# Bashar KhorbatlyUniversity of Bergen | UiB · Department of Mathematics

Bashar Khorbatly

Ph.D. Mathematics

Analysis of PDEs arising in fluid mechanics

## About

21

Publications

2,109

Reads

**How we measure 'reads'**

A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more

82

Citations

Introduction

I am employed at the Department of Mathematics at University of Bergen on a SEAS Postdoctoral Research fellowship funded by the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 101034309. My research activity targets mainly the area of partial differential equations, my focus lies on studying dynamic properties of solutions to asymptotic nonlinear water waves models and nonlinear dispersive PDEs arising in fluid mechanics.

Education

November 2016 - December 2019

November 2016 - December 2019

## Publications

Publications (21)

In this paper, we prove an orbital stability result for the Degasperis-Procesi peakon with respect to perturbations having a momentum density that is first negative and then positive. This leads to the orbital stability of the antipeakon-peakon profile with respect to such perturbations.
Cite as : B. Khorbatly, L. Molinet. On the orbital stability...

The aim of this paper is to prove that the Degasperis-Procesi antipeakon-peakon profile is asymptotically stable for all time. We start by proving the asymptotic stability of a single Degasperis-Procesi peakon and antipeakon with respect to perturbations having a momentum density that is first negative and then positive. Then this result is extende...

In the mathematical theory of water waves, this paper focuses on the hierarchy of higher order asymptotic models. The well-posedness of the medium amplitude extended Green-Naghdi model, as well as higher-ordered Boussinesq-Peregrine and Boussinesq models, is first demonstrated. Introducing a regularization term and various physical topography varia...

We derive the classical Boussinesq system to model equatorial water flows with the weak Coriolis effect due to Earth's rotation. The system is derived from the Zakharov-Craig-Sulem formulation of the water wave equations in the f-plane approximation. Then, for the obtained system, we find some exact traveling wave solutions of the form $sech^2$ and...

The aim of this paper is to provide an alternative proof of the well-posedness of the Green-Naghdi equations with the Coriolis effect established by Chen et al. (2018). We showed that an additional assumption on the initial horizontal velocity is not necessary to obtain well-posedness. Indeed, with a refined symmetrizer and appropriately scaling th...

Since the work of Stokes on steady progressive surface waves (Stokes, 1847), there has been interest in fluid particle trajectories and associated mass flux. The original result obtained by Stokes was based on linear theory, and implied that there is a net forward drift in the fluid beneath a propagating surface wave. In the non-dimensional case, t...

It is shown that the Boussinesq–Peregrine system, which describes long waves of small amplitude at the surface of an inviscid fluid with variable depth, admits a number of approximate conservation equations. Notably, this paper provides accurate estimations for the approximate conservation of the mechanical balance laws associated with mass, moment...

In the context of the initial data and an amplitude parameter ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}, we establish a local existe...

This article takes into account the Korteweg–de Vries (KdV) equation as an approximate model of long waves of small amplitude at the free surface with inviscid fluid. It is demonstrated that the mechanical balance quantities, as defined by the solution of the KdV equation, rigorously approximate those in the Euler system within the $L^\infty$ space...

This paper is a continuation of a previous work on the extended Green-Naghdi system. We prolong the system, in arbitrary dimension, with/without surface tension, and for a general bottom topography. Confining the work to the one-dimensional case, well-posedness and consistency with respect to initial data and parameters are proved taken into accoun...

Following a straightforward proof for symmetric solutions to be traveling waves by Pei (Exponential decay and symmetry of solitary waves to Degasperis‐Procesi equation. Journal of Differential Equations. 2020;269(10):7730‐7749), we prove that classical symmetric solutions of the highly nonlinear shallow water equation recently derived by Quirchmayr...

The purpose of this paper is to present the derivation and mathematical analysis of a new asymptotic model that describes the evolution of medium amplitude internal waves propagating between a flat rigid-lid and a highly variable topography. The smallness assumptions on the topographic variation parameter used in Lteif et al. (2015) and in Lteif an...

This study deals with higher-order asymptotic equations for the water-waves problem. We considered the higher-order/extended Boussinesq equations over a flat bottom topography in the well-known long wave regime. Providing an existence and uniqueness of solution on a relevant time scale of order 1/ √ ε and showing that the solution's behavior is clo...

This study deals with higher-ordered asymptotic equations for the water-waves problem. We considered the higher-order/extended Boussinesq equations over a flat bottom topography in the well-known long wave regime. Providing an existence and uniqueness of solution on a relevant time scale of order $1/\sqrt{\eps}$ and showing that the solution's beha...

The aim of this paper is to give an alternative proof for the derivation of a prior energy estimate. Consequently, this allows to define a natural energy norm of the long-term well-posedness result established by S. Israwi in [Nonlinearity 23,2889-2904 (2010)] but for the original system, in which the partial operator curl is not involved.

Green-Naghdi equations are commonly used in coastal oceanography to describe the propagation of large-amplitude surface waves. In this paper, a new convenient equivalent system to the standard two-dimensional case of these equations is presented. This system helps in studying the existence of solution and in its numerical simulations, also gives so...

In this paper, a generalized nonlinear Kawahara equation with time t>0 and x∈R space‐dependent coefficient is considered. We show that the construction of solution with a standard fixed point method can be accomplished so that the well‐posedness in Hs(R) for some s>3/2 is proved under a specific nonpositive differential condition. We introduce an a...

In this paper, we prove an orbital stability result for the Degasperis-Procesi peakon with respect to perturbations having a momentum density that is first negative and then positive. This leads to the orbital stability of the antipeakon-peakon profile with respect to such perturbations.
Cite as : B. Khorbatly, L. Molinet. On the orbital stability...

Green-Naghdi equations are commonly used in coastal oceanography to describe the propagation of large-amplitude surface waves. In this paper, a new convenient equivalent system to the standard two-dimensional case of these equations is presented. This system helps in studying the existence of solution and in its numerical simulations, also gives so...

In this paper, we will derive the two-dimensional extended Green-Naghdi system (see Matsuno [Proc. R. Soc. A 472, 20160127 (2016)] for determination in a various way) for flat bottoms of order three with respect to the shallowness parameter μ. Then we consider the one-dimentional extended Green-Naghdi equations taking into account the effect of a s...