<p>We investigate the phenomenon of Landau damping within the context of the Vlasov-Riesz system. Employing a nonlinear analysis technique developed by Ionescu, Pausader, Wang, and Widmayer, our primary objective is to extend the understanding of Landau damping to kinetic models beyond the traditional Vlasov-Poisson system. Our analysis is grounded in the use of the Gevrey-<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation> norm, and we demonstrate that the critical Gevrey index <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\gamma =-\frac{4}{3}\alpha +\frac{5}{3}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>=</mo> <mo>-</mo> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> <mi>α</mi> <mo>+</mo> <mfrac> <mn>5</mn> <mn>3</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation> is dependent on the parameter <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> from the Riesz interaction, defined as <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$U=\left( -\Delta \right) ^{-\alpha }(\rho -1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <mo>=</mo> <msup> <mfenced close=")" open="("> <mo>-</mo> <mi mathvariant="normal">Δ</mi> </mfenced> <mrow> <mo>-</mo> <mi>α</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. This study not only broadens the applicative scope of Landau damping but also deepens the theoretical understanding of its mechanisms in more complex kinetic frameworks.</p>

Landau damping for the Vlasov-Riesz system

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2026年6月3日

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Zhiwen Zhang

摘要

We investigate the phenomenon of Landau damping within the context of the Vlasov-Riesz system. Employing a nonlinear analysis technique developed by Ionescu, Pausader, Wang, and Widmayer, our primary objective is to extend the understanding of Landau damping to kinetic models beyond the traditional Vlasov-Poisson system. Our analysis is grounded in the use of the Gevrey- $\gamma $ $γ$ norm, and we demonstrate that the critical Gevrey index $\gamma =-\frac{4}{3}\alpha +\frac{5}{3}$ $γ = - \frac{4}{3} α + \frac{5}{3}$ is dependent on the parameter $\alpha $ $α$ from the Riesz interaction, defined as $U=\left( -\Delta \right) ^{-\alpha }(\rho -1)$ $U = {(- Δ)}^{- α} (ρ - 1)$ . This study not only broadens the applicative scope of Landau damping but also deepens the theoretical understanding of its mechanisms in more complex kinetic frameworks.