A note on the uniqueness of weak solutions to a class of cross-diffusion systems

The uniqueness of bounded weak solutions to strongly coupled parabolic equations in a bounded domain with no-flux boundary conditions is shown. The equations include cross-diffusion and drift terms and are coupled selfconsistently to the Poisson equation. The model class contains special cases of the Maxwell-Stefan equations for gas mixtures, generalized Shigesada-Kawasaki-Teramoto equations for population dynamics, and volume-filling models for ion transport. The uniqueness proof is based on a combination of the $H^{-1}$ technique and the entropy method of Gajewski.


Introduction
Several techniques have been developed for the analysis of nonlinear parabolic systems, including sufficient conditions for the global existence of weak or strong solutions [3,18,22,29]. However, the proof of uniqueness of weak solutions is generally much more delicate, in particular for strongly coupled systems. In this paper, we prove the uniqueness of bounded weak solutions to a class of cross-diffusion systems. The proof is based on a combination of the H −1 technique and the method of Gajewski [14], where a certain semimetric measures the distance between two solutions. It is shown that the semimetric is related to relative entropies.
For consistency, the initial datum has to satisfy the condition The diffusion coefficients A ij and drift coefficients B ij are defined by (4) A ij (u) = p(u 0 )δ ij + a j u i q(u 0 ), B ij (u) = r(u 0 )u i δ ij , i, j = 1, . . . , n, for some functions p, q, and r and numbers a j ≥ 0. Our main assumption is that these functions do not depend on the species number i. Then u 0 satisfies a nonlinear driftdiffusion equation (see (12) below), and this property allows us to initiate the uniqueness proof. We do not know how to relax this assumption in the context of weak solutions. The diffusion matrix A(u) = (A ij (u)) is not assumed to be positive definite and it may degenerate. The existence theory developed in [17] is based on the assumption that there exists a transformation of variables such that the transformed diffusion matrix becomes positive semidefinite, allowing for some degeneracy; see [18] for details.
Under some conditions, model (1), (4) can be derived formally from a master equation for a continuous-time, discrete-space random walk in the macroscopic limit [26,32] or from a fluiddynamical model in the inertia approximation [18,Section 4.2]. The variables u i may describe the density of the ith population species or the ith component of a gas mixture with electrically charged components. In the former case, φ models the environmental potential, in the latter case, it denotes the electric potential. Because of these applications, it is reasonable to assume that u i ≥ 0 in Ω, t > 0.
For special choices of A ij and B ij , including condition (4), the existence of global bounded weak solutions can be shown. We give some examples and references in section 4 below. In this paper, we are only concerned with the uniqueness of weak solutions.
1.2. State of the art. Before stating and explaining our assumptions and the main result, let us review some techniques to show the uniqueness of (weak) solutions to nonlinear parabolic equations. We focus on generalized solutions since uniqueness of strong solutions is usually proved by standard L 2 estimations.
One important technique is based on the use of the test function sign + (u (1) −u (2) ), where u (1) and u (2) are two solutions and sign + is the positive sign function (sign + (s) = 1 for s > 0 and sign + (s) = 0 else). The use of this test function can be justified by employing the technique of doubling the variables, first developed by Kružkov for hyperbolic equations [21] and later extended by Carrillo to scalar parabolic equations [8] and by Blanchard and Porretta to allow for renormalized solutions [6]. We refer to the review [4] for an extensive bibliography. All these results hold for scalar equations only.
Nonlinear semigroup methods provide powerful abstract tools for proving the uniqueness of (mild of integral) solutions; see, e.g., [5]. However, this approach seems to be generally not accessible to cross-diffusion systems.
One of the first uniqueness theorems for diffusion systems was shown by Alt and Luckhaus [2] under the assumptions that the time derivative of u i is integrable and the elliptic operator is linear. The first hypothesis was relaxed to finite-energy solutions by Otto [27], and the ellipticity condition was generalized by Agueh using methods from optimal transport [1], but in both cases for scalar equations only.
Another powerful approach is the dual method which consists in choosing a test function which satisfies an appropriate dual problem [10]. This includes the H −1 method, where a test function of an elliptic dual problem is chosen. In some sense, the uniqueness problem is reduced to an existence problem of the dual problem [23]. The dual method allows one to treat diffusion systems that are, to some extent, weakly coupled; see, e.g., [10,16,25]. Based on a dual method, Pham and Temam [28] proved recently a uniqueness result for a strongly coupled population system assuming a strictly positive definite diffusion matrix.
The uniqueness of (weak) solutions may be also proven by using an entropy method. One idea is to differentiate the relative entropy H(u (1) |u (2) ), where u (1) and u (2) are two solutions emanating from the same initial data, with respect to time and to show that d dt H(u (1) |u (2) ) ≤ CH(u (1) |u (2) ) for some constant C > 0, which implies from Gronwall's lemma that H(u (1) |u (2) ) = 0 and hence u (1) = u (2) . This approach has been used to show the weak-strong uniqueness for compressible Navier-Stokes equations [11,12] and reactiondiffusion systems (with diagonal diffusion matrix) [13]. A second idea, due to Gajewski [14], is to time-differentiate the semimetric for convex entropies H and to show that d dt d(u (1) , u (2) ) ≤ 0, implying again that u (1) = u (2) . The technique has been applied to nonlinear drift-diffusion equations for semiconductors [14] and later to cross-diffusion systems [20,32]. Compared to other methods, it has the advantage that only weak solutions are needed [18,Chapter 4.7]. The Gajewski method is related to the approach of using relative entropies; see Remark 4.

Assumptions and main result.
Our approach is to combine the H −1 technique and the method of Gajewski and to generalize the results from [20,32]. The novelty is the inclusion of the potential term and the general structure of A ij (u). With hypothesis (4), equations (1) can be formulated as This can be interpreted as a drift-diffusion equation with field term q(u 0 )∇u 0 + r(u 0 )∇φ. Since u 0 depends on all u i , this is still a cross-diffusion system. However, the drift-diffusion structure is essential in the uniqueness proof. Our main result is as follows.
The paper is organized as follows. Theorem 1 is proved in section 2. Section 3 is concerned with some comments on the techniques and the proof. Some examples satisfying conditions (7)-(8) are detailed in section 4.
Step 2. Uniqueness for u i . Let u (1) = (u n ) be two weak solutions to (1). In this step, the solutions are not required to be bounded. We set u i .

Remarks
We give two comments on the regularity of the drift term and on the relation of Gajewski's semimetric to relative entropies.
Remark 3 (Lower regularity of ∇φ). We claim that the regularity on φ can be relaxed to ∇φ ∈ L ∞ (0, T ; L α (Ω)) with α > d if p(s) + q(s)s = D = const. > 0. For simplicity, we assume that D = 1. In this case, we do not need to apply the H −1 method and can use standard L 2 estimates. Let (u (1) , φ (1) ) and (u (2) , φ (2) ) be two solutions to (2), (12) with the same boundary and initial conditions. Taking u (1) − u (2) as a test function in (12), we find that By the boundedness of u (2) i and the elliptic estimate for the Poisson equation, the second integral is estimated as where Q t = Ω × (0, t) and C 5 > 0 depends on the L ∞ norm of u (2) 0 . For the first integral I 1 , we employ the Lipschitz continuity of R, he Cauchy-Schwarz inequality, the Hölder inequality, the Gagliardo-Nirenberg inequality with θ = d/2 − d/β ∈ (0, 1), and eventually the Young inequality with parameter θ: Therefore, (13) becomes and Gronwall's lemma shows that (u 0 )(t) = 0 in Ω, t > 0. Remark 4 (Comparison of Gajewski's semimetric and relative entropies). In the second step of the proof of Theorem 1, we may work with another semimetric, based on the relative entropy as done in [13], where Setting h(u) = (h(u 1 ), ..., h(u n )) by a slight abuse of notation, we see that h : R n → R n is a convex function. Instead of the expression from [13], we use its symmetrized version to obtain a semimetric: The semimetrics (5) and (18) are strongly related although they are different. First, both expressions behave like |u − v| 2 for "small" |u − v|, since a Taylor expansion shows that both semimetrics can be estimated from below by, up to a factor, where h ′′ (ξ) is the Hessian of h at some point ξ ∈ R n . Second, when differentiating d 0 (u (1) , u (2) ) with respect to time and inserting (1), the drift terms cancel, as they do when differentiating d(u (1) , u (2) ). A formal computation shows that implying that u (1) = u (2) . In order to make this argument rigorous, we need to work as in section 2 with a regularization (replacing u i + ε). In fact, the previous argument can be generalized to the following family of semimetrics. Let d 1 (u, v) = Ω g(u, v)dx for some smooth symmetric convex function g and let u (1) and u (2) be two solutions to the scalar equation (19) ∂ t u = div(a(x)∇u + uF (x)), which resembles (15), with no-flux boundary conditions and the same initial condition. We assume that a(x) ≥ 0 and F (x) ∈ R n . Set (1) , u (2) ). Then, formally, Since g is convex, the first integral is nonnegative. If we assume that then the second integral vanishes (using the symmetry of g) and consequently, d dt d 1 (u (1) , u (2) ) ≤ 0, which implies that u (1) = u (2) . The integrands of the semimetrics (5) and (18) satisfy condition (20). This argument shows that the linearity in the diffusion term of (19) is essential for the entropy method.

Examples
Theorem 1 can be applied to some cross-diffusion systems arising in applications.

4.1.
Maxwell-Stefan equations. The first example are the Maxwell-Stefan equations [24,31] (21) where J i are the fluxes and d ij the diffusion coefficients. For a formal derivation, see [18,Section 4.2]. We assume that the sum of all concentrations is constant, n+1 i=1 u i = 1, which implies that n+1 i=1 J i = 0. In contrast to (1), the fluxes are not a linear combination of the gradients ∇u i , and we need to invert the flux-gradient relations. However, because of n+1 i=1 J i = 0, the relations cannot be directly inverted. One idea is to remove the variable u n+1 = 1 − n i=1 , ending up with n equations, formulated as ∇u ′ = A 0 J ′ [19], where u ′ = (u 1 , . . . , u n ), J ′ = (J 1 , . . . , J n ), and is invertible. The existence of global bounded weak solutions was shown in [19]. Proof. By assumption, we have A computation shows that the inverse A(u) = A −1 0 is given by This expression is of the form (4) with a i = 1 and The assumptions of Theorem 1 are satisfied since r(s) = 0 and This concludes the proof.

4.2.
Shigesada-Kawasaki-Teramoto equations. The second example is the Shigesada-Kawasaki-Teramoto system (1) arising in population dynamics [30] with coefficients (22) A ij (u) = δ ij a i0 + n j=1 a ij u j + a ij u i , B ij (u) = δ ij u i , i, j = 1, . . . , n, where a ij > 0 for i = 0, . . . , n, j = 1, . . . , n. The variables u i model population densities of interacting species subject to some environmental potential. A formal derivation was given in [18,Section 4.2]. The existence of global weak solutions was proved in [9] (with B ij = 0) under the assumption that there exists a vector (π 1 , . . . , π n ) such that the detailed-balance condition π i a ij = π j a ij for all i, j = 1, . . . , n holds or the self-diffusion a ii dominates crossdiffusion a ij (i = j). Under additional conditions and for n = 2, the weak solutions are bounded [20].
Note that under the conditions of the corollary, the detailed-balance condition is satisfied with π i = a i . The corollary follows from Theorem 1 by setting p(s) = a 0 + s ≥ a 0 > 0, q(s) = 1, and r(s) = 1.

4.3.
A volume-filling model for ion transport. The ion-transport model is defined by (23) A ij (u) = D i u i for i = j, A ii (u) = D i (1 − u 0 + u i ), where u 0 = n i=1 u i and D i > 0, z i ∈ R are some constants [7]. The variables u i represent the ion concentraton of the ith species and u n+1 := 1 − u 0 the solvent concentration. The model can be derived formally from a random-walk lattice model [26,18]. The existence of global bounded weak solutions was shown in [32] without potential and in [15] including the potential term. Formulation (4) is obtained for D i = D > 0 and z i = z ∈ R by setting a i = 1, p i (s) = D(1 − s), q i (s) = D, and r i (s) = z(1 − s). The following result was already proved in [15]. We show here that the model fits in our framework.