Cross-Diffusion Systems & Volume Filling

• Cross-diffusion systems with volume filling are nonlinear parabolic models where each species' flux depends on the entire mixture and a crowding constraint, typically defined on a simplex or saturation manifold.
• The models employ entropy structures and degeneracies near saturation to ensure admissibility and compensate for the non-symmetric, non-positive definite diffusion matrices.
• Applications span biofilm growth, tumor tissue modeling, and chemotaxis, with structure-preserving numerical schemes critical for mass conservation and stability.

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 (Zamponi et al., 2015, Daus et al., 2018, Chen et al., 2019, Jüngel et al., 2 Apr 2026).

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 (Zamponi et al., 2015, Daus et al., 2018, Jüngel et al., 2021, Heitzinger et al., 30 Sep 2025).

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 (Cancès et al., 2020, Cancès et al., 2023, Cancès et al., 2024).

A common misconception is that volume filling always means a hard saturation law $\sum_{i=1}^n u_i=1$0. The literature also contains soft-crowding models in which the transport is driven by a common pressure $\sum_{i=1}^n u_i=1$1, with $\sum_{i=1}^n u_i=1$2, but without a hard constraint $\sum_{i=1}^n u_i=1$3. In the one-dimensional mixing system with independent drifts, crowding is realized through pressure-type diffusion $\sum_{i=1}^n u_i=1$4, and the paper explicitly states that hard congestion models such as $\sum_{i=1}^n u_i=1$5 are not treated there (Mészáros et al., 19 Mar 2026).

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 $\sum_{i=1}^n u_i=1$6 and $\sum_{i=1}^n u_i=1$7 (Daus et al., 2018).

2. Governing equations and constitutive structures

The generic cross-diffusion form is

$\sum_{i=1}^n u_i=1$8

or, with reactions, $\sum_{i=1}^n u_i=1$9. In the degenerate population model derived from a random-walk lattice system, the diffusion matrix is

$D=\Big\{u\in(0,1)^n:\ \sum_{i=1}^n u_i<1\Big\},$0

where $D=\Big\{u\in(0,1)^n:\ \sum_{i=1}^n u_i<1\Big\},$1, $D=\Big\{u\in(0,1)^n:\ \sum_{i=1}^n u_i<1\Big\},$2, 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 (Zamponi et al., 2015).

A central degenerate–singular example is the biofilm model

$D=\Big\{u\in(0,1)^n:\ \sum_{i=1}^n u_i<1\Big\},$3

with total biomass $D=\Big\{u\in(0,1)^n:\ \sum_{i=1}^n u_i<1\Big\},$4 and

$D=\Big\{u\in(0,1)^n:\ \sum_{i=1}^n u_i<1\Big\},$5

Equivalently,

$D=\Big\{u\in(0,1)^n:\ \sum_{i=1}^n u_i<1\Big\},$6

This system is porous-medium degenerate as $D=\Big\{u\in(0,1)^n:\ \sum_{i=1}^n u_i<1\Big\},$7 and superdiffusion-singular as $D=\Big\{u\in(0,1)^n:\ \sum_{i=1}^n u_i<1\Big\},$8; the single-species reduction

$D=\Big\{u\in(0,1)^n:\ \sum_{i=1}^n u_i<1\Big\},$9

fixes the relation between $u_0=1-\sum_{i=1}^n u_i$0 and $u_0=1-\sum_{i=1}^n u_i$1 (Daus et al., 2018).

In the strict-filling Maxwell–Stefan setting, the constitutive law is not written directly as $u_0=1-\sum_{i=1}^n u_i$2. Instead,

$u_0=1-\sum_{i=1}^n u_i$3

with

$u_0=1-\sum_{i=1}^n u_i$4

Eliminating $u_0=1-\sum_{i=1}^n u_i$5 yields a cross-diffusion PDE with inverse mobility $u_0=1-\sum_{i=1}^n u_i$6, but the saddle-point formulation is the natural closure because $u_0=1-\sum_{i=1}^n u_i$7 and $u_0=1-\sum_{i=1}^n u_i$8 (Cancès et al., 2020).

Multiphase tissue models provide a mechanistic derivation from phase mass balances and force balances. With symmetric drag coefficients $u_0=1-\sum_{i=1}^n u_i$9, intraphase pressure coefficients $0\le u_i\le 1$0, and interphase pressure coefficients $0\le u_i\le 1$1, the fluxes satisfy

$0\le u_i\le 1$2

so the governing equation becomes

$0\le u_i\le 1$3

When all drag coefficients are equal to one, $0\le u_i\le 1$4 and the system reduces to $0\le u_i\le 1$5. 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 (Jüngel et al., 2 Apr 2026).

Specialized subclasses preserve the same occupancy logic. In the volume-filling Keller–Segel model,

$0\le u_i\le 1$6

both diffusion and chemotactic drift carry the factor $0\le u_i\le 1$7, so the mobility degenerates at $0\le u_i\le 1$8 and $0\le u_i\le 1$9 (Geltner et al., 20 May 2026). 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 (Bruna et al., 2016).

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

$\sum_i u_i\le 1$0

with entropy variables

$\sum_i u_i\le 1$1

This formula appears in general volume-filling uniqueness theory, in entropy-characterization results, and in many multiphase models. Its Hessian

$\sum_i u_i\le 1$2

couples all species through the vacancy variable (Chen et al., 2019, Heitzinger et al., 30 Sep 2025, Jüngel et al., 2 Apr 2026).

The biofilm model possesses a non-standard entropy,

$\sum_i u_i\le 1$3

and the associated entropy variables are

$\sum_i u_i\le 1$4

The singular behavior of $\sum_i u_i\le 1$5 as $\sum_i u_i\le 1$6 is part of the mechanism preventing saturation, while the lower bound on entropy dissipation controls both a weighted $\sum_i u_i\le 1$7 term and Fisher-type quantities involving $\sum_i u_i\le 1$8 (Daus et al., 2018).

In the strict-filling Maxwell–Stefan case, the entropy is the mixing entropy

$\sum_i u_i\le 1$9

so $u_0$0 and $u_0$1. The continuous entropy–entropy dissipation inequality has the form

$u_0$2

with $u_0$3. This provides a symmetric Onsager structure on the tangent space $u_0$4 even though the original closure is singular in the conserved direction (Cancès et al., 2020).

The paper by Chen and Jüngel gives a general criterion: a system has an entropy structure if there exists a strictly convex $u_0$5 such that

$u_0$6

is symmetric positive definite on the admissible set. Normal ellipticity of $u_0$7 is necessary, and it becomes sufficient when $u_0$8 is symmetric. For volume-filling matrices of the form

$u_0$9

normal ellipticity follows under positivity assumptions on $\sum_{i=1}^n u_i=1.$0; if $\sum_{i=1}^n u_i=1.$1 and $\sum_{i=1}^n u_i=1.$2 with $\sum_{i=1}^n u_i=1.$3 convex, the entropy density

$\sum_{i=1}^n u_i=1.$4

yields an entropy structure (Chen et al., 2019).

A further refinement is the augmented mobility used in weak–strong uniqueness. Writing $\sum_{i=1}^n u_i=1.$5, the augmented matrix $\sum_{i=1}^n u_i=1.$6 has row and column sums equal to zero, so it is singular in the conserved direction. Positivity holds only on the tangent subspace

$\sum_{i=1}^n u_i=1.$7

through the matrix $\sum_{i=1}^n u_i=1.$8. 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 (Heitzinger et al., 30 Sep 2025).

Not every model is an exact gradient flow. For hard-sphere excluded-volume systems, the symmetric case admits

$\sum_{i=1}^n u_i=1.$9

with explicit mobility $A=\{v\in\mathbb{R}_+^n:\ \sum_i v_i=1\},$0, but the nonsymmetric case satisfies such a formulation only up to an $A=\{v\in\mathbb{R}_+^n:\ \sum_i v_i=1\},$1 remainder. The paper terms this an asymptotic gradient flow, which isolates the order at which entropy methods remain structurally informative (Bruna et al., 2016).

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 $A=\{v\in\mathbb{R}_+^n:\ \sum_i v_i=1\},$2 via a combination of the $A=\{v\in\mathbb{R}_+^n:\ \sum_i v_i=1\},$3 method, Gajewski’s $A=\{v\in\mathbb{R}_+^n:\ \sum_i v_i=1\},$4-monotonicity, and subadditivity of the Fisher information (Zamponi et al., 2015).

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

$A=\{v\in\mathbb{R}_+^n:\ \sum_i v_i=1\},$5

The entropy inequality yields the singular integrability estimate

$A=\{v\in\mathbb{R}_+^n:\ \sum_i v_i=1\},$6

which excludes saturation. Under $A=\{v\in\mathbb{R}_+^n:\ \sum_i v_i=1\},$7 and $A=\{v\in\mathbb{R}_+^n:\ \sum_i v_i=1\},$8, the relative entropy decays algebraically and

$A=\{v\in\mathbb{R}_+^n:\ \sum_i v_i=1\},$9

Uniqueness is proved under the additional assumptions $\sum_i J_i=0$0 and

$\sum_i J_i=0$1

using an $\sum_i J_i=0$2-type argument for the total biomass and Gajewski’s semimetric for the species components (Daus et al., 2018).

Large-time asymptotics remain delicate in the genuinely degenerate case. Chen, Jüngel, Lin, and Liu consider

$\sum_i J_i=0$3

with different diffusivities $\sum_i J_i=0$4 and volume-filling function $\sum_i J_i=0$5 satisfying $\sum_i J_i=0$6. 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 $\sum_i J_i=0$7, which yields

$\sum_i J_i=0$8

without an explicit rate in the degenerate regime (Chen et al., 2023).

Weak–strong uniqueness has been established in much broader generality by Heitzinger and Jüngel. For

$\sum_i J_i=0$9

they assume a Boltzmann entropy, coercivity of $\sum_{i=1}^n u_i=1$00, boundedness of the entropy production, and a Lipschitz condition on the augmented mobility. If $\sum_{i=1}^n u_i=1$01 is a bounded weak solution and $\sum_{i=1}^n u_i=1$02 is a positive strong solution with the same initial data, then $\sum_{i=1}^n u_i=1$03 for all later times. The proof is a relative entropy argument based on projection onto the tangent subspace $\sum_{i=1}^n u_i=1$04 (Heitzinger et al., 30 Sep 2025).

A separate uniqueness mechanism applies when the equations collapse to a scalar equation for a weighted sum $\sum_{i=1}^n u_i=1$05. In the class

$\sum_{i=1}^n u_i=1$06

coupled to $\sum_{i=1}^n u_i=1$07, Jüngel and Zamponi prove uniqueness of bounded weak solutions by combining an $\sum_{i=1}^n u_i=1$08 estimate for $\sum_{i=1}^n u_i=1$09 with Gajewski’s semimetric for the vector system (Chen et al., 2017).

Qualitative behavior depends strongly on the specific saturation mechanism. In the degenerate Keller–Segel model with volume filling, global weak solutions exist for all $\sum_{i=1}^n u_i=1$10; for $\sum_{i=1}^n u_i=1$11 one also has weak–strong uniqueness and exponential convergence to the homogeneous steady state under $\sum_{i=1}^n u_i=1$12, while for $\sum_{i=1}^n u_i=1$13 the one-dimensional steady problem admits nonconstant increasing equilibria, giving a rigorous pattern-formation regime (Geltner et al., 20 May 2026). 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 (Jüngel et al., 2 Apr 2026).

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,

$\sum_{i=1}^n u_i=1$14

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=1}^n u_i=1$15, the edgewise flux constraint $\sum_{i=1}^n u_i=1$16, 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 $\sum_{i=1}^n u_i=1$17 convergence rate is $\sum_{i=1}^n u_i=1$18, and in two-dimensional simulations the relative entropy decays exponentially (Cancès et al., 2020).

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 $\sum_{i=1}^n u_i=1$19. Their scheme uses a vector-valued discrete chain rule

$\sum_{i=1}^n u_i=1$20

to handle the non-diagonal entropy Hessian caused by volume filling. The corresponding edge means are defined by scalar equations involving $\sum_{i=1}^n u_i=1$21, reducing to logarithmic means for Boltzmann entropy. Under the coercivity condition

$\sum_{i=1}^n u_i=1$22

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 (Jüngel et al., 2021).

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 (Cancès et al., 2023).

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 (Cancès et al., 2024).

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 (Daus et al., 2018). Maxwell–Stefan systems describe multicomponent gas or fluid mixtures with strict filling and pairwise drag coefficients (Cancès et al., 2020). 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 (Jüngel et al., 2 Apr 2026). Volume-filling chemotaxis couples density saturation to drift, yielding bounded Keller–Segel dynamics and asymptotic limits (Geltner et al., 20 May 2026).

Volume filling also appears outside the classical simplex setting. The one-dimensional mixing model with independent drifts uses common-pressure diffusion $\sum_{i=1}^n u_i=1$23 and removes the “total mixing” assumption on initial supports by introducing an inhomogeneous ratio variable $\sum_{i=1}^n u_i=1$24 that satisfies a linear parabolic transport equation. This yields global weak solutions with $\sum_{i=1}^n u_i=1$25, but uniqueness and higher-dimensional extensions remain open (Mészáros et al., 19 Mar 2026). In coarse-grained cell invasion, the vacancy $\sum_{i=1}^n u_i=1$26 leads to

$\sum_{i=1}^n u_i=1$27

and the one-dimensional traveling-wave analysis gives the minimal speed $\sum_{i=1}^n u_i=1$28. The model captures a pinning phenomenon when the ECM ahead is fully occupied, a feature absent from simpler reductions (Crossley et al., 2023).

Control-theoretic extensions are also present. In the bilinear optimal-control problem for a degenerate chemotaxis system with volume-filling effect,

$\sum_{i=1}^n u_i=1$29

the structural assumptions

$\sum_{i=1}^n u_i=1$30

encode two-sided degeneracy and saturation $\sum_{i=1}^n u_i=1$31. 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 (Chamoun et al., 2024).

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 (Daus et al., 2018). 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 (Jüngel et al., 2 Apr 2026). For the degenerate Keller–Segel system, higher-dimensional pattern selection and sharp thresholds in $\sum_{i=1}^n u_i=1$32 remain open (Geltner et al., 20 May 2026). For pressure-type mixing with independent drifts, the present theory is one-dimensional and does not address uniqueness (Mészáros et al., 19 Mar 2026).

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 (Chen et al., 2019, Heitzinger et al., 30 Sep 2025, Jüngel et al., 2021).