The convergence problem of the generalized Korteweg-de Vries equation in Fourier-Lebesgue space

Abstract

In this paper, we investigate the pointwise convergence problem of the generalized Korteweg-de Vries (gKdV) equation with data in the Fourier-Lebesgue space. Firstly, for the Airy equation, we show the almost everywhere pointwise convergence with data in ${\hat{H}}^{s,\frac{\alpha -1}{2}}\left(R\right),(s\ge \frac{1}{\alpha -1},5\le \alpha <\infty )$, furthermore, we show that the maximal function estimate related to the Airy equation can fail with data in ${\hat{H}}^{s,\frac{\alpha -1}{2}}\left(R\right)(s<\frac{1}{\alpha -1})$. Then, for the gKdV equation, we establish the pointwise convergence results with the data in ${\hat{H}}^{\frac{1}{\alpha -1},\frac{\alpha -1}{2}}\left(R\right)(5\le \alpha <\frac{23}{3})$, in particular, we establish the pointwise convergence results with small data in ${\hat{\dot{H}}}^{\frac{1}{4},2}\left(R\right)$, which implies that the pointwise convergence of generalized KdV equation is closely related to the pointwise convergence of linear KdV equation in the Fourier-Lebesgue spaces.