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Razvan Mosincat

Publications and source records attributed to Razvan Mosincat.

6 recordsLinked to original sources
arXiv2023

Unconditional uniqueness for the Benjamin-Ono equation

We study the unconditional uniqueness of solutions to the Benjamin-Ono equation with initial data in $H^{s}$, both on the real line and on the torus. We use the gauge transformation of Tao and two iterations of normal form reductions via integration by parts in time. By employing a refined Strichartz estimate we establish the result below the regularity threshold $s=1/6$. As a by-product of our proof, we also obtain a nonlinear smoothing property on the gauge variable at the same level of regularity.

math.AP↗
arXiv2020

Global well-posedness and scattering for the Dysthe equation in $L^2(\mathbb R^2)$

This paper focuses on the Dysthe equation which is a higher order approximation of the water waves system in the modulation (Schrödinger) regime and in the infinite depth case. We first review the derivation of the Dysthe and related equations. Then we study the initial-value problem. We prove a small data global well-posedness and scattering result in the critical space $L^2(\mathbb R^2)$. This result is sharp in view of the fact that the flow map cannot be $C^3$ continuous below $L^2(\mathbb R^2)$. Our analysis relies on linear and bilinear Strichartz estimates in the context of the Fourier restriction norm method. Moreover, since we are at a critical level, we need to work in the framework of the atomic space $U^2_S$ and its dual $V^2_S $ of square bounded variation functions. We also prove that the initial-value problem is locally well-posed in $H^s(\mathbb R^2)$, $s>0$. Our results extend to the finite depth version of the Dysthe equation.

math.AP↗
arXiv2018

Unconditional uniqueness for the derivative nonlinear Schrödinger equation on the real line

We prove the unconditional uniqueness of solutions to the derivative nonlinear Schrödinger equation (DNLS) in an almost end-point regularity. To this purpose, we employ the normal form method and we transform (a gauge-equivalent) DNLS into a new equation (the so-called normal form equation) for which nonlinear estimates can be easily established in $H^s(\mathbb{R})$, $s>\frac12$, without appealing to an auxiliary function space. Also, we prove that low-regularity solutions of DNLS satisfy the normal form equation and this is done by means of estimates in the $H^{s-1}(\mathbb{R})$-norm.

math.AP↗
arXiv2018

Stochastic nonlinear Schrödinger equations on tori

We consider the stochastic nonlinear Schrödinger equations (SNLS) posed on $d$-dimensional tori with either additive or multiplicative stochastic forcing. In particular, for the one-dimensional cubic SNLS, we prove global well-posedness in $L^2(\mathbb{T})$. As for other power-type nonlinearities, namely (i) (super)quintic when $d = 1$ and (ii) (super)cubic when $d \geq 2$, we prove local well-posedness in all scaling-subcritical Sobolev spaces and global well-posedness in the energy space for the defocusing, energy-subcritical problems.

math.AP↗
arXiv2016

Global well-posedness of the derivative nonlinear Schrödinger equation with periodic boundary condition in $H^{\frac12}$

We establish the global well-posedness of the derivative nonlinear Schrödinger equation with periodic boundary condition in the Sobolev space $H^{\frac12}$, provided that the mass of initial data is less than $4π$. This result matches the one by Miao, Wu, and Xu and its recent mass threshold improvement by Guo and Wu in the non-periodic setting. Below $H^{\frac12}$, we show that the uniform continuity of the solution map on bounded subsets of $H^s$ does not hold, for any gauge equivalent equation.

math.AP↗