arXiv subjects
Publications and source records attributed to Baoxiang Wang.
We study the wellposedness of Cauchy problem for the fourth order nonlinear Schrödinger equations i\partial_t u=-\epsΔu+Δ^2 u+P((\partial_x^αu)_{\absα\ls 2}, (\partial_x^α\bar{u})_{\absα\ls 2}),\quad t\in \Real, x\in\Real^n, where $\eps\in\{-1,0,1\}$, $n\gs 2$ denotes the spatial dimension and $P(\cdot)$ is a polynomial excluding constant and linear terms.
Considering the Cauchy problem for the modified finite-depth-fluid equation $\partial_tu-\G_δ(\partial_x^2u)\mp u^2u_x=0, u(0)=u_0$, where $\G_δf=-i \ft ^{-1}[\coth(2πδξ)-\frac{1}{2πδξ}]\ft f$, $δ\ges 1$, and $u$ is a real-valued function, we show that it is uniformly globally well-posed if $u_0 \in H^s (s\geq 1/2)$ with $\norm{u_0}_{L^2}$ sufficiently small for all $δ\ges 1$. Our result is sharp in the sense that the solution map fails to be $C^3$ in $H^s (s<1/2)$. Moreover, we prove that for any $T>0$, its solution converges in $C([0,T]; H^s)$ to that of the modified Benjamin-Ono equation if $δ$ tends to $+\infty$.
Considering the Cauchy problem for the Korteweg-de Vries-Burgers equation \begin{eqnarray*} u_t+u_{xxx}+ε|\partial_x|^{2α}u+(u^2)_x=0, \ u(0)=ϕ, \end{eqnarray*} where $0<ε,α\leq 1$ and $u$ is a real-valued function, we show that it is globally well-posed in $H^s\ (s>s_α)$, and uniformly globally well-posed in $H^s (s>-3/4)$ for all $ε\in (0,1)$. Moreover, we prove that for any $T>0$, its solution converges in $C([0,T]; H^s)$ to that of the KdV equation if $ε$ tends to 0.
We study the Cauchy problem for the generalized elliptic and non-elliptic derivative nonlinear Schrodinger equations, the existence of the scattering operators and the global well posedness of solutions with small data in Besov spaces and in modulation spaces are obtained. In one spatial dimension, we get the sharp well posedness result with small data in critical homogeneous Besov spaces. As a by-product, the existence of the scattering operators with small data is also shown. In order to show these results, the global versions of the estimates for the maximal functions on the elliptic and non-elliptic Schrodinger groups are established.
In this paper we use a unified way studying the decay estimate for a class of dispersive semigroup given by $e^{itϕ(\sqrt{-Δ})}$, where $ϕ: \mathbb{R}^+\to \mathbb{R}$ is smooth away from the origin. Especially, the decay estimates for the solutions of the Klein-Gordon equation and the beam equation are simplified and slightly improved.