arXiv math/01081842001-08-29math.AP

A bilinear Airy- estimate with application to gKdV-3

A Gruenrock

备注：: A misprint in the online abstract was corrected

英文摘要：

The Fourier restriction norm method is used to show local wellposedness for the Cauchy Problem for the generalized KdV-equation of order three with data in the usual Sobolev space H^s, s > -1/6. For real valued data in L^2 global wellposedness follows by the conservation of the L^2-norm. The main new tool is a bilinear estimate for solutions of the Airy- equation.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Refinement of Strichartz estimate for Airy equation in non-diagonal case and its application

In this paper, we give an improvement of the Strichartz estimate for Airy equation in the non-diagonal case. As an application, we prove the small data scattering and existence of a special non-scattering solutions, which are minimal in suitable sense, to the mass-subcritical generalized Korteweg-de Vries (gKdV) equation. Especially, we remove several technical restrictions on our previous work about the existence of a special non-scattering solution.

2017-04-27math.AP

Bilinear local smoothing estimate for Airy equation

In this short note, we prove a refinement of bilinear local smoothing estimate to Airy solutions, when the frequency support of two wave are separated. As an application we prove a smoothing property of a bilinear form.

2011-08-02math.AP

On well-posedness of generalized Korteweg-de Vries equation in scale critical ^L^r space

The purpose of this paper is to study local and global well-posedness of initial value problem for generalized Korteweg-de Vries (gKdV) equation in ^L^r. We show (large data) local well-posedness, small data global well-posedness, and small data scattering for gKdV equation in the scale critical ^L^r space. A key ingredient is a Stein-Tomas type inequality for the Airy equation, which generalizes usual Strichartz estimates for ^L^r-framework.

2015-07-14math.AP

Invariance of the Gibbs measures for periodic generalized Korteweg-de Vries equations

In this paper, we study the Gibbs measures for periodic generalized Korteweg-de Vries equations (gKdV) with quartic or higher nonlinearities. In order to bypass the analytical ill-posedness of the equation in the Sobolev support of the Gibbs measures, we establish deterministic well-posedness of the gauged gKdV equations within the framework of the Fourier-Lebesgue spaces. Our argument relies on bilinear and trilinear Strichartz estimates adapted to the Fourier-Lebesgue setting. Then, following Bourgain's invariant measure argument, we construct almost sure global-in-time dynamics and show inv…

2022-02-25math.AP

Analyticity of extremisers to the Airy Strichartz inequality

We prove that there exists an extremal function to the Airy Strichartz inequality, $e^{-t\partial_x^3}: L^2(\mathbb{R})\to L^8_{t,x}(\mathbb{R}^2)$ by using the linear profile decomposition. Furthermore we show that, if $f$ is an extremiser, then $f$ is extremely fast decaying in Fourier space and so $f$ can be extended to be an entire function on the whole complex domain. The rapid decay of the Fourier transform of extremisers is established with a bootstrap argument which relies on a refined bilinear Airy Strichartz estimate and a weighted Strichartz inequality.

2010-12-29math.AP

Global well-posedness for periodic generalized KdV equation

In this paper, we show the global well-posedness for periodic gKdV equations in the space $H^s(\mathbb{T})$, $s\ge \frac12$ for quartic case, and $s> \frac59$ for quintic case. These improve the previous results of I-team in 2004. In particular, the result is sharp for quintic case. The main approaches are the I-method combining with the resonance decomposition developed by Miao et al in 2010, and a bilinear Strichartz estimate in periodic setting.

2014-05-03math.AP

更多相关论文收起更多论文

一类带一阶贝塞尔正则化的Airy方程的小速度孤立波稳定性与整体适定性

Small-speed solitary-wave stability and global well-posedness for a Bessel-regularized Airy equation

Department of Mathematics, Universidad Nacional de Colombia（哥伦比亚国立大学数学系）

本研究针对带一阶贝塞尔正则化的Airy方程，证明了小速度下孤立波的轨道稳定性，建立了相关动力学框架并验证了稳定性准则的非退化性与斜率条件。

2026-10-14math.AP

An improved local wellposedness result for the modified KdV equation

The Cauchy problem for the modified KdV equation is shown to be locally well posed for data u_0 in the space \hat(H^r_s) defined by the norm ||u_0||:=||<ξ>^s \hat(u_0)||_L^r', provided 4/3 < r \le 2, s \ge 1/2 - 1/(2r). For r=2 this coincides with the best possible result on the H^s - scale due to Kenig, Ponce and Vega. The proof uses an appropriate variant of the Fourier restriction norm method and linear as well as bilinear estimates for the solutions of the Airy equation.

2003-12-11math.AP

环面上 Strichartz 估计的改进及其应用

Refinements of Strichartz estimates on tori and applications

本文在环面上证明依赖频率时间区间的三线性与新线性 Strichartz 估计，并将其用于 Airy 传播子，得到修正 KdV 方程解存在性的最优 Sobolev 正则性。

2023-12-11math.APmath.CA