New lower bounds on the radius of spatial analyticity for the KdV equation

Abstract

The radius of spatial analyticity for solutions of the KdV equation is studied. It is shown that the analyticity radius does not decay faster than t −1/4 as time t goes to infinity. This improves the works [Selberg, da Silva, Lower bounds on the radius of spatial analyticity for the KdV equation, Annales Henri Poincaré, 2017, 18(3): 1009-1023] and [Tesfahun, Asymptotic lower bound for the radius of spatial analtyicity to solutions of KdV equation, arXiv preprint arXiv:1707.07810, 2017]. Our strategy mainly relies on a higher order almost conservation law in Gevrey spaces, which is inspired by the I−method.

Improved lower bounds of analytic radius for the Benjamin-Bona-Mahony equation

“…(2) Establish an almost conservation law in G σ,1 , namely 1 This approach is introduced by Selberg and Tesfahun in [10], can be understood a variant I-method [11] in analytic spaces. The method is powerful and has been used to establish analytic radius lower bounds for KdV equations [12,13,14,15,16], KdV-BBM equations [17,18] and other dispersive equations [19,20,21,22,23,24]. For more results on the analytic radius, we refer to the survey [25].…”

The global attractor for the weakly damped KdV equation on R has a finite fractal dimension

“…In the case and , the well‐posedness of the Cauchy problems and attracts a lot of attention. We refer the readers to Colliander et al, Linares and Ponce, Killip and Vişan, for some results in Sobolev spaces, and Selberg and da Silva and Huang and Wang for results in Gevrey spaces.…”

On the radius of spatial analyticity for defocusing nonlinear Schrödinger equations

“…More precisely, if the initial data are real-analytic and have a uniform radius of analyticity σ 0 > 0, so there is a holomorphic extension of the data to a complex strip S σ0 = {x + iy : x, y ∈ R d , |y 1 |, |y 2 |, • • • , |y d | < σ 0 }, then we may ask whether or not and up to what degree the solution at some later time t preserves the initial analyticity; we would like to estimate the radius of analyticity of the solution at time t, σ(t), which is possibly shrinking. This type of question was first introduced by Kato and Masuda [16] in 1986 and there are plenty of works for nonlinear dispersive equations such as the KP equation [3], KdV type equations [4,5,24,28,14,22,2], Schrödinger equations [6,27,1], and Klein-Gordon equations [18].…”

“…Later, this exponential decay was improved to an algebraic lower bound, ct −12 , by Bona, Grujić and Kalisch [3]. See [16,20,9] for further refinements. We also refer the reader to [4,19,13,17,15] for other nonlinear dispersive equations like Schrödinger, Klein-Gordon and Dirac-Klein-Gordon equations.…”