Asymptotic lower bound for the radius of spatial analyticity to solutions of KdV equation

Abstract

It is shown that the uniform radius of spatial analyticity σ(t ) of solutions at time t to the KdV equation cannot decay faster than |t | −4/3 as |t | → ∞ given initial data that is analytic with fixed radius σ 0 . This improves a recent result of Selberg and Da Silva, where they proved a decay rate of |t | −(4/3+ε) for arbitrarily small positive ε. The main ingredients in the proof are almost conservation law for the solution to the KdV equation in space of analytic functions and space-time dyadic bilinear L 2 estimates associated with the KdV equation. 1991 Mathematics Subject Classification. 35Q53; 35L30. Key words and phrases. KdV equation; Radius of spatial analticity; Asymptotic lower bound .1 4.3. Local well-posedness in Gevery class. Define the map Φ(u)(t ) = χ(t /4)W (t ) f + B(u, u)(t ). Let s ∈ [−3/4, 0] and J = [−1, 1]. By Lemma 7(ii), (iii) and Corollary 2 we have Φ(u) X σ,s J

On the persistence of spatial analyticity for the Beam Equation

“…The method used here for proving lower bounds on the radius of analyticity was introduced in [27] in the context of the 1D Dirac-Klein-Gordon equations. This method is based on an approximate conservation laws, and has been applied to prove an algebraic lower bound (decay rate) of order t −1/α for some α ∈ (0, 1] on the radius of spatial analyticity of solutions to a number of nonlinear dispersive and wave equations (see e.g., [1,[25][26][27][29][30][31]). The optimal decay rate that can be obtained in this setting is 1/t, which corresponds to α = 1 (see e.g., [1,29,31]).…”

Cauchy theory for the water waves system in an analytic framework

“…The study of various nonlinear partial differential equations in spaces of analytic functions has also received a great attention. We can mention the well-posedness results in analytic spaces by Kato and Masuda [31] which apply to many equations in fluid dynamics, the study of the Rayleigh-Taylor instability by Sulem-Sulem [47], the study of the Cauchy problem for the semi-linear one dimensional Schrödinger equations by Bona-Grujic and Kalisch [12], Selberg-D.O.da Silva [46], the work on the KdV equation by Hayashi [20], Tesfahun [48] and on the periodic BBM equation by Himonas-Petronilho [21], the work by Kucavica-Vicol [32] on the Euler equation, the work on quasilinear wave equations and other quasilinear systems by Alinhac and Métivier [8] and Kuksin-Nadirashvili [33], the work by Matsuyama and Ruzhansky [36] on the Kirchhoff equation, Gancedo-Granero-Belinchón-Scrobogna [17] for the Muskat problem and the one of Pierre [43] for the MHD equations. We should also mention the recent works by Mouhot-Villani [37], Bedrossian-Masmoudi-Mouhot [11] and by Grenier-Nguyen-Rodnianski [19] on the Landau damping for analytic and Gevrey data.…”

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.…”