We study a basic plasma physics model-the one-fluid Euler-Poisson system on the square torus, in which a compressible electron fluid flows under its own electrostatic field. In this paper we prove long-term regularity of periodic solutions of this system in 2 spatial dimensions with small data.Our main conclusion is that on a square torus of side length R, if the initial data is sufficiently close to a constant solution, then the solution is wellposed for a time at least R/(ǫ 2 (log R) O(1) ), where ǫ is the size of the initial data.

We study a fundamental model in fluid mechanics—the 3D gravity water wave equation, in which an incompressible fluid occupying half the 3D space flows under its own gravity. In this paper we show long‐term regularity of solutions whose initial data is small but not localized. Our results include: almost global well‐posedness for unweighted Sobolev initial data and global well‐posedness for weighted Sobolev initial data with weight |x|α for any α > 0. In the periodic case, if the initial data lives on an R by R torus, and ϵ close to the constant solution, then the life span of the solution is at least R/ϵ2(logR)2. © 2021 Wiley Periodicals LLC.

We consider smooth solutions of the Burgers-Hilbert equation that are a small perturbation δ from a global periodic traveling wave with small amplitude . We use a modified energy method to prove the existence time of smooth solutions on a time scale of 1 δ with 0 < δ 1 and on a time scale of δ 2 with 0 < δ 2 1. Moreover, we show that the traveling wave exists for an amplitude in the range (0, * ) with * ∼ 0.29 and fails to exist for > 2 e .

We study a fundamental model in fluid mechanics-the 3D gravity water wave equation, in which an incompressible fluid occupying half the 3D space flows under its own gravity. In this paper we show long-term regularity of solutions whose initial data is small but not localized. Our results include: almost global wellposedness for unweighted Sobolev initial data and global wellposedness for weighted Sobolev initial data with weight |x| α , for any α > 0.I live upstream and you downstream From night to night of you I dream Unlike the strewam you're not in view Though we both drink from River Blue -Chinese Song Lyrics by Li

No abstract