T. Burak Gürel | Searcharxiv

arXiv2023

Global Well-posedness for the Biharmonic Quintic Nonlinear Schrödinger Equation on $\mathbb{R}^2$

We prove that the Cauchy problem for the 2D quintic defocusing biharmonic Schrödinger equation is globally well-posed in the Sobolev spaces $H^s(\mathbb{R}^2)$ for $\frac{8}{7}<s<2$. Our main ingredient to establish the result is the $I$-method of Colliander-Keel-Staffilani-Takaoka-Tao \cite{colliander2002almost} which is used to construct the modified energy functional that is almost conserved in time.

math.AP↗

arXiv2022

Smoothing and Global Attractors for the Hirota-Satsuma System on the Torus

We consider the Hirota-Satsuma system, a coupled KdV-type system, with periodic boundary conditions. The first part of the paper concerns with the smoothing estimates for the system. More precisely, it is shown that, for initial data in a Sobolev space, the difference of the nonlinear and linear evolutions lies in a smoother space. The smoothing gain we obtain depends very much on the arithmetic nature of the coupling parameter $a$ which determines the structure of the resonant sets in the estimates. In the second part, we address the forced and damped Hirota-Satsuma system and obtain counterpart smoothing estimates. As a consequence of these estimates, we prove the existence and smoothness of a global attractor in the energy space.

math.AP↗