## Abstract

We consider systems of reaction–diffusion equations as gradient systems with respect to an entropy functional and a dissipation metric given in terms of a so-called Onsager operator, which is a sum of a diffusion part of Wasserstein type and a reaction part. We provide methods for establishing geodesic *λ*-convexity of the entropy functional by purely differential methods, thus circumventing arguments from mass transportation. Finally, several examples, including a drift–diffusion system, provide a survey on the applicability of the theory.

## 1. Introduction

In several papers by Otto (see [1–3]), certain scalar diffusion problems were formulated as gradient flows with respect to the free energy or relative entropy and the Wasserstein metric. In [4], it was shown that general reaction–diffusion systems, with reactions satisfying the detailed balance condition, can be written as a gradient system with respect to the relative entropy. The associated dissipation metric is most easily modelled by considering its inverse , called the Onsager operator, as the sum of a diffusion part and a reaction part. The diffusion part is a vector-valued version of the Wasserstein metric used for the scalar Fokker–Planck equation in [1,3], namely where is the vector of the densities of the species and is the associated thermodynamical driving force, also called the vector of chemical potentials. Here, is a general density-dependent mobility tensor, which is symmetric and positive semidefinite. Using a symmetric and positive semidefinite matrix , we obtain the full Onsager operator and together with the entropy functional giving the thermodynamical driving force we find the gradient system which leads to a large class of reaction–diffusion systems.

The focus of this work is to provide conditions on the system such that the driving functional is geodesically *λ*-convex with respect to the metric . This means that is *λ*-convex for all arc-length parametrized geodesic curves *γ*:[*s*_{a},*s*_{b}]→*X*, i.e.
1.1
for all *θ*∈[0,1] and *s*_{0},*s*_{1}∈[*s*_{a},*s*_{b}], where *s*_{θ}=(1−*θ*)*s*_{0}+*θs*_{1}. The study of geodesic *λ*-convexity for scalar drift–diffusion equations given by
1.2
where *u*↦*E*(*u*) is convex and *u*↦*μ*(*u*) is concave, was initiated in [5] and has been studied extensively since then (e.g. [6–9]). An essential tool in this theory is the characterization of the geodesic curves in terms of mass transportation and the optimal transport problem of the Monge–Kantorovich type. Presently, such a method is not available for systems of equations or for scalar equations with reaction terms, which destroy the conservation of mass. Instead, our work relies on a differential characterization of geodesic *λ*-convexity developed in [10].

Thus, after an introduction of gradient structures for reaction–diffusion systems in §2 we provide an abstract version of the theory developed in [10]. We mainly address the abstract framework and present the estimates to obtain concrete convexity properties, while the functional analytic aspects as well as the full framework in terms of complete metric spaces are postponed to subsequent work. Moreover, we assume that our evolutionary system
1.3
generates a suitable smooth local semiflow on a scale of Banach spaces *Z*⊂*Y* ⊂*H* with dense embeddings; see §3 for the details. The main characterization involves the quadratic form
which can be seen as the form induced by the metric Hessian of . The main result is that is geodesically *λ*-convex if the estimate
1.4
holds for all suitable ** u** and

**. Our proof is a straightforward generalization of the approach in [10] that is based on the evolutionary variational inequality (EVI)**

*ξ*_{λ}given by 1.5 where and is the distance induced by . The idea is to show that (1.3) and (1.4) imply (1.5) and finally deduce (1.1).

Condition (1.4) is closely related to the Bakry–Émery conditions [11,12] and provides a strengthened version of the classical entropy–dissipation estimate (see 13 for an application to reaction–diffusion systems). In fact, introducing the quantity and the solutions ** u** of (1.3) satisfy
By (1.4), there exists

*α*≥

*λ*such that for all

**. Assuming**

*u**α*>0, in [14] the decay estimates are used to derive convergence for . We discuss further useful properties of the geodesic

*λ*-convexity in §3

*c*, also if

*λ*<0.

The main part of this work surveys possible applications of the abstract theory; see §4. We emphasize that geodesic convexity is a strong structural property of a gradient system that is rather difficult to achieve, in particular with respect to distances that are associated with the Wasserstein metric. Our examples show that there are at least some non-trivial reaction–diffusion equations or systems that satisfy this beautiful property. First, we discuss simple reaction kinetics satisfying the detailed balance conditions, i.e. ordinary differential equation systems in the form This includes the case of general reversible Markov chains , where is a stochastic generator (see also [15–17]).

In the following sections, we treat partial differential equations or systems where estimate (1.4) heavily relies on a well-chosen sequence of integrations by parts, where the occurring boundary integrals need to be taken care of. We use the fact that, for convex domains *Ω* and functions *ξ*∈H^{2}(*Ω*) with ∇*ξ*⋅*ν*=0 on ∂*Ω*, we have ∇(|∇*ξ*|^{2})⋅*ν*≤0 on ∂*Ω*; see proposition 4.2. Section 4*b* gives a lower bound for the geodesic convexity of with respect to the inhomogeneous Wasserstein metric induced by , where , thus generalizing results in [18]. Theorem 4.3 provides a new result of geodesic convexity for and from (1.2), where the concave mobility *u*↦*μ*(*u*) is allowed to be decreasing, i.e. *μ*′(*u*)<0, thus complementing results in [9].

Sections 4*d*,*e* discuss problems with reactions, namely
The first case with *f*(*u*)=*k*(1−*u*) gives geodesic *λ*-convexity with , whereas the second case gives geodesic 0-convexity. In §4*f*, a one-dimensional drift–diffusion system with charged species is considered, where the nonlinear coupling occurs via the electrostatic potential. The final example discusses cross-diffusion of the Stefan–Maxwell type for ** u**=(

*u*

_{1},…,

*u*

_{I}) under the size-exclusion condition

*u*

_{1}+⋯+

*u*

_{I}≡1.

There are further interesting applications of gradient flows where methods based on geodesic convexity can be employed, even though the system under investigation may not be geodesically *λ*-convex, e.g. the fourth-order problems studied in [19–21]. Possible applications to viscoelasticity are discussed in [22]. In [23], a diffusion equation with Dirichlet boundary conditions, which leads to absorption, is investigated.

## 2. Gradient structures for reaction–diffusion systems

In this section, we give some general background on gradient and Onsager systems. All our arguments are formal and will be made precise in the following sections.

A gradient system is a triple where *X* is the state space containing the states ** u**∈

*X*. For simplicity, we assume that

*X*is a reflexive Banach space with dual

*X**. The driving functional is assumed to be differentiable (in a suitable way) such that the potential restoring force is given by . The third ingredient is a metric tensor , i.e. is linear, symmetric and positive (semi-)definite.

The gradient flow associated with is the (abstract) force balance
2.1
where we recall that the ‘gradient’ of the functional is an element of *X* (in contrast to the differential ) and is calculated in terms of . The left-hand side of (2.1) is an abstract force balance, as can be seen as a viscous force arising from the motion of ** u**. We call the linear, symmetric and positive semidefinite operator the Onsager operator and the corresponding triple the Onsager system.

As we are mainly interested in reaction–diffusion systems we consider densities of diffusive species *X*_{1},…,*X*_{I}. The driving functional of the evolution is of the form
where is a bounded domain and *E* is a sufficiently smooth energy density. It was shown in [4] that for a wide class of reaction–diffusion systems gradient or Onsager structures can be specified. The central point is that in the Onsager form we have an additive splitting of the Onsager operator into a diffusive and a reaction part, namely , where ** ξ** is the thermodynamically conjugated force being dual to the rate . We define the diffusion part , following the Wasserstein approach to diffusion introduced by Otto in [1,3], and the reaction part as follows:
2.2
Here, and are symmetric, positive semidefinite tensors of order four and two, respectively. The evolution is described by
2.3
subjected to the no-flux boundary condition for

*x*∈∂

*Ω*. The symmetry of the tensor allows us to define the dual dissipation potential We call

*Ψ** the dual dissipation potential as it is the Legendre transform of the dissipation potential , i.e. we have In the following sections, we will specify the structure of the functional and the Onsager operator and present some illustrative examples.

### (a) Chemical reaction kinetics of mass-action type

Pure chemical reaction systems are ordinary differential equation systems , where often the right-hand side is written in terms of polynomials associated with the reaction kinetics. It was observed in [4] that under the assumption of detailed balance (also called reversibility) such systems have a gradient structure with the relative entropy
as the driving functional, where *w*_{i}>0 denotes fixed reference densities.

We assume that there are *R* reactions of mass-action type (e.g. [24–26]) between the species *X*_{1},…,*X*_{I} denoted by
where and are the backward and forward reaction rates, respectively, and the vectors contain the stoichiometric coefficients of the *r*th reaction. The associated reaction system for the densities (in a spatially homogeneous system, where diffusion can be neglected) reads
2.4
where we use the monomial notation .

The main assumption to obtain a gradient structure is that of *detailed balance*, which means that there exists a reference density vector ** w** such that all

*R*reactions are balanced individually, namely 2.5 Here, we have used the freedom to allow for reaction coefficients to depend on the densities. As in [4], we now define the Onsager matrix 2.6 and find that reaction system (2.4) takes the form 2.7 This follows easily by using the definition of

*Λ*and the rules for logarithms, namely .

### (b) Coupling diffusion and reaction

Now we consider coupled reaction–diffusion systems. The driving functional for the evolution is the total relative entropy . The Onsager operator is given by the sum with and as in (2.2). Hence, with given in (2.6) the coupled system reads where .

As an example of a reaction–diffusion system, we consider the quaternary system studied in [27,28], namely the evolution of a mixture of diffusive species *X*_{1},*X*_{2},*X*_{3} and *X*_{4} in a bounded domain *Ω* undergoing a reversible reaction of the type
2.8
For the density vector ** u**=(

*u*

_{1},

*u*

_{2},

*u*

_{3},

*u*

_{4}), we introduce the free energy functional For simplicity, we assume that

*k*

^{fw}=

*k*

^{bw}=1 and can take

*w*

_{i}=1. We have the stoichiometric vectors

**=(1,1,0,0),**

*α***=(0,0,1,1), and thus The tensor and the corresponding Onsager operator leads to the reaction–diffusion system**

*β*In fact, many reaction–diffusion systems studied in the literature (including semiconductor models involving an elliptic equation for the electrostatic potential), e.g. [29,30,31,32,33], have the structure developed above. Except for recent work [4,34,35]; 13, the gradient structure was not displayed and exploited explicitly, only the Liapunov property of the free energy was used for deriving *a priori*estimates.

### (c) Non-isothermal coupled systems

We now extend the system from §2*b* and consider the non-isothermal case when the temperature of the system is not constant but an independent field coupled to the densities ** u**. For such systems, we have two functionals, namely the total energy, which is preserved during the evolution of the system, and the total entropy, which acts as the driving force. Instead of using the temperature as an additional variable, it has certain advantages to use the internal energy as a free variable (see [35,36] for details). Thus, the functionals are
Now the Gibbs relation leads to the definition of the temperature as

*θ*=

*Θ*(

**,**

*u**e*):=1/∂

_{e}

*S*(

**,**

*u**e*), where the relation ∂

_{e}

*S*(

**,**

*u**e*)>0 is imposed.

The major advantage of the formulation in terms of (** u**,

*e*) is that the energy conservation is a linear constraint. Moreover, following [37], it is reasonable to assume that

*S*is a concave function in (

**,**

*u**e*). The equations can be written in (

**,**

*u**e*) using the Onsager operator where the fourth-order tensor has the block structure where , and are symmetric and positive semidefinite. The evolution equations for (

**,**

*u**e*) take the form and This form has the major advantage that we can read off ‘parabolicity’ in the sense of Petrovsky [38], Sect. VII.8 for the full-coupled system by assuming that is positive definite and D

^{2}

*S*is negative definite. Hence, local existence results can be obtained from [39].

Moreover, we are able to postulate suitable strongly coupled models by assuming that *S* has the form
2.9
where ** w**(

*e*) is the vector of reference densities in the detailed balance condition (2.5), which may now depend on the internal energy (i.e. on the temperature).

### (d) Drift–reaction–diffusion equations

We close this section by considering a drift–diffusion system coming from the theory of semiconductor devices. More precisely, we treat the simplest semiconductor model, namely the van Roosbroeck system. One additionally needs to take into account that the electric charge of the species generates an electric potential, whose electric field creates drift forces proportional to the charges of the species. We recite here briefly the results of [4], Sect. 4 and refer to the latter for the full discussion. Moreover, we refer to [34] for drift–diffusion systems exhibiting bulk–interface interaction.

The system’s state is described by the electron and hole densities and , respectively. The charged species generate an electrostatic potential *ϕ*_{n,p} being the unique solution of the linear potential equation
2.10a
where is a given doping profile, and *q*_{n}=−1 and *q*_{p}=1 are the charge numbers with opposite signs. The evolution of the densities *n*,*p* is governed by diffusion, drift according to the electric field ∇*ϕ*_{n,p}, and recombination according to simple creation–annihilation reactions for electron–hole pairs, namely
With mobilities *μ*_{n}(*n*,*p*),*μ*_{p}(*n*,*p*)>0 and reaction rate *κ*(*n*,*p*)>0, the drift–diffusion system reads
2.10b
For establishing a gradient structure, we define the functional as the sum of electrostatic and free energy
The thermodynamic conjugated forces, also called quasi-Fermi potentials, read and . Here, we used that *ϕ*_{n,p} solves (2.10a) and depends linearly on *n* and *p*. The Onsager operator takes the form
Thus, again we have two Wasserstein terms for the electrochemical potentials coupled with a reaction term. We immediately find that for *q*_{n}=−*q*_{p} (opposite charges of the electron and holes) it holds that . This means that the total charge is a conserved quantity, i.e. . Moreover, using that
we see that is the desired Onsager structure of van Roosbroeck system (2.10).

A similar gradient system with only one species was considered in [40] It is a gradient system for the energy and the Wasserstein operator .

### (e) On the metric induced by reaction and diffusion

As we have seen above, it is most natural to model reaction–diffusion systems in terms of the Onsager operator. Hence, we will formulate the convexity conditions in terms of and the vector field . However, from the mathematical point of view the metric and the induced distance are important as well. Following the famous Benamou–Brenier formulation [41], we can characterize our in a similar fashion
2.11
In particular, concavity of the tensors and (i.e. for all ** ξ** the mapping is concave) we find that is convex, which can be used to establish the existence of geodesic curves.

## 3. Geodesically *λ*-convex gradient systems

### (a) Abstract set-up

In this section, we provide an abstract formulation such that the theory of [10] can be applied to general systems , in particular to systems of partial differential equations, where is allowed to be a partial differential operator as well. The main point of [10] is that it is sufficient to establish the geodesic *λ*-convexity of on a dense set, where all the calculations on functions can be done rigorously. Then the abstract theory allows us to extend the geodesic *λ*-convexity of to the closure of the domain of .

We consider a set which is a closed subset of a Banach space *X*, e.g. vectors of Radon measures. For the smooth solutions and their velocities, we need smaller spaces
with dense and continuous embeddings. For *u*∈*Y* , the norm induced by the metric will be equivalent to that of the Hilbert space *H*, for which we assume
We assume that open and connected sets and exist such that
We consider a gradient system satisfying
3.1
Thus, the evolution reads
where, having in mind partial differential equations, we assume the smoothness of the vector field
3.2
which is what one would obtain composing the smoothness of and .

### (b) Geodesic curves and geodesic *λ*-convexity

The metric tensor generates a distance in the usual way: For *u*_{0},*u*_{1}∈*X*, we define the set of connecting curves via
This allows us to define the distance as follows:
3.3
Here, *γ*′ denotes the derivative with respect to the arc-length parameter *s*, and
Clearly, is symmetric and satisfies the triangle inequality. We assume positivity, i.e.
3.4
Thus, we may consider also the metric gradient system in the sense of [7]. We refer to this work or to [9] for distances in more general cases. As in any metric space , a geodesic curve connecting *u*_{0} and *u*_{1} is a curve *γ*∈** C**(

*u*

_{0},

*u*

_{1}) satisfying 3.5

### Remark 3.1

If is a convex subset of *Y* ⊂*X* and is concave for all *η*, then is (jointly) convex on . As a consequence the functional in (3.3) and hence is convex as well.

### Remark 3.2

In only a very few cases can be calculated explicitly, all relying on the Wasserstein distance *d*_{Wass} (see [7,42]). For constants *μ*>0 and *κ*≥0, consider the operator , which is affine in *u*. For *κ*=0 we have, on the set *X*={*u*∈ M(*Ω*) | *u*≥0} of non-negative Radon measures, the distance
For *κ*=0, the Onsager operator is mass preserving, hence *X* decomposes into the components *X*_{α}={*u*∈*X* | vol(*u*)=*α*}. For *μ*=0, there is no spatial interaction, and we find the explicit formula . For *μ*,*κ*>0, there are geodesic curves between all points of *X*, and we conjecture the formula
This and other characterizations of reaction–diffusion distances will be investigated in subsequent work.

For a given , a functional is called *geodesically λ-convex* with respect to the metric if for all arc-length parametrized geodesics

*γ*:[

*s*

_{a},

*s*

_{b}]→

*X*the map is

*λ*-convex, i.e. 3.6 for all

*θ*∈[0,1] and

*s*

_{0},

*s*

_{1}∈[

*s*

_{a},

*s*

_{b}], where

*s*

_{θ}=(1−

*θ*)

*s*

_{0}+

*θs*

_{1}.

### (c) Properties of geodesically *λ*-convex gradient flows

In this section, we collect some useful properties of geodesically *λ*-convex systems. We refer to [7] for the full discussion. First, we have a Lipschitz continuous dependence of the solutions *u*_{j}, *j*=1,2, on the initial data (see also [10]), namely
3.7
In particular, for *λ*≥0 we have a contraction semigroup. If *λ*>0, we obtain exponential decay towards the unique equilibrium state *u*_{*}, i.e.
Second, the time-continuous solutions can be well approximated by interpolants obtained by incremental minimizations: fixing a time step *τ*>0 we define iteratively
For geodesically *λ*-convex functionals , the minimizers are unique for any *τ*∈ ]0,*τ*_{0}[ if 1/*τ*_{0}+*λ*≥0. Moreover, if *u* is the time-continuous solution with *u*(0)=*u*_{0} and if is the left-continuous piecewise constant interpolant of , then
(see [7], theorems 4.0.9+10), where *λ*_{τ}=*λ* for *λ*<0 and for *λ*>0.

Finally, it was shown in [10], proposition 3.1 that for geodesically *λ*-convex functionals the solutions of the (differential) gradient flow (2.1) satisfy a purely metric formulation in terms of the evolutionary variational inequality (EVI_{λ})
where for a function we set . This differential form is equivalent to the integrated form of (EVI_{λ}) given by
In particular, the solutions of (EVI_{λ}) satisfy the uniform regularization bound
Moreover, the solutions are uniformly continuous in time

### (d) Completion of smooth gradient flows

In addition to (3.1) and (3.2), we now assume that ( generates a global semiflow in the form with a semigroup , i.e. The assumptions on the semiflow are 3.8 In particular, this implies that and satisfy 3.9 We define the functionals and via and obtain the following formulae.

### Proposition 3.3

For and

*v*∈C^{1}([*t*_{0},*t*_{1}];*H*), we have 3.10For all

*v*∈*Z*and*t*≥0, we have 3.11

### Proof.

Part (i) follows simply by the assumed smoothness of and the chain rule for the Fréchet derivative in Banach spaces. Part (ii) is an application of part (i) by using and .

The central idea of [10] is the transport of curves defined via
The main tool is the following relation (3.12) for the functions:
where *γ*′_{t}(*s*)=∂_{s}(*γ*_{t}(*s*))∈*Y* .

### Proposition 3.4

For every curve *γ*∈** C**(

*w*,

*u*), we have 3.12

### Proof.

We first observe that the mapping *Γ*:(*s*,*t*)↦*γ*_{t}(*s*) satisfies
In particular, using the definition of the semiflow , we have the relations
Note that we will not need an expression for *γ*′_{t}(*s*). Applying proposition 3.3(i) and the above formulae for ∂_{t}*γ*_{t}(*s*) and ∂_{t}(*γ*′_{t}(*s*)) we find
which is the desired result.

One of the main achievements of [10] was to show that identity (3.12) can be used to derive the evolutionary variational inequality (EVI_{λ}), namely
3.13
It is especially interesting that this result holds without any completeness of the space . The crucial assumption needed is that *B*(*s*,*t*) can be estimated in terms of *A*(*s*,*t*), namely in the form *B*(*s*,*t*)≥*λA*(*s*,*t*) along the curves *γ*_{t}. The following result is an abstract version of the ideas in [10].

### Theorem 3.5

*Assume that* *generates the semigroup* *and the above conditions (3.1)—(3.8) hold. If additionally
*
3.14
*then the semigroup* *satisfies (EVI*_{λ}*) given in (3.13).*

### Proof.

We apply the theory in [10], Sect. 5, where the underlying metric space (X,*d*) is not assumed to be complete. Hence, we are able to choose . Following the analysis in the proof of [10], theorem 5.1 (see (5.14) and (5.17) there), we obtain the final estimate
Dividing by *t*, using and , and taking the limit *t*→0^{+}, we obtain
As was arbitrary, it can be replaced by , and the desired (EVI_{λ}) in (3.13) follows.

As in applications the metric is often not given explicitly (see examples in §4), it is desirable to express the fundamental estimate (3.14) in terms of the Onsager operator .

### Proposition 3.6

Assume that 3.15 then estimate (3.14) holds.

### Proof.

The proof is immediate as for a given *v*∈*Y* we can use in (3.15). After using the formula for the derivative of the inverse, namely we find (3.14).

The conditions in proposition 3.6 are similar to the conditions of Bakry & Émery [11] and Bakry [12]. We now return to the metric evolution in the larger space . For this, we assume that on can be extended to a metric on such that 3.16 Moreover, is assumed to have a lower semicontinuous extension (with respect to the metric topology). Finally, is assumed to be dense, namely 3.17 Using the Lipschitz continuity (3.7), there is a unique continuous extension . Then, Daneri & Savaré [10], theorem 3.3 provides the following result.

### Theorem 3.7

*If (3.16), (3.17) and the assumptions of theorem 3.5 hold, then the semiflow* *associated with the gradient system* *satisfies (EVI*_{λ}*) (3.13) and the Lipschitz continuity (3.7) with* *replaced by* *. Moreover,* *is geodesically λ-convex on* *i.e. for every arc-length parametrized geodesic curve* *we have
*
3.18
*for s∈[0,1].*

## 4. Examples

This section surveys possible applications of the abstract methods developed above to scalar equations as well as reaction–diffusion systems. In particular, we show geodesic *λ*-convexity in a smooth setting by establishing the estimate . We generalize the known results for scalar drift–diffusion equations (with conserved mass) to systems with reaction terms (non-conserved masses). The discussion of the corresponding metric spaces is postponed to future research.

### (a) Pure reaction systems and Markov chains

In [4], an entropy gradient structure was established for general reaction systems of mass-action type that satisfy the detailed balance condition. We consider a vector of densities and *R* polynomial reactions
4.1
Here is the reference density, which is obviously a steady state and satisfies the detailed balance condition. Moreover, *k*^{r}(** u**)≥0 is the reaction coefficient (normalized with respect to

**), and the vectors are the stoichiometric vectors for the forward and backward reactions. Usually, the entries are assumed to be non-negative integers but this is not necessary here. The gradient system with and gives (4.1). We find , where is defined via see also [16]. Note that the vector field is nonlinear and that the matrices and have no homogeneity or concavity properties, in general.**

*w*We want to study a few simple cases and discuss the possibility of geodesic *λ*-convexity. For *R*=1 we drop the reaction number *r* and write ** γ**=

**−**

*α***and**

*β***=(**

*ϱ**u*

_{i}/

*w*

_{i})

_{i}. Then, The general case seems too difficult to be analysed, hence we reduce to the case

*k*(

**)≡1. Introducing the matrix**

*u**V*=diag(1/

*u*

_{i})

_{i}we have D

_{u}(

*u*^{α})[

**]=**

*γ*

*u*^{α}

**⋅**

*α**V*

**and after some elementary calculations involving the properties of the function**

*γ**Λ*[16] we find For geodesic

*λ*-convexity, we have to show that

*m*(

**)≥**

*u**λΛ*(

*ϱ*^{α},

*ϱ*^{β}), which after dividing by

*Λ*(

*ϱ*^{α},

*ϱ*^{β}) leads to the formula In the special case where

*α*

_{i}

*β*

_{i}=0 for all

*i*, we find the simpler form This formula applies to example (2.8) where

**=(1,1,0,0)**

*α*^{T}and

**=(0,0,1,1)**

*β*^{T}. Because of |

**|,|**

*α***|≥2 the infimum is**

*β**λ*=0 (by choosing

**=**

*ϱ**ε*(1,1,1,1) and letting

*ε*tend to 0).

### Example 4.1

The annihilation–creation reaction which models the recombination and generation of electron–hole pairs in semiconductors (cf. [43,44] and §2*d*) reads
4.2
The formula yields .

Discrete Markov chains can be seen as special reaction systems where only exchange reactions occur. The reaction system takes the form
4.3
We assume that there is a unique steady-state ** w** with

*w*

_{i}>0 for all

*i*(also called irreducibility). A much stronger assumption is the condition of detailed balance, which reads

*Q*

_{ij}

*w*

_{j}=

*Q*

_{ji}

*w*

_{i}for

*i*,

*j*=1,…,

*I*. According to [16,15], (4.3) is induced by the gradient system , where , and where are the unit vectors. Moreover, it is shown in [16] that for all Markov chains there is a such that is geodesically

*λ*-convex. For special classes, like tridiagonal

*Q*, explicit estimates for

*λ*are obtained.

### (b) Scalar diffusion equation

We consider a bounded, convex domain , *d*≥1, with smooth boundary. In *Ω* we are given the scalar diffusion equation
4.4
This equation is the gradient flow of the energy with respect to the Onsager operator given via
where *E* and *μ* are such that *μ*(*u*)*E*′′(*u*)=*a*(*u*) holds. In particular, we assume that and the sign conditions
We impose that solutions of (4.4) are sufficiently smooth for given smooth initial conditions such that the assumptions of the last section for the semiflow are satisfied.

In the following, we slightly deviate from the setting in §3*d* in that we consider and to be open and connected subsets of affine spaces *u*_{*}+*Y* and *u*_{*}+*Z*, where the shift is given by *u*_{*}=1/|*Ω*|. This modification allows us to extend our theory to the space of probability measures . More precisely, let be the space of Radon measures *ρ* (using that *Ω* is bounded and all moments are finite) and denote the subset of probability measures such that
The results of §3*d* can be easily adapted to this case.

The quadratic form associated with the operator defines in a natural way the spaces
Moreover, we choose *s*≥4 such that *s*>2+*d*/2 and define the spaces
The boundary condition in the definition of the set is necessary to ensure that the semiflow satisfies . In particular, for a solution holds . Obviously, we have *Z*⊂*Y* ⊂*X* and and with dense embeddings.

Our analysis is similar to that in [10], Sect. 4 with the main difference being that we have to take care of the boundary conditions when doing integrations by part. There are two crucial observations for the case with boundaries. First, the curvature of the boundary of convex bodies provides a sign for the normal derivative ∇(|∇*ξ*|^{2})⋅*ν*≤0, whenever ∇*ξ*⋅*ν*=0 holds (see proposition 4.2). Second, the test functions will satisfy *two boundary conditions*, namely

In order to show that the assumptions of proposition 3.6 hold, we have to compute the quadratic form , with
For , we use the abbreviation and obtain
where in both cases the boundary terms vanish using *a*(*u*)∇*u*⋅*ν*=0 and (*a*′(*u*)*v*∇*u*+*a*(*u*)∇*v*)⋅*ν*=0 from *v*∈*Y* and . Applying integration by parts one more time yields
where we have set and used that ∇*ξ*⋅*ν*=0. Finally, integrating by parts one last time leads to
4.5
Here, we used Bochner’s formula . The boundary integral is non-positive using the assumption *H*(*u*)≥0 and proposition 4.2 below.

Thus, we have shown that holds if we assume that *u*↦*μ*(*u*) is concave and *aμ*≥((*d*−1)/*d*)*H*≥0. Here, the latter condition is owing to the elementary estimate
Now, proposition 3.6 states that is geodesically 0-convex. As the present result will be a special case of the result in the next section, we refer to theorem 4.3 for the precise statement.

Thus, we have generalized [10], theorem 4.2 from manifolds without boundary to the case of convex domains in with smooth boundaries. The condition of convexity is quite natural in the context of optimal transport, because only convex domains are still complete metric length spaces with respect to the Euclidean distance.

We close this section with the result on the sign of ∇(|∇*ξ*|^{2})⋅*ν* on the boundary. We refer to [45, ch. 3; [20], lemma 5.2] for previous proofs, but still give an independent proof of a more general result needed in §4*g*. It involves the second fundamental form of the boundary, i.e. for two tangent vectors *τ*_{1},*τ*_{2}∈T_{x}∂*Ω* we have , where *ν* is the outer normal vector.

### Proposition 4.2

Assume that is a domain with C^{2} boundary. Then, for functions *ξ*_{1},*ξ*_{2}∈H^{3}(*Ω*) with ∇*ξ*_{1}⋅*ν*=∇*ξ*_{2}⋅*ν*=0 on ∂*Ω* we have the identity
4.6
where ∇_{∥}*ξ* denotes the tangential part of the gradient ∇_{∥}*ξ*=∇*ξ*−(∇*ξ*⋅*ν*)*ν*. In particular, if *Ω* is convex and *ξ*_{2}=*ξ*_{1}, then ∇(|∇*ξ*_{1}|^{2})⋅*ν*≤0 on ∂*Ω*.

### Proof.

Without loss of generality we assume that *ξ*_{j} is smooth. We denote by a smooth extension of the outer unit normal *ν* into *Ω*. For *x*∈*Ω* we compute
4.7
On the boundary the product vanishes identically, such that on ∂*Ω*. Hence, there are scalar functions such that on ∂*Ω*. Inserting this into (4.7) and using ∇*ξ*_{j}⋅*ν*=0 we have established (4.6).

For a convex body, the second fundamental form is positive semidefinite. Hence, formula (4.6) gives the desired result for *ξ*_{1}=*ξ*_{2}.

We end this section by mentioning that the theory can also be applied to smooth inhomogeneous systems, e.g. where the mobility depends on the spatial variable *x*∈*Ω*,
where , and there exists *α*_{0}>0 with . The appropriate boundary conditions are now for *x*∈∂*Ω*. Doing the appropriate integrations by part, we obtain the formula
where all terms involving third derivatives of *ξ* cancel, and the tensors and ** B** are given via , and . Proposition 4.2 can be generalized for spatially dependent mobilities leading to three additional terms owing to the spatial derivatives of :
If the sum of these terms is negative, using

*α*

_{0}>0 and (giving ) and pointwise minimization over provides a such that .

For an isotropic mobility matrix satisfying the boundary relation for all *x*∈∂*Ω* and *τ*∈T_{x}∂*Ω*, we obtain the simplified estimate
4.8
where again minimization with respect to D^{2}*ξ* was used in the first estimate. Here, denotes the largest eigenvalue of a symmetric matrix . In space dimensions *d*=1 and 2, we obtain
4.9a
and
4.9b
Our result can be compared with the estimates obtained in [18, theorem 1.5] with completely different methods. The results there are formulated using the Wasserstein distance *W*_{I}, whereas our results are formulated in terms of , which is called *W*_{G} there (see [18, eqn. (1.67)]). Thus, our rate may differ from the contractivity rate *α*, which takes the form in our smooth setting.

### (c) A scalar drift–diffusion equation with concave mobility

We now generalize the diffusion equation of the previous section by adding drift terms induced by a given potential *V* . Moreover, we allow the density *u* to be restricted to a bounded interval, i.e. we assume that there is a bound such that
Such restriction occurs in systems with exclusion principles. We refer to [46; 47] and §4*g*. Our work relates to [9; [21], proposition 4.6], where the entropy and the potential energy are studied concerning their geodesic *λ*-convexity. We make the result of the latter work more precise. We have the total energy and the Onsager operator
The drift–diffusion equation takes the form
4.10
where *a*(*u*)=*μ*(*u*)*E*′′(*u*). We again impose the sign conditions
4.11
In the case , we explicitly allow for the case *μ*′(*u*)<0, which occurs in the commonly used mobility *μ*(*u*)=*u*(1−*u*) on ]0,1[. We will see that the non-monotonicity of *μ* gives rise to new conditions. We emphasize that the following result does not need the condition ∇*V* ⋅*ν*=0 on ∂*Ω* employed in [21, proposition 4.6].

### Theorem 4.3

*Assume that Ω is a convex bounded domain in* *with smooth boundary. In addition to (4.11), define* *and assume
*
4.12
*If the potential* *satisfies* *then* *are geodesically λ-convex for* *where
*
*and
*

Before giving the proof of this result note that the case of a linear mobility (i.e. *μ*(*u*)=*u*) for the Wasserstein distance gives the standard result as . Moreover, simply characterizes the *λ*-convexity of *V* on the Euclidean space *Ω*. Note that in the case *μ*′(*u*)<0 we need *λ*-concavity of *V* .

### Proof.

We proceed exactly as in the previous section. We only have the new terms associated with *V* . As is independent of *V* and the vector field depends linearly on *V* , the new terms are also linear in *V* . Together with from (4.5), we have
4.13
To reach this result, we emphasize that the integrations by parts have to be done of the full vector field such that in satisfies the additional boundary condition [*w*(*a*′(*u*)∇*u*+*μ*′(*u*)∇*V*)+*a*(*u*)∇*w*]⋅*ν*=0 obtained by differentiating the boundary condition in (4.10).

While the first term in (4.13) can be immediately estimated from below by , the other terms do not have a sign. That is why in [9] it was expected that the potential energy is not geodesically convex. However, to estimate the geodesic convexity of we can use the non-negative term −*μ*′′(*a*/2)|∇*u*|^{2}|∇*ξ*|^{2} occurring in and not needed otherwise to show positivity of . Abbreviating ** U**=∇

*u*and

**=∇**

*X**ξ*, we have to estimate the following terms from below: Thus, the result is established.

We conclude by making the conditions more explicit in the case of *μ*(*u*)=*u*(1−*u*) on ]0,1[ and *E*′′(*u*)=1/*μ*(*u*), i.e. . We obtain and , where *r*_{spec} denotes the spectral radius.

### (d) A scalar nonlinear reaction–diffusion equation

In a convex, bounded and smooth domain *Ω*, we consider the reaction–diffusion equation
We assume that it is the gradient flow of the free energy and the Onsager operator defined via
4.14
Hence, we assume the relation . The reaction coefficient *κ* satisfies
4.15
The concavity of *κ* implies that of , which is the prerequisite of the convexity of ; see remark 3.1.

Similar to the previous examples, we introduce the spaces
and calculate . With and , we obtain
Integrating the first term twice in *I*_{1} (using ∇*ξ*⋅*ν*=∇*v*⋅*ν*=0 on ∂*Ω*) and inserting the definition of , we find
Similarly, we integrate the first term by parts in *I*_{2} (using ∇*u*⋅*ν*=0) and obtain
Again using proposition 4.2, we can estimate the boundary integral and obtain
where
and
Using assumption (4.15), the last term, which involves |∇*u*|^{2}*ξ*^{2}, is non-negative and can be dropped. We define *M*_{2}(*u*)=*f*(0)+*uf*′(*u*)−*f*(*u*) such that *M*_{2}′(*u*)=*m*_{2}(*u*) and *M*_{2}(0)=0. The term involving *m*_{2} can be integrated by parts (using ∇*ξ*⋅*ν*=0) via
The pointwise estimate −*M*_{2}(*u*)*ξΔξ*≥−(*u*/*d*)(*Δξ*)^{2}−(*dM*_{2}(*u*)^{2}/4*u*)*ξ*^{2} yields
Thus, we have established the following result.

### Theorem 4.4

*Let Ω, κ and M*_{j} *be given as above. Define the values
*
*and set* *. If* *then* *defined in (*4.14*) is geodesically λ***-convex.*

### Proof.

To conclude the proof, we have to establish
for all . As the first term in the above lower estimate for is non-negative, it suffices to show *M*_{1}(*u*)−*M*_{2}(*u*)≥*λ***u* and *M*_{3}(*u*)−*dM*_{2}(*u*)^{2}/(4*u*)≥*λ***κ*(*u*) for all *u*≥0. As these estimates are exactly the definitions of *λ**_{j}, the desired result is established.

The following result provides sufficient conditions on the function *κ*, satisfying (4.15), that lead to a geodesically *λ*-convex gradient system. It is posed in terms of the ansatz *κ*(*u*)=*k*(*u*)*Λ*(1,*u*) and shows that *k* can be chosen to be constant near *u*=0 giving the linear reaction term *f*(*u*)=*k*(0)(*u*−1) there. For large *u*, one may choose for *c*>0 and *p*∈[0,1] leading to the nonlinear reaction term .

### Proposition 4.5

Consider a function *κ* satisfying (4.15) and let *k*(*u*)=*κ*(*u*)/*Λ*(1,*u*)>0. If there exist and positive constants *k*_{j}, *j*=0,…,3 such that *k* satisfies the conditions
4.16a
4.16b
4.16c
4.16d
then in theorem 4.4 we have . The case *k*≡*k*_{0} gives .

### Proof.

We denote by *η*_{j}(*u*) the functions in the infima defining *λ**_{j} in theorem 4.4. As both functions are continuous on , it suffices to estimate *η*_{j} near *u*=0 and .

ad *η*_{1}: By (4.16a), we have *M*_{1}(0)−*M*_{2}(0)=−*f*(0)/2=*k*_{0}/2>0 and conclude *η*_{1}(*u*)≥0 for sufficiently small *u*. For *u*≥2, we have
Using (4.16b), we obtain *η*_{1}(*u*)≥*k*_{1}/4, for all sufficiently large *u*.

ad *η*_{2}: For *u*≤1, we have *f*(*u*)≤0; using *κ*′≥0, we conclude *M*_{3}(*u*)≥*f*′(*u*)*κ*(*u*). Moreover, from *f*(*u*)=(*u*−1)*k*(*u*) and (4.16c) we conclude *f*∈C^{1,α}([0,*u*_{0}]). Hence, satisfies |*M*_{2}(*u*)|≤*Cu*^{1+α}. Together we find
For large *u*, we use the asymptotics for given via
Using (4.16d), we find *η*_{2}(*u*)≥*k*_{2}/4−*dk*_{3}/2, which gives the desired result.

For the last statement, note that *M*_{1}(*u*)=*k*_{0}(*u*+1)/2+*κ*(*u*)≥*k*_{0}*u*/2, *M*_{2}≡0 and *M*_{3}(*u*)=*k*_{0}(*κ*(*u*)−(*u*−1)*κ*′(*u*)/2)≥*k*_{0}*κ*(*u*)/2. Here, the last estimate follows from the explicit relation (*u*−1)*κ*′(*u*)=(1−*κ*(*u*)/(*k*_{0}*u*))*κ*(*u*). (see [16, (A.3)]). □

### (e) A linear reaction–diffusion system

For ** u**=(

*u*

_{1},

*u*

_{2}), we consider the system of coupled linear equations 4.17 which is the gradient flow for the energy and the Onsager operator given via 4.18 Observe that the total mass is conserved along solutions of (4.17), i.e. . We fix a constant state , choose the Sobolev index

*s*as before and define the spaces As is linear, we compute With the shorthand , we obtain with and Integrating the first term in

*I*

_{1}by parts twice, using the boundary conditions ∇

*v*

_{i}⋅

*ν*=∇

*ξ*

_{i}⋅

*ν*=0 and finally substituting gives Similarly, we integrate the second term and obtain Thus, using again Bochner’s formula and proposition 4.2 we arrive at It was shown in [16, example 3.5] that

*m*(

**)≥2**

*u**Λ*(

**)≥0 holds.**

*u*The main task is to control the mixed terms with prefactor *kδ*_{j} that are collected in the function *G*. Unfortunately, we can estimate these terms only in the case of equal mobilities *δ*_{j}=*δ*>0. For *H*(** u**,

**)=(2/**

*ξ**δ*)

*G*(

*δ*,

*δ*,

**,**

*u***), some rearrangements yield the identity where As**

*ξ**Λ*is a concave function, we have . To estimate we integrate the very first term by parts twice (using ∇

**⋅**

*ξ**ν*=0 and ∇

*Λ*(

**)⋅**

*u**ν*=0) and find Hence, we have established the following result.

### Theorem 4.6

*If Ω is smooth and convex and δ*_{1}*=δ*_{2}*>0, then the gradient system (4.17) generated by* *from (4.18) is geodesically 0-convex.*

### (f) Drift–diffusion system in one dimension

We consider the one-dimensional version of the drift–reaction–diffusion system (2.10) for electrons and holes in a semiconductor; see §2*d*. We further simplify the system by neglecting the reaction terms (*np*−1).

To highlight the general structure, we treat a system with *I* non-negative densities *u*_{i}∈L^{1}(*Ω*) with *Ω*=]0,1[, where the species have the charge vector . The system takes the form
4.19a
4.19b
4.19c
where ′ is the partial derivative with respect to *x*. The potentials ** V**=(

*V*

_{1},…,

*V*

_{I}) are smooth functions and contain possible doping terms. The system is the gradient flow for 4.20 As we have no reaction between the species and no-flux boundary conditions, the individual masses are conserved. The electrostatic potential

*ϕ*

_{u}is a linear function of

**⋅**

*q***, namely**

*u**ϕ*

_{u}=

*L*

**⋅**

*q***. In the one-dimensional case, we have an explicit solution formula 4.21 The function spaces can be introduced as in the above examples. We only give the calculation of the operator , where now the quadratic nature of owing to the terms**

*u**u*

_{i}

*ϕ*′

_{u}has to be observed. Using the two boundary conditions for , we find Now the quadratic form can be calculated as usual and where we used the boundary conditions

*ξ*′

_{i}=0 on ∂

*Ω*. Combining the two terms and using some cancellation, we arrive at with . The first two terms can be estimated in the standard way. For the interaction via

*ϕ*

_{u}and

*L*, we note that

*g*is such that formula (4.21) can be applied. When assuming additionally that

*ε*≡

*ε*

_{0}, the third and fourth term can be rewritten as There are two cases in which this quadratic form can be estimated from below. First, in the case

*I*=1 we obviously have

*Q*

_{u}≡0. For

*I*=2, the expression simplifies to Thus, we find

*Q*

_{u}≥0 if

*q*

_{1}

*q*

_{2}≤0; this means that the particles are oppositely charged. Of course, we could add further uncharged particles (i.e.

*q*

_{j}=0), but this is useless as they do not interact with the other particles. We summarize our findings as follows.

### Theorem 4.7

*Consider the gradient system* *defined via (4.19) and (4.20) with constant ε. Assume either I=1 or I=2 and q*_{1}*q*_{2}*<0. If the potentials V* _{i} *are λ*_{i}*-convex, i.e. V ′′*_{i}*≥λ*_{i} *on Ω, then* *is geodesically λ***-convex with* .

### (g) A multi-particle system with cross-diffusion

In several applications, one is interested in reaction–diffusion systems with *I* species, where the microscopic sites are occupied exactly by one species. We refer to [48; 47]. On the macroscopic level, this means that the density vector ** u**=(

*u*

_{1},…,

*u*

_{I}) satisfies the pointwise restriction 4.22 Moreover, the mobility tensor obeys the Stefan–Maxwell law (e.g. [48]) 4.23 Using (4.22), we easily see that is positive semidefinite, namely 4.24 Thus, we consider the energy functional 4.25a and

**=(**

*V**V*

_{1},…,

*V*

_{I}) is a vector of potentials with

**⋅**

*V***≡0. Thus,**

*e***determines the equilibrium state**

*V***via**

*w**w*

_{i}= e

^{−Vi}. Moreover, the Onsager operator acts now on the vector-valued dual variables and takes the form 4.25b Taking into account the constraint (4.22) when calculating the differentials, we find the nonlinear evolutionary systems Here, the diffusion term is linear since is exactly the inverse of D

^{2}

*E*(

**) (taking the constraint into account). We see that the special choice of with negative off-diagonal terms simplifies the diffusion terms, whereas the drift terms from the potential become more involved. This approach was also used in [48; 46], while in [47] the off-diagonal terms are not used.**

*u*In particular, the mass of each component is preserved during the flow, namely
In the case *I*=2, the system reduces to a scalar equation for *u*∈[0,1] via ** u**=(

*u*,1−

*u*) of the form which is covered by the analysis treated in §4

*c*.

We now restrict ourselves to the case ** V**≡0 and leave the general case for future research. Our aim is to show that the pure (uncoupled) diffusion is geodesically 0-convex. This statement is non-trivial because the metric induced by the mobility tensor couples the densities in a non-trivial way. However, as can be estimated from above by we see that can be estimated from above by the component-wise Wasserstein distance, i.e.

### Theorem 4.8

*Consider the gradient system* *defined in (4.25) with* *V**≡0 and Ω bounded, smooth and convex. Then,* *is geodesically 0-convex with respect to* .

### Proof.

To estimate the quadratic form , we define the spaces *Z*, *Y* and *H* as before in the Sobolev space H^{s}, H^{s−2} and H^{1}, respectively. Moreover, for the functions ** u**,

**, and we have the following boundary conditions: 4.26**

*ξ*Using and giving , we have
Using (c) and (b), we can integrate the first term by parts twice. The second term will be integrated once using (a). After inserting the definition of ** v**, we arrive at
The first term will now be integrated by part once again by using (b), which also implies

*Ξ*_{u}⋅

*ν*=0. Integrating the second term will generate a boundary integral that will be non-negative by proposition 4.2, Here, the indices

*α*and

*β*denote partial derivatives with respect to

*x*

_{α}.

The first term in is positive by Bochner’s identity. To estimate *μ*, we interchange the summation indices *i* and *j* in the fourth term to find that the last two terms can be combined into the term (*u*_{i}*u*_{j})_{β}*ξ*_{iαβ}*ξ*_{jα}. Thus, integration by parts, employing proposition 4.2, and exploiting the cancellation of the terms involving *ξ*_{iααβ} gives
where . Here we used the boundary conditions (b), which give *Ξ*_{u}⋅*ν*=0, and hence on ∂*Ω*. The first term in the above integral is non-negative, while the other two terms are non-positive. However, they are dominated by the corresponding positive terms obtained earlier, e.g. . Using the same rearrangement as in (4.24), we find the final expression
where . Thus, we have finally established the desired result , and theorem 4.8 is proved.

## Funding statement.

M.L. was supported by DFG via the Matheon project D22. A.M. was partially sponsored by ERC-2010-AdG no. 267802 *AnaMultiScale*.

## Acknowledgements

The authors are grateful to Giuseppe Savaré for stimulating discussions and several helpful remarks.

## Footnotes

One contribution of 11 to a Theme Issue ‘Entropy and convexity for nonlinear partial differential equations’.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.