Do you want to publish a course? Click here

Sharp well-posedness for the cubic NLS and mKdV in $H^s(mathbb R)$

60   0   0.0 ( 0 )
 Publication date 2020
  fields
and research's language is English




Ask ChatGPT about the research

We prove that the cubic nonlinear Schrodinger equation (both focusing and defocusing) is globally well-posed in $H^s(mathbb R)$ for any regularity $s>-frac12$. Well-posedness has long been known for $sgeq 0$, see [51], but not previously for any $s<0$. The scaling-critical value $s=-frac12$ is necessarily excluded here, since instantaneous norm inflation is known to occur [11, 38, 46]. We also prove (in a parallel fashion) well-posedness of the real- and complex-valued modified Korteweg-de Vries equations in $H^s(mathbb R)$ for any $s>-frac12$. The best regularity achieved previously was $sgeq tfrac14$; see [15, 24, 32, 38]. An essential ingredient in our arguments is the demonstration of a local smoothing effect for both equations, with a gain of derivatives matching that of the underlying linear equation. This in turn rests on the discovery of a one-parameter family of microscopic conservation laws that remain meaningful at this low regularity.



rate research

Read More

We prove global well-posedness of the fifth-order Korteweg-de Vries equation on the real line for initial data in $H^{-1}(mathbb{R})$. By comparison, the optimal regularity for well-posedness on the torus is known to be $L^2(mathbb{R}/mathbb{Z})$. In order to prove this result, we develop a strategy for integrating the local smoothing effect into the method of commuting flows introduced previously in the context of KdV. It is this synthesis that allows us to go beyond the known threshold on the torus.
365 - Tristan Roy 2017
We prove global well-posedness for the $3D$ radial defocusing cubic wave equation with data in $H^{s} times H^{s-1}$, $1>s>{7/10}$.
73 - Zihua Guo , Yifei Wu 2016
We prove that the derivative nonlinear Schr{o}dinger equation is globally well-posed in $H^{frac 12} (mathbb{R})$ when the mass of initial data is strictly less than $4pi$.
82 - Shunhang Zhang 2021
In this paper, we first prove the local well-posedness of strong solutions to the incompressible Hall-MHD system with initial data $(u_0,B_0)in H^{frac{1}{2}+sigma}(mathbb{R}^3)times H^{frac{3}{2}}(mathbb{R}^3)$ and $sigmain (0,2)$. In particular, if the viscosity coefficient is equal to the resistivity coefficient, we can reduce $sigma$ to $0$ with the aid of the new formulation of the Hall-MHD system observed by cite{MR4193644}. Moreover, we establish the global well-posedness in $H^{frac{1}{2}+sigma}(mathbb{R}^3)times H^{frac{3}{2}}(mathbb{R}^3)$ with $sigmain (0,2)$ for small initial data and get the optimal time-decay rates of solutions. Our results improves some previous works.
281 - Yongming Luo 2021
We consider the large data scattering problem for the 2D and 3D cubic-quintic NLS in the focusing-focusing regime. Our attention is firstly restricted to the 2D space, where the cubic nonlinearity is $L^2$-critical. We establish a new type of scattering criterion that is uniquely determined by the mass of the initial data, which differs from the classical setting based on the Lyapunov functional. At the end, we also formulate a solely mass-determining scattering threshold for the 3D cubic-quintic NLS in the focusing-focusing regime.
comments
Fetching comments Fetching comments
Sign in to be able to follow your search criteria
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا