Ñ L2p0,T ; pH1pΩqq˚q

is well-defined, monotone, hemicontinuous, and satisfies for all u P L2p0,T ; H1pΩqq the bounds

2 hMpuq, uiL2p0,T ;pH1pΩqq˚q,L2p0,T ;H1pΩqq ě αr||∇u||L2p0,T ;pL2pΩqqnq,

||Mpuq||L2p0,T ;pH1pΩqq˚q ď Mr||∇u||L2p0,T ;pL2pΩqqnq.

Proof. The assumptions on the coefficients ai together with the Lipschitz continuity of Gε readily imply that for a.a. t P r0,T s the operator Mpt, ¨q is well-defined and satisfies (4.9). Moreover, due to (4.3) and 1 ˚ 1 Gε Lipschitz, it is also clear that Mp¨, uq : r0,T s Ñ pH pΩqq is measurable on r0,T s for all u P H pΩq, whereas for a.a. t the mapping λ ÞÑ hMpt, u ` λvq, wipH1pΩqq˚,H1pΩq is continuous on R, the latter meaning that Mpt, ¨q is hemicontinuous. Using H¨older’sinequality, the fact that Gε is Lipschitz with Lipschitz constant 1, the assumptions on the ai’s, and the properties of Rr, we compute that

hMpt, uq ´ Mpt, vq, u ´ vipH1pΩqq˚,H1pΩq

“ ∇pu ´ vq ¨ a1pt, ¨q∇pu ´ vq dx ´ pGεpRrpa3pt, ¨q∇uqq ´ GεpRrpa3pt, ¨q∇vqqq ¨ a2pt, ¨q∇pu ´ vq dx żΩ żΩ 1 2 1 1 1 1 2 ´ 2 2 ´ 2 2 ě a1 ∇pu ´ vq ´ Rr a1 a3pt, ¨q a1 ∇pu ´ vq a1 a2pt, ¨q a1 ∇pu ´ vq dx pL2pΩqqn żΩ › 1 › ˇ ´1 ´ ¯¯ˇ ˇ 1 ´ 2 ¯ˇ › ´ 2 › ˇ ´ 2 ˇ ˇ 2 ˇ ě › 1 ´ a1 a2› ˇ a1 a3 }Rr}LppL2pˇΩˇqqnq a1 ∇pu ´ vq ˇ L8p0,T ;L8pΩqq L8p0,T ;L8pΩqq pL2pΩqqn ˆ › › › › ˙ › › α ∇›u v ›2 › › › › (4.10) ě r } p› ´ q}›pL2pΩqqn › › › › for u, v P H1pΩq, which proves monotonicity. Further, taking v “ 0 in (4.10) and using Mpt, 0q “ 0 yields (4.8). Part (i) is thus proved. A proof of (ii) can be done similarly; we omit further details.

Using the properties of the averaging operators proved in Subsection 3.1 we can define weak solutions to (4.1) in a manner very similar to that for the classical, purely local case (i.e., when a2 ” 0):

Definition 4.3. Let (4.3)-(4.7) hold. We call the function cr : r0,T s ˆ Ω Ñ R a weak solution of (4.1) if:

2 1 2 2 1 ˚ (i) cr P L p0,T ; H pΩqq X Cpr0,T s; L pΩqq, Btcr P L p0,T ; pH pΩqq q;

1 (ii) cr satisfies (4.1a)-(4.1b) in the following sense: for all ϕ P H pΩq and a.a. t P p0,T q

hBtcr, ϕipH1pΩqq˚,H1pΩq “ ´ a1∇cr ¨ ∇ϕ dx ` a2Gεpa3Rrp∇crqq ¨ ∇ϕ dx ` hf, ϕipH1pΩqq˚,H1pΩq ; Ω Ω ż ż (4.11)

2 (iii) crp0, ¨q “ c0 in L pΩq. Using standard theory one readily proves the following existence result: Lemma 4.4. Let (4.3)-(4.7) hold. Then there exists a unique weak solution to (4.1) in terms of Defini- tion 4.3. The solution satisfies the following estimates:

2 2 2 2 }cr}Cpr0,T s;L2pΩqq ` αr}∇cr}L2p0,T ;pL2pΩqqnq ď C3pαr,T q }c0}L2pΩq ` }f}L2p0,T ;pH1pΩqq˚q , (4.12)

2 2 2´ ¯ }Btcr}L2p0,T ;pH1pΩqq˚q ď C4pαr,Mr,T q }c0}L2pΩq ` }f}L2p0,T ;pH1pΩqq˚q . (4.13) ´ ¯ 11 Proof. The existence of a unique weak solution to (4.1) is a direct consequence of Lemma 4.2(i) and the standard theory of evolution equations with monotone operators, see, e.g. [47, Chapter III Proposition 4.1]. It remains to check the bounds (4.12) and (4.13). Taking ϕ :“ cr in the weak formulation (4.11) and using [48, Chapter III Lemma 1.2], (4.8), and the Young inequality, we obtain that

1 d 2 2 }c } 2 ď ´ α }∇c } 2 n ` }c } 1 }f} 1 ˚ 2 dt r L pΩq r r pL pΩqq r H pΩq pH pΩqq 2 2 “ ´ αr}cr}H1pΩq ` αr}cr}L2pΩq ` }cr}H1pΩq}f}pH1pΩqq˚

1 2 2 1 ´1 2 ď ´ α }c } 1 ` α }c } 2 ` α }f} 1 ˚ , 2 r r H pΩq r r L pΩq 2 r pH pΩqq which yields (4.12) due to the Gronwall lemma. Finally, using (4.9), we obtain from the weak formulation (4.11) that

2 2 2 2 }Btcr}L2p0,T ;pH1pΩqq˚q ď2Mr }∇cr}L2p0,T ;pL2pΩqqnq ` 2}f}L2p0,T ;pH1pΩqq˚q. Together with (4.12) this implies (4.13).

5 Nonlocal models involving averaging operators Tr and Sr In this section we study the following model IBVP:

` Btcr “ ∇ ¨ pDcpcr, vrq∇cr ´ crχpcr, vrqRrp∇gpcr, vrqqq ` fcpcr, vrq in R ˆ Ω, (5.1a) ` Btvr “ Dv∆vr ` fvpcr, vrq in R ˆ Ω, (5.1b) ` Dcpcr, vrqBν cr ´ crχpcr, vrqRrp∇gpcr, vrqq ¨ ν “ DvBν vr “ 0 in R ˆ BΩ, (5.1c) crp0, ¨q “ c0, vrp0, ¨q “ v0 in Ω. (5.1d)

Here, as in the previous section, Rr stands for any of the two averaging operators:

Rr P tTr, Sru.

We assume that the diffusion coefficient Dv is either a positive number, or it is zero. Equations (5.1a)-(5.1b) are closely related to (1.1) and (1.4) in Section1 , the difference being that the terms involving the adhesion velocity/non-local gradient are now replaced by those including the averaging operators Tr/Sr from Section3 . Our motivation for introducing this change is twofold. First of all, due to (3.5) and (3.8) it affects the points in the boundary layer ΩzΩr, at the most. On the other hand, Example 3.3 indicates that including, e.g., Ar can lead to limits with unexpected blow-ups on the boundary of Ω. System (5.1) is a non-local version of the hapto-/chemotaxis system

` Btc “ ∇ ¨ pDcpc, vq∇c ´ cχpc, vq∇gpc, vqq ` fcpc, vq in R ˆ Ω, (5.2a) ` Btv “ Dv∆v ` fvpc, vq in R ˆ Ω, (5.2b) ` Dcpc, vqBν cr ´ cχpc, vqBν gpc, vq “ DvBν v “ 0 in R ˆ BΩ, (5.2c) cp0, ¨q “ c0, vp0, ¨q “ v0 in Ω. (5.2d)

In this case, the actual diffusion and haptotactic sensitivity coefficients are

Dcpc, vq “ Dcpc, vq ´ cχpc, vqBcgpc, vq,

χpc, vq “ χpc, vqBvgpc, vq, r so that in the classical formulation (5.2a) takes the form r ` Btc “ ∇ ¨ Dcpc, vq∇c ´ cχpc, vq∇v ` fcpc, vq. in R ˆ Ω. ´ ¯ The main goal of this Section isr to establish, underr suitable assumptions on the system coefficients which are introduced in Subsection 5.1, a rigorous convergence as r Ñ 0 of solutions of the nonlocal model family (5.1) to those of the local model (5.2), see Theorem 5.8. This is accomplished in the final Subsection 5.4. Since we are dealing here with a new type of nonlocal system, we establish for (5.1) the existence of nonnegative solutions in Subsections 5.2 and 5.3.

12 5.1 Problem setting and main result of the section We begin with several general assumptions about the coefficients of system (5.1).

` ` ` 1 ` ` Assumptions 5.1. Let Dv P R0 , Dc, χ P CbpR0 ˆ R0 q, and g, fc, fv P C pR0 ˆ R0 q satisfy for some s ě 0:

` ` C5 ď Dc ď C6 in R0 ˆ R0 for some C5,C6 ą 0, 8 ` ` 2 ∇pc,vqg, ∇pc,vqfv P pL pR0 ˆ R0 qq ,

fcp0, ¨q ” 0,

fvp¨, 0q ” 0.

Assume that the coefficients satisfy the following bounds:

C12 :“ sup c|χpc, vq| ă 8, (5.3) c,vě0

C13 :“ sup |Bcgpc, vq| ă 8. (5.4) c,vě0 Further, we assume that the initial values satisfy

2 0 ď c0 P L pΩq, 1 0 ď v0 P H pΩq. (5.5)

Remark 5.2. If Dv ą 0, then assumption (5.5) can be replaced by a weaker one, such as

2 v0 P L pΩq.

We keep (5.5) in order to simplify the exposition.

In addition, we will later choose one of the following assumptions on fc and the nonlocal operator:

Assumptions 5.3 (Further assumptions on fc). One of the following conditions holds: (a) 8 ` ` 2 ∇pc,vqfc P L pR0 ˆ R0 q (b) ` ˘

s ` ` |fcpc, vq| ď C7p1 ` |c| q in R0 ˆ R0 for some C7 ě 0, (5.6) s`1 ` ` cfcpc, vq ď C8 ´ C9c in R0 ˆ R0 for some C8 ě 0,C9 ą 0.

Assumptions 5.4 (Assumptions on Rr). One of the following holds:

(a) for a given fixed r P p0, r0s

C12C13 C10p}Rr}q :“ 1 ´ }Rr}LppL2pΩqqnq ą 0 C5

(b)

C12C13 C11 :“ ă 1. (5.7) C5

Example 5.5. Let

Dv “ 0, ´rρ Frpρq :“ pn ` 1qe ,

Sccc ` Scvv gpc, vq :“ for some constants Scc,Scv ą 0, 1 ` c ` v 1 ` c Dcpc, vq :“ , 1 ` c ` v

13 b χpc, vq :“ , b ą 0, 1 ` c ` v c fcpc, vq :“ µc pKc ´ c ´ ηcvq for some constants Kc, ηc ą 0, µcą0, 1 ` c2 c fvpc, vq :“ µvvpKv ´ vq ´ λvv for some constants Kv, λv ą 0, µv ě 0, 1 ` c and assume that

0 ď v0 ď Kv.

Then, it holds a priori that

0 ď v ď Kv

` for any v which solves (5.1b). Therefore it suffices to consider the coefficient functions in R0 ˆ r0,Kvs. ` For Dc it holds on R0 ˆ r0,Kvs that 1 ` c 1 Dcpc, vq ě ě “: C5 1 ` c ` Kv 1 ` Kv and

Dcpc, vq ď 1 “: C6.

8 ` ` 2 Moreover, ∇pc,vqg, ∇pc,vqfv P pL pR0 ˆ R0 qq , due to

|Sccp1 ` vq ´ Scvv| C13 “ sup |Bcgpc, vq| “ max max 2 c,vě0 0ďvďKv cě0 p1 ` c ` vq

Scc ScvKv “ max Scc, ´ , 1 ` K p1 ` K q2 " ˇ v v ˇ* ˇ ˇ ˇ |Scvp1 ` cq ´ Sccc| ˇ sup |Bvgpc, vq| “ max maxˇ 2 ˇ c,vě0 0ďvďKv cě0 p1 ` c ` vq |S p1 ` cq ´ S c| “ max cv cc ă 8, cě0 p1 ` cq2

sup |Bcfvpc, vq| “λvKv c,vě0 and c sup |Bvfvpc, vq| “ sup µvpKv ´ 2vq ´ λv ă 8. 1 ` c c,vě0 c,vě0 ˇ ˇ ˇ ˇ ˇ ˇ ` ` For C7:“µcpKc ` 1 ` ηcKvq, C8:“µcpKc ` 1q andˇ C9:“µc we can estimateˇ on R0 ˆ R0 that

|fcpc, vq| ď C7, c cfcpc, vq ď µc Kc ` ´ c ď C8 ´ C9c. 1 ` c2 ˆ ˙ Further, bc C12 “ sup “ b cě0 1 ` c holds. Thus, Assumptions 5.1, 5.3(b) and 5.4(b) are fulfilled if

Scc ScvKv p1 ` Kvqb max Scc, ´ ă 1. 1 ` K p1 ` K q2 " ˇ v v ˇ* ˇ ˇ ˇ ˇ This choice of coefficient functions can be usedˇ to describe a populationˇ of cancer cells which interact among themselves and with the surrounding extracellular matrix (ECM) tissue. Both interaction types are due to adhesion, whether to other cells (cell-cell adhesion) or to the matrix (cell-matrix adhesion). The

14 interaction force Frpρq is taken to diminish with increasing interaction range ρ and/or of the sensing radius r: cells too far apart/out of reach hardly interact in a direct way. Function gpc, vq characterises effective interactions. Here the coefficients Scc and Scv represent cell-cell and cell-matrix adhesion strengths, respectively. Our choice of g accounts for some adhesiveness limitation imposed by high local cell and tissue densities. It is motivated by the fact that overcrowding may preclude further adhesive bonds, e.g. due to saturation of receptors. The diffusion coefficient Dcpc, vq is chosen to be everywhere positive and increase with a growing population density, thus enhancing diffusivity under population pressure, but, further, limited by excessive cell-tissue interaction. The latter also applies to the choice of the sensitivity function χ. Indeed, there is evidence that tight packing of cells and ECM limits diffusivity and the advective effects of haptotaxis [37]. Thereby the constant b ą 0 is assumed to be rather small. Finally, fc and fv describe growth of cells and tissue limited by concurrence for resources. Next, we introduce weak-strong solutions to our problem. The definition is as follows:

` Definition 5.6. Let Assumptions 5.1 hold. Let r P r0, r0s. We call a pair of functions pcr, vrq : R0 ˆΩ Ñ ` ` R0 ˆ R0 a global weak-strong solution of (5.1) if for all T ą 0:

2 1 2 1 1,8 ˚ (i) cr P L p0,T ; H pΩqq X Cwpr0,T s; L pΩqq, Btcr P L p0,T ; pW pΩqq q;

1 2 2 2 2 (ii) vr P Cpr0,T s; H pΩqq, Btvr P L p0,T ; L pΩqq, Dvvr P L p0,T ; H pΩqq;

1 1 2 2 (iii) fcpcr, vrq P L p0,T ; L pΩqq, fvpcr, vrq P L p0,T ; L pΩqq;

1 (iv) pcr, vrq satisfies (5.1) in the following weak-strong sense: for all ϕ P C pΩq and a.a. t P p0,T q

hBtcr, ϕipW 1,8pΩqq˚,W 1,8pΩq “ ´ pDcpcr, vrq∇cr ´ crχpcr, vrqRrp∇gpcr, vrqqq ¨ ∇ϕ dx żΩ ` fcpcr, vrqϕ dx, (5.8a) żΩ

2 crp0, ¨q “ c0 in L pΩq, (5.8b)

and

Btvr “ Dv∆vr ` fvpcr, vrq a.e. in p0,T q ˆ Ω, (5.8c)

DvBν vr “ 0 a.e. in p0,T q ˆ BΩ, (5.8d) 1 vrp0, ¨q “ v0 in H pΩq. (5.8e)

Remark 5.7. Observe that for r “ 0 we obtain a corresponding solution definition for the local system (5.2). Our main result now reads:

Theorem 5.8. Let Assumptions 1.1, 5.1, 5.3, and 5.4(b) hold. Then, there exists a sequence rm Ñ 0 as m Ñ 8 and solutions pcrm , vrm q and pc, vq in terms of Definition 5.6 corresponding to r “ rm and r “ 0, respectively, s.t.

2 2 cr Ñ c in L p0,T ; L pΩqq, m mÑ8 2 2 vr Ñ v in L p0,T ; L pΩqq. m mÑ8 This Theorem is proved in Subsection 5.4. Notation 5.9. Dependencies upon such parameters as the space dimension n, domain Ω, function c, the norms of the initial data c0 and v0, norms and bounds for the coefficient functions are mostly not indicated in an explicit way.

15 5.2 Global existence of solutions to (5.1): the case of fc Lipschitz

In this Subsection we address the existence of solutions to the nonlocal model (5.1) for the case when fc satisfies Assumptions 5.3(a). The main result of the Subsection is as follows: Theorem 5.10. Let Assumptions 1.1, 5.1, and 5.3(a) hold and let r satisfy Assumptions 5.4(a). Then 2 1 ˚ there exists a global weak-strong solution to (5.1) in terms of Definition 5.6 with Btcr P L p0,T ; pH pΩqq q. Since we aim at constructing nonnegative solutions, it turns out to be helpful to consider first the following family of approximating problems:

Btcrε “ ∇ ¨ Dcpcrε, vrεq∇crε ´ crεχpcrε, vrεq GεpRrpBcgpcrε, vrεq∇crεqq

´ ´ ` ` GεpRrpBvgpcrε, vrεq∇vrεqq ` fcpcrε, vrεq in R ˆ Ω, (5.9a) ¯¯ ` Btvrε “ Dv∆vrε ` fvpcrε, vrεq in R ˆ Ω, (5.9b)

Dcpcrε, vrεq∇crε ´ crεχpcrε, vrεq GεpRrpBcgpcrε, vrεq∇crεqq

´ ` ` GεpRrpBvgpcrε, vrεq∇vrεqq ¨ ν “ DvBν vrε “ 0 in R ˆ BΩ, (5.9c)

crεp0, ¨q “ c0, vrεp0, ¨q “ v0 ¯ in Ω, (5.9d) where Gε was defined in (4.2). In order to obtain existence for the original problem, i.e., for ε “ 0, we first prove existence of nonnegative solutions for the cases when ε, Dc ą 0. This corresponds to a chemotaxis problem with a nonlocal flux-limited drift. Weak-strong solutions to (5.9) are understood as in Definition 5.6, with the obvious modification of the weak formulation, which now reads:

hBtcrε, ϕipH1pΩqq˚,H1pΩq “ ´ Dcpcrε, vrεq∇crε ¨ ∇ϕ dx żΩ ` crεχpcrε, vrεqGεpRrpBcgpcrε, vrεq∇crεqq ¨ ∇ϕ dx żΩ ` crεχpcrε, vrεqGεpRrpBvgpcrε, vrεq∇vrεqq ¨ ∇ϕ ` fcpcrε, vrεqϕ dx. (5.10) żΩ Lemma 5.11. Let Assumptions of Theorem 5.10 be satisfied. Assume further that

ε, Dv ą 0.

2 1 ˚ Then there exists a global weak-strong solution to (5.9) with Btcrε P L p0,T ; pH pΩqq q. Proof. To begin with, we extend the coefficients:

Dcpc, vq :“ Dcp´c, vq, pχ, g, fc, fvqpc, vq :“ ´pχ, g, fc, fvqp´c, vq for c ă 0.

These coefficients still satisfy Assumptions 5.1, 5.3(a), and 5.4(a) if we consider all suprema over c P R ` instead of c P R0 . Our approach to proving existence is based on the classical Leray-Schauder principle [52, Chapter 6, §6.8, Theorem 6.A]. In order to apply this theorem we first ’freeze’ crε in the system coefficients of (5.9), replacing it byc ¯rε. Correspondingly, we obtain the following weak formulation in place of (5.10): For all ϕ P H1pΩq and a.a. t ą 0

hBtcrε, ϕipH1pΩqq˚,H1pΩq “ ´ Dcpc¯rε, vrεq∇crε ¨ ∇ϕ dx żΩ ` c¯rεχpc¯rε, vrεqGεpRrpBcgpc¯rε, vrεq∇crεqq ¨ ∇ϕ dx żΩ ` c¯rεχpc¯rε, vrεqGεpRrpBvgpc¯rε, vrεq∇vrεqq ¨ ∇ϕ ` fcpc¯rε, vrεqϕ dx, (5.11a) żΩ

2 crεp0, ¨q “ c0 in L pΩq (5.11b) and

Btvrε “ Dv∆vrε ` fvpc¯rε, vrεq a.e. in p0,T q ˆ Ω, (5.11c)

16 DvBν vrε “ 0 a.e. in p0,T q ˆ BΩ, (5.11d) 1 vrεp0, ¨q “ v0 in H pΩq. (5.11e)

2 2 Let T ą 0 and letc ¯rε P L p0,T ; L pΩqq. Since fv is assumed to be Lipschitz, we can make use of the standard theory [33] which implies that the semilinear parabolic initial boundary value problem (5.11c)- 2 2 2 2 (5.11e) possesses a unique global strong solution 0 ď vrε P L p0,T ; H pΩqq with Btvrε P L p0,T ; L pΩqq, and satisfying the estimate

2 2 2 2 2 }vrε}L8p0,T ;H1pΩqq ` }vrε}L2p0,T ;H2pΩqq ` }Btvrε}L2p0,T ;L2pΩqq ďC14pT q }v0}H1pΩq ` }c¯rε}L2p0,T ;L2pΩqq . ´ (5.12)¯

Here and further in the proof we omit the dependence of constants upon Dv. Set

a1 :“ Dcpc¯rε, vrεq, a2 :“ c¯rεχpc¯rε, vrεq, a3 :“Bcgpc¯rε, vrεq,

hf, ϕipH1pΩqq˚,H1pΩq :“ c¯rεχpc¯rε, vrεqGεpRrpBvgpc¯rε, vrεq∇vrεqq ¨ ∇ϕ ` fcpc¯rε, vrεqϕ dx. żΩ Due to our assumptions about Dc, χ, g, and fc, these coefficients ai and f satisfy the requirements of Lemma 4.2. Consequently, there exists a unique global weak solution cr to problem (4.1) with these coefficients. We estimate for the corresponding constants αr and Mr introduced in Lemma 4.2:

αr ěC5C10prq “: C15prq, (5.13)

Mr ďC6 ` C12C13 }Rr}LppL2pΩqqnq “: C16prq, (5.14) and, due to (5.12),

}f}L2p0,T ;pH1pΩqq˚q ď}∇vrε}L2p0,T ;pL2pΩqqnq||Bvg|| 8 ` ` }Rr} 2 n C12 L pR0 ˆR0 q LpL pΩqq q

` }Bcfc} 8 ` ` }vrε}L2p0,T ;L2pΩqq ` }c¯rε}L2p0,T ;L2pΩqq L pR0 ˆR0 q

ďC17pr,T q 1 ` }c¯rε}L`2p0,T ;L2pΩqq . ˘ (5.15)

Combining (4.12)-(4.13) and (5.13)-(5.15),` we obtain the following˘ bounds for cr:

2 2 2 }crε}Cpr0,T s;L2pΩqq ` αr}∇crε}L2p0,T ;L2pΩqq ď C18pr,T q 1 ` }c¯rε}L2p0,T ;L2pΩqq , (5.16)

2 2 ´ ¯ }Btcrε}L2p0,T ;pH1pΩqq˚q ď C19pr,T q 1 ` }c¯rε}L2p0,T ;L2pΩqq . (5.17) ´ ¯ Now consider the mapping

Φ :c ¯rε ÞÑ crε.

Thanks to (5.16) and (5.17), Φ is well-defined in L2p0,T ; L2pΩqq and

2 2 2 1 2 1 ˚ Φ: L p0,T ; L pΩqq Ñ tu P L p0,T ; H pΩqq : Btu P L p0,T ; pH pΩqq qu maps bounded sets on bounded sets. (5.18)

Due to the Lions-Aubin lemma, (5.18) implies that

Φ: L2p0,T ; L2pΩqq Ñ L2p0,T ; L2pΩqq maps bounded sets on precompact sets. (5.19)

2 2 2 2 Next, we verify that Φ is closed in L p0,T ; L pΩqq. Consider a sequence tc¯rεmu Ă L p0,T ; L pΩqq s.t.

2 2 c¯rεm Ñ c¯rε in L p0,T ; L pΩqq, (5.20) mÑ8 2 2 Φpc¯rεmq “:crεm Ñ crε in L p0,T ; L pΩqq. (5.21) mÑ8 We need to check that

Φpc¯rεq “ crε.

Due to (5.20) we have (by switching to a subsequence, if necessary) that

c¯rεm Ñ c¯rε a.e. (5.22) mÑ8

17 Further, (5.18) and (5.21) together with the Banach-Alaoglu theorem imply that

2 1 crεm á crε in L p0,T ; H pΩqq, (5.23) mÑ8 2 1 ˚ Btcrεm á Btcrε in L p0,T ; pH pΩqq q. (5.24) mÑ8

1 By the definition of Φ we have thatc ¯rεm and crεm satisfy: for all ϕ P H pΩq and a.a. t P p0,T q

hBtcrεm, ϕipH1pΩqq˚,H1pΩq “ ´ Dcpc¯rεm, vrεmq∇crεm ¨ ∇ϕ dx żΩ ` c¯rεmχpc¯rεm, vrεmqGεpRrpBcgpc¯rεm, vrεmq∇crεmqq ¨ ∇ϕq dx żΩ ` c¯rεmχpc¯rεm, vrεmqGεpRrpBvgpc¯rεm, vrεmq∇vrεmqq ¨ ∇ϕ żΩ ` fcpc¯rεm, vrεmqϕ dx, (5.25a)

2 crεmp0, ¨q “ c0 in L pΩq (5.25b) and

Btvrεm “ Dv∆vrεm ` fvpc¯rεm, vrεmq a.e. in p0,T q ˆ Ω, (5.25c)

DvBν vrεm “ 0 a.e. in p0,T q ˆ BΩ, (5.25d) 1 vrεmp0, ¨q “ v0 in H pΩq. (5.25e)

2 2 From (5.12) and (5.20) we conclude that the sequence tvrεmu is uniformly bounded in L p0,T ; H pΩqq 2 2 and Btvrεm P L p0,T ; pL pΩqq. Hence the Lions-Aubin lemma and the Banach-Alaoglu theorem imply that there exists vrε s.t. (after switching to a subsequence, if necessary)

2 2 vrεm á vrε in L p0,T ; H pΩqq, mÑ8 2 2 Btvrεm á Btvrε in L p0,T ; L pΩqq, mÑ8 2 1 vrεm Ñ vrε in L p0,T ; H pΩqq and a.e. in p0,T q ˆ Ω, (5.26) mÑ8 and this vrε satisfies equation (5.11c) forc ¯rε as well as the initial and boundary conditions in the required sense. Further, due to (5.23) and (5.24) we have in the usual way that

2 crεmpt, ¨q á crεpt, ¨q in L pΩq for all t ą 0. (5.27) mÑ8 In particular,

crεmp0, ¨q “ c0, i.e. the initial condition is satisfied. It remains now to pass to the limit in (5.25a). For this purpose we use the Minty-Browder method. To shorten the notation, we introduce for m P N Y t8u

hMmpuq, ϕiL2p0,T ;pH1pΩqq˚q,L2p0,T ;H1pΩqq T :“ Dcpc¯rεm, vrεmq∇u ¨ ∇ϕ ´ GεpRrpBcgpc¯rεm, vrεmq∇uqqc¯rεmχpc¯rεm, vrεmq ¨ ∇ϕ dxdt, ż0 żΩ hfm, ϕiL2p0,T ;pH1pΩqq˚q,L2p0,T ;H1pΩqq T :“ c¯rεχpc¯rεm, vrεmqGεpRrpBvgpc¯rεm, vrεmq∇vrεmqq ¨ ∇ψ ` fcpc¯rεm, vrεmqψ dxdt, ż0 żΩ where

c¯rε8 :“ c¯rε, v¯rε8 :“ v¯rε.

Due to Lemma 4.2(ii) and (5.14) each operator Mm is monotone, hemicontinuous, and satisfies

||Mmpcrεmq||L2p0,T ;pH1pΩqq˚q ď C16prq}crεm}L2p0,T ;H1pΩqq ď C20prq.

18 Consequently, there is η P L2p0,T ; pH1pΩqq˚q s.t.

2 1 ˚ Mmpcrεmq á η in L p0,T ; pH pΩqq q. (5.28)

Next, from (5.22) and (5.26) we conclude using the boundedness and continuity of functions Gε, ∇g, ∇fc, ` 2 and pc, vq ÞÑ cχpc, vq over R ˆ R0 and of operator Rr in L pΩq and the dominated convergence theorem that

2 1 ˚ fm Ñ f8 in L p0,T ; pH pΩqq q. (5.29) mÑ8 A similar argument yields

2 1 ˚ Mmpwq Ñ M8pwq, in L p0,T ; pH pΩqq q mÑ8 so that due to (5.23) and the compensated compactness

hMmpwq, crεmi 2 1 ˚ 2 1 Ñ hM8pwq, crεi 2 1 ˚ 2 1 . L p0,T ;pH pΩqq q,L p0,T ;H pΩqq mÑ8 L p0,T ;pH pΩqq q,L p0,T ;H pΩqq Observe that the weak formulation (5.25a) is equivalent to

1 ˚ Btcrεm “ ´Mmpcrεmq ` fm in pH pΩqq . (5.30)

Combining (5.24), (5.28), and (5.29) we can pass to the weak limit in (5.30) and obtain

1 ˚ Btcrε “ ´η ` f8 in pH pΩqq . (5.31)

2 1 For w P L p0,T ; H pΩqq and m P N we have due to the monotonicity of Mm that

Xm :“ hMmpcrεmq ´ Mmpwq, crεm ´ wipH1pΩqq˚,H1pΩq ě 0. (5.32)

Moreover, setting ϕ “ crεm in (5.25) and inserting the obtained term into the definition of Xm, we conclude that

Xm “ ´ hMmpcrεmq, wiL2p0,T ;pH1pΩqq˚q,L2p0,T ;H1pΩqq ´ hMmpwq, crεm ´ wiL2p0,T ;pH1pΩqq˚q,L2p0,T ;H1pΩqq

1 2 1 2 ` }c } 2 ´ }c pT q} 2 ` hf , c i 2 1 ˚ 2 1 . (5.33) 2 0 L pΩq 2 rεm L pΩq m rεm L p0,T ;pH pΩqq q,L p0,T ;H pΩqq Combining (5.27) for t “ T ,(5.21), (5.23), (5.28), (5.32), and (5.33), we obtain

0 ď lim sup Xm ď ´ hη, wiL2p0,T ;pH1pΩqq˚q,L2p0,T ;H1pΩqq ´ hM8pwq, crε ´ wiL2p0,T ;pH1pΩqq˚q,L2p0,T ;H1pΩqq mÑ8 1 2 1 2 ` }c } 2 ´ }c pT q} 2 ` hf , c i 2 1 ˚ 2 1 . 2 0 L pΩq 2 rε L pΩq 8 rε L p0,T ;pH pΩqq q,L p0,T ;H pΩqq

2 1 As crε satisfies (5.31), it follows from the last equation that for all w P L p0,T ; H pΩqq it holds that

0 ď hη ´ M8pwq, crε ´ wiL2p0,T ;pH1pΩqq˚q,L2p0,T ;H1pΩqq .

Since M8 is monotone and hemicontinuous, Minty’s lemma implies that it is maximal monotone. Con- sequently, η “ M8pcrεq. Altogether, we conclude that pcrε, vrεq satisfies (5.11) forc ¯rε, meaning that Φpc¯rεq “ crε holds, i.e. Φ is a closed operator. Together with (5.19), this implies that

Φ: L2p0,T ; L2pΩqq Ñ L2p0,T ; L2pΩqq is a compact operator. (5.34)

Since we aim to apply the Leray-Schauder principle [52, Chapter 6, §6.8, Theorem 6.A], it is necessary to consider for λ P p0, 1q the system which corresponds to cr “ λΦpcrq. The corresponding weak-strong formulation reads:

hBtcrε, ϕipH1pΩqq˚,H1pΩq “ ´ Dcpcrε, vrεq∇crε ¨ ∇ϕ dx żΩ ´1 ` crεχpcrε, vrεqλGεpλ RrpBcgpcrε, vrεq∇crεqq ¨ ∇ϕ dx żΩ

19 ` λ GεpRrpBvgpcrε, vrεq∇vrεqq ¨ crεχpcrε, vrεq∇ϕ ` fcpcrε, vrεqϕ dx, Ω ż (5.35a)

2 crεp0, ¨q “ λc0 in L pΩq (5.35b) and

Btvrε “ Dv∆vrε ` fvpcrε, vrεq a.e. in p0,T q ˆ Ω, (5.35c)

DvBν vrε “ 0 a.e. in p0,T q ˆ BΩ, (5.35d) 1 vrεp0, ¨q “ v0 in H pΩq. (5.35e)

Taking ϕ :“ crε in (5.35) and estimating the right-hand side by using Assumptions 5.1 and 5.4(a), the H¨olderinequality, and the fact that |Gεpxq| ď |x|, we obtain that

1 d 2 ||c || 2 2 dt rε L pΩq 2 ď ´ C5C10p}Rr}q }∇crε}pL2pΩqqn 2 ` λ C12||Bvg|| 8 ` ` }Rr}LppL2pΩqqnq }∇crε} 2 n ||∇vrε||pL2pΩqqn `}Bcfc} 8 ` ` ||crε|| 2 L pR0 ˆR0 q pL pΩqq L pR0 ˆR0 q L pΩq ´ 2 ¯ ď ´ C5C10p}Rr}q }∇crε}pL2pΩqqn 2 ` C12||Bvg|| 8 ` ` }Rr}LppL2pΩqqnq }∇crε} 2 n ||∇vrε||pL2pΩqqn `}Bcfc} 8 ` ` ||vrε|| 2 L pR0 ˆR0 q pL pΩqq L pR0 ˆR0 q L pΩq holds for a.e. t P p0,T q. Further, performing estimates similar to the proof of Theorem 5.13 below and using (5.12), we conclude that the set

2 2 cr P L p0,T ; L pΩqq : cr “ λΦpcrq for λ P p0, 1q is uniformly bounded. Consequently, for all ε P p0, 1q the Leray-Schauder principle( implies that Φ has a fixed point crε, which together with the corresponding vrε, satisfies (5.9) in the weak-strong sense on the interval r0,T s. Since T ą 0 was arbitrary, the standard prolongation argument yields the existence of a global solution. It remains to check that crε is nonnegative. Taking ϕ :“ ´pcrεq´ “ mintcrε, 0u in (5.10) and using fcp0, ¨q ” 0, the boundedness of Gε,Dc, Bcfc, and pc, vq ÞÑ cχpc, vq, along with the H¨olderand Young inequalities, yields

1 d 2 }pc q } 2 2 dt rε ´ L pΩq 2 “ ´ Dcp´pcrεq´, vrεq |∇pcrεq´| dx ´ GεpRrpBcgpcrε, vrεq∇crεqq ¨ pcrεq´χp´pcrεq´, vrεq∇pcrεq´ dx żΩ żΩ ´ GεpRrpBvgpcrε, vrεq∇vrεqq ¨ pcrεq´χp´pcrεq´, vrεq∇pcrεq´ dx ` fcp´pcrεq´, vrεqpcrεq´ dx żΩ żΩ 2 2 2 ď ´ C5}∇pcrεq´}pL2pΩqqn ` C12}pcrεq´}L2pΩq}∇pcrεq´}pL2pΩqqn ` }Bcfc}L8 ` ` }pcrεq´}L2pΩq ε pR0 ˆR0 q 2 ďC21}pcrεq´}L2pΩq.

Since crεp0, ¨q “ c0 ě 0, the Gronwall inequality implies that pcrεq´ “ 0, i.e. that crε ě 0.

Remark 5.12. Observe that crε cannot be replaced by ´pcrεq´ inside the nonlocal operator. This is why we introduced the flux-limitation. Now we are ready to prove Theorem 5.10. Proof. (of Theorem 5.10). We start with the case

Dv ą 0.

Lemma 5.11 gives the existence of solutions pcrε, vrεq to (5.9). Setting ϕ “ crε in (5.10), using the facts that fc is Lipschitz and |Gεpxq| ď |x|, we can estimate similarly to Theorem 5.13 below and obtain upper bounds of the form (5.40)-(5.46), which are independent from ε (with p “ q “ 2 there). Applying the

20 Lions-Aubin lemma and the Banach-Alaoglu theorem, we conclude the existence of a pair of nonnegative functions cr and vr having the regularity stated in Definition 5.6 and such that for a sequence εm Ñ 0 mÑ8 it holds that

2 2 crε Ñ cr in L p0,T ; L pΩqq and a.e. in p0,T q ˆ Ω, (5.36) m mÑ8 2 1 vrε Ñ vr in L p0,T ; H pΩqq and a.e. in p0,T q ˆ Ω, (5.37) m mÑ8 2 1 crε á cr in L p0,T ; H pΩqq. (5.38) m mÑ8

x|x| Consider an arbitrary measurable set E Ă p0,T q ˆ Ω. Using Gεpxq ´ x “ ´ε 1`ε|x| , we can estimate for every component i P t1, . . . , nu:

pGεm pRrpBcgpcrεm , vrεm q∇crεm qq ´ RrpBcgpcrεm , vrεm q∇crεm qqi dx dt ˇżE ˇ ˇ T ˇ ˇ 2 ˇ ďεˇm |RrpBcgpcrεm , vrεm q∇crεm | dx dt ˇ 0 Ω ż ż 2 2 n ďεm}Rr}LppL pΩqq qC13}∇crεm }L2p0,T ;pL2pΩqqnq, where the last term tends to 0 as εm Ñ 0. As the term inside the integral is moreover bounded in mÑ8 2 2 L p0,T ; L pΩqq by a constant independent from εm, we conclude by using a result from [21, p. 6] that

2 2 n Gε pRrpBcgpcrε , vrε q∇crε qq ´ RrpBcgpcrε , vrε q∇crε q á 0 in L p0,T ; pL pΩqq q. m m m m m m m mÑ8

2 2 n From this and the boundedness of }∇crεm }L p0,T ;pL pΩqq q,(5.36)-(5.38), Lemma 3.5 or 3.7 (i) and (ii), respectively, the fact that |Gεpxq| ď |x|, the continuity of Bcg, χ,(5.3), (5.4), compensated compactness, the dominated convergence theorem, and the H¨olderinequality, we obtain that for all ψ P L2p0,T ; H1pΩqq it holds that

T

Gεm pRrpBcgpcrεm , vrεm q∇crεm qq ¨ crεm χpcrεm , vrεm q∇ψ dx dt 0 Ω ż T ż Ñ RrpBcgpcr, vrq∇crq ¨ crχpcr, vrq∇ψ dx dt. mÑ8 ż0 żΩ The convergence to the remaining terms in (5.8a) and the rest of (5.8) can be obtained in a way either completely analogous or very similar to the corresponding parts of the proof of Lemma 5.11. In order to prove existence for the case Dv “ 0 consider a family of solutions pcrDv , vrDv q corresponding to Dv P p0, 1q. Estimating similarly to the proof of Theorem 5.13 below and performing a standard limit procedure based on the Banach-Alaoglu theorem, the dominated convergence theorem, the Lions lemma [35, Lemma 1.3], and the compensated compactness, one readily obtains a solution pcr0, vr0q for Dv “ 0 in the sense of Definition 5.6. Observe that this time the gradient of v-component enters linearly, so that no strong convergence is required. We omit further details.

5.3 Global existence of solutions to (5.1): the case of fc dissipative In this Subsection we provide an extension of the existence Theorem 5.10 from Subsection 5.2: Theorem 5.13. Let Assumptions 1.1, 5.1, and 5.3(b) hold and let r satisfy Assumptions 5.4(a). Set

s ` 1 q q :“ min 2, . q˚ :“ . (5.39) s q ´ 1 " * Then there exists a global weak-strong solution to (5.1) in terms of Definition 5.6, with q 1,q˚ ˚ Btcr P L p0,T ; pW pΩqq q and satisfying the following estimates: For all T ą 0

||cr||L8p0,T ;L2pΩqq ď C22pT, }Rr}LppL2pΩqqnqq, (5.40)

||∇cr||L2p0,T ;pL2pΩqqnq ď C22pT, }Rr}LppL2pΩqqnqq, (5.41)

21 ||Btcr||Lq p0,T ;pW 1,q˚ pΩqq˚q ď C22pT, }Rr}LppL2pΩqqnqq, (5.42)

||vr||L8p0,T ;L2pΩqq ď C22pT, }Rr}LppL2pΩqqnqq, (5.43)

||∇vr||L8p0,T ;pL2pΩqqnq ď C22pT, }Rr}LppL2pΩqqnqq, (5.44)

||Btvr||L2p0,T ;L2pΩqq ď C22pT, }Rr}LppL2pΩqqnqq, (5.45)

}fcpcr, vrq}Lq p0,T ;Lq pΩqq ď C22pT, }Rr}LppL2pΩqqnqq, (5.46)

}fvpcr, vrq}L2p0,T ;L2pΩqq ď C22pT, }Rr}LppL2pΩqqnqq. (5.47)

Proof. For k P N set fckpc, vq :“ fcpc, vqηkpcq, where ηk is a cut-off function:

8 ηk P C0 pBkp0qq with ηk ” 1 in Bk´1p0q and 0 ď ηk ď 1. (5.48)

Since fck is Lipschitz, Theorem 5.10 implies the existence of a solution pcrk, vrkq in terms of Definition 5.6 2 1 ˚ with Btcrk P L p0,T ; pH pΩqq q, which corresponds to fc “ fck. Our next aim is to prove that pcrk, vrkq satisfies the same bounds as in the statement of the Theorem with some constant C22pT, }Rr}LppL2pΩqqnqq which does not depend upon k. Set C23p}Rr}q :“}Rr}LppL2pΩqqnq.

Taking ϕ :“ crk in (5.8a) written for crk and using Assumptions 5.1, 5.3(b), 5.4(a) and the H¨olderand Young inequalities, we compute

1 d 2 }c } 2 2 dt rk L pΩq

“ ´ pDcpcrk, vrkq∇crk ´ crkχpcrk, vrkqRrp∇gpcrk, vrkqqq ¨ ∇crk ` crkfckpcrk, vrkq dx Ω ż ´ ¯ 2 1`s ď ´ C5 }∇crk}pL2pΩqqn ` C12 }∇crk}L2pΩq }Rrp∇gpcrk, vrkqq}pL2pΩqqn ` pC8 ´ C9crk qηkpcrkq dx żΩ 2 1`s ď ´ C5 }∇crk}pL2pΩqqn ` C12C23p}Rr}q }∇crk}pL2pΩqqn }∇gpcrk, vrkq}pL2pΩqqn ` C24 ´ C9 crk ηkpcrkq dx Ω 2 ż ď ´ C5 }∇crk}pL2pΩqqn ` C12C23p}Rr}q }∇crk}pL2pΩqqn }Bcgpcrk, vrkq∇crk}pL2pΩqqn

1`s ` C12C23p}Rr}q }∇crk}pL2pΩqqn }Bvgpcrk, vrkq∇vrk}pL2pΩqqn ` C24 ´ C9 crk ηkpcrkq dx Ω 2 ż ď ´ C5C10p}Rr}q }∇crk} 2 n ` C12C23p}Rr}q }Bvg} 8 ` ` }∇crk} 2 n }∇vrk} 2 n pL pΩqq L pR0 ˆR0 q pL pΩqq pL pΩqq 1`s ` C24 ´ C9 crk ηkpcrkq dx żΩ 2 2 1`s ď ´ 2C25p}Rr}q }∇crk}pL2pΩqqn ` C26p}Rr}q }∇vrk}pL2pΩqqn ` C24 ´ C9 crk ηkpcrkq dx. (5.49) żΩ Next, we estimate vrk. If Dv ą 0, then standard theory [33] yields that for all 0 ă t ď T

2 2 2 2 2 }vrk}L8p0,t;H1pΩqq ` }vrk}L2p0,t;H2pΩqq ` }Btvrk}L2p0,t;L2pΩqq ďC27pT q }v0}H1pΩq ` }crk}L2p0,t;L2pΩqq . ´ (5.50)¯

Here and further in the proof we omit the dependence of constants upon Dv. If Dv “ 0, then we get the ODE

Btvrk “fvpcrk, vrkq. (5.51)

Hence, the assumptions on fv and the solution components together with the chain rule imply that

2 1 Btvrk PL p0,T ; H pΩqq.

Computing the gradient on both sides of (5.51), multiplying by ∇vrk throughout, integrating over Ω, and using Assumptions 5.1 and the Young inequality, we obtain that

1 d 2 2 }∇v } 2 n “ B f pc , v q|∇v | ` B f pc , v q∇c ¨ ∇v dx 2 dt rk pL pΩqq v v rk rk rk c v rk rk rk rk żΩ ` ˘ 22 2 ď }Bvfv} 8 ` ` }∇vrk} 2 n ` }Bcfv} 8 ` ` }∇crk}pL2pΩqqn }∇vrk}pL2pΩqqn L pR0 ˆR0 q pL pΩqq L pR0 ˆR0 q 2 2 ďC28}∇vrk}pL2pΩqqn ` C29 }∇crk}pL2pΩqqn . (5.52)

Applying the Gronwall inequality to (5.52) yields

2 2 2 }∇vrk}L8p0,t;L2pΩqq ďC30pT q }∇v0}L2pΩq ` }∇crk}L2p0,t;L2pΩqq . (5.53) ´ ¯ Multiplying (5.51) by vrk we obtain in a similar fashion that

2 2 2 }vrk}L8p0,t;L2pΩqq ďC30pT q }v0}L2pΩq ` }crk}L2p0,t;L2pΩqq . (5.54) ´ ¯ Adding (5.53) and (5.54) together yields

2 2 2 }vrk}L8p0,t;H1pΩqq ďC30pT q }v0}H1pΩq ` }crk}L2p0,t;H1pΩqq . (5.55) ´ ¯ Estimating the right-hand side of (5.51) by using (5.54) implies

2 2 2 }Btvrk}L2p0,T ;L2pΩqq ďC30pT q }v0}L2pΩq ` }crk}L2p0,T ;L2pΩqq . (5.56) ´ ¯ Further, combining (5.49) with (5.50) if Dv ą 0 and with (5.55) if Dv “ 0 and using the Gronwall inequality yields for crk the same estimates as (5.40) and (5.41), and the estimate

T 1`s crk ηkpcrkq dxdt ď C31pT, }Rr}q. (5.57) ż0 żΩ

From (5.6) and (5.57), the embedding of Lebesgue spaces, and ηk P r0, 1s we conclude that

s }fckpcrk, vrkq}Lq p0,T ;Lq pΩqq ďC32pT q ` C33||crkηkpcrkq|| s`1 s`1 L s p0,T ;L s pΩqq s T s`1 1`s ďC32pT q ` C33 crk ηkpcrkq dx dt ˜ż0 żΩ ¸ ďC34p}Rr},T q. so that (5.46) holds for fckpcrk, vrkq. Combining (5.40) and (5.41) for crk with (5.50) or (5.55) and (5.56) (depending on the sign of Dv) and using the equation for vrk yields such bounds as (5.43)-(5.45) and (5.47) for crk and vrk. Finally, combining Assumptions 5.1 with bounds on ∇crk, ∇vrk, and fckpcrk, vrkq, the weak formulation (5.8a), and estimating in a standard way yields (5.42) for Btcrk. Since pcrk, vrkq satisfy (5.40)-(5.47) uniformly in k, a standard limit procedure based on the Banach- Alaoglu theorem, the dominated convergence theorem, the Lions lemma, and the compensated compact- ness yields the existence of a weak-strong solution pcr, vrq to (5.8) which satisfies (5.40)-(5.47).

5.4 Limiting behaviour of the nonlocal model (5.1) as r Ñ 0 In this Subsection we finally prove our main result concerning convergence for r Ñ 0. Proof. (of Theorem 5.8) Due to (5.7) and Lemma 3.5 (iv) or 3.7 (iv), respectively, there exists a sequence rm Ñ 0 as m Ñ 8 such that 1 sup }Rrm }LppL2pΩqqnq ă . mPN C11

Since for each such rm the Assumptions 5.4(a) are satisfied, Theorem 5.13 is applicable and yields the exis- 2 n tence of solutions pcrm , vrm q which satisfy (5.40)-(5.47). Replacing }Rr} by C11 in C22pT, }Rr}LppL pΩqq qq makes the constant in (5.40)-(5.47) independent of m. Using the Lions-Aubin lemma and the Banach- Alaoglu theorem we conclude (by possibly switching to a subsequence) that

2 2 cr Ñ c, vr Ñ v in L p0,T ; L pΩqq, a.e. in p0,T q ˆ Ω (5.58) m mÑ8 m mÑ8 2 1 cr á c, vr á v in L p0,T ; H pΩqq. (5.59) m mÑ8 m mÑ8

23 Using standard arguments based on the Banach-Alaoglu theorem, the dominated convergence theorem, the Lions lemma, and assumptions on χ and g we conclude from (5.58) and (5.59) that

2 2 cr χpcr , vr q Ñ cχpc, vq in L p0,T ; L pΩqq, (5.60) m m m mÑ8 2 1 gpcr , vr q á gpc, vq in L p0,T ; H pΩqq. (5.61) m m mÑ8

Observe that for any ψ P L8p0,T ; W 1,8pΩqq the following estimate holds:

T 2 |Rrm pcrm χpcrm , vrm q∇ψq ´ cχpc, vq∇ψ| dx dt (5.62) ż0 żΩ T 2 ď2 |Rrm pcrm χpcrm , vrm q∇ψq ´ Rrm pcχpc, vq∇ψq| dx dt ˜ż0 żΩ T 2 ` |Rrm pcχpc, vq∇ψq ´ cχpc, vq∇ψ| dx dt . (5.63) ż0 żΩ ¸ Now, using (5.60) together with Lemma 3.5(i) and (iii) and (3.6) or Lemma 3.7(i) and (iii) and (3.9), respectively, we conclude that the right hand side of (5.63) tends to zero, hence

2 2 n Rr pcr χpcr , vr q∇ψq Ñ cχpc, vq∇ψ in L p0,T ; pL pΩqq q. (5.64) m m m m mÑ8 Thus, using Lemma 3.5(ii) or Lemma 3.7(ii), respectively, and compensated compactness, we obtain from (5.61) and (5.64) that

T T

crm χpcrm , vrm qRrm p∇gpcrm , vrm qq ¨ ∇ψ dx dt “ ∇gpcrm , vrm q ¨ Rrm pcrm χpcrm , vrm q∇ψq dxdt 0 Ω 0 Ω ż ż ż T ż Ñ ∇gpc, vq ¨ cχpc, vq∇ψ dxdt. mÑ8 ż0 żΩ The convergence in the remaining terms, equations, and conditions follows by means of a standard limit procedure based on the Banach-Alaoglu theorem, the dominated convergence theorem, the Lions lemma, and the compensated compactness. We omit these details.

6 Numerical simulations in 1D

We perform numerical simulations to investigate on the one hand the effect of differences between hitherto choices of nonlocal operators and our novel ones proposed in Section3 , and on the other hand convergence between nonlocal and local formulations. For compactness, our current study restricts to the prototypical nonlocal model for cellular adhesion (1.1), its reformulation as (5.1), and the corresponding local model (5.2). Thus, for (5.1) we take the operator form Rr “ Tr, with Tr as in (3.2). These models can be interpreted in the context of a population of cells invading an adhesion-laden ECM/tissue environment and, with this in mind, we initially concentrate cells at the centre of a one-dimensional domain Ω “ r0,Ls and impose an initially homogeneous ECM. Specifically, we set for the ECM

v0pxq “ 1, x P Ω (6.1) and consider for the cell population a Gaussian-shaped aggregate

2 c0pxq “ exp ´αpx ´ xcq , x P Ω, (6.2) where we set xc “ L{2 or xc “ 0. ` ˘ The numerical scheme follows that described in [23], which we refer to for details. Briefly, a Method of Lines approach is invoked whereby equations are first discretised in space (in conservative form, via a finite volume method) to yield a high-dimensional system of ODEs, which are subsequently integrated in time. Discretisation of advective terms follows a third order upwinding scheme, augmented by flux limiting to preserve positivity of solutions and the resulting scheme is (approximately) second-order accurate in space. Time integration has been performed with standard Matlab ODE solvers: our default is “ode45” with absolute and relative error tolerances set at 10´6, but simulations have been compared for

24 Figure 1: Comparison between nonlocal formulations (1.1) and (5.1). (a-b) Cell and matrix densities for the models (1.1) and (5.1) at t “ 2.5 and t “ 5. (c) Difference between the solutions. For these simulations we take α “ 10, r “ 1, Dc “ 0.01, χ “ 1, Fr “ 2, fc “ 0 and fvpc, vq “ ´cv, along with (a-c) gpc, vq “ 10v, (d-f) gpc, vq “ 2.5c ` 10v. varying space discretisation step, ODE solver, and error tolerances. To measure the difference between two distinct solutions over time we define a distance function as follows:

dpu1px, tq, u2px, tqqptq “ - |u1px, tq ´ u2px, tq| dx , ż Ω where u1 and u2 denote the two solutions that are being compared.

6.1 Comparison of nonlocal operator representations We first explore the correspondence between forms of nonlocal operator representation: we choose the prototypical nonlocal model for cell/matrix adhesion (1.1) and its reformulation (5.1), therefore taking for the latter the operator form Rr “ Tr with Tr as in (3.2). In what follows, solutions to (1.1) are denoted cA and vA and those for (5.1) denoted cT and vT . For simplicity we restrict in this section to a minimalist formulation in which Dc “ constant, χ “ 1, fc “ 0. Cell-matrix interactions are defined by gpc, vq “ Sccc ` Scvv and fvpc, vq “ ´µcv, where Scc and Scv respectively represent cell-to-cell and cell- to-matrix adhesion strengths and fv simplistically describes (direct) proteolytic degradation of matrix by cells parametrised by degradation rate µ. Figure1 shows the computed solutions under (a-c) negligible cell-cell adhesion (Scc “ 0) and (d-f) moderate cell-cell adhesion (Scc “ Scv{4). The equivalence of the two formulations is revealed through the negligible difference between solutions, with the distance magnitude attributable to the subtly distinct numerical implementation. Both simulations describe an invasion/infiltration process, in which matrix degradation by the cells generates an adhesive gradient that pulls cells into the acellular surroundings. The impact of cell-cell adhesion is manifested in the compaction of cells at the leading edge into a tight aggregate. However, as pointed out in Section3 , differences in the nonlocal formulations can emerge in the vicinity of boundaries. To highlight this we consider an equivalent formulation to Figure1 (a-c), but with the cells initially placed at the left boundary (xc “ 0 in (6.2)), e.g. suggesting a tumor mass which is concentrated there and whose cells are expected to detach and migrate into the considered 1D domain, travelling from left to right. As stated earlier we impose zero-flux boundary conditions at x “ 0 (and x “ L), and further suppose c “ v “ 0 and ∇c “ ∇v in the extradomain region (RzΩ). Representative simulations are shown in Figure2 . They are in agreement with our observation in Example 3.3. Indeed,

25 Figure 2: (a-c) Comparison between nonlocal formulations (1.1) and (5.1) near boundaries. Model as in Figure1 (a-c), but with the cells initially concentrated at the boundary. (d-f) Comparison of the two forms of nonlocal operator corresponding to the simulations represented in (a-c). The operators are practically identical sufficiently far from the boundary, but can diverge significantly for distances ă r from the boundaries. for this scenario, in the prototypical nonlocal model (1.1)-(1.2) there is a very large adhesion velocity modulus at x “ 0; the cells are crowded within the tumor mass and their mutual interactions are maintained during the invasion process in a sufficiently strong manner to ensure a collective shift of the still concentrated cell aggregate, with a correspondingly strong tissue degradation in its wake. In the reformulation (5.1)-(3.2), rather, the adhesion magnitude at x “ 0 is for the same initial condition much lower - suggesting a tumor whose cells are readier to detach and migrate individually. This results in a more diffusive spread, with accordingly less degradation of tissue, and with cell mass remaining available at the original site over a larger time span. The latter scenario is different from the former one, but it seems nevertheless reasonable, as a tumor mass would very often not move as a whole from its original location to another in a relatively short time; moreover, the active cells in a sufficiently large tumor (releasing substantial amounts of acidity) are known to preferentially adopt a migratory phenotype and perform EMT (epithelial-mesenchymal transition), see e.g., [27, 41, 44], which supports the idea of cells moving in a loose way rather than in compact, highly aggregated assemblies 2. As such, our simulations suggest that, within this particular function- and parameter setting, choosing the adhesion operator in the form (1.2) instead of (3.2) might possibly overestimate the tumor invasion speed and associated healthy tissue degradation, thereby predicting a spatially concentrated tumor and neglecting regions with lower cell densities which can nevertheless trigger tumor recurrence if untreated.

6.2 Comparison between nonlocal and local formulation Having compared together the original, (1.1), and the new, (5.1), nonlocal formulations, we next consider the extent to which their dynamics can be captured by the classical local formulation (5.2). Note that for nonlocal model simulations we will restrict to the original formulation (1.1), so that we can avail ourselves of an already well-established efficient (in terms of computational time) numerical scheme [23]. Here we use cL and vL to denote solutions to the local formulation and cAr and vAr to denote solutions to the nonlocal model with sensing radius r. We remark that a large number of related local and nonlocal models have been numerically studied to describe the invasion-type process considered here (e.g. [42,1, 24, 40]): here the specific focus is to explore the convergence of nonlocal to local form as r Ñ 0, which, as far as we are aware, has not been systematically investigated.

2unless environmental influences dictate conversion to a collective type of motion

26 Figure 3: Convergence between nonlocal and local/classical formulations under negligible cell-cell adhe- sion, Scc “ 0, Scv “ 10. Functional forms as proposed in Example 5.5, with modifications specified in the subfigures. (a) Solutions for r “ 0.1, 0.3, 1.0 at (a1) t “ 2, (a2) t “ 4 and (a3) t “ 8; (a4) Distance between local/nonlocal solutions as a function of time. For these simulations, we take a “ 0.01, b “ 1, µc “ 0.01, Kc “ 2, ηc “ 1, µv “ 0, λv “ 1. (b) Solutions for r “ 0.1, 0.3, 1.0 at (b1) t “ 2, (b2) t “ 4 and (b3) t “ 8; (b4) Distance between local/nonlocal solutions as a function of time. Parameters as in (a) except µv “ 1, Kv “ 1. (c) Solutions for fc “ 0 and fvpc, vq “ ´cv, with the other parameters as in (a).

As in the first test we use the initial values (6.1) and (6.2), choosing xc “ L{2, α “ 10 in the latter, and consider the coefficients and functions as proposed in Example 5.5. Under these choices the resul- tant nonlinear diffusion coefficient for the c-equation in the classical local formulation (compare (5.2a)) becomes 2 2 2 a p1 ` cq p1 ` c ` vq ´ bcp1 ` cvqpScc ` pScc ´ Scvqvq D˜ cpc, vq “ . (6.3) p1 ` cvq2p1 ` c ` vq2

Notably, this potentially becomes negative under an injudicious combination of adhesive strengths Scc, Scv, and of a, b. Likewise, the actual haptotaxis sensitivity function takes the form

S ` pS ´ S qc χ˜pc, vq “ b cv cv cc . (6.4) p1 ` cvqp1 ` c ` vq2

Again, depending on the relationship between Scc and Scv, this can become negative, which would lead to repellent haptotaxis: cells effectively moving away from regions with large ECM gradients, a rather unexpected behaviour. This suggests that cell-tissue adhesions should dominate over cell-cell adhesions,3 as ’usual’ haptotaxis, i.e. towards the increasing tissue gradient, is known to be an essential component of cell migration, this applying to several types of cells moving through the ECM (tumor cells, mesenchymal stem cells, fibroblasts, endothelial cells, etc.) see e.g. [34, 43, 51] and references therein.

Simulations are plotted in Figure3 where we show cell densities for the local model (cL) and nonlocal model under three sensing radii (cAr“0.1, cAr“0.3, cAr“1.0). In this first set of simulations we assume negligible cell-cell adhesion (Scc “ 0), which automatically ensures positivity for the diffusion coefficient of the equivalent local model, D˜ cpc, vq. We note that matrix renewal is absent (µv “ 0) in the left- hand column and present (µv ą 0) in the central column. In the right-hand column we show the greater generality of the results under vastly simplified kinetics, specifically setting fcpc, vq “ 0 and fvpc, vq “ ´cv (with the other functional forms as in Example 5.5). Simulations highlight the convergence between local and nonlocal models as r Ñ 0: for r “ 0.1, the solution differences become negligible. However,

3An analogous behaviour was suggested by the two-scale structured population model with adhesion introduced in [19].

27 Figure 4: Time restricted convergence under moderate cell-cell adhesion, Scc “ 2.5, Scv “ 10. Top row shows solutions across the full spatial region (r0, 20s), the bottom row magnifies a relevant portion for clarity. Solutions to local and nonlocal models under the functional forms proposed in Example 5.5 for r “ 0.01, 0.1, 0.3, 1.0 at (a) t “ 3, (b) t “ 3.5 and (c) t “ 5. In (a) solutions to the local model continue to exist and we observe convergence between local and nonlocal formulations. In (b-c) the solutions to the local model are noncomputable. Nonlocal models, however, can destabilise into a pattern of aggregates. Parameters: a “ 0.01, b “ 1, µc “ 0.01, Kc “ 2, ηc “ 1, µv “ 0, λv “ 1 and adhesion parameters as above. distinctions emerge for large r, where we can expect significant discrepancy between the solutions. This suggests that the local model fails to accurately predict the behaviour in cases where cells sample over relatively large regions of their local environment. Next, we extend to include a degree of cell-cell adhesion, setting functions and parameters as in Figure3 , except now Scc ą 0. Notably this raises the possibility of a negative diffusion coefficient in the classical formulation and subsequent illposedness. Solutions under a representative set of parameters are shown in Figure4. For t below some critical time we observe convergence as before, with the nonlocal formulation converging to solutions of the local model as r Ñ 0. However, continued matrix degradation further depletes v, with the result that (6.3) can become negative. At this point (in this case t « 3.2 ...) the local model becomes illposed and its solutions become incomputable (implying nonexistence of solutions). However, the nonlocal formulation appears to preserve wellposedness, consistent with previous theoretical studies where extending to a nonlocal formulation regularises a singular local model (e.g. [29]). Solutions to the nonlocal model instead destabilise into a quasi-periodic pattern of cell aggregations, maintained through the cell-cell adhesion, and with a wavelength shrinking as r Ñ 0. Finally, we remark that convergence of solutions extends beyond the specific functional forms and, as a representative example, we consider a minimalist setting based on linear/constant forms. Specifically, we set Dc “ a (constant), χ “ 1, fc “ 0, gpc, vq “ Sccc ` Scvv and fvpc, vq “ ´µcv. In this scenario, the diffusion and haptotaxis coefficients for the classical local formulation (5.2) reduce to

D˜ cpc, vq “ a ´ Sccc andχ ˜pc, vq “ Scv. (6.5)

Positivity is only guaranteed under appropriate parameter selection. Such a case is illustrated in Figure5 (a) where we assume negligible cell-cell adhesion (Scc “ 0). Clearly, we observe convergence between the nonlocal and local formulations as r Ñ 0. Inappropriate parameter selection, however, generates backward diffusion in the local model and solutions are consequently incomputable. Under such scenarios, however, solutions to the nonlocal model appear to exist: Figure5 (b) plots the behaviour under shrinking r. In all cases considered in this test the cells do not reach the boundary region where the difference between the nonlocal formulations (1.1) and (5.1) can play a role. Thus, we expect the same solution if reformulation (5.1) is applied instead.

28 Figure 5: Convergence between nonlocal and local/classical formulations under a set of minimalistic linear functional forms (Dc “ 0.01, χ “ 1, fc “ 0, gpc, vq “ Sccc ` Scvv, fvpc, vq “ ´µcv). (a) Negligible cell-cell adhesion, Scc “ 0,Scv “ 10: solutions shown at (left) t “ 2.5 and (middle) t “ 5, with the distance between solutions to the nonlocal and local model shown in the right panel.

7 Discussion

In this work we provide a rigorous limit procedure which links nonlocal models involving adhesion or a nonlocal form of chemotaxis gradient to their local counterparts featuring haptotaxis, respectively chemotaxis in the usual sense. As such, our paper closes a gap in the existing literature. Moreover, it offers a unified treatment of the two types of models and extends the previous mathematical framework to settings allowing for more general, solution dependent, coefficient functions (diffusion, tactic sensitivity, adhesion velocity, nonlocal taxis gradient, etc.). Finally, we provide simulations illustrating some of our theoretical findings in 1D. Our reformulations in terms of Tr and Sr reveal the tight relationship between the nonlocal operators Ar and ∇˚r and the (local) gradient. This suggests that both nonlocal descriptions (adhesion, chemotaxis) actually encompass the dependence on the signal gradients rather than on the signal concentration/density itself, which is in line with the biological phenomenon. Indeed, through their transmembrane elements (e.g. receptors, ion channels etc.) the cells are mainly able to perceive and respond to differences in the signal at various locations or within more or less confined areas rather than measure effective signal concentrations. Along with the mentioned solution dependency of the nonlocal model coefficients, the influence of the gradient possibly reflects into contributions of the adhesion/nonlocal chemotaxis to the (nonlinear) diffusion in the local setting obtained through the limiting procedure. The set Ωr (as introduced in Section2 ) can be regarded as the ’domain of restricted sensing’, meaning that there cells a priori sense only what happens inside Ω, the domain of interest. The measure of this subdomain is a decreasing function of the sensing radius r. When r Ñ 0 the set Ωr tends to cover the whole domain Ω, whereas as r increases the cells can sense at increasingly larger distances; correspondingly, Ωr shrinks. For r ą diampΩq the restricted sensing domain is empty: everywhere in Ω the cells can perceive signals not only from any point within Ω but potentially also from the outside. In this paper, however, we look at models with no-flux boundary conditions. This corresponds, e.g., to the impenetrability of the walls of a Petri dish or that of comparatively hard barriers limiting the areas populated by migrating cells, e.g. bones or cartilage material. As a result, the cells in the boundary layer ΩzΩr have a much reduced ability to stretch their protrusions outside Ω and thus gain little information from without. To simplify matters, we assume in this work that there is no such information or it is insufficient to trigger any change in their behaviour. In the definitions of Tr and Sr this corresponds to the integrands being set to zero in ΩzΩr. It is important to note that for points x P ΩzΩr the influence of a signal p in a direction y P S1 is not taken into account by ∇˚r at all if x ` ry R Ω. If Sr is used instead, then its contribution to the average is given by

1 y˜ :“ n χΩ∇ppx ` rsyq ds ¨ y y. ˆż0 ˙ Thus, thanks to integration w.r.t. s, the resulting vectory ˜ assembles the impact of those parts of the segment connecting x and x ` ry which are contained in Ω. It is parallel to y, and it may have the same or the opposite orientation. In particular this means that although for a certain range of directions large parts of the sensing region of a cell are actually outside Ω, this may still strongly influence the speed and

29 actual direction of the drift. The effect of integration w.r.t. s in Tr is less obvious, since in this case the average w.r.t. y is computed over the ball B1. This already achieves the covering of the whole sensing region by allowing a cell to gather information about the signal not only in any direction y{|y|, but also at any distance less than r. The additional integration over the path x ` rsy, s P r0, 1s, appears to mean that cells at x P Ωr are able to measure the average of the signal gradient all along such line segment rather than its value directly at the ending point. Indeed, from a biological viewpoint this description seems to make more sense, as cells do not jump from one position to another, nor do they send out their protrusions in a discontinuous way bypassing certain space points along a chosen direction. Averages over cell paths are then averaged w.r.t. y, which finally determines the direction of population movement. Example 3.4 indicates that the effect of even an extremely concentrated signal gradient is mollified by averaging. This agrees with our expectations from using non-locality. In higher dimensions n ě 2, the two-stage averaging in Tr (w.r.t. s and y) produces a direction field which is smooth away from the concentration point and also weakens but still keeps the singularity there. In contrast, averaging only w.r.t. y leads instead to jump discontinuities at a unit distance from the accumulation point. Moreover, we remark that without integrating w.r.t. s in Trp∇¨q one cannot regain Ar. The effect observed in Example 3.3 further supports the conjecture that the nonlocal operators which act directly on the signal gradients might actually be a more appropriate modelling tool. While inside the subdomain Ωr there is no difference (recall Lemmas 3.1 and 3.2), inside the boundary layer ΩzΩr the limiting behaviour as r Ñ 0 is qualitatively distinct. Indeed, Example 3.3 shows that using, e.g., Ar, leads, for r Ñ 0, to unnatural sharp singularities at the boundary of Ω even in the absence of signal gradients, whereas this does not happen if Tr is used instead. Simulations in Subsection 6.1 (see Figure2 ) confirm our theoretical findings and show a substantial difference between the solutions obtained with the two nonlocal formulations involving (1.2) and (3.2), respectively. The choice (3.2) is motivated above all from a mathematical viewpoint (as it enables a rigorous, well-justified passage to the limit for r Ñ 0), but it also seems to make sense biologically, as our above comments and the simulations performed for the particular setting in Subsection 6.1 suggest.

Acknowledgement

MK and CS acknowledge the partial support of the research initiative Mathematics Applied to Real World Challenges (MathApp) of the TU Kaiserslautern. CS also acknowledges funding by the Federal Ministry of Education and Research (BMBF) in the project GlioMaTh.

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