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Engin Başakoğlu

Publications and source records attributed to Engin Başakoğlu.

10 recordsLinked to original sources
arXiv2026

Hyperbolic nonlinear Schrödinger equations on $\mathbb{R}\times \mathbb{T}$

In this paper, we consider the hyperbolic nonlinear Schrödinger equations (HNLS) on $\mathbb{R}\times\mathbb{T}$. We obtain the sharp local well-posedness up to the critical regularity for cubic nonlinearity and in critical spaces for higher odd nonlinearities. Moreover, when the initial data is small, we prove the global existence and scattering for the solutions to HNLS with higher nonlinearities (except the cubic one) in critical Sobolev spaces. The main ingredient of the proof is the sharp up to the endpoint local/global-in-time Strichartz estimates.

math.AP↗
arXiv2025

Global Well-posedness and Scattering for Stochastic generalized KdV Equations with additive noise

We study the defocusing stochastic generalized Korteweg-de Vries equations (sgKdV) driven by additive noise, with a focus on mass-critical and supercritical nonlinearities. For integers $k \geq 4$, we establish local well-posedness almost surely up to scaling critical regularity. We also prove global well-posedness and scattering in $L^{2}_{x}(\mathbb{R})$ for the mass-critical equation with small initial data; also in $H^{1}_{x}(\mathbb{R})$ for the mass supercritical equation. In particular, we prove oscillatory integral estimates associated with more general dispersion relations, which are of independent interest; and we make use of a special case of these estimates as a main ingredient for the necessary bounds on the tail of the stochastic convolution for sgKdV, which is crucial to conclude scattering results.

math.AP↗
arXiv2025

Well-posedness for the periodic Hyperbolic nonlinear Schrödinger equations

We establish local well-posedness for the hyperbolic nonlinear Schrodinger equation (HNLS) in the critical spaces. Following the approach of Killip and Visan, we derive scale-invariant Strichartz estimates for HNLS on both rational and irrational tori, thereby removing the epsilon-loss of derivative present in the hyperbolic Strichartz estimates of Bourgain and Demeter.

math.AP↗
arXiv2025

Sharp unconditional well-posedness of the 2-$d$ periodic cubic hyperbolic nonlinear Schrödinger equation

We study semilinear local well-posedness of the two-dimensional periodic cubic hyperbolic nonlinear Schrödinger equation (HNLS) in Fourier-Lebesgue spaces. By employing the Fourier restriction norm method, we first establish sharp semilinear local well-posedness of HNLS in Fourier-Lebesgue spaces (modulo the endpoint case), including almost scaling-critical Fourier-Lebesgue spaces. Then, by adapting the normal form approach, developed by the second author with Guo and Kwon (2013) and by the second and third authors (2021), to the current hyperbolic setting, we establish sharp unconditional uniqueness of HNLS within the semilinear local well-posedness regime. As a key ingredient to both results, we establish sharp counting estimates for the hyperbolic Schrödinger equation. As a byproduct of our analysis, we also obtain sharp unconditional uniqueness of the (usual) two-dimensional periodic cubic nonlinear Schrödinger equation in Fourier--Lebesgue spaces for $p \ge 3$.

math.AP↗
arXiv2024

Scattering for Stochastic Nonlinear Schrödinger Equations with additive noise

We study the scattering for the energy-subcritical stochastic nonlinear Schrödinger equation (SNLS) with additive noise. In particular, we examine the long-time behavior of solutions associated with the noise $ϕ(x)g(t,ω)dB(t,ω)$ formed by a Schwartz function $ϕ$, and an adapted process $g(t,ω)$ satisfying certain decay. Essentially, the aim of the current paper is to prove almost sure scattering in the spaces $L^2$ and the pseudo-conformal space $Σ$ for an initial data in $Σ$; also in $H^1$ for an initial data in $H^1$.

math.AP↗
arXiv2024

Local well-posedness for the periodic Boltzmann equation with constant collision kernel

We study the Boltzmann equation with the constant collision kernel in the case of spatially periodic domain $\mathbb{T}^d$, $d\geq 2$. Using the existing techniques from nonlinear dispersive PDEs, we prove the local well-posedness result in $L^{2,r}_vH^s_x$ for $s>\frac{d}{2}-\frac{1}{4}$ and $r>\frac{d}{2}$. To reach the result, the main tool we establish is the $L^4$ Strichartz estimate for solutions to the corresponding linear equation.

math.AP↗
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↗
arXiv2021

Global Smoothing for the Davey-Stewartson System on $\mathbb{R}^2$

In this paper we study the regularity properties of solutions to the Davey-Stewartson system. It is shown that for initial data in a Sobolev space, the nonlinear part of the solution flow resides in a smoother space than the initial data for all times. We also obtain that the Sobolev norm of the nonlinear part of the evolution grows at most polynomially. As an application of the smoothing estimate, we study the long term dynamics of the forced and weakly damped Davey-Stewartson system. In this regard we give a new proof for the existence and smoothness of the global attractors in the energy space.

math.AP↗