--- title: Cross-Diffusion Systems & Volume Filling url: https://www.emergentmind.com/topics/cross-diffusion-systems-with-volume-filling type: topic --- # Cross-Diffusion Systems & Volume Filling Cross-diffusion systems with volume filling are nonlinear parabolic systems for densities or volume fractions in which the flux of each component depends on the full composition and on a crowding constraint, typically through a void fraction \(u_0=1-\sum_{i=1}^n u_i\) or through the strict saturation identity \(\sum_{i=1}^n u_i=1\). They arise in population dynamics, biofilm growth, Maxwell–Stefan mixtures, thin-film deposition, tumor-growth and tissue models, chemotaxis, and coarse-grained cell invasion. Their characteristic analytical difficulty is that the diffusion matrix is generally neither symmetric nor positive definite in the original variables, while admissibility is encoded by an entropy structure on a simplex-type state space and by degeneracies or singularities near vacuum or saturation [1502.05617, 1805.02106, 1908.06873, 2604.01827]. ## 1. Admissible states, occupancy constraints, and modeling regimes A standard state space is the open simplex \[ D=\Big\{u\in(0,1)^n:\ \sum_{i=1}^n u_i<1\Big\}, \] with solvent or vacancy fraction \(u_0=1-\sum_{i=1}^n u_i\). This formulation appears in degenerate population models, volume-filling biofilm systems, entropy-based finite-volume schemes, and general weak–strong uniqueness theory. It encodes finite-size exclusion by requiring \(0\le u_i\le 1\) and \(\sum_i u_i\le 1\), so that the free volume \(u_0\) remains nonnegative [1502.05617, 1805.02106, 2105.05476, 2509.25978]. A distinct regime is strict filling, where the mixture occupies all available volume and \[ \sum_{i=1}^n u_i=1. \] This is the natural constraint in the Maxwell–Stefan formulation for volume fractions, in one-dimensional moving-interface deposition models, and in several multiphase tissue and gas-mixture models. In that setting, admissible states belong to \[ A=\{v\in\mathbb{R}_+^n:\ \sum_i v_i=1\}, \] and the fluxes satisfy \(\sum_i J_i=0\), reflecting incompressibility or vanishing total flux [2007.09951, 2303.15817, 2407.15457]. A common misconception is that volume filling always means a hard saturation law \(\sum_i u_i=1\). The literature also contains soft-crowding models in which the transport is driven by a common pressure \(P(\rho)\), with \(\rho=\sum_i u_i\), but without a hard constraint \(0\le \rho\le 1\). In the one-dimensional mixing system with independent drifts, crowding is realized through pressure-type diffusion \(J_i=-\rho_i\partial_x(P(\rho)+V_i)\), and the paper explicitly states that hard congestion models such as \(P(\rho)=-\log(1-\rho)\) are not treated there [2603.18770]. Boundary conditions are likewise model-dependent. No-flux conditions dominate in population, Maxwell–Stefan, tissue, chemotaxis, and moving-interface settings, while the biofilm system of Daus, Milišić, and Zamponi also admits mixed Dirichlet–Neumann data together with a compatibility condition \(M_D=\sum_i u_{D,i}<1\) and \(\sup_x\sum_i u_{i,0}(x)<1\) [1805.02106]. ## 2. Governing equations and constitutive structures The generic cross-diffusion form is \[ \partial_t u_i=\operatorname{div}\Big(\sum_{j=1}^n A_{ij}(u)\nabla u_j\Big) \] or, with reactions, \(\partial_t u_i-\operatorname{div}(\sum_j A_{ij}(u)\nabla u_j)=r_i(u)\). In the degenerate population model derived from a random-walk lattice system, the diffusion matrix is \[ A_{ij}(u)=\delta_{ij}p_i(u)q(u_{m+1})+u_i\frac{\partial p_i(u)}{\partial u_j}q(u_{m+1})+u_ip_i(u)q'(u_{m+1}), \] where \(u_{m+1}=1-\sum_{i=1}^m u_i\), \(q(0)=0\), and the last term is the explicit volume-filling contribution. The fluxes vanish as the free volume tends to zero, which produces degeneracy at crowding [1502.05617]. A central degenerate–singular example is the biofilm model \[ \partial_t u_i-\sum_{j=1}^n \operatorname{div}\!\big(A_{ij}(u)\nabla u_j\big)=r_i(u), \] with total biomass \(M=\sum_i u_i\le 1\) and \[ A_{ij}(u)=\alpha_i\delta_{ij}p(M)q(M)+\alpha_i u_i\big(p(M)q'(M)-p'(M)q(M)\big). \] Equivalently, \[ \partial_t u_i-\alpha_i\,\operatorname{div}\!\left(p(M)^2\nabla\Big(\frac{u_iq(M)}{p(M)}\Big)\right)=r_i(u). \] This system is porous-medium degenerate as \(M\to0\) and superdiffusion-singular as \(M\to1\); the single-species reduction \[ \partial_t M-\operatorname{div}\!\left(\frac{M^a}{(1-M)^b}\nabla M\right)=0,\qquad a,b>1, \] fixes the relation between \(p\) and \(q\) [1805.02106]. In the strict-filling Maxwell–Stefan setting, the constitutive law is not written directly as \(J=-A(u)\nabla u\). Instead, \[ \partial_t u_i+\operatorname{div}J_i=0,\qquad \nabla u+A(u)J=0,\qquad \langle 1,J\rangle=0, \] with \[ A_{ii}(u)=\sum_{j\neq i} c_{ij}u_j,\qquad A_{ij}(u)=-c_{ij}u_i\quad (i\neq j). \] Eliminating \(J\) yields a cross-diffusion PDE with inverse mobility \(A(u)^{-1}\), but the saddle-point formulation is the natural closure because \(\ker A(u)=\operatorname{Span}\{u\}\) and \(\operatorname{Ran}A(u)=\{w:\langle1,w\rangle=0\}\) [2007.09951]. Multiphase tissue models provide a mechanistic derivation from phase mass balances and force balances. With symmetric drag coefficients \(k_{ij}>0\), intraphase pressure coefficients \(q_{ij}\), and interphase pressure coefficients \(r_{ij}\), the fluxes satisfy \[ K(u)J=-A(u)\nabla u, \] so the governing equation becomes \[ \partial_t u=\operatorname{div}\big(K(u)^{-1}A(u)\nabla u\big). \] When all drag coefficients are equal to one, \(K(u)=I\) and the system reduces to \(\partial_t u=\operatorname{div}(A(u)\nabla u)\). This derivation unifies Maxwell–Stefan diffusion, thin-film solar-cell systems, tumor-growth models, and bounded variants of SKT- and Busenberg–Travis-type population systems [2604.01827]. Specialized subclasses preserve the same occupancy logic. In the volume-filling Keller–Segel model, \[ \partial_t \rho=\nabla\cdot\Big((1-\rho)\rho^{m-1}\nabla\rho-\chi\rho(1-\rho)\nabla c\Big),\qquad \tau\partial_t c=\eta\Delta c-c+\rho, \] both diffusion and chemotactic drift carry the factor \(\rho(1-\rho)\), so the mobility degenerates at \(\rho=0\) and \(\rho=1\) [2605.21296]. In the excluded-volume hard-sphere model, the exact gradient-flow structure survives only in the symmetric case of equal sizes and diffusivities, while the general case admits only an asymptotic gradient-flow structure up to the order of the matched-asymptotic derivation [1607.07044]. ## 3. Entropy formulations and entropy variables The dominant analytical mechanism is an entropy structure. In the simplex setting with explicit free volume, the canonical Boltzmann entropy is \[ h(u)=\sum_{i=0}^n u_i(\log u_i-1),\qquad u_0=1-\sum_{i=1}^n u_i, \] with entropy variables \[ w_i=\frac{\partial h}{\partial u_i}=\log\frac{u_i}{u_0}. \] This formula appears in general volume-filling uniqueness theory, in entropy-characterization results, and in many multiphase models. Its Hessian \[ h''(u)_{ij}=\frac{\delta_{ij}}{u_i}+\frac{1}{u_0} \] couples all species through the vacancy variable [1908.06873, 2509.25978, 2604.01827]. The biofilm model possesses a non-standard entropy, \[ h(u)=\sum_{i=1}^n \big(u_i\log u_i-u_i+1\big)+\int_0^M\log\!\left(\frac{q(s)}{p(s)}\right)\,ds, \qquad M=\sum_{i=1}^n u_i, \] and the associated entropy variables are \[ w_i=\log\!\left(\frac{u_iq(M)}{p(M)}\right)-\log\!\left(\frac{u_{D,i}q(M_D)}{p(M_D)}\right). \] The singular behavior of \(\log(q/p)\) as \(M\to1\) is part of the mechanism preventing saturation, while the lower bound on entropy dissipation controls both a weighted \(|\nabla M|^2\) term and Fisher-type quantities involving \(\nabla\sqrt{u_i}\) [1805.02106]. In the strict-filling Maxwell–Stefan case, the entropy is the mixing entropy \[ E(u)=\int_\Omega \sum_{i=1}^n u_i\log u_i\,dx, \] so \(w_i=\log u_i\) and \(\nabla u=M(u)\nabla w\). The continuous entropy–entropy dissipation inequality has the form \[ \frac{d}{dt}E(u(t)) +\frac{\alpha}{2}\int_\Omega \sum_{i=1}^n |\nabla \sqrt{u_i}|^2\,dx +\frac{c^*}{2}\int_\Omega \sum_{i=1}^n |J_i|^2\,dx \le 0, \] with \(\alpha=4/(c^*+2\bar c)\). This provides a symmetric Onsager structure on the tangent space \(\{w:\langle1,w\rangle=0\}\) even though the original closure is singular in the conserved direction [2007.09951]. The paper by Chen and Jüngel gives a general criterion: a system has an entropy structure if there exists a strictly convex \(h\) such that \[ H(u)A(u)=D^2h(u)\,A(u) \] is symmetric positive definite on the admissible set. Normal ellipticity of \(A(u)\) is necessary, and it becomes sufficient when \(H(u)A(u)\) is symmetric. For volume-filling matrices of the form \[ A_{ij}(u)=d_{ij}p_i(u)q_i(u_0)+u_ip_i(u)q_j(u_0)+u_ig_i(u_0)\frac{\partial p_i}{\partial u_j}(u), \] normal ellipticity follows under positivity assumptions on \(p_i,q_i,q_i'\); if \(q_i=q\) and \(p_i=\exp(\partial_{u_i}\chi)\) with \(\chi\) convex, the entropy density \[ h(u)=\sum_i u_i(\log u_i-1)+\int^{u_0}\log q(s)\,ds+\chi(u) \] yields an entropy structure [1908.06873]. A further refinement is the augmented mobility used in weak–strong uniqueness. Writing \(\bar u=(u_0,u_1,\dots,u_n)\), the augmented matrix \(\bar B(\bar u)\) has row and column sums equal to zero, so it is singular in the conserved direction. Positivity holds only on the tangent subspace \[ L(\bar u)=\{Y\in\mathbb{R}^{n+1}:\ \sqrt{\bar u}\cdot Y=0\}, \] through the matrix \(G(\bar u)=\bar B_{ij}(\bar u)/\sqrt{u_i u_j}\). This subspace positivity is sufficient for the relative entropy method and clarifies why full positive definiteness is unnecessary in volume-filling systems with conservation laws [2509.25978]. Not every model is an exact gradient flow. For hard-sphere excluded-volume systems, the symmetric case admits \[ \partial_t \binom{r}{b}=\nabla\cdot\!\left(M(r,b)\nabla \binom{u}{v}\right), \] with explicit mobility \(M(r,b)\), but the nonsymmetric case satisfies such a formulation only up to an \(O(\epsilon^{2d})\) remainder. The paper terms this an asymptotic gradient flow, which isolates the order at which entropy methods remain structurally informative [1607.07044]. ## 4. Existence, uniqueness, and long-time behavior Global weak solvability via entropy methods is now available for several major subclasses. The degenerate population model with volume filling admits global-in-time bounded weak solutions for arbitrary species number in a bounded domain with no-flux boundary conditions. The proof uses approximation, entropy dissipation, and nonlinear Aubin–Lions compactness lemmas adapted to degenerate weights. Under additional assumptions, the same framework yields exponential convergence to the constant steady state and uniqueness when \(p_i\equiv1\) via a combination of the \(H^{-1}\) method, Gajewski’s \(E\)-monotonicity, and subadditivity of the Fisher information [1502.05617]. For the biofilm system of Daus, Milišić, and Zamponi, one has global-in-time weak solutions under general reactions and mixed Dirichlet–Neumann data, together with \[ u_i\ge0,\qquad M=\sum_i u_i<1\quad\text{a.e. in }Q_T. \] The entropy inequality yields the singular integrability estimate \[ \int_0^T\!\!\int_\Omega (1-M)^{1-b-\kappa}\,dx\,dt\le C, \] which excludes saturation. Under \(\lambda_r=C_r=0\) and \(b\ge2\), the relative entropy decays algebraically and \[ \sum_{i=1}^n \|u_i(t)-u_{D,i}\|_{L^2(\Omega)}^2\le \frac{C}{1+t}. \] Uniqueness is proved under the additional assumptions \(\alpha_i=1\) and \[ r_i(u)=r_i^{(0)}(M)+r^{(1)}(M)u_i, \] using an \(H^{-1}\)-type argument for the total biomass and Gajewski’s semimetric for the species components [1805.02106]. Large-time asymptotics remain delicate in the genuinely degenerate case. Chen, Jüngel, Lin, and Liu consider \[ \partial_t u_i=\operatorname{div}\Big(\sum_j A_{ij}(u)\nabla u_j\Big) \] with different diffusivities \(D_i>0\) and volume-filling function \(q(u_0)\) satisfying \(q'(0)=0\). Standard logarithmic Sobolev estimates do not directly relate entropy and entropy production. Their key device is to apply convex Sobolev inequalities to modified entropy densities involving \(q\circ q\), which yields \[ u_i(t)\to u_i^\infty \quad\text{strongly in }L^p(\Omega)\ \text{for all }12\) the one-dimensional steady problem admits nonconstant increasing equilibria, giving a rigorous pattern-formation regime [2605.21296]. In multiphase tissue models, equal drag coefficients lead to global bounded weak solutions and exponential convergence to the spatial average, whereas unequal drag yields only positive stability and local-in-time existence in general; global entropy-based existence is recovered when the drag coefficients are sufficiently close to one, or in explicit two-species parameter regimes [2604.01827]. ## 5. Numerical discretization and structure preservation Structure-preserving finite-volume methods mirror the continuous entropy mechanics. For the strict-filling Maxwell–Stefan system, Cancès, Ehrlacher, and Monasse propose a two-point flux approximation in which interface values are chosen as logarithmic means, \[ u_{i,\sigma}= \frac{u_{i,K}-u_{i,L}}{\log u_{i,K}-\log u_{i,L}}, \] whenever both traces are positive. The discrete unknowns satisfy implicit Euler mass balances and discrete Maxwell–Stefan flux relations in entropy variables. The resulting scheme preserves positivity, species masses, the strict volume-filling constraint \(\sum_i u_i=1\), the edgewise flux constraint \(\sum_i J_i=0\), and a discrete entropy–entropy dissipation inequality closely matching the continuous one. Under mesh refinement, the approximate solutions converge to weak solutions; in one space dimension the observed \(L^1(Q_T)\) convergence rate is \(O(N^{-2})\), and in two-dimensional simulations the relative entropy decays exponentially [2007.09951]. A more general discrete boundedness-by-entropy framework was developed by Bessemoulin-Chatard, Chainais-Hillairet, Filbet, and Jüngel for source-free cross-diffusion systems with simplex constraint \(u_0=1-\sum_i u_i\). Their scheme uses a vector-valued discrete chain rule \[ H(u_\sigma)\,D_{K,\sigma}u=D_{K,\sigma}h'(u) \] to handle the non-diagonal entropy Hessian caused by volume filling. The corresponding edge means are defined by scalar equations involving \(h_i''\), reducing to logarithmic means for Boltzmann entropy. Under the coercivity condition \[ z^\top H(u_\sigma)A_\sigma(u_\sigma)z\ge c_A\sum_i u_{i,\sigma}^{2(s-1)}z_i^2, \] the method yields existence of discrete solutions, positivity, preservation of the cellwise volume-filling constraint, a discrete entropy inequality, and convergence to weak solutions. The framework covers Maxwell–Stefan, thin-film solar-cell, and certain tumor-growth models; the one-dimensional Maxwell–Stefan tests again show second-order spatial convergence, while thin-film relative entropies decay exponentially [2105.05476]. Moving-interface problems require additional geometric care. The cut-cell finite-volume scheme for vapor deposition updates the interface position through Butler–Volmer-type reaction fluxes while preserving cellwise saturation, mass conservation, nonnegativity, and free-energy decay. Its discrete free energy is a sum of solid- and gas-phase entropies weighted by the deformed control-volume lengths, and the interface motion is coupled consistently to the discrete species fluxes. Numerical tests show monotone free-energy decay, convergence toward the stationary two-phase state, and first-order spatial accuracy, with the moving-interface treatment limiting the order relative to fixed-interface schemes [2303.15817]. The later moving-mesh formulation of the same class of interface-coupled systems keeps the entropy structure explicit in both phases. It uses logarithmic means, mobility-form fluxes, and local mesh deformation around the interface. The scheme preserves mass conservation, nonnegativity, the volume-filling constraints, decay of the free energy, and the asymptotic approach to stationary states. Numerical experiments show exponential decay in cases with a unique two-phase equilibrium and finite-time collapse to a one-phase stationary state when the equilibrium condition fails [2407.15457]. ## 6. Related model families, applications, and open directions The range of applications is broad. Biofilm growth is modeled by degenerate–singular multi-species equations with mixed boundary conditions and general reactions [1805.02106]. Maxwell–Stefan systems describe multicomponent gas or fluid mixtures with strict filling and pairwise drag coefficients [2007.09951]. Multiphase tissue models derive cross diffusion from force balances combining interphase pressures and viscous drag, and recover Maxwell–Stefan, tumor-growth, thin-film solar-cell, and bounded population systems as special cases [2604.01827]. Volume-filling chemotaxis couples density saturation to drift, yielding bounded Keller–Segel dynamics and asymptotic limits [2605.21296]. Volume filling also appears outside the classical simplex setting. The one-dimensional mixing model with independent drifts uses common-pressure diffusion \(J_i=-\rho_i\partial_x(f'(\rho)+V_i)\) and removes the “total mixing” assumption on initial supports by introducing an inhomogeneous ratio variable \(\phi^\eta\) that satisfies a linear parabolic transport equation. This yields global weak solutions with \(r=\rho_1/\sigma\in L^\infty(0,T;BV(\Omega))\), but uniqueness and higher-dimensional extensions remain open [2603.18770]. In coarse-grained cell invasion, the vacancy \(v=1-c-e\) leads to \[ \partial_t c=\nabla\cdot(v\nabla c-c\nabla v)+cv,\qquad \partial_t e=-\lambda ec, \] and the one-dimensional traveling-wave analysis gives the minimal speed \(c_{\min}=2(1-m_0)\). The model captures a pinning phenomenon when the ECM ahead is fully occupied, a feature absent from simpler reductions [2302.11345]. Control-theoretic extensions are also present. In the bilinear optimal-control problem for a degenerate chemotaxis system with volume-filling effect, \[ \partial_t N-\nabla\cdot(a(N)\nabla N)+\nabla\cdot(x(N)\nabla C)=0, \] the structural assumptions \[ a(0)=a(1)=0,\qquad x(0)=x(1)=0 \] encode two-sided degeneracy and saturation \(0\le N\le1\). The paper proves well-posedness of the state system, existence of an optimal control, first-order optimality conditions, and weak solvability of the degenerate adjoint equations [2407.07519]. Several open problems are explicit in the literature. In the biofilm model, numerical simulations show faster convergence than the algebraic rate presently proved, and extending the entropy method to exponential decay remains open [1805.02106]. In the multiphase tissue framework, the gap between equal-drag entropy theory and general-drag positive stability is explained by a mismatch between degenerate pressure forces and nondegenerate drag forces, leaving truly global entropy methods for arbitrary drag coefficients unresolved [2604.01827]. For the degenerate Keller–Segel system, higher-dimensional pattern selection and sharp thresholds in \((m,\chi)\) remain open [2605.21296]. For pressure-type mixing with independent drifts, the present theory is one-dimensional and does not address uniqueness [2603.18770]. Taken together, these developments show that “volume filling” is not a single constitutive law but a structural principle: admissible states lie on a simplex or saturation manifold, diffusion degenerates at the crowded boundary, and well-posedness is typically recovered not from uniform ellipticity but from entropy, relative entropy, or free-energy dissipation. The modern theory therefore hinges less on symmetry of the original diffusion matrix than on the availability of the correct entropy variables, tangent-space coercivity, and structure-preserving discretizations [1908.06873, 2509.25978, 2105.05476]. Source: https://www.emergentmind.com/topics/cross-diffusion-systems-with-volume-filling