# Nonlinear echoes and Landau damping with insufficient regularity

**Authors:** Jacob Bedrossian  
**arXiv:** [1605.06841](https://arxiv.org/abs/1605.06841)  
**Date:** 2016-05-22  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/1605.06841v3/

## Key points

- Jacob Bedrossian investigates the extension of Mouhot and Villani's theorem on Landau damping near equilibrium for Vlasov-Poisson equations to high Sobolev spaces in the case of gravitational interactions.
- The author proves that this extension is not possible in general, as there exist background distributions for every Sobolev space that can produce arbitrarily small perturbations with an arbitrary number of isolated nonlinear oscillations in density.
- These oscillations are known as plasma echoes.
- The author also analyzes electrostatic interactions and shows a sequence of small background distributions and asymptotically smaller perturbations that exhibit similar nonlinear echoes.
- The findings suggest that any extension of Mouhot and Villani's theorem to Sobolev spaces would require additional non-resonance effects from the background distribution, unlike in the case of Gevrey-$\nu$ regularity where results are uniform in the size of small backgrounds.
- Overall, this study highlights limitations in extending existing results on Landau damping near equilibrium for Vlasov-Poisson equations to high Sobolev spaces when considering gravitational interactions.

## Summary

In the paper "Nonlinear Echoes and Landau Damping with Insufficient Regularity," Jacob Bedrossian investigates the extension of the theorem of Mouhot and Villani on Landau damping near equilibrium for the Vlasov-Poisson equations on $\mathbb{T}_x \times \mathbb{R}_v$ to high Sobolev spaces in the case of gravitational interactions. The author demonstrates that this extension is not possible in general. Bedrossian proves that for every Sobolev space, there exist background distributions such that one can construct arbitrarily small perturbations that exhibit an arbitrary number of isolated nonlinear oscillations in the density. These oscillations are commonly referred to as plasma echoes in the physics community. By analyzing electrostatic interactions, the author also shows a sequence of small background distributions and asymptotically smaller perturbations in $H^s$ that display similar nonlinear echoes. The findings indicate that any extension of Mouhot and Villani's theorem to Sobolev spaces would have to depend crucially on some additional non-resonance effect coming from the background. This is unlike the case of Gevrey-$\nu$ with $\nu < 3$ regularity, where results are uniform in the size of small backgrounds. Therefore, it is concluded that the uniform dependence on small background distributions obtained in Mouhot and Villani's theorem in Gevrey class does not hold true in Sobolev spaces. Overall, this study highlights limitations in extending existing results on Landau damping near equilibrium for Vlasov-Poisson equations to high Sobolev spaces when considering gravitational interactions. It emphasizes the importance of considering additional non-resonance effects from background distributions when studying these phenomena.