Continuous dependence for NLS equations in Sobolev spaces $H^s$

报告题目:Continuous dependence for NLS equations in Sobolev spaces $H^s$.


报告人:戴蔚  博士(中科院应用数学所)






摘要:In recent years, by applying harmonic analysis and other analysis tools, a great progress has been made in understanding the nature of solutions to Schr\"{o}dinger equation and other related nonlinear problems. In this work, we will apply the methods and techniques from harmonic analysis to the study of well-posedness of Cauchy problem for nonlinear Schr\"{o}dinger equations in Sobolev spaces, in particular, to the study of the continuous dependence of solutions on the initial data. Many mathematicians have pay attention to this problem, such as T. Kato, T. Cazenave and T. Tao, their works solve the cases $0< s< \min\{1,N/2\}$ and $s=0,1$. In this talk, we consider the Cauchy problem for the NLS equation $i \partial_{t}u+ \Delta u=\lambda_{0}u+\lambda_{1}|u|^\alpha u$ in $\mathbb{R}^{N}$, where $\lambda_{0},\lambda_{1}\in\mathbb{C}$, in $H^s$ subcritical and critical case: $0<\alpha\leq\frac{4}{N-2s}$ when $1< s< \frac{N}{2}$ and $0<\alpha<+\infty$ when $s\geq\frac{N}{2}$, we show in virtually all the cases where local existence of solution to NLS is known that the solution depends continuously on the initial value in the standard sense in $H^{s}(\mathbb{R}^{N})$ if $\alpha$ satisfies certain assumptions, that is, the local solution flow is continuous $H^{s}\rightarrow H^{s}$. Furthermore, if $\alpha$ satisfies more restrictive conditions, we show that the flow is locally Lipschitz.









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