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<p>We study the existence and the rate of equilibration of weak solutions to a two-component system of non-linear diffusion-aggregation equations, with small cross diffusion effects. The aggregation term is assumed to be purely attractive, and in the absence of cross diffusion, the flow is exponentially contractive towards a compactly supported steady state. Our main result is that for small cross diffusion, the system still converges exponentially, at a slightly lower rate, to a deformed but still compactly supported steady state.</p><p>Our approach relies on the interpretation of the PDE system as a gradient flow in a two-component Wasserstein metric. The energy consists of a uniformly convex part responsible for self-diffusion and non-local aggregation, and a totally non-convex part that generates cross diffusion; the latter is scaled by a coupling parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\varepsilon &gt;0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. The core idea of the proof is to perform an <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\varepsilon \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation>-dependent modification of the convex/non-convex splitting and establish a control on the non-convex terms by the convex ones.</p>

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Convergence to equilibrium for cross diffusion systems with nonlocal interaction

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年2月17日

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  • Daniel Matthes,
  • Christian Parsch

摘要

We study the existence and the rate of equilibration of weak solutions to a two-component system of non-linear diffusion-aggregation equations, with small cross diffusion effects. The aggregation term is assumed to be purely attractive, and in the absence of cross diffusion, the flow is exponentially contractive towards a compactly supported steady state. Our main result is that for small cross diffusion, the system still converges exponentially, at a slightly lower rate, to a deformed but still compactly supported steady state.

Our approach relies on the interpretation of the PDE system as a gradient flow in a two-component Wasserstein metric. The energy consists of a uniformly convex part responsible for self-diffusion and non-local aggregation, and a totally non-convex part that generates cross diffusion; the latter is scaled by a coupling parameter \(\varepsilon >0\) ε > 0 . The core idea of the proof is to perform an \(\varepsilon \) ε -dependent modification of the convex/non-convex splitting and establish a control on the non-convex terms by the convex ones.