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 self-consistently 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{-1}$$\end{document} 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.

Model equations
The equations describe the evolution of the concentrations u i , in a bounded domain ⊂ R d (d ≥ 1), where u = (u 1 , . . . , u n ) and φ is a potential solving the Poisson equation where u = n i=1 a i u i for some constants a i ≥ 0, and f (x) is a given background concentration. We complement the equations by no-flux boundary and initial conditions, n j=1 A i j (u)∇u j · ν = ∇φ · ν = 0 on ∂ , u(0) = u 0 in , i = 1, . . . , n. (3) For consistency, the initial datum has to satisfy the condition The diffusion coefficients A i j and drift coefficients B i j are defined by 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 satisfies a nonlinear driftdiffusion equation (see (12)), 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 i j (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) and (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 fluid dynamical 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 , t > 0.
For special choices of A i j and B i j , including condition (4), the existence of global bounded weak solutions can be shown. We give some examples and references in Sect. 4. In this paper, we are only concerned with the uniqueness of weak solutions. 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) ) ≤ C H(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 reaction-diffusion systems (with diagonal diffusion matrix) [13]. A second idea, due to Gajewski [14], is to time differentiate the semimetric d(u (1) , u (2) ) = H (u (1) ) + H (u (2) 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 i j (u). With hypothesis (4), equations (1) can be formulated as This can be interpreted as a drift-diffusion equation with field term q(u)∇u +r (u)∇φ.
Since u depends on all u i , this is still a cross-diffusion system. However, the driftdiffusion structure is essential in the uniqueness proof. Our main result is as follows.
Then (u, φ) is unique in the class of solutions satisfying φdx = 0, ∇φ ∈ L ∞ (0, T ; L ∞ ( )), and In the case r ≡ 0, the boundedness of u is not needed, provided that REMARK 2.
1. The regularity assumption on the potential can be relaxed to ∇φ ∈ The idea of the proof is first to show the uniqueness of (u, φ). Indeed, multiplying (1) by a i and summing over i = 1, . . . , n leads to a nonlinear drift-diffusion equation for u, Vol. 18 (2018) A note on the uniqueness of weak solutions 809 coupled with Poisson equation (2), where Since the diffusion operator in (10) may degenerate, it is natural to apply the H −1 technique. Indeed, given two solutions (u (1) , φ (1) ), (u (2) , φ (2) ) with the same initial data, we use the test function φ := φ (1) − φ (2) in (12), which solves the dual problem − φ = u (1) − u (2) in , ∇φ · ν = 0 on ∂ . Then, using conditions (7)- (8), it can be shown that d dt (2) and φ (1) = φ (2) . In this step, we need the regularity ∇φ (2) The second step is to prove the uniqueness of (6). For this, we employ the method of Gajewski [14], based on an estimation of the semimetric n ) be two weak solutions to (6). A formal computation shows that d dt d(u (1) , u (2) ) ≤ 0 and hence d(u (1) , u (2) ) = 0. The convexity of h implies that u (1) = u (2) . In order to make this argument rigorous, we need to regularize the entropy, since terms with log u i may be not defined on sets where u i = 0. We discuss in Remark 4 the applicability of the Gajewski method.
The paper is organized as follows. Theorem 1 is proved in Sect. 2. Section 3 is concerned with some comments on the techniques and the proof. Some examples satisfying conditions (7)-(8) are detailed in Sect. 4.
Therefore, we can differentiate t → d ε (u (1) Vol. 18 (2018) A note on the uniqueness of weak solutions 813 Rearranging the terms, we end up with d dt d ε (u (1) , u (2) Since for suitable functions u, v, the first term is nonpositive. Then, integrating in time and observing that d ε (u (1) Expanding h ε u (1) i and h ε u (2) i at u (1) i + u (2) i /2 up to second order and summing the resulting expressions, we find that, for some θ i } + 1 dx.
Since F · ∇u j i ∈ L 1 (Q T ) for j = 1, 2, we may apply the dominated convergence theorem giving, as ε → 0, dxds → 0, j = 1, 2. Therefore, (16) becomes is finite a.e. in . If r ≡ 0 and u is not bounded, then we need integrability (9) to make the computations rigorous. This concludes the proof of Theorem 1.

Remarks
We give two comments on the regularity of the drift term and on the relation of Gajewski's semimetric to relative entropies.

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 . . , 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: 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 d dt d 0 (u (1) , u (2) implying that u (1) = u (2) . In order to make this argument rigorous, we need to work as in Sect. 2 with a regularization (replacing u . In fact, the previous argument can be generalized to the following family of semimetrics. Let d 1 (u, v) = 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 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) ), g 12 = ∂ 2 g ∂u∂v (u (1) , u (2) ), g 22 = ∂ 2 g ∂v 2 (u (1) , u (2) ).
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 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.

Maxwell-Stefan equations
The first example is the Maxwell-Stefan equations [24,31] where J i are the fluxes and d i j 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 u i , 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]. Vol. 18 (2018) A note on the uniqueness of weak solutions 817 COROLLARY 5 (Maxwell-Stefan model). Let d i j = D 0 and d i,n+1 = D for i, j = 1, . . . , n. Then from Maxwell-Stefan system (3), (21) has at most one weak solution.
Proof. By assumption, we have A computation shows that the inverse A(u) = A −1 0 is given by This concludes the proof.

Shigesada-Kawasaki-Teramoto equations
The second example is Shigesada-Kawasaki-Teramoto system (1) arising in population dynamics [30] with coefficients A i j (u) = δ i j a i0 + n j=1 a i j u j + a i j u i , B i j (u) = δ i j u i , i, j = 1, . . . , n, (22) where a i j > 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 i j = 0) under the assumption that there exists a vector (π 1 , . . . , π n ) such that the detailed-balance condition π i a i j = π j a i j for all i, j = 1, . . . , n holds or the self-diffusion a ii dominates cross-diffusion a i j (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.