\section{Introduction}
In this work we consider the following reaction-diffusion system for the evolution on an interval $x\in\Omega$ of two species with population densities $\rho,\eta\geq0$ :
\begin{equation}\label{eq:main}
\begin{cases}
\partial_t \rho = \partial_x\left(\rho\partial_x \chii(\rho+\eta)\right)+\rho F_1(\rho,\eta)+\eta G_1(\rho,\eta), \\
\partial_t \eta = \partial_x\left(\eta \partial_x\chii(\rho+\eta)\right)+\eta F_2(\rho,\eta)+\rho G_2(\rho,\eta),
  \end{cases}
\end{equation}
where $\chi : \left[0,\infty\right)\to \left[0,\infty\right)$ is a $C^1$ super-linear function modelling nonlinear diffusion and $F_i$ and $G_i$, $i=1,2$ model the reaction phenomena. Systems of this type appear naturally in mathematical biology. A fundamental biological phenomenon in interactions among different biological species is the \emph{inhibition} or \emph{activation} of growth whenever two populations occupy the same habitat. One species may promote or suppress the proliferation of the other species. In models involving cells or bacteria, the limited growth of different cell types can be attributed to volume or size constraints of the individual cells forming the different populations. The diffusive part in \eqref{eq:main} was originally introduced in the seminal papers \cite{GP84} and \cite{BGHP85,BGH87} and exhibits an intriguing phenomenon: segregated densities remain separated at all times.

In fact, nonlinear diffusions are natural ways to include volume filling effects into mathematical biology models, see \cite{HPVolume,CCVolume} in the case of the classical Keller-Segel system. They help to avoid blow-up in these aggregation models in a biologically meaningful way and lead generically to asymptotic stabilisation.
In the absence of reactions, the system leads to the nonlinear diffusion equation
\begin{equation}
	\label{eq:diff_sigma}
	\partial_t \sigma = \partial_x\left(\sigma\partial_x\chii(\sigma)\right) = \partial_x^2 \beta(\sigma).
\end{equation}
with $\sigma=\rho+\eta$, in which $\chii(\sigma)$ models the resistance to compression of the whole group of individuals $\sigma$. The natural assumption on \eqref{eq:diff_sigma} in order to be a diffusion equation is $\beta'(s)>0$ for $s>0$, possibly degenerating at $s=0$, or equivalently $\chi''(s)>0$ for $s>0$. The particular relevant case of $\chi(s)=s^2/2$ can be understood as the mean-field limit of interacting particles with very localised repulsion, see \cite{Ol,BCM,BV,Klar}.

Related reaction-diffusion models to \eqref{eq:main} appear in tissue growth models where cell adhesion and volume effects are important factors determining cell sorting in heterogeneous cell populations, see \cite{MurTog, CHS17, BDFS, CCS17}, and zebrafish lateral line patterning \cite{VS}. They are also basic building bricks for a variety of cancer invasion models in the literature \cite{Chaplain.2005,  Preziosi.2003, Stinner.2015, Johnston.2010,BerDaPMim,ArmPaiShe1,ArmPaiShe2,ArmPaiShe3,GerCha,DomTruGerCha,SfaKolHelLuk} in which the coupling with other biologically meaningful modelling factors such as extracellular matrix, enzymatic activators and other substances are taken into account. These works usually involve drift terms due to long range attraction and/or repulsion between individuals leading to related mathematical difficulties with respect to \eqref{eq:main}, see for instance \cite{KM17}.

The nonlinear diffusion equation \eqref{eq:diff_sigma} is well-studied, see \cite{Vaz}, and it can be understood as a gradient flow, see \cite{Otto,AGS,Vil_1,Snt_B}, of the energy functional
\begin{equation*
 \eE(\sigma)=\int_\Omega \chi(\sigma) \d x,
\end{equation*}
in the metric space of probability measures endowed with a suitable topology induced by the euclidean transport distance denoted by $d_2$. The wellposedness of solutions to the nonlinear diffusion equation \eqref{eq:diff_sigma} was obtained in \cite{Otto} by means of the so called \emph{JKO-scheme}, \emph{cf.} \cite{JKO98}, which is a particular case of the minimising movement scheme by De Giorgi, see \cite{Amin} and the references therein. The idea of such a scheme is to recursively construct a sequence by solving a minimisation problem in a certain metric space $(X,d)$. Given some initial condition $\bar\sigma$ for Eq. \eqref{eq:diff_sigma} and a fixed time step $0< \tau < 1$ we set $\sigma_\tau^0=\bar{\sigma}$, and then recursively define
\begin{equation}\label{eq:JKO_intro}
  \sigma_\tau^{n+1}\in \argmin\limits_{\sigma\in X} \left\{\frac{1}{2\tau}d^2(\sigma,\sigma_\tau^{n})+\eE(\sigma)\right\},
\end{equation}
for $n\in\N$.
The seminal work of R.~J.~McCann \cite{McC97} shows that $\eE(\sigma)$ is displacement convex or geodesically convex on the metric space of probability measures on the line endowed with $d_2$ as soon as $\beta$ is nondecreasing on $(0, + \infty)$, called the McCann's condition in short. We also refer to \cite[Chap. 9]{AGS}, \cite[p. 26]{CJMTU}, or \cite[Chap. 17]{Vil_2} for this classical notion, and \cite{BC} for related issues.
Upon choosing a proper time interpolation $\sigma_\tau$, it can be proven that the sequence $\left\{\sigma_\tau\right\}_\tau$ converges to a weak solution of Eq. \eqref{eq:diff_sigma}, see \cite{Vil_1,AGS,Vil_2,Snt_B} and the references therein.

In this work, we propose a variational splitting scheme in order to construct weak solutions to the reaction-diffusion system \eqref{eq:main}. More precisely, we solve in an inner time stepping the diffusive part of the system by the JKO scheme related to the nonlinear diffusion equation \eqref{eq:diff_sigma}, and then we transport both densities $\rho,\eta$ through the flow generated by the equation for the total population $\sigma$.
Note that in this step the total and individual masses of the populations are unchanged in time. In the second inner time stepping of the splitting scheme, we solve the system of ODEs parameterised by the spatial variable $x\in\Omega$ leading to the final approximation of our new population densities after a time step. This variational splitting scheme will be written in details in Section 2. The splitting between reaction and diffusion steps is natural from the numerical analysis view point as it has already been used for variations of Keller-Segel models where the diffusion step is solved by the JKO scheme \cite{Carrillo.2008,CKL17} in the case of a single population density coupled with a system of reaction-diffusion equations.

Our main result shows the convergence of the splitting variational scheme towards weak solutions of the system \eqref{eq:main}. The main mathematical difficulty here arises from the cross-diffusion term allowing for segregation fronts to form in the solutions. This phenomenon was proven in \cite{BGHP85} in the case of initial data with separated supports for the populations. More precisely, while \cite{GP84} constructs a source solution of the system without reactions similar to the well-known Barenblatt-Pattle profiles \cite{Vaz} for nonlinear diffusions, \cite{BGHP85} constructs a solution to the system without reactions by formulating it as a free boundary problem for a single effective equation, and by characterising the segregation front through this free boundary. This approach can only work in case the support of both populations are at a positive distance to each other initially.
Later \cite{BerDaPMim, BHIM12} combined both the nonlinear diffusion and the reaction to obtain a system similar to \eqref{eq:main} showing similar segregation phenomena by regularisation techniques. However, their approach heavily relies on the absence of vacuum as they assume that $\sigma_0$ is bounded below by a positive constant.

These remarkable results have severe consequences --  initially smooth solutions lose their regularity when both densities meet each other. In fact, they become discontinuous at the contact interface immediately. This phenomenon legitimises our functional space choice as bounded functions of bounded variation, see the precise notion of weak solution and assumptions on the initial data in the next section.

In contrast to \cite{BGHP85,BerDaPMim, BHIM12}, we show the convergence of our variational splitting scheme for general initial data even in the presence of vacuum and for general nonlinearities.
Moreover, we recover their result in \cite{BGHP85,BerDaPMim} about segregation fronts by showing that initial data which are initially segregated remain segregated for all times. In fact, we are even able to drop their restrictive assumption of strict boundedness away from zero, \emph{i.e.} absence of vacuum. An important technical point in our proof relies on displacement convexity of an auxiliary functional that allows to obtain further regularity on the approximate solutions in order to pass to the limit in the nonlinear diffusion terms. This auxiliary functional imposes a slightly more restricted set of nonlinear diffusions satisfying some integrability condition at the origin, see the precise conditions in the next section.

The rest of the paper is organised as follows: In Section \ref{sec:main_result} we introduce the variational splitting scheme, present the main result, and explain the strategy of the proof. Section \ref{sec:proof_main_result} is dedicated to deriving all estimates necessary for proving the existence theorem as well as the segregation theorem, and finally, in Section \ref{sec:numerics} we conclude by illustrating the result with some numerical examples.

\section{\label{sec:main_result}Preliminaries and main result}
As already mentioned, our main aim is to study the existence of weak solutions for the following  one-dimensional two species cross-diffusion and reaction system:
\begin{equation}\label{eq:complete}
\begin{cases}
\partial_t \rho = \partial_x\left(\rho\partial_x \chii(\rho+\eta)\right)+\rho F_1(\rho,\eta)+\eta G_1(\rho,\eta) ,&\quad \mbox{in } \left[0,T\right]\times\Omega, \\
  \partial_t \eta = \partial_x\left(\eta \partial_x\chii(\rho+\eta)\right)+\eta F_2(\rho,\eta)+\rho G_2(\rho,\eta), &\quad \mbox{in } \left[0,T\right]\times\Omega,\\
  \partial_x\chi'\left(\rho+\eta\right)(x,t)=0, & \quad \mbox{on } \left[0,T\right]\times\partial\Omega,\\
  \rho(\cdot,0)=\rho_0,\quad \eta(\cdot,0)=\eta_0, & \quad \mbox{in } \Omega,
\end{cases}
\end{equation}
where $\Omega\subset \R$ is an open bounded interval, $T>0$.  Moreover,  $\chi$ denotes an internal energy density and $F_i$ and $G_i$, $i=1,2$ model the reaction phenomena. As mentioned before the space of bounded functions with bounded variation is a natural functional setting.

\begin{definition}[\label{def:BV_space}Space of functions of bounded variations]
	Let $f:\bar \Omega \rightarrow \R$. We define its variation with respect to a partition $P:=\{x_1 < x_2 < \cdots < x_{|P|}\}\subset \Omega$ by
	\begin{align*}
		V_P(f):=\sum\limits_{i=1}^{|P|-1}|f(x_{i+1})-f(x_i)|.
	\end{align*}
	We call $f$ a function of bounded variation if its total variation $\sup_P V_P(f)<\infty$ is finite. Here the supremum is taken over all partitions of $\Omega$.
	We denote by $BV(\Omega)$ the set of functions whose variation is bounded. Equipped with the norm
	\begin{align*}
		\BV{f}:= \sup\limits_{P} V_P(f),
	\end{align*}
	the set $BV(\Omega)$ is a vector space.
\end{definition}

We shall see in the remainder of this section that the vector space of bounded function with bounded variation is a good choice to construct solutions. In our analysis we will exploit the following property.
\begin{lemma}[\label{lem:Ralgebra}$BV(\Omega)\cap L^\infty(\Omega)$ is an $\R$-algebra.]
	The vector space $BV(\Omega)\cap L^\infty(\Omega)$ equipped with the pointwise multiplication
	\begin{align*}
		\big(BV(\Omega)\cap L^\infty(\Omega)\big)^2\ni(f,g)\mapsto f g\in BV(\Omega)\cap L^\infty(\Omega),
	\end{align*}
	is a real algebra.
\end{lemma}

The proof of the previous result is standard. Notice that in one dimension, $BV$-regularity implies boundedness. We prefer to write $BV(\Omega)\cap L^\infty(\Omega)$ for the sake of clarity and possible future generalisations.

\subsection{\label{sec:metric_structure}Metric structure}
Consider $\Omega\subseteq \R$ an open bounded interval, and denote by $\curlyM_+(\Omega)$ the set of positive and finite measures. Throughout this paper we will make use of the following notation
\begin{align*}
	\curlyP^m(\Omega):=\left\{\mu \in \curlyM_+(\Omega)\, \big|\, \mu(\Omega)= m\right\},
\end{align*}
that is, the set of positive measures with mass $m>0$.  Consider a measure $\mu\in\curlyP^m(\Omega)$ and a Borel map $\curlyT:\R\rightarrow\R$. We denote by  $\nu = \curlyT_{\#}\mu\in\curlyP^m(\Omega)$ the push-forward measure of $\mu$ through $\curlyT$, defined by
\begin{equation*}
    \int_{\Omega}f(y)\d \curlyT_{\#}\mu(y)=\int_{\Omega}f(\curlyT(x))\d\mu(x),
\end{equation*}
for all Borel functions $f$ on $\Omega$. We call $\curlyT$ a \emph{transport map} pushing $\mu$ to $\nu$. We endow the space $\curlyP^m(\Omega)$ with the $p-$Wasserstein distance, $p\geq 1$,
\begin{equation*
     \Wp(\mu_1,\mu_2)=\inf_{\gamma\in \Pi^m(\mu_1,\mu_2)}\left\{ \int_{\Omega\times\Omega}|x-y|^{p}\d \gamma(x,y)\right\}.
\end{equation*}
Here, $\Pi^m(\mu_1,\mu_2)$ is the set of all transport plans between $\mu_1$ and $\mu_2$, that is the set of positive measures of fixed mass, $\gamma \in\curlyP^m(\Omega\times\Omega)$, defined on the product space such that $\pi^{i}_{\#}\gamma=\mu_{i}$, for $i=1,2$,  where $\pi^{i}$ denotes the projection operator on the $i$-th component of the product space. If $\mu_1$ is absolutely continuous with respect to the Lebesgue measure the optimal transport map, $\gamma$, is unique and can be written as $\gamma = (\mathrm{id}, \curlyT)_\#\mu$. In addition there exists a \emph{Kantorovich potential}, $\varphi$, that is linked to the transport map, $\curlyT$, in the following way
\begin{align}
	\label{eq:rel_T_KantorovichPotential}
	\curlyT(x) = (\id - \partial_x \varphi)(x).
\end{align}
We refer to \cite{Vil_1,AGS,Vil_2,Snt_B} and the references therein for a good account of the properties of transport distances and the state of the art in gradient flows/steepest descents of functionals in metric spaces of probability measures. While transport distances are an incredibly powerful tool for dealing with transport PDEs exhibiting a gradient flow structure, it is not applicable in the presence of source terms. This is owing to the fact that it is only defined  for two measures of the same mass.

To resolve this shortcoming we will make use of the Bounded-Lipschitz distance $\Bl$, classically used for the derivation of the Vlasov equation, see \cite{Neun,Spo,CCR,CCC,GM} and the references therein. The Bounded-Lipschitz distance $\Bl$, also frequently called \emph{flat metric}, is defined as follows
\begin{align*}
\Bl(\mu,\nu):=\sup \left\{\int_\Omega f \d\left(\mu-\nu\right)|\, \,\|f\|_{L^{\infty}(\Omega)},\|f'\|_{L^{\infty}(\Omega)}\leq 1\right\} = \|\mu-\nu\|_{({W^{1,\infty}(\Omega)})^\ast}.
\end{align*}
Since our problem is posed on the product space we extend the metric setting to the space $\mathcal{M}_+\times\mathcal{M}_+$, the product space of nonnegative measures in the canonical way. For $d\in\{\Bl, d_p\}$, $1\leq p<\infty$, we define the product metric (still denoted $d$) as
\begin{align}
	\label{eq:BL_dist}
	d(U,\tilde{U}):=d(\rho,\tilde{\rho})+d(\eta,\tilde{\eta})\,,
\end{align}
where $U=(\rho,\eta),\tilde{U}=(\tilde{\rho},\tilde{\eta})\in\curlyM_+\times\curlyM_+$.

\begin{proposition}[Properties of $\Bl$]
Let $\mu, \nu \in L_+^1(\Omega)$ be two densities. Then the following properties hold true:
\begin{enumerate}[(i.)]
\item $\Bl(\mu,\nu)\leq \|\mu-\nu\|_{L^1(\Omega)},$
\item $\Bl(\mu,\nu) \leq \Wo(\mu,\nu)$, whenever $\mu(\Omega) = \nu(\Omega)$.
\end{enumerate}
\end{proposition}
\begin{proof}
Given $\mu,\nu\in L_+^1(\Omega)$ be arbitrary and $f \in W^{1,\infty}(\Omega)$ with $\|f\|_{W^{1,\infty}(\Omega)} \leq 1$. Hence we may write
\begin{align*}
	\left|\int_\Omega f \d (\mu-\nu)\right| \leq \int_\Omega |f|\d |\mu-\nu| =\|\mu-\nu\|_{L^1(\Omega)}.
\end{align*}
Taking the supremum over all such functions $f$ we get the first statement by using definition \eqref{eq:BL_dist}. For the second statement, we additionally assume that $\mu(\Omega) = \nu(\Omega)$. We recall the dual definition of $\Wo$,
\[
\Wo(\mu,\nu)=\sup\left\{\int_\Omega f \d (\mu-\nu)\,|\,f\in \mathrm{Lip}(\Omega)\mbox{ s.t. } \|f\|_{\mathrm{Lip}(\Omega)} \leq  1\right\},
\]
where $\mathrm{Lip}(\Omega)$ is the set of Lipschitz-continuous functions, see \cite{Vil_1,Snt_B}. Then property (ii.) is a consequence of this formulation, and this concludes the proof.
\end{proof}
\begin{corollary}\label{cor:triangular:_inequality}
	Let $U_i=(\mu_i,\nu_i)\in \mathcal{M}_+(\Omega)^2$ for $i=1,2,3$. Furthermore assume $\mu_i,\nu_i\in L^1(\Omega)$ for $i=1,2$ as well as $\mu_2(\Omega)=\mu_3(\Omega)$ and $\nu_2(\Omega)=\nu_3(\Omega)$.
	Then
	\begin{align*}
		\Bl\big(U_1, U_3\big) \leq \|U_1 -U_2\|_{L^1(\Omega)} + \Wo\big(U_2,U_3\big).
	\end{align*}
\end{corollary}

Throughout two quantities are crucial for our analysis -- the sum of the two densities, $\sigma$, and the ratio, $r$, between one density, say $\rho$, and the sum
\begin{equation*}
 \sigma = \rho + \eta, \qquad \text{and} \qquad r = \frac{\rho}{\sigma},
\end{equation*}
where we assume no vacuum, \textit{i.e.} $\sigma>0$. A straightforward computation shows these functions satisfy the following system of PDEs
\begin{align}\label{eq:transf_sys}
\left\{
\begin{array}{l}
	\partial_t \sigma = \partial_x\left(\sigma\partial_x\chii(\sigma)\right)+\sigma \left(r(\tilde F_1 +\tilde G_2) +(1-r)\left(\tilde G_1 + \tilde F_2\right)\right), \\[0.7em]
	\partial_t r =\partial_x r\partial_x \chii(\sigma) + r(1-r)\left(\tilde{F}_1-\tilde{F}_2\right)+(1-r)^2\tilde{G}_1-r^2\tilde{G}_2,
\end{array}
\right.
\end{align}
where we used $\tilde{F}_i(\sigma,r)=F_i(r\sigma,(1-r)\sigma)$ and $\tilde{G}_i(\sigma,r)=G_i(r\sigma,(1-r)\sigma)$, for $ i=1,2$, to denote the reaction terms in the transformed variables. In order to simplify the analysis in Section \ref{sec:proof_main_result}, let us introduce the more concise notation
\begin{align*}
	A_1(r,\sigma): = \tilde F_1 + \tilde G_2,\quad A_2(r,\sigma) := \tilde G_1 + \tilde F_2, \quad \mathrm{and} \quad  A_3(r,\sigma):= \tilde F_1 - \tilde F_2.
\end{align*}
Note that these functions are Lipschitz and bounded as they are linear combinations of $BV\cap L^\infty$ functions. Thus the transformed system \eqref{eq:transf_sys} can be rewritten in the more compact form
\begin{align}\label{eq:transf_sys1}
\left\{
\begin{array}{l}
	\partial_t \sigma = \partial_x\left(\sigma\partial_x\chii(\sigma)\right)+\sigma \big(r A_1 +(1-r)A_2\big),\\[0.5em]
	\partial_t r =\partial_x r\partial_x\chii(\sigma) + r(1-r) A_3 +(1-r)^2\tilde{G}_1-r^2\tilde{G}_2.
\end{array}
\right.
\end{align}
\noindent
Let us note here, that taking $F_i=G_i=0$ in \eqref{eq:main}, yields the following nonlinear cross-diffusion system
\begin{align}\label{eq:cross_diff}
\left\{
\begin{array}{l}
	\partial_t \rho = \partial_x\left(\rho\partial_x\chii(\sigma)\right),\\[0.5em]
	\partial_t \eta = \partial_x\left(\eta\partial_x\chii(\sigma)\right),
\end{array}
\right.
\end{align}
where the sum $\sigma$ satisfies the nonlinear diffusion equation \eqref{eq:diff_sigma}.

In order to be useful for our purposes, we need some properties on the internal energy density. The role of these properties will be clearer after the statement of our main result.
\begin{definition}[Internal energy density]
	\label{def:feasible_NLD}
	A function $\chi:[0,\infty] \rightarrow \R$ is an internal energy density if
	\begin{enumerate}[({NL}-i)]
		\item $\chi \in C^0([0,\infty],\R) \cap C^2((0,\infty),\R)$ with $\chi''>0$,
		\item $\lim_{h\downarrow 0} \chi'(h) = 0$.
		\item the integrals $	\kappa(x) := \int_1^x \frac{\chi''(s)}{s}\d s$, and $K(\sigma):= \int_0^\sigma \kappa(x)\d x$ exist.
	\end{enumerate}
\end{definition}

As mentioned in the introduction, the space of probability measures endowed with $d_2$ has proven to be an exceptional choice of a metric space. In this case the minimiser in Eq. \eqref{eq:JKO_intro} satisfies the so-called optimality condition
\begin{align}
	\label{eq:optimality_condition}
	\frac{\varphi}{\tau} + \chi'(\sigma^{n+1}) = const.,
\end{align}
\emph{cf.} \cite[Proposition 7.20]{Snt_B}, for instance. Here $\varphi$ denotes the associated Kantorovich potential \cite[Theorem 1.17]{Snt_B}. Notice that this optimality is only obtained in those references for non-degenerate problems, where $\beta'(0^+)>0$. However, similar approximation techniques as those developed in \cite[p. 156]{Otto} and \cite[p. 27]{CJMTU} allow to overcome the difficulties associated to degenerate diffusions, where $\beta'(0^+)=0$.
In the rest of the paper, we will proceed as if we were dealing with nonlinearities leading to nondegenerate diffusions since by this standard approximation procedure, the same result can be obtained for the degenerate ones. We are now ready to introduce our notion of weak solutions.

\begin{definition}[Notion of weak solutions]\label{def:weak_sol}
A couple $\rho,\eta\in  C\big( 0,T; BV(\Omega)\cap L^{\infty}(\Omega)\big)$ is a weak solution to system \eqref{eq:complete} if $\sigma=\rho+\eta\in L^2(0,T;H^1(\Omega))$  and there holds
\begin{align*}
 \int_\Omega(\rho(t)-\rho(s))\,\zeta\d x &= \int_s^t\int_\Omega -\rho\partial_x\chi'(\rho+\eta)\partial_{x} \zeta + \left(\rho F_1(\rho,\eta)+\eta G_1(\rho,\eta)\right) \zeta \d x \d\bar\tau,\\
 \int_\Omega(\eta(t)-\eta(s))\,\xi\d x &= \int_s^t\int_\Omega -\eta\partial_x\chi'(\rho+\eta)\partial_{x} \xi + \left(\eta F_2(\rho,\eta)+\rho G_2(\rho,\eta)\right) \xi \d x \d\bar\tau,
\end{align*}
for any two test functions $\zeta, \xi \in C_c^\infty(\Omega)$.
\end{definition}

\begin{remark}
\mbox{}
\\
\indent
\label{rem:K_geodesic_convex}
\textit{1.} Notice that the functional
	\begin{align*}
		\curlyK(\sigma) := \int_\Omega K(\sigma)\d x,
	\end{align*}
associated to $\chi$, as in Definition \ref{def:feasible_NLD}, satisfies the McCann condition since $K''(s)=\tfrac{\chi''(s)}s>0$, and is therefore displacement convex. This fact will be crucial in Section \ref{sec:combi_estim} in order to prove Lemma \ref{lem:uniformH1_bound}. Let us just state here that it is necessary to obtain additional regularity from the dissipation of this functional on the nonlinear diffusion term, thus allowing us to pass to the limit in the approximating sequence.

\textit{2.}
	Let us note that we can also allow for nonlocal reaction terms, \emph{i.e.}
	\begin{align*}
		F_i = W_{1, i}\star \rho + W_{2,i}\star \eta,
	\end{align*}
	and similarly for $G_i$, for $i=1,2$. The only assumption we need to impose on the kernels is that they are smooth and integrable. These models are found in modelling  pattern formation, as for instance the kernel-based Turing pattern system \cite{Kon17} or the nonlinear aggregation-diffusion system \cite{VS}.

\textit{3.} Similarly, we can -- at least formally -- interpret the system \eqref{eq:cross_diff} as a gradient flow of the functional
\begin{equation*}
	\eE(\rho, \eta)=\int_\Omega \chi(\rho+\eta) \d x,
\end{equation*}
in the product Wasserstein space.
\end{remark}

\subsection{Splitting scheme}
We are now ready to introduce our splitting scheme for equation \eqref{eq:complete}. Let some initial data $\rho_0,\eta_0\in BV(\Omega)\cap L^{\infty}(\Omega)$ be given such that there exists a function $r_0 \in BV(\Omega)$ such that
\begin{align*}
	\sigma_0:=\rho_0+\eta_0,\qquad\mbox{ and } \qquad  \frac{\rho_0}{\sigma_0}=r_0\big|_{\{\sigma_0>0\}},
\end{align*}
and $0\leq r_0 \leq1$. Furthermore we assume $F_i$ and $G_i$, $i=1,2$ are bounded and Lipschitz with respect to $\rho$ and $\eta$ and we impose $G_1(0,\cdot)\geq0$ and $G_2(\cdot,0)\geq0$ to ensure positivity of solutions.\\

We fix $0<\tau<1$ and $n\in \{1,\ldots, N\}$, with $N\in \N$ such that $N\tau=T$. We then recursively construct the piecewise constant approximation to the system as follows. We impose
\[
 (\rho_\tau^0,\eta_\tau^0)=(\rho_0,\eta_0),
\]
and then construct $\Unp=(\rhonp,\etanp)$ by the following scheme. We split the equation into a reaction step and a diffusion step on the time interval $[t^{n}, t^{n+1})$, with $t^n = n \tau$, for all $1\leq n\leq N$.

\subsubsection{Reaction step}
The reaction phase consists of solving the system of ordinary differential equations
\begin{align*}
\left\{
\begin{array}{l}
	\partial_t \sigma = \Sigma(\sigma, r):= \sigma \big(r A_1 +(1-r)A_2\big),\\[0.5em]
	\partial_t r = R(\sigma, r):=r(1-r) A_3 +(1-r)^2\tilde{G}_1-r^2\tilde{G}_2,\\[0.5em]
	\sigma(t^n)=\sigma^n, \quad\mbox{and}\quad r(t^n)=r^n,
\end{array}
\right.
\end{align*}
in the time interval $[t^n, t^{n+1})$.  We then set
\begin{subequations}
\begin{align}
\label{eq:r_step}
\rhohalf := r\sigma\big|_{t= (n+1)\tau}, \quad \mbox{and}\quad \etahalf := (1-r)\sigma\big|_{t=(n+1)\tau}.
\end{align}
A straightforward computation reveals
\begin{align}
	\label{eq:rsigma_eqn}
	\begin{split}
	\partial_t (r \sigma) &= r\sigma \tilde F_1 + (1-r)\sigma \tilde G_1,\\
	\partial_t ((1-r) \sigma) &= (1-r)\sigma \tilde F_2 + r\sigma \tilde G_2,
	\end{split}
\end{align}
verifying that $r\sigma$ and $(1-r)\sigma$ solve the reaction part of Eq. \eqref{eq:complete}.

\subsubsection{Diffusion step}
After the reaction phase we solve
\begin{equation}\label{eq:JKO}
  \Unp\in \argmin\limits_{U\in\curlyP^{m_1}(\Omega)\times \curlyP^{m_2}(\Omega)}\left\{\frac{1}{2\tau}\Wt\left(U,\Uhalf\right)+\eE(U)\right\},
\end{equation}
where $m_1=\rhohalf(\Omega)$ and $m_2 = \etahalf(\Omega)$. Let $\curlyT$ denote the associated transport map pushing, \emph{i.e.} $\curlyT_\#\Unp=\Uhalf$. We define
\begin{align}
	\label{eq:JKOrsigma}
	\sigmanp:=\rhonp+\etanp, \qquad\mbox{ and }\qquad \rnp:=\rhalf\circ\curlyT.
\end{align}
\end{subequations}
While the definition of $\sigmanp$ is somewhat natural, the definition of $\rnp$ seems a bit surprising. Let us note here that, indeed, $\rnp = \rhonp/\sigmanp$ where $\sigmanp>0$. For the precise argument we refer to Section \ref{sec:estim_diffu}, Eq. \eqref{eq:composition_with_monotone_T}.

\subsubsection{Combination of both steps -- construction of a solution}
Throughout this paper we refer to Eq. \eqref{eq:r_step} as \emph{reaction step} and to Eq. \eqref{eq:JKO} as \emph{diffusion step}, respectively.
\begin{definition}\label{def:piec_con}[Piecewise constant interpolation]
	Let $(\rn, \sigman)_{n\in\N}$ be the sequen\-ce obtained from the splitting scheme. Then we define the piecewise constant interpolations by
	\begin{align*}
    	r_\tau(t,x) = \rn(x), \quad \text{and} \quad \sigma_\tau(t,x) = \sigman (x),
    \end{align*}
as well as
    \begin{align*}
    	\rho_\tau(t,x) = \rn(x)\sigman(x), \quad \text{and} \quad \eta_\tau(t,x) = (1-\rn (x))\sigman(x),
    \end{align*}
    for all $(x,t)\in \Omega \times [t^n,t^{n+1})$. Furthermore we write $U_\tau := (\rho_\tau,\eta_\tau)$.
\end{definition}

We will say that two densities $\rho,\eta\in BV(\Omega)\cap L^\infty(\Omega)$ are segregated if the intersection of the interior of their supports is empty. We are now ready to state our main result.

\begin{theorem}[Convergence to weak solutions] Let $\rho_0,\eta_0\in BV(\Omega)\cap L^\infty(\Omega)$ and assume there exists a function $r_0\in BV(\Omega)$ such that $r_0 = \rho_0/(\rho_0+\eta_0)$ on $\{\rho_0+\eta_0>0\}$ and $0\leq r_0\leq 1$. Then, upon the extraction of a subsequence, the piecewise constant approximations $(\rho_\tau)_{\tau>0}$ and $(\eta_\tau)_{\tau>0}$ converge to a weak solution of system \eqref{eq:complete} in the sense of Definition \ref{def:weak_sol}. Moreover, if initially the two densities $\rho_0,\eta_0$ are segregated, then the limit densities $\rho(t,\cdot),\eta(t,\cdot)$ remain segregated for all times.
\end{theorem}

\section{\label{sec:proof_main_result}Proof of the main result}
This section is dedicated to proving the main result of the paper -- the convergence of the approximation obtained by the splitting scheme to a solution of the system. It is organised as follows: in Section \ref{sec:estim_reac} and \ref{sec:estim_diffu} we establish the crucial BV-estimates and $L^\infty$-bounds. In Section \ref{sec:combi_estim} we combine the estimates from the previous sections in order to get uniform estimates for a whole iteration. Finally, in Section \ref{sec:convergence} we show how to extract a convergent subsequence and identify its limit as a weak solution to system \eqref{eq:complete}.

\subsection{Estimates for reaction step\label{sec:estim_reac}}
Since the right-hand sides, $\Sigma(\sigma,r), R(\sigma,r)$ are Lipschitz continuous in both components we note that the solution of the reaction system is unique.
\begin{proposition}[$L^\infty$ estimates of the reaction step]
\label{prop:linfty_reaction}
Let $(\rn, \sigman)$ be given by our splitting scheme. Then there holds
\begin{align*}
	 0\leq\sigmahalf \leq \Linfty{\sigman} \exp(c\tau), \mbox{ and } 0\leq \rhalf \leq 1.
\end{align*}
\end{proposition}
\begin{proof}
We show first that there holds $\rhalf \in [0,1]$. Assume the contrary, \emph{i.e.} $\rhalf<0$ or $\rhalf>1$. If $\rhalf<0$ then, by continuity, there exists a time $t^\star\in(t^n,t^{n+1})$ such that $r(t^\star)=0$ and $\partial_t r(t^\star) < 0$. However, this is absurd as
\begin{align*}
	0 > \partial_t r(t^\star) = R\big(\sigma(t^\star), r(t^\star)\big) = \tilde G_1\big(\sigma(t^\star), 0\big) \geq 0.
\end{align*}
Analogously, it can be shown that $\rnp\leq1$. Finally, for the positivity of $\sigma$ we can use a similar argument. Let us assume $\sigman\geq0$ and $\sigmahalf<0$. Then, there exists another $t^\star$ such that
\begin{align*}
	0> \partial_t\sigma(t^\star) = \Sigma\big(\sigma(t^\star), r(t^\star)\big) = 0,
\end{align*}
which clearly is a contradiction. For the $L^\infty$-bound we simply apply Gronwall's lemma and the fact that $r\in[0,1]$.
\end{proof}

Next we address the BV-estimates during the reaction step.

\begin{proposition}[Bounded variation of $\rhalf$ and $\sigmahalf$]
\label{prop:bv_reaction}
Let us consider $(\rn, \sigman)$ as initial data for our splitting scheme. Then,  the reaction step is BV-stable in the following sense.
	\begin{align*}
		\BV{\rhalf} + \BV{\sigmahalf} \leq (\BV{\rn} + \BV{\sigman})\,\exp(c \tau),
	\end{align*}
    for some positive $K$, depending only on the Lipschitz constants of $F_i,G_i$ and the $L^\infty$-bounds on $F_i,G_i$, for $i=1,2$.
\end{proposition}
\begin{proof}
	Using the transformed system, Eqs.\eqref{eq:transf_sys1}, $r$ and $\sigma$ satisfy the following equations in the reaction step
    \begin{align*}
    	\partial_t \sigma = \Sigma(\sigma, r), \quad \mbox{ and }\quad\partial_t r = R(\sigma, r).
    \end{align*}
    Upon integrating in time we get
    \begin{align*}
    	\sigma(t) = \sigma(s) + \int_s^t \Sigma\big(\sigma(\bar\tau), r(\bar \tau)\big) \d \bar \tau, \quad \mbox{and} \quad     	r(t) = r(s) + \int_s^t R\big(\sigma(\bar\tau), r(\bar \tau)\big) \d \bar \tau.
    \end{align*}
    Now, let $P\subset \Omega$ be an arbitrary partition. We compute the variation of $\sigma$ and $r$ with respect to $P$ and obtain
    \begin{align*}
    	Q(t):= V_P(\sigma(t)) + V_P(r(t)) \leq V_P(\sigma(s)) + V_P(r(s)) + \int_s^t V_P(\Sigma(\bar \tau)) + V_P(R(\bar \tau))\d \bar \tau,
    \end{align*}
    whence
	\begin{align*}
		Q(t) \leq Q(s) + c\int_s^t Q(\bar \tau) \d \bar \tau,
	\end{align*}
	where $c$ only depends on the $L^\infty$-bounds and the Lipschitz-continuity of $A_i$, for $\in\{1,2,3\}$ and the $L^\infty$-bounds of $\sigma$ and $r$.
	Applying Gronwall's lemma we finally obtain
	\begin{align*}
		Q(t)\leq Q(s)\exp\big(c(t-s)\big).
	\end{align*}
	Passing to the supremum on the right-hand side and then on the left-hand side yields the result.

\end{proof}

\subsection{Estimates for diffusive step\label{sec:estim_diffu}}
This section is devoted to establishing $BV$-estimates and $L^\infty$-bounds in the diffusive step. To this end we will make use of the following remark.
\begin{remark}[Same optimality conditions]\label{rem:opt}
Let $\rhonp$ and $\etanp$ be given by the JKO step for $\mathcal{E}$, \emph{cf.} \eqref{eq:JKO}. Then, the optimality conditions, Eq. \eqref{eq:optimality_condition}, take the following form:
\begin{equation*
	\frac{\delta \mathcal{E}}{\delta \rho} + \frac{\varphi_\rho}{\tau} = c_1, \quad \frac{\delta \mathcal{E}}{\delta \eta} + \frac{\varphi_\eta}{\tau} = c_2, \quad \text{and} \quad
  \mbox{with }  \frac{\delta \mathcal{E}}{\delta \rho} = \frac{\delta \mathcal{E}}{\delta \eta} = \chii(\sigma^{n+1}),
\end{equation*}
for some constants $c_1$ and $c_2$, and the respective Kantorovich potentials $\varphi_\rho$ (from $\rhonp$ to $\rhohalf$) and $\varphi_\eta$ (from $\etanp$ to $\etahalf$), see \cite{Snt_B}. Notice that the previous identities imply that $\varphi_\rho=\varphi_\eta=\varphi$ up to an additive constant. Moreover, the uniqueness of the optimal map implies that $\curlyT(x) = (\id - \partial_x \varphi)(x)$ is also the optimal map from $\sigmanp$ to $\sigmahalf$, and $\varphi$ is the corresponding Kantorovich potential.
\end{remark}

\begin{proposition}[$L^\infty$ stability of the diffusive step]
\label{prop:linfty_diffusion}
Let $(\rhalf, \sigmahalf)$ be given by the splitting scheme \eqref{eq:r_step}. Then these quantities satisfy
\begin{align*}
	0 \leq \sigmanp \leq \Linfty{\sigmahalf},\quad \text{ and } \quad 0 \leq \rnp \leq 1,
\end{align*}
after the diffusion step for any $1\leq n \leq N$.
\end{proposition}
\begin{proof}
Choose $x_0 \in \argmax (\sigmanp)$. Then, by the optimality condition, Eq. \eqref{eq:optimality_condition},

\begin{align*}
	\chi'(\sigmanp) + \frac{\varphi}{\tau} = c,
\end{align*}
we have $x_0 \in \argmin(\varphi)$ where we used the fact that $\chi''\geq 0$, \emph{cf.} Definition \ref{def:feasible_NLD}, $(N\!L-iii)$. Hence $\varphi''(x_0)\geq 0$ and consequently, by passing to the derivative in Eq. \eqref{eq:rel_T_KantorovichPotential} we get
\begin{align}
	\label{eq:T_prime_less_than_one}
	\curlyT'(x_0) = 1 - \partial_{xx} \varphi(x_0) \leq 1.
\end{align}
where $\curlyT$ is the transport map from $\sigmanp$ to $\sigmahalf$.  After a change of variables we get
\begin{align*}
	\sigmanp(x) &\leq \|\sigmanp\|_{L^\infty}= \sigmanp(x_0)=\curlyT'(x_0) \sigmahalf(\curlyT(x_0))\!\stackrel{\eqref{eq:T_prime_less_than_one}}{\leq} \sigmahalf(\curlyT(x_0))\\
    & \leq  \|\sigmahalf\|_{L^\infty},
\end{align*}
for any $x\in \Omega$. For the non-negativity we observe that $\curlyT'\geq 0$ by Brenier's theorem \cite{Bre87, Bre91, Vil_1, Vil_2}. Thus
\begin{align*}
	\sigmanp(x)= \sigmahalf\circ \curlyT(x) \curlyT'(x)\geq 0.
\end{align*}
Finally the bounds for $\rnp$ follow from its definition, \emph{cf.} Eq. \eqref{eq:JKOrsigma}, as the composition with a monotone function does not change  the infimum and the supremum of a function. This concludes the proof.
\end{proof}

\begin{proposition}[Bounded variation of $\rnp$ and $\sigmanp$]
\label{prop:bv_diffusion}
Let $(\rhalf, \sigmahalf)$ be given. After the diffusive step they satisfy the following estimate.
\begin{align*}
	\BV{\sigmanp} \leq \BV{\sigmahalf}, \qquad \text{and} \qquad \BV{\rnp} \leq \BV{\rhalf}.
\end{align*}
\end{proposition}
\begin{proof}
The result for the BV-norm of the minimiser, $\sigmanp$, is shown analogously to the proof of  Theorem 1.1., \emph{cf.} \cite{DPMSV16}.
Now we need to show that the BV-norm of the ratio $r$ does not increase. Recall the definition of $r^{n+1}$, \emph{cf.} Eq. \eqref{eq:JKOrsigma}, as
\begin{align*}
	r^{n+1}:=r^{n+1/2} \circ \curlyT,
\end{align*}
where $\curlyT$ is the transport map such that $\rhohalf=\curlyT_\#\rhonp$ and $\sigmahalf = \curlyT_\#\sigmanp$. Note that it is indeed the same function and there holds $\curlyT'\geq 0$, by Brenier's theorem \cite{Bre87, Bre91, Vil_1, Vil_2} .

Now, let  $P\subset\Omega$  be any partition of $\Omega$. There holds
\begin{align*}
	V_P(r^{n+1}) = V_P(r^{n+1/2} \circ \curlyT) = V_{P'}(r^{n+1/2}) \leq \BV{r^{n+1/2}},
\end{align*}
where $P'$ is another partition induced by the monotone map $\curlyT$. Taking the supremum over all partitions $P$, we get
\begin{align*}
	\BV{r^{n+1}} \leq \BV{r^{n+1/2}}.
\end{align*}

Finally note that, indeed,
\begin{align*}
	\rnp(x)  = \frac{\rhonp(x)}{\sigmanp(x)},
\end{align*}
on $\supp(\sigmanp)$ as we shall see now. According to Remark \ref{rem:opt}, the  same transport map $\curlyT$ pushes $\rhonp$ onto $\rhohalf$ and $\sigmanp$ onto $\sigmahalf$. As a consequence the densities satisfy
\begin{align*}
	\rhonp(x) = \rhohalf\big(\curlyT(x)\big) \, \curlyT'(x), \quad \text{and} \quad \sigmanp(x) = \sigmahalf\big(\curlyT(x)\big)\, \curlyT'(x),
\end{align*}
whence
\begin{align}
\label{eq:composition_with_monotone_T}
	\rnp(x) :=  \rhalf\circ\curlyT(x)=\frac{\rhohalf}{\sigmahalf}\circ\curlyT (x) =\frac{\rhonp}{\sigmanp}(x).
\end{align}
\end{proof}

\subsection{Combined estimates for an entire splitting step\label{sec:combi_estim}}
We have now garnered all information necessary to pass to the limit. Let us combine the estimates from the previous section in the following lemma.
\begin{lemma}[\label{lem:BVLinfty}$BV$-estimates and $L^\infty$-estimates]
The sequence $(r_\tau, \sigma_\tau)_{\tau>0}$ \ obtained by the splitting scheme is uniformly bounded in $BV(\Omega)$ and $L^\infty(\Omega)$. More precisely there holds
\begin{align*}
	\sup\limits_{t\in [0,T]} \Linfty{r_\tau} \leq C, \qquad \mbox{and} \qquad \sup\limits_{t\in[0,T]} \Linfty{\sigma_\tau} \leq C,
\end{align*}
and
\begin{align*}
	\sup\limits_{t\in [0,T]}  \BV{r_\tau}\leq C, \qquad\mbox{ and }\qquad \sup\limits_{t\in [0,T]} \BV{\sigma_\tau} \leq C,
\end{align*}
for some positive constant $C<\infty$ only depending on $T$.
\end{lemma}
\begin{proof}
	The uniform $L^\infty$-bound is a consequence of combining Propositions \ref{prop:linfty_reaction} and \ref{prop:linfty_diffusion}. We use these uniform $L^\infty$-bounds in the estimates for the $BV$-norm, \emph{cf.} Propositions \ref{prop:bv_reaction} and \ref{prop:bv_diffusion}. Combining both for the reaction and diffusion step we also obtain the uniform $BV$-bounds.
\end{proof}

As a result of Lemma \ref{lem:BVLinfty} and Lemma \ref{lem:Ralgebra} we obtain the following corollary.
\begin{corollary}[$BV$-estimates and $L^\infty$-estimates for $\rho, \eta$]
The sequences of approximated densities $(\rho_\tau)_{\tau>0}$, $ (\eta_\tau)_{\tau>0}$ are uniformly bounded in  $BV(\Omega)\cap L^\infty(\Omega)$.
\end{corollary}

Finally, we need to prove an estimate on the cross-diffusion term to be able to pass to the limit later. This estimate is achieved in Lemma \ref{lem:uniformH1_bound} which is preceded by two technical lemmas. We exploit the existence of an auxiliary functional guaranteed by Definition \ref{def:feasible_NLD}. Note that in the absence of the reaction part this would indeed be an entropy in the classical sense, \emph{i.e.} that it is decayed along solutions. Since we are interested in a uniform estimate we shall begin by proving a control of this functional during each reaction phase.

\begin{lemma}[\label{lem:H1_part1}Control of the auxiliary functional in the reaction step]
	$\curlyK$ increases at most at a constant rate independent of $n$. More precisely there holds
	\begin{align*}
		\curlyK(\sigma^{n+1/2}) \leq \curlyK(\sigma^n)+c\tau,
	\end{align*}
	for any $n\in\N$.
\end{lemma}
\begin{proof}
	A straight forward computation yields
	\begin{align*}
		\ddt\int_\Omega K(\sigma)\d x &=\int_\Omega K(\sigma) \partial_t \sigma \d x = \int_\Omega K(\sigma) \sigma (A+ r B)\d x\leq c,
	\end{align*}
	using the uniform $L^\infty$-bound on $\sigma$ and the fact that $|\Omega|<\infty$.
	Hence,
	$$
		\curlyK(\sigma^{n+1/2}) \leq \curlyK(\sigma^n)+c\tau,
	$$
	where $c$ is independent of $n$.
\end{proof}

Next, we address the diffusion step. As mentioned earlier the auxiliary functional, $\curlyK$, is an entropy for the diffusive part and from its dissipation we obtain the  necessary regularity, as asserted in the following lemma.
\begin{lemma}[$H^1$-bound for $\sigma^{n+1}$]
\label{lem:H1_part2}
The minimiser of the JKO step satisfies the following estimate
\begin{align*}
	\tau \|\partial_x \sigma^{n+1}\|_{L^2(\Omega)}^2 \leq  \left( \curlyK(u^{n+1/2}) - \curlyK(u^{n+1})\right),
\end{align*}
for each $1\leq n\leq N$.
\end{lemma}

\begin{proof}
Let
$$
	(\rho^{n+1}, \eta^{n+1})\in \argmin \left\{\frac{1}{2\tau}\Wt(\cdot,U^{n+1/2}) + \frac12 \int_\Omega \eE(U)\d x \right\}.
$$
Let $\sigma_s = (\curlyT_s)_\# \sigma^{n+1} $ be the geodesic interpolation between $\sigma_s|_{s=0} = \sigma^{n+1}$ and $\sigma_s|_{s=1} = \sigma^{n+1/2}$, given by
\begin{align*}
	\curlyT_s = (1-s)\id + s \curlyT,
\end{align*}
and
\begin{align*}
	\curlyT = \id - \partial_x \varphi,
\end{align*}
for the associated Kantorovich potential, $\varphi$, \emph{cf.} Eq. \eqref{eq:rel_T_KantorovichPotential}. As a consequence the velocity field is given by
\begin{align*}
	v_s = (\curlyT-\id) \circ \curlyT_s^{-1},
\end{align*}
satisfying the following continuity equation,
\begin{align*}
	\partial_s \rho_s = \partial_x(\rho_s v_s).
\end{align*}
We differentiate the entropy along the geodesic and  obtain
\begin{align*}
	\label{eq:entropyevolution1}
	\frac{\d }{\d s}\int_\Omega K(\sigma_s) \d x &= \int_\Omega \kappa(\sigma_s) \partial_s \sigma_s \d x = - \int_\Omega \kappa'(\sigma_s)\partial_x\sigma_s \sigma_s v_s\d x\\
	&= - \int_\Omega \chi''(\sigma_s) \partial_x \sigma_s v_s\d x = - \int_\Omega\partial_x \chi'(\sigma_s) v_s \d x.
\end{align*}
Thus, at $s=0$, the evolution of the entropy Eq. \eqref{eq:entropyevolution1} becomes
\begin{align*}
	\ddt \int_\Omega K(\sigma_s) \d x\bigg|_{s=0} = \int_\Omega \partial_x \chi'(\sigma^{n+1})  \partial_x \varphi \d x.
\end{align*}
Using the optimality condition, Eq. \eqref{eq:optimality_condition}, we obtain
\begin{align*}
	 \tau \int_\Omega \big|\partial_x \chi'( \sigma^{n+1})\big|^2 \d x  &=  -\int_\Omega \partial_x \chi'(\sigma^{n+1})\partial_x \varphi \d x\\
     &=\ddt \int_\Omega K(\sigma_s) \d x\bigg|_{s=0}\\
     &\leq \curlyK(\sigma^{n+1/2}) - \curlyK(\sigma^{n+1}),
\end{align*}
where the last inequality is a consequence of the geodesic convexity of the entropy, \emph{cf.} Remark \ref{rem:K_geodesic_convex}. This concludes the proof.
\end{proof}

We combine the previous lemmas to obtain the desired estimate for a full iteration and finally for the piecewise constant interpolation, $\sigma_\tau$.

\begin{lemma}[\label{lem:uniformH1_bound}Uniform $L^2((0,T)\times\Omega)$-bound for $\partial_x \sigma_\tau$]
There holds
\begin{align*}
	\|\partial_x \chi'(\sigma_\tau)\|_{L^2((0,T)\times\Omega)} \leq C,
\end{align*}
for some positive constant depending only on $T$.
\end{lemma}
\begin{proof}
	This statement is a consequence of combining Lemma \ref{lem:H1_part1} and Lemma \ref{lem:H1_part2} to get
	\begin{align*}
		\tau \|\partial_x \chi'(\sigma^{n+1})\|_{L^2(\Omega)}^2 &\leq \curlyK(\sigma^{n+1/2}) - \curlyK(\sigma^{n+1})\\
        &\leq c\tau  +\curlyK(\sigma^{n})-\curlyK(\sigma^{n+1}).
	\end{align*}
	Summing over $n=0\ldots N-1$ gives
	\begin{align*}
		\|\partial_x \chi'(\sigma_\tau)\|_{L^2((0,T)\times\Omega)}^2 \leq cT + \curlyK(\sigma^0) - \inf\limits_{\sigma}\curlyK(\sigma) \leq C,
	\end{align*}
     which yields the statement of the lemma.
\end{proof}

\begin{lemma}[Total-square $2-$Wasserstein distance estimates]
\label{lem:total_square_estimate}
For every $n\in\{1,\ldots,N\}$ consider two consecutive steps for \eqref{eq:JKO},  $\Uhalf=(\rhohalf,\etahalf)$ and $\Unp=(\rhonp,\etanp)$, then there exists a constant $C$ such that
\begin{align*}
 \frac{1}{2\tau}\sum_{n=0}^{N}\Wt (\Uhalf,\Unp)\leq C.
\end{align*}
\end{lemma}
\begin{proof}
Using the minimising property of $\Unp$ we have
\[
 \frac{1}{2\tau}\Wt (\Uhalf,\Unp)\leq \eE(\Uhalf)-\eE(\Unp).
\]
Adding and subtracting $\eE(\Un)$ on the right-hand side, and considering that
\begin{align}
	\label{eq:intermediate_eqn1}
	&\eE(\Uhalf)-\eE(\Un)\\
	&=\int_\Omega \chi(\sigmahalf) \d x-\int_\Omega \chi(\sigman) \d x\nonumber\\
	& = \int_\Omega \chi\left(\sigman + \int_{t^n}^{t^{n+1}}\sigma(r A_1+ (1-r)A_2)\d s \right) \d x -\int_\Omega \chi(\sigman) \d x\nonumber\\
	&\leq \chi'(\xi) \int_{t^n}^{t^{n+1}} \sigma(r A_1+ (1-r)A_2) \d s,\nonumber
\end{align}
due to a Taylor expansion of $\chi$, where $\xi\in[\sigman, \sigmahalf]$. Using the $L^\infty$-bounds on $\sigma, r$ and $F_i,G_i$, we obtain
\begin{align*}
	&\eE(\Uhalf)-\eE(\Un)\leq C\tau,
\end{align*}
whence
\begin{equation*
 \frac{1}{2\tau}\Wt (\Uhalf,\Unp)\leq \eE(\Un)-\eE(\Unp)+C\tau,
\end{equation*}
for a full time step.
Finally, summing  over  $n$ gives  the result:
\begin{align*}
 \frac{1}{2\tau}\sum_{n=1}^{N}\Wt (\Uhalf,\Unp) & \leq \sum_{n=0}^{N}\left(\eE(\Un)-\eE(\Unp)+C\tau\right)\\
 &=\eE(U^0)-\eE(U_\tau^N)+N\left(C\tau\right)\\
 & \leq \eE(U^0)-\inf\limits_{U\in \mathcal{M}_+\times\mathcal{M}_+}\eE(U)+CN \tau\\
 &\leq \bar{C}.
\end{align*}
\end{proof}
\subsection{Convergence\label{sec:convergence}}
We now prove the convergence result.
\begin{proposition}\label{prop:ArzelaAscoli}
The piecewise constant interpolations defined in Definition \ref{def:piec_con} admit subsequences converging uniformly to absolutely continuous curves  $\bar{\rho},\bar{\eta}$ with  values in $\curlyM_+(\Omega)$. Moreover, $\bar\rho$ and $\bar\eta$ are $\Bl$-continuous functions on $[0,T]$.
\end{proposition}
\begin{proof}
The proof is based on the application of a generalised version of the Ascoli-Arzel\`a theorem, \emph{cf.} Ref. \cite{AGS}, Section 3.

We begin by establishing `almost continuity' of the approximation. To this end let $0\leq s<t\leq T$ be two time instances. Then there exist two uniquely determined integers, $m,k$, satisfying
\begin{align*}
	s\in((m-1)\tau,m\tau], \quad \mbox{and}\quad  t\in((k-1)\tau,k\tau],
\end{align*}
such that
\begin{align}
	\label{eq:holder_cty_estimate}
	\Bl(U_\tau(s),U_\tau(t)) &\leq \sum_{n=m}^{k-1}\Bl(\Un,\Unp)\leq\left(\sum_{n=m}^{k-1}\Bl^2(\Un,\Unp)\right)^{\frac12}|k-m|^{\frac12},
\end{align}
where $U_\tau(t)=\left(\rho_\tau(t),\eta_\tau(t)\right)$ as defined  in Definition \ref{def:piec_con}. It becomes apparent that we need to address the bounded Lipschitz term next. To this end we use the triangulation established in Corollary \ref{cor:triangular:_inequality} to estimate it by the $L^1$-distance in the reaction step and the $W_1$-distance in the diffusion step. Hence
\begin{align*}
	\sum_{n=m}^{k-1}\Bl^2(\Un, \Unp) \leq 2 \sum_{n=0}^{N}  \|\Un -  \Uhalf\|_{L^1(\Omega)}^2 + 2 \sum_{n=0}^{N} \Wo^2(\Uhalf,\Unp).
\end{align*}
For the reaction step an argument similar to Eq. \eqref{eq:intermediate_eqn1} yields
\begin{align*}
	\|\Un -  \Uhalf\|_{L^1(\Omega)}^2\leq C\tau^2,
\end{align*}
and, using that the $p$-Wasserstein distances are ordered, we even have
\begin{align*}
	\sum_{n=m}^{k-1}\Bl^2(\Un, \Unp) &\leq 2 \sum_{n=0}^{N}\|\Un -  \Uhalf\|_{L^1(\Omega)}^2 + 2 \sum_{n=0}^{N}\Wt(\Uhalf,\Unp)\\
	&\leq C\tau,
    \end{align*}
    where we used the total-square estimate, Lemma \ref{lem:total_square_estimate}.
Therefore Eq. \eqref{eq:holder_cty_estimate} becomes
\begin{align*}
		d_{BL}(U_\tau(s),U_\tau(t))
		&\leq\left(\sum_{n=m}^{k-1}\Bl^2(\Un,\Unp)\right)^{\frac12}|k-m|^{\frac12}\\
		&\leq C\sqrt{\tau} \left(\frac{|t-s|}{\tau}+1\right)^{\frac12}\\
		&\leq C (\sqrt{|t-s|} + \sqrt{\tau}).
\end{align*}
Thus we get the `almost $\frac{1}{2}$-H\"{o}lder continuity' for the curve $U_\tau(t)$ and we obtain the uniform narrow compactness on compact time intervals by using the refined version of Ascoli-Arzel\`a's theorem, \emph{cf.} \cite{AGS}, Section 3.
\end{proof}

\begin{corollary}[\label{cor:strong_convergence}Strong convergence in $L^p(0,T;L^q(\Omega))$]
	Let $1 \leq p,q < \infty$ and  $(\rho_\tau)_{\tau>0}$ and $(\eta_\tau)_{\tau>0}$ be the sequences of the piecewise constant interpolations as in Definition \ref{def:piec_con}. Then there exist two functions $\rho,\eta\in L^p(0,T;L^q(\Omega))$ and subsequences, again denoted by $(\rho_\tau)_{\tau>0}$ and $(\eta_\tau)_{\tau>0}$, such that
	\begin{align*}
		\rho_\tau \rightarrow \rho, \qquad \mbox{ and } \qquad \eta_\tau \rightarrow \eta,
	\end{align*}
	strongly in $L^p(0,T;L^q(\Omega))$ as $\tau \rightarrow 0$. Moreover the convergence holds pointwise in time, \emph{i.e.} for all $t\in [0,T]$ there holds
	\begin{align*}
		\rho_\tau(t) \rightarrow \rho(t), \qquad \mbox{ and } \qquad \eta_\tau(t) \rightarrow \eta(t),
	\end{align*}
	strongly in $L^q(\Omega)$.
\end{corollary}
\begin{proof}
	Note that it suffices to show the result for $(\rho_\tau)_{\tau>0}$ as the same argument applies for $(\eta_\tau)_{\tau>0}$. By Proposition \ref{prop:ArzelaAscoli} we can extract a subsequence, still denoted the same, such that
	\begin{align*}
		\rho_\tau(t)\rightharpoonup \rho,
	\end{align*}
	in $\curlyM_+(\Omega)$ for all $t\in [0,T]$.
Furthermore, from the uniform $BV\cap L^\infty$-bounds,
for any $t\in [0,T]$ the sequence converges strongly in $L^1(\Omega)$. Using the uniform $L^\infty$
-bounds and the dominated convergence theorem, we obtain the pointwise-in-time convergence in any $L^q(\Omega)$.

	Now let us apply the same argument on the whole domain, $[0,T]\times \Omega$. The pointwise convergence and the uniform $L^\infty(0,T;L^\infty(\Omega))$-bound imply
	\begin{align*}
		\rho_\tau \rightarrow \rho,
	\end{align*}
	strongly in $L^p(0,T;L^q(\Omega))$ by the dominated convergence theorem. This concludes the proof.
\end{proof}

\begin{lemma}[Identification of the limit]
The sequence constructed in \eqref{eq:r_step}, \eqref{eq:JKO} converges to a weak solution of Eqs. \eqref{eq:main}.
\end{lemma}
 \begin{proof}
 The proof consists of two parts -- the diffusion part and the reaction part. We write them in their respective weak formulation and combine them to obtain the complete approximation of the weak formulation. Here we only show the argument for $\rho$ as the corresponding result for $\eta$ is shown analogously. We begin with the diffusion part. \\

\noindent
\textbf{Diffusion part.}
We consider the two steps before and after the application of the JKO scheme \eqref{eq:JKO},
\begin{align*}
	\Uhalf=(\rhohalf,\etahalf)	, \qquad\mbox{and}\qquad\Unp=(\rhonp,\etanp).
\end{align*}
For a given test function $\zeta \in C_c^\infty(\Omega)$, let $\curlyT$ be the optimal transport map from  $\rhonp$ to  $\rhohalf$. Upon integration over $\Omega$ we get
\begin{equation*}
\frac{1}{\tau}\int_\Omega\left(\rhonp(x)-\rhohalf(x)\right)\zeta(x)\d x=\frac{1}{\tau}\int_\Omega\rhonp(x)\left(\zeta(x)-\zeta (\curlyT(x))\right)\d x.
\end{equation*}
Taylor expanding $\zeta(T(x))$ around $x$ yields
\begin{align*}
	\zeta(\curlyT(x)) &= \zeta(x + (\curlyT(x)-x))\\
		&=\zeta(x) + \partial_x\zeta(x)(\curlyT(x)-x) + \curlyO\left(|\curlyT(x)-x|^2\right).
\end{align*}
Moreover, using the fact that $\curlyT(x)=x-\partial_x \varphi(x)$, where $\varphi$ is the Kantorovich potential associated to the optimal map $\curlyT$, the integral above can be rewritten as
\begin{align*}
\frac{1}{\tau}\int_\Omega&\left(\rhonp(x)-\rhohalf(x)\right)\zeta(x)\d x\\
 & =-\frac{1}{\tau}\int_\Omega\rhonp(x)\left(\curlyT(x)-x\right)\partial_x\zeta(x)\d x+O\left(\Wt(\rhonp,\rhohalf)\right)\\
  & =\frac{1}{\tau}\int_\Omega\rhonp(x)\partial_x\varphi(x)\partial_x\zeta(x)\d x+O\left(\Wt(\rhonp,\rhohalf)\right).
\end{align*}
Thanks to the optimality condition, Remark \ref{rem:opt},
we get
\begin{align}
	\begin{split}
	\label{eq:diffusion_step_identification}
\frac{1}{\tau}\int_\Omega&\left(\rhonp-\rhohalf\right)\zeta \d x\\
&=-\int_\Omega\rhonp\partial_x\chi'(\sigmanp)\,\partial_x\zeta \d x+\curlyO\left(\Wt(\rhonp,\rhohalf)\right).
	\end{split}
\end{align}

\noindent
\textbf{Reaction part.}
Note that
\begin{align}
	\label{eq:reaction_step_identification}
	\begin{split}
	 \int_\Omega\frac{\rhohalf-\rhon}{\tau}\zeta \d x
	 &= \int_\Omega \frac{\rhalf\sigmahalf - \rn\sigman}{\tau}\zeta\d x\\
	 &=\int_\Omega\int_{t^n}^{t^{n+1}}\frac{1}{\tau} \partial_t \big(r(\bar\tau) \sigma(\bar \tau)\big)\d \bar \tau \d x,\\
	 &=\int_\Omega \left(\rhalf\sigmahalf \tilde F_1 + (1-\rhalf)\sigmahalf\tilde G_1\right) \d x + \curlyO(\tau)\\
	 &=\int_\Omega \left(\rhohalf F_1^{n+1/2}+\etahalf G_1^{n+1/2}\right) \zeta \d x+\curlyO(\tau),
	 \end{split}
\end{align}
with the shortcuts $F_1^{n+1/2}=F_1(\rhohalf,\etahalf)$ and $G_1^{n+1/2}=G_1(\rhohalf,\etahalf)$, having used Eq. \eqref{eq:rsigma_eqn}.

\noindent
\textbf{Combination of both steps.}
Let us combine the reaction step and the diffusion step of the splitting scheme. Upon summing up Eqs.(\ref{eq:diffusion_step_identification}, \ref{eq:reaction_step_identification})  we obtain
\begin{align}
\label{eq:almost_weak_form}
\begin{split}
 0 =& \frac{1}{\tau}\int_\Omega\zeta(x)(\rhonp(x)-\rhon(x))\d x+\curlyO(\Wt(\rhonp,\rhohalf))\\
 &+ \int_\Omega \rhonp\partial_x\chi'(\rhonp(x)+\etanp(x))\partial_{x} \zeta(x)\d x\\
 &  - \int_\Omega\left(\rhohalf F_1(\rhohalf,\etahalf)+\etahalf G_1(\rhohalf,\etahalf)\right) \zeta \d x\\
 &+\curlyO(\tau).
\end{split}
\end{align}
We rewrite this equation in terms of the  piecewise constant interpolation, Definition \ref{def:piec_con}. For any $0<s<t <T$  there are two uniquely determined integers $m,k$ such that
\[
	s\in \left(m\tau,(m+1)\tau\right], \quad \mbox{and} \quad t\in\left(k\tau,(k+1)\tau\right].
\]
Multiplying Eq. \eqref{eq:almost_weak_form} by $\tau$ and summing from  $m$ to $k-1$, we obtain
\begin{align*}
 0 =& \int_\Omega\zeta(x)(\bar{\rho}_\tau(t,x)-\bar{\rho}_\tau(s,x))\d x+\curlyO(\tau)\\
 &+ \int_s^t\int_\Omega \bar{\rho}_\tau(\bar\tau,x)\partial_x\chi'\big(\bar{\rho}_\tau(\bar\tau,x)+\bar{\eta}_\tau(\bar\tau,x)\big)\partial_{x} \zeta(x)\\
 &  - \bigg(\bar{\rho}_\tau(\bar\tau,x) F_1\big(\bar{\rho}_\tau(\bar\tau,x),\bar{\eta}_\tau(\bar\tau,x)\big)+\bar{\eta}_\tau(\bar\tau,x) G_1\big(\bar{\rho}_\tau(\bar\tau,x),\bar{\eta}_\tau(\bar\tau,x)\big)\bigg) \zeta(x) \d x d\bar\tau.
\end{align*}
We are now ready to pass to the limit. The strong convergence obtained in Corollary \ref{cor:strong_convergence} allows us to pass to the limit in the nonlinear reaction terms. Moreover the cross-diffusion term converges due to the weak-strong $L^2(0,T;L^2(\Omega))$-duality, by Lemma \ref{cor:strong_convergence} and Corollary \ref{cor:strong_convergence}. Passing  to the limit $\tau\to 0$ we obtain  the weak formulation for $\rho$. The same argument for $\eta$ yields the statement.
\end{proof}

We end the paper with a stunning result which can be seen as a generalisation of the result of Bertsch \emph{et al.}, \emph{cf.} \cite{BGHP85,BGH87}. In their papers they prove that initially segregated species stay segregated at all times. We can drop their assumption that
\begin{align*}
	\supp(\rho_0) < \supp(\eta_0),
\end{align*}
\emph{i.e.} that both species are ordered and prove the following, more general theorem.
\begin{theorem}[\label{thm:segregation}Segregation in the case of no cross-reaction.]
	Assume no cross-reaction terms, \emph{i.e.} $G_1\equiv G_2\equiv 0$ and that both species are initially segregated, \emph{i.e.}
	\begin{align*}
		\int_\Omega \rho_0(x)\,\eta_0(x)\d x = 0.
	\end{align*}
	Then, there exists a solution such that both species stay segregated at all times.
\end{theorem}
\begin{proof}
	It suffices to show this property at the level of the discrete scheme since, once it is established, we use the strong $L^2$-convergence of $\rho_\tau$ and $\eta_\tau$ to show segregation is also kept in the limit:
	\begin{align*}
		0 = \lim_{\tau \to 0} \int_\Omega \rho_\tau(t,x)\eta_\tau(t,x) \d x = \int_\Omega \rho(t,x) \eta(t,x)\d x.
	\end{align*}
	Thus let us now show the property at the level of the approximation. It is clear that in the reaction step segregation is kept as it does not change the support of both $\rho$ and $\eta$. Hence we only need to prove that segregation is kept in the diffusion step. This is done by contradiction. Let us assume there exists an $1\leq n \leq N$ such that
	\begin{align}
		\label{eq:no_overlap}
		\int_\Omega \rhohalf(x) \etahalf(x)\d x = 0,
	\end{align}
	while
	\begin{align*}
		\int_\Omega \rhonp(x) \etanp(x) \d x > 0.
	\end{align*}
	Then there exists $\delta > 0$ and a set $B$ with $|B|>0$, such that
	\begin{align*}
		 |\curlyT'(x)|<\infty\,, \qquad \rhonp(x)> \delta\,, \quad\mbox{and}\quad \etanp(x) > \delta,
	\end{align*}
	for almost all $x\in B$. As both species have the same transport map $\curlyT$ in common there exists a set $A$ such that $A = \curlyT(B)$ and
	\begin{align*}
		\begin{split}
			0 &< \delta < \rhonp(x) =   \rhohalf(\curlyT(x))\curlyT'(x),\\
			0 &< \delta <  \etanp(x)=  \etahalf(\curlyT(x))\curlyT'(x),
		\end{split}
	\end{align*}
	which is absurd, for we assumed \eqref{eq:no_overlap}. Thus segregation is kept at each iteration which concludes the proof of the theorem.
\end{proof}

\section{\label{sec:numerics}Numerical simulations}
In this section we illustrate our main result by showing some numerical simulations using a different splitting scheme. This splitting scheme is based on the finite volume scheme introduced in \cite{BF,CCH15,CHS17}. We solve the cross-diffusion inner time step with this finite volume scheme instead of the optimal mass transport approach. Then we solve the reaction step along the ODEs. We refer to \cite{CKL17} for the numerical difficulties related to the implementation of the mass transport approach in a splitting fashion.

\begin{figure}[ht!]
	\centering
	\subfigure[Initial data.]{
		\includegraphics[width=0.45\textwidth]{graphs/1_init}
	}
	\subfigure[Final time.]{
		\includegraphics[width=0.45\textwidth]{graphs/1_final}
	}
	\caption{System \eqref{eq:lotka_volterra} initialised with two indicator functions. Both species diffuse individually while increasing according to the reaction term. They remain segregated in agreement with Theorem \ref{thm:segregation} and reach a constant stationary state after some time.}
	\label{fig:lotka_volterra_init_n_final}
\end{figure}

\subsection{Lotka-Volterra type reaction}
Let us consider the system
\begin{align}
	\label{eq:lotka_volterra}
	\begin{split}
	\partial_t \rho &= \partial_x (\rho \partial_x(\sigma)) + \rho(1-\sigma),\\
	\partial_t \eta &= \partial_x (\eta \partial_x(\sigma)) + \eta(1-\sigma),
	\end{split}
\end{align}
modelling two species avoiding overcrowding due to the nonlinear cross-diffusion term as well. In addition the growth term is of Lotka-Volterra type, \emph{cf.} Figures \ref{fig:lotka_volterra_init_n_final} \& \ref{fig:lotka_volterra_intermediate}.

\begin{figure}[ht!]
 	\centering
 		\includegraphics[width=0.45\textwidth]{graphs/1_evol}
 	\caption{Evolution in time of system \eqref{eq:lotka_volterra}.}
	\label{fig:lotka_volterra_intermediate}
\end{figure}

Let us consider now a little modification of system \eqref{eq:lotka_volterra}, as in \cite{BerDaPMim},
\begin{align}
	\label{eq:lotka_volterra_BerDaPMim}
	\begin{split}
	\partial_t \rho &= \partial_x (\rho \partial_x(\sigma)) + \rho(1-\sigma),\\
	\partial_t \eta &= \partial_x (\eta \partial_x(\sigma)) + \eta(1-\sigma/2).
	\end{split}
\end{align}
Using the same parameters and domain as in \cite{BerDaPMim} Figures \ref{fig:lotka_volterra_BerDaPMim_init_n_final} \& \ref{fig:lotka_volterra_BerDaPMim_intermediate} show complete segregation in agreement with their result and moreover numerically validate our main result in Theorem \ref{thm:segregation}.
 \begin{figure}[ht!]
 	\centering
 	\subfigure[Initial data.]{
 		\includegraphics[width=0.45\textwidth]{graphs/2_init}
 	}
 	\subfigure[Final time.]{
 		\includegraphics[width=0.45\textwidth]{graphs/2_final}
 	}
 	\caption{System \eqref{eq:lotka_volterra_BerDaPMim} initialised with two indicator functions. Immediately both species start to grow due to the reaction terms while diffusing independently. In addition they remain segregated as proven by Theorem \ref{thm:segregation}.}
\label{fig:lotka_volterra_BerDaPMim_init_n_final}
\end{figure}

\begin{figure}[ht!]
  	\centering
  		\includegraphics[width=0.5\textwidth]{graphs/2_evol}
  	\caption{Evolution in time of system \eqref{eq:lotka_volterra_BerDaPMim}.}
 	\label{fig:lotka_volterra_BerDaPMim_intermediate}
 \end{figure}

In the second case, Figures \ref{fig:lotka_volterra_BerDaPMim_two_init_n_final} \& \ref{fig:lotka_volterra_BerDaPMim_two_intermediate}, we see the competition between both species clearly as one invades the other.

\begin{figure}[ht!]
 	\centering
 	\subfigure[Initial data.]{
 		\includegraphics[width=0.45\textwidth]{graphs/3_init}
 	}
 	\subfigure[Final time.]{
 		\includegraphics[width=0.45\textwidth]{graphs/3_final}
 	}
 	\caption{System \eqref{eq:lotka_volterra_BerDaPMim} is initialised with $\rho(x)=\mathds{1}_{[-\pi/3,\pi/3]} (1+\cos(3x))$ and $\eta(x)= 1-\rho(x)$. Although both species intermingle initially the first species, $\rho$, grows rapidly and pushes aside the second species, $\eta$. After some time it seems that $\rho$ has completely eliminated $\eta$ on its support.}
 \label{fig:lotka_volterra_BerDaPMim_two_init_n_final}
 \end{figure}

\begin{figure}[ht!]
  	\centering
  		\includegraphics[width=0.5\textwidth]{graphs/3_evol}
  	\caption{Evolution in time of system \eqref{eq:lotka_volterra_BerDaPMim}.}
 	\label{fig:lotka_volterra_BerDaPMim_two_intermediate}
 \end{figure}


\section{Introduction}
Following the development of intense, coherent laser-based sources of terahertz radiation \cite{hoffmann11}, the past decade has witnessed an increased interest in the use of this type of radiation to coherently control the properties of materials on the sub-picosecond time scale. Terahertz photons, with energies in the meV range, can drive nonlinear dynamics without significantly increasing the entropy of the system \cite{trigo08,liu12,daranciang12,mankowsky14,staub14,beaud14,kubacka14,dakovski15,bonetti16,
dean16,henighan16,grubel16,chen16}. In the field of condensed matter physics, the investigations of ultrafast dynamics driven by strong terahertz fields are frequently performed using terahertz-pump (usually in the 1 - 10 THz range, 300 to 30 $\mu$m in wavelength) and visible or near-infrared probing light (typically a sub-100 fs pulse).

To study the effects in the strong-field limit, the strength of the terahertz field can be locally enhanced exploiting near-field effects in meta-materials \cite{chen06,razzari11,werley12,liu12,zhang15,savoini16,kozina17}, which typically consist of micrometer-sized metallic structures deposited on the sample surface. However, since the area of the sample is often significantly smaller than the area of metallic structures in meta-materials, the reflectivity in the visible or near-infrared frequency range of the probe is dominated by the latter. As a consequence, it is extremely challenging to isolate the sample response, despite the enhancement provided by the meta-material. This problem can be mitigated by using dielectric and absorbing coatings, for instance  to enhance the magneto-optical activity in magnetic thin films, and to reduce the background reflections \cite{balasubramanian88,atkinson91,qureshi04,qureshi05,barman06,wang07}. This solution greatly boosts the signal up to a point where single nano-structures can be measured. The drawback of this approach is that it imposes constraints on the choice of layers underneath the target structure. This limitation can become crucial if these underlayers are utilized  to tune the important properties of the studied thin films. In this case, a more suitable solution is to deposit an anti-reflection (AR) coating only on the metal structures forming the meta-material, to minimize the reflection from those areas, which is the main factor affecting the strength of the measured signal. At the same time, the AR layer should not perturb the terahertz radiation that still needs to be enhanced by the metal layers.

In this work, we propose a simple, but until now unexplored, single-layer anti-reflection coating design that can be implemented on arbitrary meta-material structures comprising highly conducting and reflective metallic layers. The coating suppresses the near-infrared reflection typically utilized to probe the response of the sample, without noticeably affecting the terahertz radiation at much larger wavelengths. We have performed transfer matrix method calculations, as well as measurements of the reflectivity both in the near-infrared and terahertz range, to demonstrate the functionality of our design. We have also investigated, using time-domain finite element simulations, the near-field enhancement properties of a dipole antenna - a template for the terahertz meta-materials - covered with the anti-reflection coating. Finally, we experimentally measured the magneto-optical Kerr effect from a magnetic wire placed in the gap of the antenna.

\section{Anti-reflection coating design}
\begin{figure}[t]
\centering\includegraphics[width=0.8\textwidth]{figure1}
\caption{Design of the dipole antenna for terahertz near-field enhancement in the gap between two metallic electrodes, covered with an anti-reflection coating for near-infrared and visible radiation. A single-cycle of the terahertz field, with the suitable polarization for the optimal coupling to the antenna, is sketched. The pink arrows schematically show the working principle of the anti-reflection coating for a metal, where destructive interference (zig-zag arrows within the top layer) is combined with the dielectric losses to compensate for the forbidden transmission through the metallic electrodes (crossed-out arrows in the metal layer), as described in detail in the text.}
\label{fig1}
\end{figure}

Terahertz meta-materials can be formed by depositing metallic (typically gold) layers that can locally enhance the electromagnetic field of incident radiation. One of the simplest realization of such structure consists of two metallic strips separated by a small gap, i.e. a dipole antenna \cite{biagioni12}. For a suitable geometry of the antenna and polarization of the incident radiation, opposite charges can be induced by the electromagnetic field at the opposite edges of the gap, producing a strong local enhancement of electric field within the gap. Intuitively, but incorrectly, this charge motion is often attributed to the current driven by the electric field. However, the correct explanation is that the local electric field enhancement in the gap is caused by the screening of the \emph{magnetic} field, which induces a current flow in the metal, in the direction orthogonal to it (parallel to the electric field). This current flow, known as the eddy current, can penetrate within the skin depth of the material (75 nm at 1 THz for gold). In contrast, the \emph{electric} field component of the radiation is screened virtually instantaneously at the surface of the conductor by the charge redistribution, and at terahertz frequencies provides a negligible contribution to the net current flow in the bulk of the material, even in non-ideal metals.

In the standard AR coatings, designed to minimize the reflection from dielectric materials, one exploits the phenomenon of destructive interference of the waves reflected at the two interfaces, to cancel the total electric field that propagates in the backward direction. Since the energy of the electromagnetic wave is conserved, the transmission through the dielectric is maximized. However, this mechanism cannot be implemented for coatings on metals, since the wave cannot propagate through the metal, and hence the reflection cannot be eliminated.

For an AR coating to work for a metal, it is necessary to create destructive interference (to suppress Fresnel reflections), and to simultaneously absorb the radiation, as shown schematically in Fig.~\ref{fig1}. In other words, the dielectric layer needs to be sufficiently lossy in the visible/near-infrared region, so that the wave decays after multiple reflections at the interfaces. This idea was proposed decades ago by Hass et al. \cite{hass56}, who demonstrated that lossy double dielectric layers can suppress the reflectivity of aluminum and copper in the visible range, while maintaining high-reflectivity in the mid-infrared range, up to the wavelength of 10 $\mu$m. Moreover, they also highlighted the fact that absorption in single-layer AR coatings is necessary to reduce the high reflectance of metals in the visible range. In a later related work by Yoshida \cite{yoshida79}, a single-layer AR coating for metals was described mathematically. He first considered a non-absorbing dielectric layer with real refractive index $n_1>1$ and thickness $d_1$, deposited on top of a metallic substrate characterized by $\tilde{n}_2=n_2+ik_2$. The reflectance $R$ for the monochromatic light with wavelength $\lambda$, impinging at normal incidence on the three-layers stack composed of the air ($n_0=1$), the non-absorbing dielectric coating, and the metal substrate, is given by \cite{yoshida79}
\begin{equation}
R~=~\left| \dfrac{r_{01}+r_{12}\exp\left(2i\delta_1\right)}{1+r_{01}r_{12}\exp\left(2i\delta_1\right)} \right|^2,
\end{equation}
where $r_{01}=\left(1-n_1\right)/\left(1+n_1\right)$ is the Fresnel reflection coefficient for the air-dielectric interface, $r_{12}=\left(n_1-\tilde{n}_2\right)/\left(n_1+\tilde{n}_2\right)$ is the Fresnel reflection coefficient for the dielectric-metal interface, and $\delta_1=2\pi n_1d_1/\lambda$. The reflectance reaches a minimum when \cite{park64},
\begin{equation}
n_1d_1~=~\dfrac{\lambda}{2}\left[(m+1)-\dfrac{\alpha_{12}}{2\pi}\right],
\label{optpath}
\end{equation}
where $m$ is an integer value, and
\begin{align}
R_{min}~&=~\left(\dfrac{r_{01}+\rho_{12}}{1+r_{01}\rho_{12}}\right)^2, \\
{\rho_{12}}~&=~\left|r_{12}\right|~=~\left[\dfrac{\left(n_1-n_2\right)^2+{k_2}^2}{\left(n_1+n_2\right)^2+{k_2}^2}\right]^{\frac{1}{2}}, \\
\alpha_{12}~&=~\mathrm{Arg}\left(r_{12}\right)~=~\arctan\left(\dfrac{-2n_1k_2}{{n_1}^2-{n_2}^2-{k_2}^2}\right). \label{alpha_12}
\end{align}
The minimum reflectance $R_{min}$ is zero when $\rho_{12}=|r_{01}|$, giving
\begin{equation}
n_1~=~\left(n_2+\dfrac{{k_2}^2}{n_2-1}\right)^{\frac{1}{2}}.
\label{n1_noabs}
\end{equation}
If $k_2=0$, which corresponds to a dielectric coating on top of a dielectric substrate, Eq.~\eqref{n1_noabs} gives $n_1={n_2}^{1/2}$. Since $n_2>n_1$, the solution for $\alpha_{12}=\arctan\left(0\right)$ in Eq.~\eqref{alpha_12} should be $\alpha_{12}=\pi$, because of the $\pi$ phase shift introduced by the dielectric coating-dielectric substrate interface in this case. Thus, Eq.~\eqref{optpath} gives $n_1d_1=\lambda/4$ for $m=0$, defining a \emph{quarter-wave} coating, in which the reflectance is minimized by the destructive interference in the coating layer. Moreover, Eq.~\eqref{n1_noabs} imposes a constraint on the values of $n_2$ and $k_2$, since $n_1>1$. The effect of this constraint is that, according to Yoshida \cite{yoshida79}, ``zero reflection cannot be achieved with a single dielectric film coating for metals'' with large extinction coefficient $k\gtrapprox3$, such as silver and gold. However, in this case, zero reflection can be obtained by allowing the dielectric coating to be slightly absorbing.

In the following, we experimentally confirm that the reflection from gold, and hence from any good metal, can be suppressed by using a single layer of sputtered amorphous silicon ($\alpha$-Si). In the visible/near-infrared range, a thin $\alpha$-Si film acts as a dielectric with a relatively large imaginary part of the refractive index, since the electronic states are not characterized by well-defined momentum, enhancing the radiation absorption in $\alpha$-Si as compared to its crystalline form \cite{adachi00}. On the other hand, low absorption in the terahertz range  ($\lambda\sim100$ $\mu$m), and the small thickness compared to the radiation wavelength, make these layers practically invisible, thus maintaining the high-reflectivity characteristics of gold in this range.

\begin{figure}[t]
\centering\includegraphics[width=8cm]{figure2}
\caption{ (Solid curves: Calculated reflectance at the wavelength of 800 nm for the Air/$\alpha$-Si/Au/Si(substrate)/Air multilayer, as a function of the $\alpha$-Si thickness, for two different Au thicknesses at normal incidence. Dashed curve: Calculated reflectance at a wavelength of 800 nm for an ideal dielectric on top of a 100 nm Au layer, characterized by $n=3.9$ and zero imaginary part of the refractive index.}
\label{fig2}
\end{figure}

We first used the transfer matrix method (TMM) \cite{bornwolf} to simulate the feasibility of this approach. We simulated the Air/$\alpha$-Si/Au/Si(substrate)/Air multilayer, where the outermost Air layers were semi-infinite, and the substrate was 500 $\mu$m thick. The radiation was assumed to be monochromatic with a wavelength of 800 nm, the typical center-wavelength of a Ti:sapphire laser, at a normal incidence to the multilayer stack. We used the refractive indices $n_{\rm Air}=1$, $n_{\alpha-\rm{Si}}\approx 3.90 + 0.11j$, $n_{\rm Au}\approx 0.15 + 4.91j$, and $n_{\rm Si}\approx 3.681 + 0.005j$ \cite{refrindex}.

In Fig.~\ref{fig2} we plot the reflectance of the stack as a function of the $\alpha$-Si thickness, for two gold layers with different thickness. For thin gold (20 nm), part of the radiation can be transmitted into the substrate, and $\approx30$ nm of amorphous silicon on top of it can efficiently suppress the reflectivity. For thick gold (100 nm), enough to prevent any transmission, a thicker amorphous silicon layer ($\approx230$ nm) is needed to achieve the same suppression.

In the same figure, we also plot the reflectivity (dashed lines) of a fictitious dielectric layer with zero imaginary part of the refractive index, and its magnitude equal to that of the the amorphous silicon, representing a conventional dielectric with negligible losses. It is evident that such a layer on top of a 100-nm thick gold layer cannot efficiently suppress the reflectivity, demonstrating that the cumulative losses after multiple reflections are necessary to realize the anti-reflection configuration.

We note that a single $\alpha$-Si anti-reflection coating remains efficient over a broad range of incidence angles. We used TMM to check that, when varying the incidence angle from 0 to 37.5 degrees, the optimal thickness for the $\alpha$-Si layer varies by less than 2\%. Furthermore, at the optimal thickness, the 800 nm reflectance remains below 0.05 (an acceptable value for the coating to properly work), at angles of incidence as high as 50 degrees. This can be understood as a consequence of the large refractive index of silicon, which causes the electromagnetic wave to strongly refract when entering the AR layer. As a result, the optical path in the silicon layer noticeably increases only at very large angles of incidence.

\section{Experimental and numerical verification}
\begin{figure}[t]
\centering\includegraphics[width=8cm]{figure3}
\caption{Experimental (symbols) and calculated (line) reflectance for a $\alpha$-Si/Au/Si(substrate) sample based on a 20 nm-thick Au layer, at wavelengths of 800 nm (magenta) and 300 $\mu$m  (black), the latter corresponding to the radiation frequency of 1 THz.}
\label{fig3}
\end{figure}

In Fig.~\ref{fig3}, we plot the calculated and the measured reflectance for several Air/$\alpha$-Si($t$)/Au(20 nm)/Si(substrate)/Air multilayers, as a function of $t$, both for 800 nm and for the terahertz radiation impinging on the sample at 10 degrees incidence. The 800 nm radiation was produced by the Ti:sapphire-based regenerative amplifier (Coherent Legend) in  40 fs pulses, with the 30 nm FWHM bandwidth around the center 800 nm wavelength, as measured by a grating spectrometer. The reflectance at 800 nm was measured directly using a photodiode. The signal was scaled using the known reflectance value of a commercial gold mirror. The reflectance $R$ of the terahertz radiation, generated by optical rectification in a OH1 organic crystal \cite{oh1}, was determine from the measured transmittance $T$, using $R=1-A-T$, where the absorption $A$ was calculated based on the TMM. The transmittance was taken to be proportional to the square of the normalized amplitude of the maximum electro-optical sampling signal in a 100 $\mu$m thick, 110-cut GaP crystal.

The excellent agreement between the data and the calculations directly demonstrates the functionality of our design in suppressing the near-infrared reflectivity, with the appropriate thickness of $\alpha$-Si, 27 nm and 127 nm in the studied case of 800 nm radiation. We emphasize that our experiment demonstrates efficient suppression of the broadband 40 fs pulses of 800 nm radiation. This is not surprising, considering that the bandwidth to carrier ratio is less than 4\%. This result confirms the suitability of our design for the conventional ultrafast experiments. On the other hand, the terahertz reflectivity is unchanged by the $\alpha$-Si layer, suggesting that the terahertz near-field enhancement is also likely unaffected the silicon layer. However, since the measured reflectivity is a far-field property, and near-field properties are in general very sensitive to interface effects, one needs to perform a more detailed investigation of the possible effects of the anti-reflection coating on the terahertz radiation in the near-field regime.

\begin{figure}[t]
\centering\includegraphics[width=8cm]{figure4}
\caption{(a) Frequency-domain, finite element analysis of the enhancement map for a monochromatic electromagnetic field with frequency $f=1$ THz incident on two gold plates separated by a gap. (b) Time-domain finite element simulations of the $x$-component of the electric field for a single-cycle (broadband) terahertz field at the center of the gap, without  Au plates (solid black curve), with Au plates (solid gray curve) and with $\alpha$-Si/Au plates (filled gray dots). In all the calculations, the electric field of the incident radiation is polarized along the $x-$axis, and the propagation direction is along the $z-$axis, normal to the sample plane.}
\label{fig4}
\end{figure}

To analyze the near-field effects of the coating, we have performed finite-element numerical calculations using COMSOL Multiphysics\textregistered software~\cite{comsol}. In Fig.~\ref{fig4}(a), we plot the electric field enhancement at a frequency of 1 THz, for a set of two infinitely long, 65 $\mu$m wide, 20 nm thick gold plates separated by a gap of 2 $\mu$m. The terahertz electric field is applied along the $x-$axis in this Figure. The field enhancement is computed by dividing the electric field value in the gap region by the electric field value at the air/silicon interface, in the absence of the gold plates. In Fig.~\ref{fig4}(b), we plot time-dependence of the $x$-component of the terahertz electric field in the middle of the gap ($x=0$), and at $z=10$ nm above the silicon substrate. The simulation used the experimental time profile of the impinging terahertz field,  measured by electro-optical sampling, with the peak value of $\approx300$ kV/cm.

For the bare silicon-air interface, the terahertz field is reduced to approximately half of its free-space magnitude, consistent with the relative amplitude of the transmitted wave at an air/silicon interface computed as $t=2/(n+1)$, with $n\approx3.4$ \cite{footnote3}. The presence of the gold plates introduces a slight temporal shift, and enhances the amplitude of the terahertz field to more than 500 kV/cm, consistent with the $\approx4$ times enhancement observed in the frequency domain simulations of Fig.~\ref{fig4}(a) \cite{footnote2}. Most importantly, the addition of the $\alpha$-Si layer on top of the gold plates does not noticeably affect the field, confirming our intuitive conclusions based on the negligible effect of the $\alpha$-Si layer on the terahertz reflectivity.

To test the functionality of the AR coating in a practical configuration, we measured the polar magneto-optical Kerr effect (MOKE) \cite{qiu00} from the 3 nm-thick CoNi film patterned into a 1 $\mu$m-wide, 100 $\mu$m-long wire. The CoNi stack is formed by a Ta(2)|Cu(2)|[Co(0.2)|Ni(1)]$_{3}$|Ni(0.5)|Ta(3) multilayer (thicknesses in nm) with perpendicular magnetic anisotropy. The wire is located in the 2 $\mu$m-wide gap between two 100 $\mu$m long and 65 $\mu$m wide gold plates, coated with 27 nm of $\alpha$-Si. By analyzing the magneto-optic response, we can unambiguously identify the signal coming from the embedded CoNi wire, with no contribution from the non-magnetic electrodes. As the MOKE signal typically results in a tiny intensity variation on top of a large background, this specific system implementation also demonstrates the general suitability of our design for the detection of small effects other than magneto-optical ones.

\begin{figure}[t]
\centering\includegraphics[width=8cm]{figure5}
\caption{Polar Kerr ellipticity as a function of the magnetic field applied orthogonal to the sample plane, for the 800 nm and 550 nm wavelengths of the probing light. Symbols: average of 25 hysteresis loops for a  1 $\mu$m-wide, 100 $\mu$m-long CoNi wire. Solid curves:  average of 4 hysteresis loops hysteresis loops for a 100 $\mu$m $\times$ 100 $\mu$m CoNi square, at  the same wavelengths. Inset: Zoom in on the 550 nm wavelength hysteresis loop for the CoNi wire.}
\label{fig5}
\end{figure}

The polar MOKE loops from the embedded wire are plotted in the main panel of Fig.~\ref{fig5} for two different wavelengths of the probing radiation, 550 nm and 800 nm. At 800 nm, the AR coating optimized for this wavelength is expected to completely suppress the reflectivity of the gold electrodes, while at 550 nm, a substantial reflection from the metallic pads is expected. Using radiation of different wavelengths is geometrically equivalent to studying samples with different AR coating thickness, with the advantage that the very same sample can be used and the wavelength can be tuned very accurately.

The plotted MOKE signals reflect the change in the polarization ellipticity of the probing light, determined with a suitable polarization analyzing system. In these measurements, we utilized a polarization modulation technique \cite{vavassori00} to provide the sensitivity necessary for testing the effectiveness of the proposed AR coating. In this setup, the polarization of the incident light was modulated at the frequency $\omega$, while both the total measured intensity $I_0$ and its variation $I_\omega$ at the modulation frequency
were simultaneously recorded. The polarization ellipticity was determined from $I_\omega$ normalized by the total intensity $I_0$ reflected from the sample. While the variations of $I_\omega$ were affected only by the magnetic structure, the magnitude of $I_0$ is determined by the whole probed area, including the gold plates.

The data plotted with symbols in Fig.~\ref{fig5} clearly show that the AR coating significantly enhances the signal-to-background ratio, resulting in a more than a ten-fold increase of the realtive amplitude of the loop at the design wavelength. We have checked that the increase in the relative signal is not caused by the difference between the magneto-optical constants of CoNi between 800 and 550 nm wavelengths, by measuring the MOKE signal from a 100 $\mu$m $\times$ 100 $\mu$m CoNi square of 3 nm thickness, with no gold electrodes surrounding the structure. The resulting hysteresis loops are shown with solid curves in the the same Fig.~\ref{fig5}.

To quantitatively analyze our observations, we note that if the areas and the reflectivity of different reflecting regions are known, one can predict the difference in the total measured Kerr ellipticity $\epsilon_K$ between the CoNi square larger than the probing spot, and the wire-shaped sample, according to \cite{qureshi05}
\begin{equation}
\dfrac{\epsilon_{K, \rm wire}}{\epsilon_{K, \rm square}} = \dfrac{A_{\rm wire}}{A_{\rm wire} + A_{\rm Si}\dfrac{R_{\rm Si}}{R_{\rm CoNi}} + A_{\rm CoatedAu}\dfrac{R_{\rm CoatedAu}}{R_{\rm CoNi}}},
\label{eq_model}
\end{equation}
where $A_m$ is the total area occupied by a certain material $m$ illuminated by the laser beam, and $R_m$ the corresponding reflectivity that can be measured experimentally or calculated using Fresnel equations.

Table~\ref{table1} summarizes the relationship between the ellipticity of the wire and the ellipticity of the square, assuming that the probing light is focused in to a uniform circular spot with diameter $\phi=75$ $\mu$m, and the various probed areas are $A_{\rm wire}=A_{\rm Si}\approx\phi h$ $(h=1~\mu\rm m)$, $A_{\rm CoatedAu}\approx\pi(\phi/2)^2-2\phi h$. The area occupied by the wire is therefore about 2\% of the total area. Indeed, for the wavelength of 550 nm, at which the Au reflectivity is not suppressed by the AR coating, the ellipticity signal for the wire is about 1.6\% of that for the large square, both theoretically and experimentally. In contrast, for the 800 nm wavelength, we expect and observe an increase of this ratio by an order of magnitude. The deviation between the theoretically expected value (24\%) and the experimental value (19\%) can be explained by a combination of a few-nm uncertainty in the deposited material thickness, deviations of the optical properties of different layers from their nominal values, and by the effects of the nanowire and gold electrode edges, whose scattering properties were not taken into account in the calculations reported in Table~\ref{table1}.

\begin{table}[t]
\caption{Summary of the ellipticity ratio between a CoNi wire and a CoNi square calculated according to Eq.~(\ref{eq_model}).}
\begin{center}
    \begin{tabular}{| l | c | c | c | c |}
    \hline
     &  $R_{\rm CoNi}$ & $R_{\rm Si}$ & $R_{\rm CoatedAu}$ & $\epsilon_{K, \rm wire}/\epsilon_{K, \rm square}$\\ \hline \hline
    550 nm (theor.) & 0.49 & 0.40 & 0.50 & $\mathbf{0.017}$\\ \hline
    550 nm (exp.) & 0.46 & 0.40 & 0.50 & $\mathbf{0.016}$\\ \hline\hline
    800 nm (theor.) & 0.47 & 0.36 & 0.020 & $\mathbf{0.24}$\\ \hline
    800 nm (exp.) & 0.49 & 0.34 & 0.031 & $\mathbf{0.19}$\\ \hline
    \end{tabular}
\end{center}
\label{table1}
\end{table}

\section{Conclusion}
In summary, we have designed and experimentally demonstrated an anti-reflection coating for highly reflective metals, which are typically utilized in the fabrication of terahertz meta-materials. The anti-reflection coating can efficiently suppress the reflection of light in the visible and infrared ranges, typically used in the studies of ultrafast phenomena by  pump-probe techniques. At the same time, the coating does not perturb the propagation of terahertz radiation, and does not affect the near-field enhancement in the meta-materials. Our results are expected to open a path for time-resolved experiments aimed at probing the ultrafast dynamics driven in nano-scale structures by strong terahertz fields, by using table-top femtosecond near-infrared laser sources.

\section*{Funding}
Swedish Research Council (Grant E0635001); Marie Sk\l{}odowska Curie Actions, Cofund (Project INCA 600398s); European Research Council (ERC) (Starting Grant 715452 ``MAGNETIC-SPEED-LIMIT''); H2020 European Union Programme, FETOPEN-2016-2017 (Project 737093 ``FEMTOTERABYTE'');  Spanish Ministry of Economy, Industry and Competitiveness under the Maria de Maeztu Units of Excellence Programme - MDM-2016-0618; US National Science Foundation (NSF) (Grant ECCS-1503878, Grant DMR-1504449); U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Award No. 2015-SLAC-100238-Funding.

\end{document}