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Well-posedness of aggregation-diffusion systems with irregular kernels

José A. Carrillo, Yurij Salmaniw, Jakub Skrzeczkowski

TL;DR

This work establishes a robust well-posedness theory for aggregation-diffusion systems with irregular, bounded kernels, including kernels of bounded variation that are not differentiable or positive-definite. It proves global existence of weak solutions in two regimes (small initial mass or symmetric BV kernels) and, under additional structural assumptions (such as detailed balance and a mixed regularity condition $\nabla K*K\in L^2$), shows uniqueness and higher regularity leading to strong and classical solutions, also extending to $n$-species systems. The analysis combines entropy dissipation, gradient-flow structure, and maximal parabolic regularity, with a regularization/compactness framework to handle non-smooth interaction kernels. Numerical simulations in 1D using a positivity-preserving scheme illustrate the qualitative behavior, including stationary states and species segregation under irregular kernels, providing insight into long-time dynamics and supporting the theoretical results.

Abstract

We consider aggregation-diffusion equations with merely bounded nonlocal interaction potential $K$. We are interested in establishing their well-posedness theory when the nonlocal interaction potential $K$ is neither differentiable nor positive (semi-)definite, thus preventing application of classical arguments. We prove the existence of weak solutions in two cases: if the mass of the initial data is sufficiently small, or if the interaction potential is symmetric and of bounded variation without any smallness assumption. The latter allows one to exploit the dissipation of the free energy in an optimal way, which is an entirely new approach. Remarkably, in both cases, under the additional condition that $\nabla K\ast K$ is in $L^2$, we can prove that the solution is smooth and unique. When $K$ is a characteristic function of a ball, we construct the classical unique solution. Under additional structural conditions we extend these results to the $n$-species system.

Well-posedness of aggregation-diffusion systems with irregular kernels

TL;DR

This work establishes a robust well-posedness theory for aggregation-diffusion systems with irregular, bounded kernels, including kernels of bounded variation that are not differentiable or positive-definite. It proves global existence of weak solutions in two regimes (small initial mass or symmetric BV kernels) and, under additional structural assumptions (such as detailed balance and a mixed regularity condition ), shows uniqueness and higher regularity leading to strong and classical solutions, also extending to -species systems. The analysis combines entropy dissipation, gradient-flow structure, and maximal parabolic regularity, with a regularization/compactness framework to handle non-smooth interaction kernels. Numerical simulations in 1D using a positivity-preserving scheme illustrate the qualitative behavior, including stationary states and species segregation under irregular kernels, providing insight into long-time dynamics and supporting the theoretical results.

Abstract

We consider aggregation-diffusion equations with merely bounded nonlocal interaction potential . We are interested in establishing their well-posedness theory when the nonlocal interaction potential is neither differentiable nor positive (semi-)definite, thus preventing application of classical arguments. We prove the existence of weak solutions in two cases: if the mass of the initial data is sufficiently small, or if the interaction potential is symmetric and of bounded variation without any smallness assumption. The latter allows one to exploit the dissipation of the free energy in an optimal way, which is an entirely new approach. Remarkably, in both cases, under the additional condition that is in , we can prove that the solution is smooth and unique. When is a characteristic function of a ball, we construct the classical unique solution. Under additional structural conditions we extend these results to the -species system.
Paper Structure (21 sections, 22 theorems, 188 equations, 6 figures)

This paper contains 21 sections, 22 theorems, 188 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Hypotheses
  4. Statement of Main Results
  5. The Scalar Equation
  6. Preliminaries
  7. Apriori estimates
  8. Improved estimates
  9. Existence of weak solutions (proof of Theorems \ref{['thm:existsmallmassscalar']}-\ref{['thm:existarbmassscalar']})
  10. Higher order estimates
  11. Uniqueness
  12. Existence of unique strong and classical solutions (proof of Theorems \ref{['thm:scalarfinal']}-\ref{['thm:scalarclassicalsolution']})
  13. The General -Species System
  14. Apriori estimates
  15. Improved estimates and existence of weak solutions
  16. ...and 6 more sections

Key Result

Theorem 2.1

Assume h1 holds and that where $m := \left\Vert u_0\right\Vert_{L^1(\mathbb{R}^d)}$. Then, there exists a global weak solution $u \geq 0$ in $Q_T$ solving problem eq:mainscalar in the sense of Definition def:weaksoln. Moreover, if $u_0 \in L^p({\mathbb{R}^d})$ then $u \in L^\infty(0,T; L^p({\mathbb{R}^d}))$ for any $1 \leq

Figures (6)

  • Figure 1: Simulation of problem \ref{['application:scalartophat']} with $\alpha = 2.0$ and numerical domain size of $24$ ($L=12$). A supplementary video of this simulation is hosted on figshare found here: https://doi.org/10.6084/m9.figshare.25942969.v1
  • Figure 2: Simulation of problem \ref{['application:scalartophat']} with $\alpha = 30$ and numerical domain size of $12$ ($L=6$). A supplementary video of this simulation is hosted on figshare found here: https://doi.org/10.6084/m9.figshare.25934431.v1
  • Figure 3: Simulation of problem \ref{['application:scalartophat']} with $\alpha = 20$ and numerical domain size of $24$ ($L=12$). A supplementary video of this simulation is hosted on figshare found here: https://doi.org/10.6084/m9.figshare.25934425.v1
  • Figure 4: Simulation of problem \ref{['application:scalartophat']} with $\alpha =-20$ and numerical domain size $32$ ($L=16$). A supplementary video of this simulation is hosted on figshare found here: https://doi.org/10.6084/m9.figshare.25934428.v1
  • Figure 5: Simulation of problem \ref{['application:systemtophat']} with $\alpha_{11} =20$, $\alpha_{12} = \alpha_{21} = -10$, and $\alpha_{22} = 2$, with numerical domain length $20$ ($L=10$). A supplementary video of this simulation is hosted on figshare found here: https://doi.org/10.6084/m9.figshare.25943101.v1
  • ...and 1 more figures

Theorems & Definitions (47)

  • Theorem 2.1: Existence of global weak solution for scalar equation with small mass
  • Theorem 2.2: Existence of global weak solution for scalar equation with arbitrary mass
  • Theorem 2.3: Existence of unique strong solution
  • Theorem 2.4: Existence of unique classical solution
  • Definition 3.1: weak solution
  • Lemma 3.2
  • proof
  • Definition 3.3: strong solution
  • Definition 3.4: classical solution
  • Lemma 3.5: Apriori estimates under \ref{['h1']} and small-mass condition \ref{['const:c1']}
  • ...and 37 more