双线性Airy估计及其在gKdV-3中的应用

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.

Airy方程的双线性局部光滑估计

Bilinear local smoothing estimate for Airy equation

本文证明了当两个波的频率支集分离时，Airy解的双线性局部光滑估计的一个改进。作为应用，我们证明了双线性型的一个光滑性质。

广义Korteweg-de Vries方程在尺度临界L^r空间中的适定性

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

研究了广义Korteweg-de Vries（gKdV）方程在L^r空间中的初值问题的局部和整体适定性.在尺度临界^L^r空间中，我们证明了gKdV方程的（大数据）局部适定性、小数据全局适定性和小数据离散性.一个关键的成分是一个Stein-Tomas型不等式的Airy方程，它推广了通常的Eschenhartz估计的^L^r-框架。

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…

Airy Schihartz不等式极值解的解析性

Analyticity of extremisers to the Airy Strichartz inequality

利用线性剖面分解，证明了Airy Schahartz不等式的极值函数$e^{-t\partial_x^3}: L^2(\mathbb{R})\to L^8_{t,x}(\mathbb{R}^2)$的存在性.进一步证明了，如果$f$是一个极值，则$f$在傅里叶空间中衰减非常快，因此$f$可以扩展为整个复域上的整函数.利用自举方法建立了极值点的傅里叶变换的快速衰减，该方法依赖于一个改进的双线性Airy-Schahartz估计和一个加权Schahartz不等式.

周期广义KdV方程的整体适定性

Global well-posedness for periodic generalized KdV equation

本文证明了周期gKdV方程在空间$H^s(\mathbb{T})$、四次空间$s\ge \frac12$和五次空间$s> \frac59$中的整体适定性.这改进了I-team在2004年的结果。特别地，对于五次情形，结果是尖锐的。主要的方法是结合Miao等人在2010年提出的共振分解的I-方法，以及周期设置下的双线性Eschhartz估计。