We study the spatially homogeneous granular medium equation <Equation ID="Equ32"> <EquationSource Format="TEX">$\begin{aligned} \partial _t\mu =\textrm{div}(\mu \nabla V)+\textrm{div}(\mu (\nabla W *\mu ))+\Delta \mu \,, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>∂</mi> <mi>t</mi> </msub> <mi>μ</mi> <mo>=</mo> <mtext>div</mtext> <mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mi mathvariant="normal">∇</mi> <mi>V</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mtext>div</mtext> <mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mi>W</mi> <mrow /> <mo>∗</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>μ</mi> <mspace width="0.166667em" /> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>within a large and natural class of the confinement potentials V and interaction potentials W. The considered problem do not need to assume that <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\nabla V$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">∇</mi> <mi>V</mi> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\nabla W$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">∇</mi> <mi>W</mi> </mrow> </math></EquationSource> </InlineEquation> are globally Lipschitz. With the aim of providing particle approximation of solutions, we design efficient forward-backward splitting algorithms. Sharp convergence rates in terms of the Wasserstein distance are provided.

Convergence rates of particle approximation of forward-backward splitting algorithm for granular medium equations

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2026年2月9日

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Matej Benko,
Iwona Chlebicka,
Jørgen Endal,
Błażej Miasojedow

摘要

We study the spatially homogeneous granular medium equation $\begin{aligned} \partial _t\mu =\textrm{div}(\mu \nabla V)+\textrm{div}(\mu (\nabla W *\mu ))+\Delta \mu \,, \end{aligned}$ $\begin{matrix} \partial_{t} μ = div (μ \nabla V) + div (μ (\nabla W * μ)) + Δ μ, \end{matrix}$ within a large and natural class of the confinement potentials V and interaction potentials W. The considered problem do not need to assume that $\nabla V$ $\nabla V$ or $\nabla W$ $\nabla W$ are globally Lipschitz. With the aim of providing particle approximation of solutions, we design efficient forward-backward splitting algorithms. Sharp convergence rates in terms of the Wasserstein distance are provided.