# Cross-Diffusion Systems with Excluded-Volume Effects and Asymptotic Gradient Flow Structures

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## Abstract

In this paper, we discuss the analysis of a cross-diffusion PDE system for a mixture of hard spheres, which was derived in Bruna and Chapman (J Chem Phys 137:204116-1–204116-16, 2012a) from a stochastic system of interacting Brownian particles using the method of matched asymptotic expansions. The resulting cross-diffusion system is valid in the limit of small volume fraction of particles. While the system has a gradient flow structure in the symmetric case of all particles having the same size and diffusivity, this is not valid in general. We discuss local stability and global existence for the symmetric case using the gradient flow structure and entropy variable techniques. For the general case, we introduce the concept of an asymptotic gradient flow structure and show how it can be used to study the behavior close to equilibrium. Finally, we illustrate the behavior of the model with various numerical simulations.

## Keywords

Nonlinear parabolic equations Interacting particle systems Asymptotic expansion Cross diffusion Size exclusion Entropy techniques and gradient flow structure## Mathematics Subject Classification

35K55 35C20 60J70 82C22 35A01 37B25## 1 Introduction

Systems of interacting particles can be observed in biology (e.g., cell populations), physics or social sciences (e.g., animal swarms or large pedestrian crowds). Macroscopic models describing the individual interactions of these particles among themselves as well as their environment lead to complex systems of differential equations (cf. e.g., Bendahmane et al. 2009; Bruna and Chapman 2012a, b; Burger et al. 2010, 2016, 2012; Di Francesco and Fagioli 2016; Painter 2009; Schlake 2011; Simpson et al. 2009). In microscopic models, the dynamics of each particle is accounted for explicitly, while the macroscopic models typically consist of partial differential equations for the population density. Passing from the microscopic model to the macroscopic equations in a systematic way is, in general, very challenging, and often one relies on closure assumptions, which can be made rigorous under certain scaling assumptions on the number and size of particles. In particular, when crowding due to the finite size of particles is included in the model, the limiting process is quite subtle and, using different assumptions and closure relations, a variety of macroscopic equations have been derived. For instance, the macroscopic equations of a two-species system where particles undergo a simple exclusion process on a lattice can be derived using formal Taylor expansions, see for example Burger et al. (2010), Simpson et al. (2009). The case when particles are not confined to a regular lattice and undergo instead a Brownian motion with hardcore interactions was considered in Bruna and Chapman (2012a) using matched asymptotic expansions. Cross-diffusion is a common feature of all these models and poses a particular challenge for the analysis since maximum principles do not hold. Classical examples of cross-diffusion systems are reaction diffusion systems or systems describing multicomponent gas mixtures. These quasi-linear parabolic systems were analyzed by Ladyzhenskaia et al. (1968) or Amann (1985, 1989), which however rely on strong parabolicity assumptions that break down for the degenerate cross-diffusion systems derived from the interacting particle systems mentioned above.

*M*is a mobility matrix and \(\partial _\mathrm{r} E\) and \(\partial _\mathrm{b} E\) denote the functional derivative of an entropy function

*E*with respect to

*r*and

*b*, respectively. The gradient flow formulation provides a natural framework to study the analytic behavior of such systems, cf. e.g., Ambrosio et al. (2008). It has been used to analyze existence and long-time behavior of systems, see for example Carrillo et al. (2014), Jüngel and Zamponi (2014), Liero and Mielke (2013), Zinsl and Matthes (2015). As a result, being able to express a PDE system as gradient flows of an entropy is a very desirable feature; yet, this is not possible in general. The lack of the gradient flow structure on the PDE level can result from the approximations made when passing from the microscopic description to the macroscopic equations. This is the case of the cross-diffusion system derived in Bruna and Chapman (2012a), which was derived using the method of matched asymptotics. There has been a lot of research on the passage from microscopic models to the continuum equations, for example in the hydrodynamic limit (Kipnis and Landim 2013). More recently, the microscopic origin of entropy structures, which connects gradient flows and the large deviation principle, was analyzed in Adams et al. (2011), Liero et al. (2015).

In this paper, we introduce the idea of an asymptotic gradient flow structure as a generalization of a standard or, as we also call it, full gradient flow structure for systems derived as an asymptotic expansion such as that in Bruna and Chapman (2012a). In this paper, we provide several analytic results for these cross-diffusion systems and introduce the notion of asymptotic gradient flows. We discuss how the closeness of these asymptotic gradient flow structures can be used to analyze the behavior of the system close to equilibrium. Furthermore, we present a global in time existence result in the case of particles of same size and diffusivity (in which the system has a full gradient flow structure). The existence proof is based on an implicit Euler discretization and Schauder’s fixed point theorem. We study the linearized system with an additional regularization term in the entropy to ensure boundedness of the solutions and deduce existence results for the unregularized system in the limit (similar to the deep quench limit for the Cahn Hilliard equation Elliott and Garcke 1996). This is, to the authors’ knowledge, the first global in time existence result for this system so far. We note, however, that it is only valid if the total density stays strictly below a certain threshold. It relates to the fact that the model assumptions break down if the maximum density is reached. We discuss this problem in more detail in Sect. 3.

This paper is organized as follows: we introduce the mathematical model in Sect. 2 and discuss the cases for which the system has either a full or an asymptotic gradient flow structure. In Sect. 3, we define the notion of asymptotic gradient flows formally and discuss how they can be used to analyze the behavior of stationary solutions close to equilibrium. Several numerical examples illustrating the deviation of stationary solutions from the equilibrium solutions for asymptotic gradient flows are presented in Sect. 4. Finally, we give a global in time existence result in the case of particles of same size and diffusivity in Sect. 5.

## 2 The Mathematical Model

*i*th particle. We define the total number of particles in the system by \(N = N_\mathrm{r} + N_\mathrm{b}\), and the distance at contact between a red and blue particles by \(\epsilon _\mathrm{br}=(\epsilon _\mathrm{r} + \epsilon _\mathrm{b})/2\). The situation detailed above can be described by the overdamped Langevin SDEs

*i*th particle and \(\mathbf{W}_i\) a

*d*-dimensional standard Brownian motion. We assume that \(\Omega \) is a bounded domain. The boundary conditions due to collisions between particles and with the domain walls are

*d*-dimensional ball of diameter \(\epsilon \), then the volume fraction in the system is

*asymptotic gradient flow*, motivated by the underlying structure of the general system (7).

### 2.1 Cross-Diffusion System for Particles of the Same Size and Diffusivity

This cross-diffusion system can be used to describe a mixture of particles that are physically identical but that are driven by different potentials \(V_\mathrm{r}\) and \(V_\mathrm{b}\) (for example cells that are attracted to different food sources, or pedestrians that want to move in different directions). Moreover, it can also be used to model the scenario where the red and the blue particles are in fact identical, but one has knowledge about the initial distributions of each sub-population, \(r_0\) and \(b_0\). This is the scenario in many experimental setups that use noninvasive fluorescent tagging. On the other hand, if the red and blue particles are identical and initially indistinguishable, then one has that \(r/N_\mathrm{r} = b/N_\mathrm{b} := p\) for all times. In this case, both Eqs. (11a) and (11b) reduce to the same equation, which coincides with the equation for the evolution of a single population of hard spheres as expected (Bruna and Chapman 2012b).

### 2.2 Cross-Diffusion System for Particles of Different Size and Diffusivity

*G*(

*r*,

*b*) is the vector

Finally, we note that the system (19) coincides with the gradient flow structure in the case \(D_\mathrm{r} = D_\mathrm{b}\) and \(\epsilon _\mathrm{r} = \epsilon _\mathrm{b}\), see (15) and (18). Note that \(G \equiv 0\) for the parameter values of the simpler system (11), as expected. Specifically, we find that if \(D_\mathrm{r} = D_\mathrm{b}\) and \(\epsilon _\mathrm{r} = \epsilon _\mathrm{b}\), then \(\theta _ r= \theta _\mathrm{b} = 0\). A natural question to ask is whether there are other parameter values for which \(G(r,b) \equiv 0\) for all *r*, *b*. Imposing that \(\theta _ r= \theta _\mathrm{b} = 0\) leads to the condition \(a_\mathrm{br}^2 = a_\mathrm{r} a_\mathrm{b}\), which in turn leads to \(\epsilon _\mathrm{r} = \epsilon _\mathrm{b}\), and thus that \(D_\mathrm{r} = D_\mathrm{b}\). Therefore, the only case for which (19) is an exact gradient flow for the system is the case which we have already studied, that is when the particle sizes and diffusivities are equal.

## 3 Gradient Flows and Asymptotic Gradient Flows Close to Equilibrium

In the following, we provide a more detailed discussion on gradient flow structures and implications for the behavior close to equilibrium.

### 3.1 Full Gradient Flow Structure Case

In this subsection, we analyze the behavior of system (17) close to equilibrium. We follow the strategy outlined in the previous subsection, by proving uniqueness of equilibrium solutions and studying the stability and well posedness of the system close to this equilibrium solution.

*M*is positive definite in the case of a strictly convex entropy functional

*E*, cf. Schlake (2011). We assume from now on:

- (AI)
Let \(V_\mathrm{r}, V_\mathrm{b} \in H^1(\Omega )\cap L^{\infty }(\Omega )\).

*M*defined in (18) is given by

*M*is positive definite if \(\rho < 1 / \bar{\gamma }\). This constraint gives a local bound on the total local volume density (using (13)), namely \(2 \phi < 1\). This is consistent with the asymptotic assumption that \(\phi \ll 1\). Hence, we define the set

### Theorem 3.1

(Linear stability) The stationary solutions of the system (17) are unique and linearly stable with respect to small perturbations \(\xi , \eta \in L^2(0,T;H^1(\Omega ))\) with zero mean.

### Proof

Due to the gradient flow structure, any stationary solution of (17) is a minimizer of the entropy subject to the constraints of given mass and \((r(x),b(x)) \in {\mathcal S}\) almost everywhere. Due to the strict convexity of the entropy and the convexity of the constraint set, the minimizer is unique.

*r*,

*b*) and in the entropy variables (

*u*,

*v*) , i.e., \(u=u_\infty +\xi , v=v_\infty +\eta \). In the latter setting, we obtain the following first-order approximation

*M*(

*r*,

*b*) are positive definite for (

*r*,

*b*) in the interior of

*S*, which is guaranteed everywhere for the stationary solution \((r_\infty ,b_\infty )\). Stability of this linear system is equivalent to nonpositivity of all the real parts of eigenvalues \(\lambda \) in

*M*(

*r*,

*b*) are positive definite, we conclude that \(\lambda <0\), which implies linear stability. \(\square \)

Note that we assumed \(\xi , \eta \) with zero mean, which corresponds to the mass conservation property of the system. Next, we consider the well posedness close to equilibrium. We shall make use of the following auxiliary lemma:

### Lemma 3.1

### Proof

### Theorem 3.2

*R*is a constant depending on \(\kappa \) and \(T>0\) only.

### Proof

*L*denote the solution of (30) for a given right-hand side. Then the fixed point operator is given by the concatenation of

*L*and

*F*, that is

*F*defined in (30) maps from \(X\times X\) into \(Y\times Y\), where

*J*is self-mapping into the ball \(B_\mathrm{r}\) and contractive. The self-mapping property follows from the fact that

*R*such that \(R<\frac{1}{C}\), we can apply Banach’s fixed point theorem which guarantees the existence of unique solutions \((u,v) \in B_\mathrm{r}\). \(\square \)

### 3.2 Asymptotic Gradient Flow Structure

*asymptotic gradient flow structure*, motivated by the fact that it was derived from an asymptotic expansion in \(\epsilon \). For further motivation, consider a gradient flow structure for the density

*w*of the form

*M*and the entropy

*E*depend on a small parameter \(\delta > 0\). With an expansion of

*M*and

*E*in terms of \(\delta \) as

*k*does not yield a gradient flow structure in general, but up to terms of order \(\delta ^{k}\) it coincides with the gradient flow structure with mobility \( \sum _{j=0}^k \delta ^j M_j(w)\) and entropy \( \sum _{j=0}^k \delta ^j E_j(w)\). In our case, we deal with the example \(k=1\) (with \(\delta = \epsilon ^d\)), where we have

### Definition 3.3

*k*if there exist densely defined operators \({\mathcal G}_j\), \(j={k+1},\ldots ,2k\) such that for \(\delta \in (0,\delta _*)\)

*w*.

*k*are available, it seems natural to perform a separate expansion to derive a lower-order model that is a gradient flow as well. For complicated models and types of expansions as in Bruna and Chapman (2012a) or Bruna and Chapman (2012b), it seems not suitable to derive such however. Hence, we shall work with the asymptotic gradient flow concept below. Note that with the above notations we can rewrite (32) as

*w*. In the case of (31), it typically means that \(E'(w_\infty ^\delta ;\delta )\) is constant. In order to prove the existence of a stationary solution of (32), one can then try the following strategy: first of all compute \(w_\infty ^\delta \) (or prove at least its existence and uniqueness by variational principles) and then use the equation

*w*on the right-hand side. Since the terms on the right-hand side are of high order in \(\delta \) or of second order in terms of \(w-w_\infty ^\delta \), there is some hope of contractivity of the fixed point operator close to equilibrium \(w_\infty ^\delta \). Such an approach can also yield some structural insight into the stationary solution, since it will be a higher-order perturbation of \(w_\infty ^\delta \). The same idea can be employed to analyze transient solutions of (32), since

*w*at fixed \(\delta \). Due to positive definiteness of \({\mathcal E}''(w_*^\delta ;\delta )\), this system can be interpreted as a linear equation for the linearized entropy variable \(\tilde{z} = {\mathcal E}''(w_*^\delta ;\delta ) \tilde{w}\), which is equivalent to considering linear stability directly in the transformed equation for the entropy variable

*z*as performed in Schlake (2011). Using the simplified notation \({\mathcal A}={\mathcal E}''(w_*^\delta ;\delta )^{-1}\) and \({\mathcal B}={\mathcal M}(w_\infty ^\delta ;\delta )\), we obtain

The application of the above strategies to prove existence of solutions and linear stability to a concrete model obviously depends on an appropriate choice of topologies. In the remaining part of this section, we focus on the analysis of the asymptotic gradient flow of the general model.

### 3.3 Asymptotic Gradient Flow Structure Case

First, we study the existence of stationary solutions to (21). Then, we discuss stability of stationary states following the ideas presented in Sect. 3.2.

Note that for \(\epsilon =0\), the equilibrium solutions are given by \((r_\infty ,b_\infty )=(C_\mathrm{r} e^{-V_\mathrm{r}},C_\mathrm{b} e^{-V_\mathrm{b}})\), with constants \(C_\mathrm{r}\) and \(C_\mathrm{b}\) depending on the initial masses only. Hence \((r_{\infty }, b_{\infty })\) are bounded for \(V_\mathrm{r}\) and \(V_\mathrm{b}\) satisfying assumption 3.1. For \(\epsilon >0\), the equilibrium solutions are a \(\mathcal {O}(\epsilon ^d)\) perturbation in \(L^{\infty }\) and therefore also uniformly bounded.

### Theorem 3.4

*R*depending on \(\epsilon \) and \(T>0\) only.

### Proof

*L*denote the solution operator to (36) for a given right-hand side

*F*(

*u*,

*v*). Then, the fixed point operator is constructed by:

*F*maps from \(X\times X\) into \(Y\times Y\), where \(Y=H^1(\Omega )\). Employing results about the elliptic operator, cf. Gilbarg and Trudinger (2015) or Evans (1998), we obtain that the solution \((\tilde{u},\tilde{v})\) to Eq. (36) is in \(X\times X\).

*J*is self-mapping into the ball \(B_\mathrm{r}\) and contractive. The self-mapping property follows from the fact that

*R*and \(\epsilon \) such that

A direct consequence of the proof is the closeness of the stationary solution \((u_*,v_*)\) to the gradient flow solution \((u_\infty ,v_\infty )\):

### Corollary 3.1

### Proof

## 4 Numerical Investigations of Steady States

*E*in (15). If the mobility matrix (18) is positive definite (which it is under the assumptions), the equilibrium states can be computed by finding constants \(\chi _\mathrm{r} \in \mathbb {R}\) and \(\chi _\mathrm{b} \in \mathbb {R}\) such that

For the general case (7), we only obtain an asymptotic gradient flow structure with the entropy \(E_\epsilon \); if we use (40) to solve for the stationary solutions we will be committing an order \(\epsilon ^{2d}\) error. Instead, we compute the exact stationary states \((r_*, b_*)\) of the general system by solving the time-dependent problem (7) for long-times, until the system has equilibrated. To solve (7), we use a second-order accurate finite-difference scheme in space and the method of lines with the inbuilt MATLAB ode solver ode15s in time.

We set \(d=2\) and consider one-dimensional external potentials \(\tilde{V}_\mathrm{r} = \tilde{V}_\mathrm{r}(x)\) and \(\tilde{V}_\mathrm{b} = \tilde{V}_\mathrm{b}(x)\) so that the stationary states will be also one dimensional. In particular, we take linear potentials \(\tilde{V}_\mathrm{r} = v_\mathrm{r} x\) and \(\tilde{V}_\mathrm{b} = v_\mathrm{b} x\) and solve for the full system (7) and for the minimizers (40) in \([-1/2,1/2]\), which is split into 200 intervals. The Newton solver is initialized with the stationary state solution in the case of point particles and terminated if \(||F(r,b,\chi _\mathrm{r}, \chi _\mathrm{b}) ||_{L^2(0,1)} \le 10^{-8}\).

### Example 1

First we consider the case: \(\epsilon _\mathrm{r} = \epsilon _\mathrm{b}\) and \(D_\mathrm{r} = D_\mathrm{b}\), that is particles of the same size and diffusivity. In this case, system (11) has a full gradient flow structure and hence we expect that the stationary states computed with the two approaches to be the same. We plot the two pairs, \((r_*, b_*)\) computed as the long-time limit of (11), and \((r_\infty , b_\infty )\), computed from (40) in Fig. 1. The parameters are \(D_\mathrm{r} = D_\mathrm{b} = 1\), \(\epsilon _\mathrm{r} = \epsilon _\mathrm{b} = 0.01\), \(N_\mathrm{b} = N_\mathrm{r} = 200\) and \(v_\mathrm{r} = 2\), \(v_\mathrm{b} = 1\). As expected, the solutions are identical.

### Example 2

From Corollary 3.1, we expect the stationary solutions corresponding to the case of an asymptotic and a full gradient flow equation agree up to order \(\mathcal {O}(\epsilon ^d)\). To investigate this, we again compare the solutions \((r_*, b_*)\) and \((r_\infty , b_\infty )\) as we move away from the case with an exact gradient flow structure [which corresponds to \(\theta _\mathrm{r} = \theta _\mathrm{b} = 0\), see (21)–(23)]. We recall that the parameters \(\theta _\mathrm{r}\) and \(\theta _\mathrm{b}\) are only zero in the symmetric case with identical particles, so that can think of these as a measure of the asymmetry in the system (either in size, diffusivity, or both). In particular, we do a one-parameter sweep with \(\theta _\mathrm{r}\), increasing it from 0 (as in Fig. 1) to \(9 \cdot 10^{-5}\), while keeping \(\epsilon _\mathrm{r} = \epsilon _\mathrm{b} =0.01\) and \(D_\mathrm{b} = 1\) fixed. This ensures that when \(\theta _\mathrm{r} = 0\) then \(\theta _\mathrm{b} = 0\). The reds diffusivity \(D_\mathrm{r}\) is varied according to (23). We plot the result for \(\theta _\mathrm{r} = 8\cdot 10^{-5}\) in Fig. 2. As expected, the error between the stationary solutions is apparent.

## 5 Global Existence for the Full Gradient Flow System

### Theorem 5.1

*r*,

*b*and \(\rho \in \mathcal {S}^\circ \), then the matrix

*M*is positive definite.

*r*,

*b*) at time \(\tau (k-1)\), we want to find \(( r_k, b_k) \in \mathcal {S}\) solving the regularized time discrete problem

Finally, uniform a priori estimates in \(\tau \) and the use of a generalized version of the Aubin–Lions lemma allow to pass to the limit \(\tau \rightarrow 0\) leading to the existence of (41). Note that the compactness results are sufficient for \(1-\bar{\gamma }\rho >0\) to pass to the correct limit in the flux terms \(J_\mathrm{r}\) and \(J_\mathrm{b}\), i.e., leading to the global existence of weak solutions to system (14).

### Lemma 5.1

### Proof

*u*is nonincreasing and as \(u(0)>0\) and \(u\left( \frac{1}{\bar{\gamma }}\right) =0\), there exists a unique fixed point \(0<z_0<\frac{1}{\bar{\gamma }}\) such that \(u(z_0)=z_0\). Then, we define \( r=e^x(1-\bar{\gamma }z_0)>0\) and \( b =e^y(1-\bar{\gamma }z_0)>0\). It holds that \( r+ b=(e^x+e^y)(1-\bar{\gamma }z_0)=z_0<\frac{1}{\bar{\gamma }}\). So, \(( r, b)\in \mathcal {S}^\circ \). Then, we define the function \(f=\tilde{h}'\circ g^{-1}:\mathbb {R}^2\rightarrow \mathbb {R}^2\). Since \(\tilde{h}''\) and \(g'\) are nonsingular matrices for \(( r , b)\in \mathcal {S}^\circ \), the Jacobian of

*f*is also nonsingular for \(( r , b)\in \mathcal {S}^\circ \). Furthermore, we have that

*Df*allow us to apply Hadamard’s global inverse theorem showing that

*f*is invertible. So, also \(\tilde{h}'\) is invertible. \(\square \)

### 5.1 Time Discretization and Regularization of System (43)

*a*is coercive since the positive semidefiniteness of

*M*(

*r*,

*b*) implies that

*S*:

- (i)
maps a convex, closed set onto itself,

- (ii)
is compact,

- (iii)
is continuous.

*M*only contains sums and products of

*r*and

*b*, we have that \(M(\tilde{r}_k,\tilde{b}_k)\rightarrow M(\tilde{r},\tilde{b})\) strongly in \(L^2(\Omega ,\mathbb {R}^2)\). The positive semidefiniteness of the matrix

*M*for \(( r, b)\in \mathcal {S}\) provides a uniform bound for \((\tilde{u}_k,\tilde{v}_k)\) in \(H^1(\Omega ;\mathbb {R}^2)\). Hence, there exists a subsequence with \((\tilde{u}_k,\tilde{v}_k)\rightharpoonup (\tilde{u},\tilde{v})\) weakly in \(H^1(\Omega ;\mathbb {R}^2)\). The \(L^{\infty }\) bounds of \(M(\tilde{r}_k,\tilde{b}_k)\) and the application of a density argument allow us to pass from test functions \((\Phi _1,\Phi _2)\in W^{1,\infty }(\Omega ,\mathbb {R}^2)\) to test functions \((\Phi _1,\Phi _2)\in H^1(\Omega ,\mathbb {R}^2)\). So, the limit \((\tilde{u},\tilde{v})\) as the solution of problem (48) with coefficients \((\tilde{r},\tilde{b})\) is well defined. Due to the compact embedding \(H^1(\Omega ,\mathbb {R}^2)\hookrightarrow L^2(\Omega ,\mathbb {R}^2)\), we have a strongly converging subsequence of \((\tilde{u}_k,\tilde{v}_k)\) in \(L^2(\Omega ,\mathbb {R}^2)\). Since the limit is unique, the whole sequence converges. From Lemma 5.1 we know that \(( r, b)=h'^{-1}(\tilde{u},\tilde{v})\) is Lipschitz continuous, which yields continuity of

*F*.

Hence, we can apply Schauder’s fixed point theorem, which assures the existence of a solution \(( r, b)\in \mathcal {S}\) to (48) with \((\tilde{r},\tilde{b})\) replaced by (*r*, *b*).

### 5.2 Entropy Dissipation

### Lemma 5.2

### Proof

*r*,

*b*and \(\rho \in \mathcal {S}\) and \(\nabla V_\mathrm{r}, \nabla V_\mathrm{b}\in L^1(\Omega )\), we deduce (50). \(\square \)

### 5.3 The Limit \(\tau \rightarrow 0\)

*K*denotes a generic constant.

### Lemma 5.3

### Lemma 5.4

### Proof

*b*which concludes the proof. \(\square \)

In order to identify the limit terms, we multiply Eq. (63) by \((1-\overline{\gamma }\rho )\).

### Lemma 5.5

- (i)
\((1-\bar{\gamma }\rho _\tau )^2\nabla r_\tau \rightharpoonup (1-\bar{\gamma }\rho )^2\nabla r \) weakly in \(L^2(\Omega _T)\)

- (ii)
\((1-\bar{\gamma }\rho _\tau )(\bar{\alpha }+\bar{\gamma }) r_\tau \nabla \rho _\tau \rightharpoonup (1-\bar{\gamma }\rho )(\bar{\alpha }+\bar{\gamma }) r \nabla \rho \) weakly in \(L^2(\Omega _T)\)

- (iii)
\((1-\bar{\gamma }\rho _\tau )r_\tau \nabla V_\mathrm{r} \rightarrow (1-\bar{\gamma }\rho )r\nabla V_\mathrm{r}\) strongly in \(L^2(\Omega _T)\),

- (iv)
\((1-\bar{\gamma }\rho _\tau )\bar{\gamma }\nabla (V_\mathrm{b}-V_\mathrm{r}) r_\tau b_\tau \rightarrow (1-\bar{\gamma }\rho )\bar{\gamma }\nabla (V_\mathrm{b}-V_\mathrm{r}) r b\) strongly in \(L^2(\Omega _T)\),

- (v)
\((1-\bar{\gamma }\rho _\tau )\frac{\tau \bar{\gamma }^2r_\tau }{1-\bar{\gamma }\rho _\tau }\nabla \rho _\tau =\tau \bar{\gamma }^2r_\tau \nabla \rho _\tau \rightarrow 0\) strongly in \(L^2(\Omega _T)\).

### Proof

The strong convergences of (iii) and (iv) can be shown by applying (70) in (iii) and the generalized Aubin–Lions lemma with \(f(r_\tau ,b_\tau )=r_\tau b_\tau \) in (iv).

Finally, as \( r_\tau \nabla \rho _\tau \) is bounded in \(L^2(\Omega _T)\) and \(\tau \rightarrow 0\), we can deduce (v).

Analogous results hold for Eq. (64) which allows us to perform the limit \(\tau \rightarrow 0\) giving a weak solution to system (41).

*E*is convex and continuous, it is weakly lower semicontinuous. Because of the weak convergence of \((r_\tau (t),b_\tau (t))\),

*r*being replaced by

*b*. Because of the \(L^{\infty }\)-bounds and the bounds in (5.3), we obtain \(\nabla ((1-\overline{\gamma }\rho _\tau )\sqrt{r_\tau }), \nabla ((1-\overline{\gamma }\rho _\tau )\sqrt{b_\tau })\in L^2(\Omega _T)\), which implies

## 6 Conclusion

Gradient flow techniques provide a natural framework to study the behavior of time evolving systems that are driven by an energy. This energy is decreasing along solutions as fast as possible, a property inherent in nature. Hence, many partial differential equation models exhibit this structure. Most of these systems arise in the mean field limit of a particle system, which has a gradient structure itself. Passing from the microscopic level to the macroscopic equations often relies on closure assumptions and approximations, which perturb the original gradient flow structure.

In this paper, we studied a mean field model for two species of interacting particles which was derived using the method of matched asymptotics in the case of low volume fraction. This asymptotic expansions results in a cross-diffusion system which has a gradient flow structure up to a certain order. We therefore introduce the notion of asymptotic gradient flows for systems whose gradient flow structure is perturbed by higher-order terms. We show that this ‘closeness’ to a classic gradient flow structure allows us to deduce existence and stability results for the perturbed or as we call them asymptotic gradient flow system. We expect that the presented approach would also be extendable to a higher number of species. However, one would have to consider all types of pairwise interactions which would result in much more complicated systems.

While the presented results on linear stability (Theorem 3.1), well posedness (Theorem 3.2) and existence of stationary solutions (Theorem 3.4) also hold on unbounded domains, the proof of the global existence result in Sect. 5 uses embeddings which do not hold on unbounded domains in general, e.g., \(H^2(\Omega )\) is compactly embedded in \(L^2(\Omega )\).

The presented work is a first step toward the development of a more general framework for asymptotic gradient flows. It provides the necessary tools to understand the impact of high-order perturbations on the energy dissipation as well as the behavior of solutions and opens interesting directions for future research.

## Notes

### Acknowledgements

The work of MB was partially supported by the German Science Foundation (DFG) through Cells-in-Motion Cluster of Excellence (EXC 1003 CiM), Münster. MTW and HR acknowledge financial support from the Austrian Academy of Sciences ÖAW via the New Frontiers Group NST-001. The authors thank the Wolfgang Pauli Institute (WPI) Vienna for supporting the workshop that led to this work.

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