A well-posedness result for a system of cross-diffusion equations

This work's major intention is the investigation of the well-posedness of certain cross-diffusion equations in the class of bounded functions. More precisely, we show existence, uniqueness and stability of bounded weak solutions under the assumption that the system has a dominant linear diffusion. As an application, we provide a new well-posedness theory for a cross-diffusion system that originates from a hopping model with size exclusions. Our approach is based on a fixed point argument in a function space that is induced by suitable Carleson-type measures.


Introduction
Systems of partial differential equations with cross-diffusion have developed into a large field of research in the last decades. Cross diffusion, the phenomenon in which the gradient in the concentration of a species causes a flux of another species, appears in various applications as the modelling of population dynamics, e.g., [7-9, 21, 33] or electrochemistry, e.g., [5]. Another important biological field that is mathematically described by systems with cross-diffusion are cell-sorting or chemotaxis-like problems, e.g., [29,30]. Chemotaxis denotes the process of cell movement provoked by chemical signals. Classical examples involve pattern formation of bacteria, e.g., [22,36], or biomedical processes as tumour invasion, e.g., [14,16]. For more detailed background information regarding the biological and modelling processes we refer the reader to [28].
In the present work we study cross-diffusion systems that are dominated by linear diffusion. More precisely, we study a system of diffusion equations that are coupled through nonlinear reaction terms The reason, why we choose to work in the setting of bounded functions is particularly motivated by the following specific example, which apparently belongs to the class of cross-diffusion systems modelled in (1), (2). We study a cross-diffusion system that can be modelled by a multidimensional advection-diffusion equation with linear drift and diffusion matrices, This system describes the evolution of different species, and ( , ) plays the role of the density or volumic fraction of the th species at time and point . The 's are the cross-diffusion coefficients, which relate the gradient of the th species' concentration with the flux of the th species' concentration. To illustrate the structure of (5), we note that the evolution of the th species can be rewritten as the linear conservative advection-diffusion equation in which the diffusion coefficient is proportional to the concentration of the concurrent species, while the advecting velocity field is linearly dependent on their concentration gradients. The system can be derived as a formal limit from a hopping model with size exclusion, see [6]. It was recently studied mathematically in [4].
Since the solution for = 1, … , represents the volumic fraction of the th species, it is reasonable to demand the solutions to partition unity, The same condition has thus to be satisfied by the initial data = ( 1 , … , ), that is, In order to ensure that the partiton condition in (6) is satisfied, even on a formal level, it is necessary to impose that the diffusion coefficients are symmetric in the sense that Even for this specific model, proving uniqueness and pointwise bounds as in (6) is rather challenging. In the following, inspired by [4], we will restrict our attention to the case, in which the cross-diffusion coefficients satisfy certain closeness assumptions. This way, despite the constraint in (6), we are in a situation in which our system under consideration is equivalent to that in (1), (2), and thus Theorem 1.1 applies.
To be more specific, thanks to the partition condition in (6), we can elegantly generate a linear diffusion term in (5), for any positive constant . In order to treat the right-hand side as a perturbation, we have to assume that the coefficients are sufficiently close to each other. This is achieved, for instance, by choosing and demanding that This assumption enables us to translate (5) or (9) into a diffusion-dominant system, see Section 2. Theorem 1.1 provides us, due to scaling argument, with a unique solution to (9) in the class of functions satisfying In fact, we will see that this system can be transferred back into the original cross-diffusion system (5), (6). We thus have the following well-posedness result.
Theorem 1.2. Suppose that the coefficients are symmetric and sufficiently close to each other in the sense of (10) and (8). Then, for every set of initial data 1 , … , satisfying (7), there exists a smooth solution 1 , … , to the cross-diffusion system (5), (6). This solution is unique in the class of functions satisfying (11). Moreover, solutions are stable in the sense of (4).

Remark 1.
We remark that Theorem 1.2 (as Theorem 2.1 and Theorem 2.3 below) is valid also for more general classes of cross-diffusion coefficients that vary in space and time, = ( , ), as long as (10) and (8) remain true.
We note that solutions are automatically bounded thanks to the modelling assumption (6), which makes ∞ a natural space for the study of well-posedness. Moreover, the gradient estimate (3) or (11) is natural in this perturbative setting (10), as it is the standard gradient estimate for the homogeneous heat equation with ∞ data-observe that the control over the gradient deterioates as → 0 with a rate proportional to the diffusion length. In this sense, we consider the conditions for well-posedness imposed in the present paper as optimal. Our well-posedness result for the system under consideration improves upon earlier results which require the solutions and data to be of higher regularity [4].
We finally remark that, in general, the analytic treatment of many cross-diffusion problems in the form of can be very challenging, since the diffusion matrix ( ) neither has to be symmetric nor positive definite, which makes it hard to ensure such modelling assumptions as in (6). Another difficulty lies in the absence of a maximum principle or general parabolic regularity theory, if the diffusion matrix is not diagonal. Sufficient conditions for the global existence of weak or strong solutions of nonlinear parabolic equations are obtained, for instance, in [1,11,20,26,31]. The problem of uniqueness is in general much harder. For mildly coupled cross-diffusion equations uniqueness has been proved by duality methods [13,19,27]. In some situations, the structure of the equations also allows for the application of entropy methods [10,21,37]. We finally mention results on weak-strong uniqueness in [4,11,15].
The paper is organized as follows: In Section 2, we introduce and discuss the precise function spaces in which we establish well-posedness. Section 3 is devoted to the study of the linear problem in these spaces. In Section 4 we come back to the nonlinear problem and provide the proofs of the main theorems.

Reformulation and results
The systems that we investigate in this work can be considered as nonlinear perturbations of multi-dimensional heat equations. Moreover, the particular (semilinear) structure of the nonlinearity considered in (1), more precisely, the properties formulated in (2), which are in turn motivated my the particular example mentioned in (5) or (9), lead to the study of bounded solutions to the respective equations in a natural way. Indeed, for any well-behaved norm ‖ ⋅ ‖ for which we have maximal regularity estimates for the heat equation, we expect that by the virtue of (2), and the nonlinear term on the right-hand side can be absorbed into the left-hand side provided that ‖ ‖ ∞ is sufficiently small. We are thus led to considering ‖ℎ‖ = ‖ℎ‖ ∞ in the case of the initial datum-a choice that is consistent with the partition of unity condition imposed in (6), (7). The space-time maximal regularity norm has to be accordingly scale-invariant. Motivated by [25], we use the following (semi-)norms, that are motivated by Carleson-measure characterizations of the BMO space, see Theorem 3 of Chapter 4.4 in [34]. Given If necessary, we identify or with its spatial periodic extension. Based on these norms we define two Banach spaces and by The underlying concept of using such norms was introduced in [25], in order to prove wellposedness for the Navier-Stokes equations with small initial data in BMO −1 . This concept was further developed in order to establish existence and uniqueness results for various (degenerate) parabolic equations, including geometric flows with rough data [24,35], the porous medium equation [23], the thin film equation [18,32], and the Landau-Lifshitz-Gilbert equation [17]. By a slight abuse of notation, we generalize these norms and spaces to vector or matrix valued functions by setting We are now in the position to present our first result (Theorem 1.1) in a more precise manner.
Under additional assumptions concerning the nonlinearity , we are able to show higher regularity of the solutions.
In the analytic case there exist constants Γ > 0 and > 0 independent of and such that sup , for every ∈ ℕ 0 and every multiindex ∈ ℕ 0 .
We finally turn to the explicit system given in (5), (6) and show how it fits into the general framework considered in Theorems 2.1 and 2.2. We have already seen that under the partition condition in (6), (5) is equivalent to (9). Our goal is to transfer the latter into a diffusiondominated system with small initial data. By rescaling time, the diffusivity constant on the lefthand side can be absorbed into the cross-diffusion coefficients, that is, we consider with coefficients ∶= − 1. At this point we note that the scaling factor has to be positive.
The closeness condition (10) now translates into the smallness condition ∶= max 1≤ ≠ ≤ | | ≪ 1 on the new coefficients. We now use the nonlinearity of the equation to shift the smallness condition further to the initial datum. This is achieved by setting ∶= and ℎ ∶= . The new partition conditions are thus and and the cross-diffusion equations become where the 's are given by , and are thus bounded, | | ≤ 1.

Remark.
To be accurate, we have to exclude the case ≡ . Since we would obtain = 0, the change of variables would not be permitted. However, this is not a significant restriction, because the the cross-diffusion system would untangle into a system of independent heat equations, which is much easier to solve.
We see that (17) has the same structure as our general model (1), where ( ) is given by Apparently, (16) provides an upper bound for the initial data and the nonlinearity ( ) depends analytically on . We are thus allowed to apply Theorems 2.1 and 2.2 and obtain well-posedness for (17) in the class together with analyticity in time and space and analytic dependence on the initial data. It only remains to verify that solutions obey the partition condition (15), the argument of which we provide in Section 4, following [4]. Our result for the cross-diffusion system (5), (6), or equivalently, (17), (15), is thus the following. Theorem 2.3. Suppose that the coefficients are symmetric in the sense of (10). Let > + 2 be given. There exist 0 > 0 and > 0 such that for every ≤ 0 and every ℎ ∈ ∞ with (16), there exists a unique solution to the system (17), (15) in the class ‖ ‖ ≤ . Moreover, the solution depends analytically on time, space and the initial data, and estimates (14) and (12) hold.

Linear theory
Our proof of Theorem 2.1 is based on a fixed point argument. We will thus start with the study of the linear problem. Our goal in this section is the following maximal regularity estimate.

Proposition 3.1. Let be a solution of the inhomogeneous heat equation
Then it holds that The argumentation for establishing this proposition is similar to those in [18,23,24,32]. It will be convenient to translate the problem onto the full space by extending all involved functions periodically from to ℝ . It is clear that the corresponding norms remain unchanged under periodic extension.
We denote the heat kernel in ℝ by Φ, i.e., Φ( , ) = (4 ) − 2 − | | 2 4 , so that solutions to (18) have the representation We will estimate the homogeneous part̃ and the inhomogeneous part̂ separately. Before doing so, we recall a standard estimate on the gradient of the heat kernel.

Lemma 3.2. For every ∈ [1, ∞], it holds that
We provide the simple proof for the convenience of the reader.
PROOF. For any > 0, the function − 2 is bounded on [0, ∞) and thus Using = | | 2 4 and = 1 2 , we thus obtain the pointwise estimate This proves the case = ∞. For smaller values of , using a chance of variables, we compute This proves the lemma. We first turn to the estimate of the solution to the homogeneous problem̃ .

Lemma 3.3. It holds that
PROOF. The maximum principle for the heat equation immediately implies the bound on the ∞ norm of̃ . In order to estimate the Carleson measure part of the norm, we observe that due to Lemma 3.2. Using this estimate we get which proves the Lemma.

Lemma 3.4. It holds that ‖̂ ‖ ≲ ‖ ‖ .
PROOF. We start with the bound on the ∞ -norm of̂ . We set = √ and split the space-time integral into a diagonal and an off-diagonal part, To bound the diagonal part, we use Hölder's inequality and get ≤ ‖∇Φ‖ for any Hölder conjugates and . We have to choose small enough such that the -norm of ∇Φ is finite. From Lemma 3.2 we get The right-hand side is finite if and only if − 2 − 2 + 2 > −1, which is equivalent to requiring that > + 2, as in the assumption of Theorem 2.1. We evaluate the integral on the right-hand side and obtain for the diagonal part of̂ that Let us now consider the off-diagonal term . Applying elementary arguments, we observe In order to control the term on the right by the Carleson measure expression which defines the norm, we have to invoke a covering argument. Using the triangle inequality and the fact that ∑ ∈ ⋅ℤ − | −̃ | is controlled by a constant only depending on the dimension , we notice that Now we claim, that there exists a constant 0 < < 1 independent of , such that This estimate directly implies that ≲ ‖ ‖ as a conclusion from the geometric series' convergence, which in turn establishes the control of the ∞ norm as desired.
To prove the claim in (20), we have to refine the spatial covering. Indeed, we cover the set ( 2 ⋅ 2 −( +1) , 2 ⋅ 2 − ) × (̃ ) by about 2 2 many cylinders ( ) of the form ( ) ∶= Since ‖ ‖ 1 ≤ ‖ ‖ by Jensen's inequality, we see that = 2 − 1 2 is a valid constant. It remains to estimate the Carleson measure part of the norm. Again, we consider separately the diagonal and the off-diagonal contribution, this time, however, by distinguishing the two cases Up to a factor 1∕ , the term on the right-hand side is precisely the term that we hat to bound in our previous argument for , see (19). We thus find and averaging over the cylinder as desired. Case 2: We assume supp( ) ⊆̌ ( ). Our argumentation for this case is based on the maximal regularity estimate for the heat equation with forcing in divergence form, Restriction on the support of the forcing, we get ‖∇̂ ‖ ( ( )) ≲ ‖ ‖ (̌ ( )) . We can coveř ( ) by ( ) ∪ 2 ( ) ∪ √ 2 ( ) and thus we obtain Maximizing in and on the left-hand side yields the missing estimate.

The nonlinear problem
In this section we want to prove Theorems 2.1, 2.2, and 2.3. Our first concern is the wellposedness of the system (1) under the assumption (2) on the nonlinearity, which we derive by a fixed point argument. To apply this argument we need the following lemma.

Lemma 4.1. It holds that
and PROOF. Since ‖ ( , ∇ )‖ is defined as the maximum of ‖ ‖ ( , ∇ ) ‖ ‖ , it suffices to show the statements of the lemma for some component of . We restrict our attention to the proof of the Lipschitz estimate (21). The argument for (22) is similar and even shorter. By the definition of the nonlinearity and an application of the triangle inequality, it holds that We make now use of the assumptions on the reaction matrix in (2) and the fact that ‖∇ ‖ = ‖ ‖̇ to estimate This proves (21).
We now have all prerequisites to prove Theorem 2.1. PROOF OF THEOREM 2.1. Let ∈ be given, and let [ℎ, ] be the solution to the linear problem (18) with inhomogeneity ∇ ⋅ ( , ∇ ) and initial data ℎ. By Proposition 3.1 we obtain the estimate ‖ [ , ℎ]‖ ≲ ‖ℎ‖ ∞ + ‖ ( , ∇ )‖ . Applying Lemma 4.1 and using the assumptions on ℎ we furthermore have ‖ℎ‖ ∞ + ‖ ( , ∇ )‖ ≲ + ‖ ‖ +1 , and thus, combining both estimates, we get the following bound on the solution of the linear problem for some constant that we keep fixed for a moment. In order to define a contraction map, we for any ∈ 2 (0) ⊆ , provided that ≤ 0 . Hence, for every such and every ℎ fixed, the function (ℎ, ⋅) maps the set 2 (0) ⊆ into itself. Furthermore, by a similar argument, given 1 and 2 , the linearity and Lemma 4.1 yield ‖ , for some constant̃ . Choosing 0 even smaller-if necessary-, we thus find the contraction estimate ‖ , for any 1 , 2 ∈ 2 (0) ⊆ and some < 1 fixed. An application of Banach's fixed point theorem thus provides a unique solution * in 2 (0) ⊆ to the equation [ℎ, * ] = * , which is nothing but (1). As a by-product, we also have the stability estimate (12).
The idea how to prove the regularity of the solution was introduced in [2,3] and is commonly referred to as Angenent's trick. PROOF OF THEOREM 2.2. To show that the dependence of the solution on the initial datum is of class , ∞ or , we consider the operator ∶ ∞ ( , ℝ ) × → defined by [ℎ, ] = − [ℎ, ], where is the fixed point map introduced in the proof of Theorem 2.1 above. Defined on (0)× 2 (0) ⊆ ∞ ( , ℝ )× , this map is of the same differentiability class as the nonlinearity ( , ∇ ) through ( ), and so is the operator by definition. Indeed, if, for instance, is 1 , we notice that and the right-hand side is a (‖ 1 − 2 ‖ ) term, and the derivative of the fixed-point map [ℎ, ] with respect to is given by the solution of the heat equation with with inhomogeneities ∇ ⋅ ( ( , ∇ ) + ∇ ( , ∇ )∇ ). Next we observe, that [0, 0] = 0 holds and [0, 0] = is invertible. We are thus in the position to apply the (analytic) implicit function theorem (see for example [12]) to deduce the existence of balls ̂ (0) ⊆ ∞ ( , ℝ ) and This shows, that [ℎ] and thereby * as well is of class , class ∞ or analytic in space and time for every ∈ and every 0 < < ∞. (13).
To cover the analytic case, it only remains to recall the elementary fact that we can estimate arbitrary derivatives of an analytic function locally by | ( ) ( )| ≤ ! Γ ‖ ‖ ∞ for some positive reals and Γ. This concludes the proof of Theorem 2.2.
We finally turn to the proof of Theorem 2.3. Thanks to the results obtained so far for the general systems, it is enough to show that solutions to (17) satisfy the partition of unity condition (15). For this purpose, it is convenient to truncate the nonlinearities. Inspired by [4], we consider with nonlinearitieŝ wherê is obtained from by restriction to the range [0, ], i.e.,̂ ∶= max 0, min( , ) . We have to show that solutions to the truncated problem satisfy (15) and that̂ = to deduce statement of Theorem 2.3. PROOF OF THEOREM 2.3. The general well-posedness result of Theorem 2.1 applies to the modified problem (23), and we see that 0 has to be chosen much smaller than 1∕ by a closer inspection of the proof. We denote the unique solution to (23) by * .
Our goal is to show, that * fulfils the partition of unity condition (15). Therefore we start by adding up all equations of (23). Due to the symmetry condition = imposed in (8), which is inherited by the 's, this leads to considering the homogeneous heat equation for ∶= ∑ =1,…, * , which is solved by = .