<p>We investigate the impact of weak collisions on Landau damping in the Vlasov-Poisson-Fokker-Planck system on a torus, specifically focusing on its proximity to a Maxwellian distribution. In the case where the Gevrey index satisfies <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\frac{1}{s}\le 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mn>1</mn> <mi>s</mi> </mfrac> <mo>≤</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, we establish the global stability and enhanced dissipation of small initial data, which remain unaffected by the small diffusion coefficient <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\nu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ν</mi> </math></EquationSource> </InlineEquation>. For Gevrey index <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\frac{1}{s}>3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mn>1</mn> <mi>s</mi> </mfrac> <mo>></mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, we prove the global stability and enhanced dissipation of initial data, whose size is on the order of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$O(\nu ^\frac{1-3s}{3-3s})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>ν</mi> <mfrac> <mrow> <mn>1</mn> <mo>-</mo> <mn>3</mn> <mi>s</mi> </mrow> <mrow> <mn>3</mn> <mo>-</mo> <mn>3</mn> <mi>s</mi> </mrow> </mfrac> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Our analysis provides insights into the effects of enhanced dissipation and plasma echoes.</p>

Weak collision effect on Landau damping for the Vlasov-Poisson-Fokker-Planck system

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发表日期：

2026年1月14日

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Yue Luo

摘要

We investigate the impact of weak collisions on Landau damping in the Vlasov-Poisson-Fokker-Planck system on a torus, specifically focusing on its proximity to a Maxwellian distribution. In the case where the Gevrey index satisfies $\frac{1}{s}\le 3$ $\frac{1}{s} \leq 3$ , we establish the global stability and enhanced dissipation of small initial data, which remain unaffected by the small diffusion coefficient $\nu $ $ν$ . For Gevrey index $\frac{1}{s}>3$ $\frac{1}{s} > 3$ , we prove the global stability and enhanced dissipation of initial data, whose size is on the order of $O(\nu ^\frac{1-3s}{3-3s})$ $O (ν^{\frac{1 - 3 s}{3 - 3 s}})$ . Our analysis provides insights into the effects of enhanced dissipation and plasma echoes.