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Superlinear nonlocal diffusion systems for multispecies populations: well-posedness and discrete chain rules
Peter Hirvonen, Ansgar Jüngel, Annamaria Pollino
TL;DR
The paper addresses analysis of nonlocal population-diffusion systems that approximate the SKT model. It develops discrete chain-rule inequalities to obtain entropy estimates and proves global existence and uniqueness of strong solutions. Its estimates also identify restrictions on diffusion conditions and on the long-time behavior that can be expected.
Problem
The paper seeks a well-posedness analysis for nonlocal cross-diffusion systems approximating the SKT population model.
Source: §1 · pp. 1–2Method
Discrete chain-rule inequalities establish an entropy inequality and a priori estimates for the nonlocal system.
Source: §1 · p. 2, §1.2 · p. 4Results
The system has global existence and uniqueness of strong solutions under the stated assumptions.
Source: §1.3 · p. 5Takeaways & Limitations
The discrete chain-rule inequalities also provide a basis for structure-preserving finite-volume schemes for SKT systems.
Source: §1 · p. 2The discrete chain-rule condition is more restrictive than the corresponding continuous condition.
Source: §1.2 · p. 4
Abstract
from arXiv · showhide
A nonlocal diffusion system describing the dynamics of multi-species populations and approximating the Shigesada-Kawasaki-Teramoto (SKT) model is analyzed in a bounded domain. The model consists of a system of integro-differential equations of Andreu-Maz\'on-Rossi-Toledo type, in which a nonlocal cross-diffusion operator acts on nonlinear diffusion potentials. We establish the global existence and uniqueness of strong solutions. The existence proof relies on suitable a priori estimates derived from an entropy inequality, whose proof requires the development of novel discrete chain-rule inequalities. As a by-product, these inequalities provide the basis for the construction of structure-preserving finite-volume schemes for the corresponding local SKT system. Uniqueness of solutions is established by means of a duality argument. Finally, one-dimensional numerical simulations illustrate the nonlocal-to-local limit and the qualitative behavior of solutions for superlinear and sublinear diffusion potentials.
1. Introduction
The paper studies a nonlocal cross-diffusion system approximating the SKT population model and develops discrete chain-rule inequalities for its analysis. It establishes global existence and uniqueness of strong solutions, while noting limits on the diffusion conditions and long-time results.
1. Introduction
- The system uses nonlocal integral operators based on pairwise differences rather than second-order derivatives to model population dispersal.The paper motivates this framework for long-range movements that classical local diffusion does not adequately represent.Source: §1 · p. 1
1.2. Key ideas.
- New discrete chain-rule inequalities yield the entropy inequality and a priori bounds needed for existence, and also underpin structure-preserving finite-volume schemes for SKT systems.The inequalities replace chain-rule equalities that do not extend to the present nonlocal model.Source: §1 · p. 2, §1.2 · p. 4
1.1. Model equations.
- The model extends an earlier two-species, s = 1 case to superlinear pressure functions with s > 1 and an arbitrary number of species.It is presented as a nonlocal approximation of the local SKT equations.Source: §1.1 · p. 2
1.2. Key ideas.
- The discrete chain-rule condition is more restrictive than the corresponding continuous condition, and the authors consider the case s < 1 unresolved because suitable inequalities are unclear.They also state that the discrete inequalities are generally not optimal compared with their continuous counterparts.Source: §1.2 · p. 4, §1.3 · p. 5
1.3. Main results.
- Under the stated assumptions, the system has a global nonnegative strong solution and that solution is unique.The existence theorem provides bounds for the solution; the uniqueness theorem applies under the same assumptions.Source: §1.3 · p. 5
- The paper identifies s = 2 as the only case in which its estimates support expecting a long-time asymptotics result.The stated lower bounds require both s ≥ 2 and 1 < s ≤ 2.Source: §1.3 · p. 6
2. Discrete chain rules
The section develops discrete inequalities that replace continuous chain-rule equalities and support the entropy estimates used in the existence analysis.
Discrete chain rules
- The preceding inequality is a discrete counterpart of the continuous chain rule relating ∇u^β·∇u^γ to |∇u^((β+γ)/2)|^2.Source: §2 · p. 6
- New discrete inequalities provide analogues of the chain-rule identity (1 −1/s)∇u^s = u∇u^(s−1).They are introduced as tools for deriving approximate entropy inequalities.Source: §2 · p. 6
- For 1 < s ≤ 2, inequality (12) bounds (a^s − b^s)^2 below by (a^(s−1) − b^(s−1))^2(a + b)^2.Source: §2 · p. 6
- Inequalities (12) and (13) hold for all a, b ≥ 0.Source: §2 · p. 6
3. Existence of a global strong solution
The proof constructs approximate solutions, derives entropy-based uniform estimates, and passes through time-discretization and kernel-regularization limits to obtain a solution on any finite time interval.
3.1. Approximate problem.
- Each implicit-Euler step has a unique solution for sufficiently small τ_k; the solution is nonnegative, while step sizes may need to shrink over time.Source: §3.1 · pp. 8–9
3.2. Approximate entropy inequalities.
- Under either Assumption (A5a) or (A5b), the approximate solutions satisfy entropy inequalities.Source: §3.2 · pp. 9, 13
3.3. Further uniform estimates.
- The estimates include uniform L1 and Ls bounds and uniform L∞ and W1,∞ bounds, which support extraction of convergent subsequences.Source: §3.3 · pp. 14–15, §3.4 · p. 17
3.4. Limit τ →0.
- As τ→0, the approximants converge strongly in L∞ and almost everywhere; the limit satisfies the weak formulation and entropy inequality.Source: §3.4 · p. 17
3.5. Limit η →0.
- The η→0 limit solves (1)–(2) with the stated time regularity and BV spatial regularity; continuation extends the solution to arbitrary T > 0.Source: §3.4 · p. 17, §3.5 · p. 18
4. Uniqueness of solutions
The duality method proves uniqueness by showing that the difference between two strong solutions vanishes. A fixed-point construction supplies the dual problem’s solution, while the section also reports exponential convergence to a constant steady state when s = 2.
4. Uniqueness of solutions
- Testing the equation for the difference of two strong solutions with an appropriate dual solution yields that their difference vanishes, proving uniqueness.The dual problem is solved using a Banach fixed-point argument, first on short time intervals and then on [0, T].Source: §4 · pp. 18–20
- The dual problem’s operator is a contraction on sufficiently short intervals, and iterating the construction gives a unique solution on [0, T].The contraction condition is T0 < 1/C, with C independent of T0.Source: §4 · p. 20
- For s = 2, solutions with no reaction terms and κ0 > 0 converge exponentially to the constant steady state.The relative entropy is bounded by its initial value times e^(−κ0t/C_P).Source: §4 · pp. 20–21
5. Numerical experiment
One-dimensional simulations examine the steady-state behavior, localization limit, and diffusion-exponent effects in a two-species model. They show that small ε approximates the local model and that the solutions behave similarly for the tested exponents.
5. Numerical experiment
- The experiments use a one-dimensional two-species model on Ω = (0, 1), with reaction terms neglected and a semi-implicit discretization.The spatial grid size is ∆x = 0.02 and the time step is ∆t = 0.004.Source: §5 · p. 21
- For small ε, the nonlocal solution accurately approximates the local solution; moreover, both solutions approach the constant steady state as time tends to infinity, for s = 1.2.Source: §5 · p. 21
- With ε = 0.08, simulations suggest solutions behave similarly for different exponent values s, indicating analogous analytical results should also hold for s < 1.Source: §5 · p. 21
Appendix A. Entropy-dissipating finite-volume schemes
The appendix applies discrete chain-rule inequalities to construct structure-preserving finite-volume schemes for cross-diffusion systems. The resulting scheme has a discrete entropy inequality with constants independent of cell size and time step.
Appendix A. Entropy-dissipating finite-volume schemes
- Discrete chain rules provide the basis for structure-preserving finite-volume schemes for cross-diffusion systems with initial and no-flux boundary conditions.The scheme is a two-point flux approximation method.Source: Appendix A · p. 21
- The scheme is formulated as an implicit Euler finite-volume method on an admissible triangulation with internal and external edges.Its discrete entropy inequality is derived using convexity and discrete integration by parts.Source: Appendix A · pp. 21–24
- The proposed scheme preserves the local flux, total mass, and entropy structure without edge values or evaluation of a Stolarsky-type mean.The paper notes that evaluating such a mean may be computationally delicate and identifies these as advantages over.Source: Appendix A · p. 24
- The finite-volume solution satisfies a discrete entropy inequality under Assumption (A5a) or (A5b), with constants independent of cell size and ∆t.The inequality is presented as the analog of the nonlocal entropy inequality.Source: Appendix A · p. 24