Long-Term Regularity of the Periodic Euler–Poisson System for Electrons in 2D

Abstract

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.

Long-term regularity of 3D gravity water waves

“…Proof. Lemma 1.4 of [95] shows (i)-(iii) for the (essentially same) case of α = 2/3. To get (iv) we use the vector-valued version of the Corollary to Theorem 2 in Section V.4 of [72], as detailed in Section I.6.4 of loc.…”

“…Therefore extension of such Strichartz estimates is crucial to the proof of the (almost) global results in Theorem 1.3 and Theorem 1.4, though we will work at the level r = 6, much higher than the sharp results cited above, because we are not aiming at optimal regularity in this paper. Similar ideas were used to show long-term regularity of the periodic Euler-Poisson equation in 2D by the author [95]. 1.3.4.…”

“…Following [95], we will take advantage of the absence of resonance in time, but will not care about space resonance. The proof combines three parts.…”

“…We do have global wellposedness, however, if we insert some weight (for example, |x| in [37]). This weight is most likely not optimal, for the author showed in [95] that for the Euler-Poisson equation, a similar quasilinear dispersive equation without resonance, the weight can be reduced to |x| 2/3 . Naturally one wonders how much the weight can be reduced for global wellposedness to remain true.…”

“…Fortunately more careful decomposition of the profile yields additional decay, a technique already present in [95]. We take a slightly different approach in this paper to adapt to the new bootstrap assumption.…”