scholarly journals Convergence problem of Schrödinger equation in Fourier-Lebesgue spaces with rough data and random data

2021 ◽  
Author(s):  
Xiangqian Yan ◽  
Yajuan Zhao ◽  
Wei Yan
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Lebesgue Spaces ◽  
Random Data ◽  
Convergence Problem

scholarly journals Normal form approach to the one-dimensional periodic cubic nonlinear Schrödinger equation in almost critical Fourier-Lebesgue spaces

2021 ◽  
Author(s):  
Tadahiro Oh ◽  
Yuzhao Wang
Keyword(s):  
Normal Form ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Lebesgue Spaces ◽  
One Dimensional ◽  
Form Equation ◽  
The One ◽  
Cubic Nonlinear Schrödinger Equation ◽  
Form Approach

AbstractIn this paper, we study the one-dimensional cubic nonlinear Schrödinger equation (NLS) on the circle. In particular, we develop a normal form approach to study NLS in almost critical Fourier-Lebesgue spaces. By applying an infinite iteration of normal form reductions introduced by the first author with Z. Guo and S. Kwon (2013), we derive a normal form equation which is equivalent to the renormalized cubic NLS for regular solutions. For rough functions, the normal form equation behaves better than the renormalized cubic NLS, thus providing a further renormalization of the cubic NLS. We then prove that this normal form equation is unconditionally globally well-posed in the Fourier-Lebesgue spaces ℱLp($${\cal F}{L^p}(\mathbb{T})$$ ℱ L p ( T ) ), 1 ≤ p < ∞. By inverting the transformation, we conclude global well-posedness of the renormalized cubic NLS in almost critical Fourier-Lebesgue spaces in a suitable sense. This approach also allows us to prove unconditional uniqueness of the (renormalized) cubic NLS in ℱLp($${\cal F}{L^p}(\mathbb{T})$$ ℱ L p ( T ) ) for $$1 \leq p \leq {3 \over 2}$$ 1 ≤ p ≤ 3 2 .

scholarly journals Pointwise convergence along a tangential curve for the fractional Schrödinger equation with 0 < m < 1

2021 ◽  
Author(s):  
Jiye Yuan ◽  
tengfei Zhao
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Pointwise Convergence ◽  
Convergence Problem ◽  
Fractional Schrödinger Equation ◽  
Fractional Schrodinger Equation

In this article, we study the pointwise convergence problem about solution to the fractional Schrödinger equation with 0 < m < 1 along the tangential curve and estimate the capacitary dimension of the divergence set. We extend the results of Cho and Shiraki in [8] for the case m > 1 to the case 0 < m < 1, which is sharp up to the endpoint.

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