关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

The Cauchy Problem

The One-Dimensional Heat Equation: Preface

Author(s): Koch, Herbert; Tataru, Daniel | Abstract: We consider the cubic Nonlinear Schrodinger Equation in one space dimension, either focusing or defocusing. We prove that the solutions satisfy a-priori local in time H^s bounds in terms of the H^s size of the initial data for s greater than or equal to -1/6.

/pdf/a-priori-bounds-for-the-1d-cubic-nls-in-negative-sobolev-34rt5zkwx3.pdf

A Priori Bounds for the 1D Cubic NLS in Negative Sobolev Spaces

We consider the Schrodinger map initial-value problem ¤ @ t = on R d R; (0) = 0; where : R d R! S 2 ,! R 3 is a smooth function. In all dimensions d 2, we prove that the Schrodinger map initial-value problem admits a unique global smooth solution 2 C(R : H 1 Q ), Q2 S 2 , provided that the data 02 H 1 Q is smooth and satises the smallness condition

http://annals.math.princeton.edu/wp-content/uploads/annals-v173-n3-p05-p.pdf

Global Schrödinger maps in dimensions $d≥ 2$: Small data in the critical Sobolev spaces

We prove local in time well-posedness for the Zakharov system in two space dimensions with large initial data in L^2 x H^{-1/2} x H^{-3/2}. This is the space of optimal regularity in the sense that the data-to-solution map fails to be smooth at the origin for any rougher pair of spaces in the L^2-based Sobolev scale. Moreover, it is a natural space for the Cauchy problem in view of the subsonic limit equation, namely the focusing cubic nonlinear Schroedinger equation. The existence time we obtain depends only upon the corresponding norms of the initial data - a result which is false for the cubic nonlinear Schroedinger equation in dimension two - and it is optimal because Glangetas-Merle's solutions blow up at that time.

On the 2d Zakharov system with L^2 Schr\"odinger data

Convolutions of singular measures and applications to the Zakharov system

We prove local in time well-posedness for the Zakharov system in two space dimensions with large initial data in L2 × H−1/2 × H−3/2. This is the space of optimal regularity in the sense that the data-to-solution map fails to be smooth at the origin for any rougher pair of spaces in the L2-based Sobolev scale. Moreover, it is a natural space for the Cauchy problem in view of the subsonic limit equation, namely the focusing cubic nonlinear Schrodinger equation. The existence time we obtain depends only upon the corresponding norms of the initial data—a result which is false for the cubic nonlinear Schrodinger equation in dimension two—and it is optimal because Glangetas–Merle's solutions blow up at that time.

https://iopscience.iop.org/article/10.1088/0951-7715/22/5/007/pdf

On the 2D Zakharov system with L2 Schrödinger data

Global well-posedness and scattering for the cubic Dirac equation with small initial data in the critical space $${{H^{\frac{1}{2}}} (\mathbb{R}^{2}}$$
 ) is established. The proof is based on a sharp endpoint Strichartz estimate for the Klein–Gordon equation in dimension n = 2, which is captured by constructing an adapted systems of coordinate frames.

/pdf/the-cubic-dirac-equation-small-initial-data-in-h-1-2-r-2-387cqnq7nc.pdf

Ioan Bejenaru

Papers

Global Schrödinger maps in dimensions $d≥ 2$: Small data in the critical Sobolev spaces

On the 2d Zakharov system with L^2 Schr\"odinger data

Convolutions of singular measures and applications to the Zakharov system

On the 2D Zakharov system with L2 Schrödinger data

The Cubic Dirac Equation: Small Initial Data in H-1/2 (R-2)