论文标题
精制的双线性strichartz估计值适用于周期性广义KDV型方程的适当性
Refined bilinear Strichartz estimates with application to the well-posedness of periodic generalized KdV type equations
论文作者
论文摘要
我们改善了以前的结果[L. Molinet和T. Tanaka，某些非线性周期性一维色散方程的无条件良好性，J。函数。肛门。 283（2022），109490]关于库奇问题的一维色散方程，在周期性环境中具有相当一般的非线性。在相同的假设下，色散操作员以高频作为傅立叶乘数（$ i | e |ξ|^αξ$）带有$ 1 \leα\ le 2 $，而非线性术语是$ \ partial_x f（u）的形式，其中$ f（u）$ f $ a $ f $是围绕taylor a的真实分析功能，是taylor a ins the taylor contrention in in in inim ins inime lim contrance linipe ly croments lius contrance lw croments lwius contrentip lw consectiz $ h^s（\ mathbb {t}）$ for $ s \ ge 1- \fracα{4} $带有$ s> 1/2 $的cauchy问题。值得注意的是，鉴于限制$ s> 1/2 $，对于连续注入$ h^s（\ mathbb {t}）$，该结果在$α= 2 $（广义kDV方程）的情况下是最佳的。我们的主要新成分是在治疗最坏的谐振相互作用时用精制的双线性估计替代了精致的Strichartz估计值。这种精致的双线性估计已经出现在Hani的工作中，在Schrödinger方程的背景下，在紧凑的歧管上。最后，主要定理在[4/3,2] $中得出$α\的全球存在结果。
We improve our previous result [L. Molinet and T. Tanaka, Unconditional well-posedness for some nonlinear periodic one-dimensional dispersive equations, J. Funct. Anal. 283 (2022), 109490] on the Cauchy problem for one dimensional dispersive equations with a quite general nonlinearity in the periodic setting. Under the same hypotheses that the dispersive operator behaves for high frequencies as a Fourier multiplier by $ i |ξ|^αξ$ with $ 1 \le α\le 2 $, and that the nonlinear term is of the form $ \partial_x f(u) $ where $f $ is a real analytic function whose Taylor series around the origin has an infinite radius of convergence, we prove the unconditional LWP of the Cauchy problem in $H^s(\mathbb{T}) $ for $ s\ge 1-\fracα{4} $ with $ s>1/2 $. It is worth noting that this result is optimal in the case $α=2$ (generalized KdV equation) in view of the restriction $ s>1/2 $ for the continuous injection of $ H^s(\mathbb{T}) $ into $ L^\infty(\mathbb{T}) $. Our main new ingredient is the replacement of refined Strichartz estimates with refined bilinear estimates in the treatment of the worst resonant interactions. Such refined bilinear estimates already appeared in the work of Hani in the context of Schrödinger equations on a compact manifold. Finally, the main theorem yields global existence results for $ α\in [4/3,2] $.
