Seminar: Nonlinear Problems of Mathematical Physics
 Date: January 11, 2024
 Speaker: Halit Sevki Aslan, University of Sao Paulo (USP), Ribeirao Preto, Brazil
 Title: Some recent developments on wave models with timedependent coefficients
 Abstract: In this talk, first we discuss some recent developments on linear and semilinear wave models with timedependent coefficients. Then, we are going to present some results on a Cauchy problem for scaleinvariant semilinear wave models with timedependent mass, dissipation and integrable timedependent propagation speed. Our goal is on the one hand to prove global (in time) existence of small data Sobolev solutions. On the other hand, we want to prove blowup results of local (in time) classical small data solutions.
 Tarih: 30 Kasım 2023
 Konuşmacı: Erhan Pişkin (Dicle Üniversitesi)
 Konuşma Başlığı: Gecikmeli Terim İçeren Değişken Üslü Kirchhoff tipli bir Denklemin Çözümlerinin Patlaması
 Özet: Bu konuşmada öncelikle değişken üslü denklemler ve gecikmeli terim içeren denklemler ile ilgili bazı temel bilgiler verilecektir. Daha sonra değişken üslü gecikmeli terim içeren Kirchhoff tipli bir denklemin çözümlerinin sonlu zamanda patlaması ile ilgili teorem ispatlanacaktır.
 Zaman : Nov 23, 2023 06:00 PM Istanbu
 Konuşmacı: Mansur I. Ismailov, Gebze Teknik Üniversitesi, Matematik Bölümü
 Başlık: Zaremba tipli sınır koşullarına sahip iki boyutlu ısı denklemi için katsayı bulma ters problemleri
 Özet: Bu konuşmada, Zaremba tipli sınır koşulları eşliğinde iki boyutlu ısı denkleminde zamana bağlı bir ısı kaynağının belirlenmesi ters problemini sunuluyor. Sınırın kısmen soğuran (Dirichlet), kısmen yansıtan (Neumann) ve kısmen ısı depolama kapasitesine sahip (Wentzell) olarak tanımlanması durumunda zamana bağlı kaynağın toplam enerjiye göre kontrol edilmesi modeli pratik açıdan önem arz etmektedir. Dikkate alınan uzaysal bölge, sınır koşulu spektral parametre içeren iki boyutlu Laplace operatörü için bir yardımcı spektral problemin özfonksiyonları cinsinden Fourier serisi analizinin uygulanmasına izin veren bir dikdörtgendir. Bu özdeğerözfonksiyon probleminin yardımıyla hem düz hem de ters problemlerin iyi tanımlılığı inceleniyor.
 Date: Nov 16, 2023
 Speaker : Varga Kalantarov, Department of Mathematics, Koç University, Istanbul
 Title :“Blow up of solutions of nonlinear wave and parabolic equations ”
 Abstract: The talk will be devoted to the problem of blow up of solutions to the Cauchy problem and initial boundary value problems for nonlinear wave equations. Some recent results on blow up of solutions of initial boundary value problems for nonlinear wave equations will be discussed.
 Date : May, 2023
 Speaker: Aynur Bulut, Department of Mathematics, University of Alabama, USA
 Title: Convex integration above the Onsager exponent for the forced Euler equations
 Abstract: In recent years, convex integration techniques have led to a number of striking results concerning nonuniqueness and anomolous dissipation for C^\alpha solutions of the incompressible Euler equations, with the recent resolution of the Onsager conjecture, showing that energy conservation fails when \alpha<1/3, playing a key role. In this talk, we give an overview of this area, and describe a new alternating convex integration framework for C^\alpha solutions of the forced Euler equations, which leads to nonuniqueness results valid for all \alpha<1/2, crossing above the Onsager threshold for the first time. This is joint work with Stan Palasek (UCLA) and Manh Khang Huynh (Georgia Institute of Technology).
 Date: April 2023
 Speaker: Sevcen Hakkayev, Trakya University, Edirne
 Title: ON THE GENERATION OF STABLE KERR FREQUENCY COMBS IN THE LUGIATO–LEFEVER MODEL OF PERIODIC OPTICAL WAVEGUIDES
 Abstract: We consider the Lugiato–Lefever (LL) model of optical fibers. We construct a two parameter family of steady state solutions, i.e., Kerr frequency combs, for small pumping parameter h > 0 and the correspondingly (and necessarily) small detuning parameter, α > 0. These are O(1) waves, as they are constructed as a bifurcation from the standard dnoidal solutions of the cubic nonlinear Schr¨odinger equation. We identify the spectrally stable ones, and more precisely, we show that the spectrum of the linearized operator contains the eigenvalues 0, −2α, while the rest of it is a subset of {µ : : Re µ = −α}. This is in line with the expectations for effectively damped Hamiltonian systems, such as the LL model. ( Joint work with M. Stanislavova and A. Stefanov)
 Date: Mar 10, 2023
 Speaker : Mete Demircigil, Department of Mathematics, Koç University, Istanbul
 Title : Aerotactic Waves in Dictyostelium discoideum : When SelfGenerated Gradients interact with Expansion by Cell Division.
 Abstract : Using a selfgenerated hypoxic assay, it is shown that Dictyostelium discoideum displays a remarkable collective aerotactic behavior: when a cell colony is covered, cells quickly consume the available oxygen and form a dense ring moving outwards at constant speed and density.We propose a simple, yet original PDE model, that enables an analytical qualitative and quantitative study of the phenomenon and reveals that the collective migration gives rise to traveling wave solutions, whose propagation can be explained through the interplay between cell division and the modulation of aerotaxis. The modeling and its conclusions supplement and are confirmed by an experimental investigation of the cell population behavior.This approach also gives rise to an explicit and novel formula of the collective migration speed of cells that encapsulates a surprising combination of expansion by cell division, such as described by the Fisher/KPP equation, and aerotaxis. The conclusions of this model appear to extend to more complex models.Interestingly the structure of the model resembles in many ways the Burger’sF/KPP Equation and the monostable ReactionDiffusion Equation with reaction term f(u)=u(1u)(1+Bu). In order to investigate further the analogies between these models, we propose a modified version of the PDE model, that will enable us to give a better characterization of the asymptotic behaviors as well as an alternative viewpoint on the inside dynamics of the traveling waves.
 Date: March 2023
 Speaker: Atanas Stefanov, Department of Mathematics, University of Alabama, USA
 Title: On the stability of solitary waves in the NLS system of the thirdharmonic generation
 Abstract: We consider the NLS system of he thirdharmonic generation, which was introduced recently by Sammut et.al. Our interest is in solitary wave solutions and their stability properties. In a recent work, Oliveira and Pastor discussed global wellposedness vs. finite time blow up, as well as other aspects of the dynamics. These authors have also constructed solitary wave solutions, via the method of mountain pass/Nehari manifold, in an appropriate range of parameters. Specifically, the waves exist only in spatial dimensions n=1,2,3. They also establish some stability/instability results for these waves. We construct the waves in the largest possible parameter space, and we provide a complete classification of their stability. In dimension one, we show stability, whereas in n=2,3, they are generally spectrally unstable, except for a small region, where they do enjoy an extra pseudoconformal symmetry. Finally, we discuss instability by blowup. In the 3 D case, and for some more restrictive set of parameters, we use virial identities methods to derive the strong instability, in the spirit of M. Ohta’s approach. In 2D, the virial identities reduce matters, via conservation of mass and energy, to the initial data. Our conclusions mirror closely the wellknown results for the scalar cubic focusing NLS, while the proofs are much more involved. Joint work with Abba Ramadan, University of Alabama.
 Date : February 24, 2023
 Time: 06:00 PM Istanbul
 Speaker : Anvarbek Meirmanov, Institute of the Ionosphere, Ministry of Education and Science, Almaty, Kazakhstan.
 Title: Mathematical models of insitu leaching.
 Abstract: Heap and insitu leaching, an important technological process to extract precious metals, nickel, copper and other compounds. We develop a general mathematicalapproach to treat the problem on a macroscopic level. Main point here are:
 exact microscopic model based on the Newton’s laws of classical continuum mechanics at the pore
 scale (10100 micron);
 free (unknown) boundaries between liquid and solid phases;
 new boundary conditions at the microscopic scale, which express mass conservation laws;
 exact mathematical models at the macroscopic scale (110 meters) as a rigorous homogenization of the corresponding microscopic models.
 Time : Jan 28, 2022 05:30
 Speaker: Ilker Koçyigit, Department of Mathematics, Koç University, Istanbul
 Title: Improved reconstruction conditions for array imaging
 Abstract : In this talk, we first give some examples from the field of inverse problems and some of their applications. Then, we will talk about array imaging which has various applications such as radar/sonar imaging and medical imaging.We will discuss its formulation as a minimization problem and talk about some known sufficient conditions for exact recovery. Then, we will present an improved recovery condition based on coherence conditions. We compare this method to the wellknown restricted isometry property and coherence conditions. We will discuss resolution estimates using it when the exact recovery of the image is not possible.
 Date January 13 2023. Speaker: : A.M. Tuffaha, Department of Mathematics and Statistics , American University of Sharjah Title: On the wellposedness of an inviscid fluidstructure interaction model
Abstract :We consider the Euler equations on a domain with free moving interface. The motion of the interface is governed by a 4th order linear EulerBernoulli beam equation. The fluidstructure interaction dynamics are realized through normal velocity matching of the fluid and the structure in addition to the aerodynamic forcing due to the fluid pressure.We derive apriori estimates and construct localintime solutions to the system in the Sobolev space H^r, with r>5/2. We also establish uniqueness in the Sobolev space H^r with r>3. An important consequence of the existence theorem is that the TaylorRayleigh instability does not occur. This is joint work with Igor Kukavica.
 Date : December 23 2022 . Speaker: Mansur I. Ismailov Gebze Technical University Title: Inverse scattering method for nonlinear Klein–Gordon equation coupled with a scalar field
Abstract : A class of negative order AblowitzKaup NewellSegur (AKNS) nonlinear evolution equations are obtained by applying the Lax hierarchy of the generalized ZakharovShabat (ZS) system. The inverse scattering problem on the whole axis is examined in the case where the ZS system consists of two equations and admits a real symmetric potential. Referring to these results, the Nsoliton solutions for the integrodifferential version of the nonlinear Klein–Gordon equation coupled with a scalar field are obtained by using the inverse scattering method. (Joint work with Cihan Sabaz
 Date: December 16 2022
 Speaker: Secen Hakkayev, Trakya University, EdirneTitle: On the stability of the compacton waves for the degenerate KdV ad NLS modelsAbstract: In this talk, we consider the degenerate semilinear Schrödinger and Korteweg deVries (KdV) equations in one spatial dimension. We construct special solutions of the two models, namely standing wave solutions of nonlinear Schrödinger equation and traveling waves for KdV equation, which turn out to have compact support, compactons. We show that the compactons are unique bellshaped solutions of the corresponding PDE’s and for appropriate variational problems as well. We provide a complete spectral characterization of such waves, for all values of p. Namely, we show that all waves are spectrally stable for 2 < p ≤ 8, while a single mode instability occurs for p > 8. This extends previous work of Germain, HarropGriffits and Marzuola, who have previously established orbital stability for some specific waves, in the range p < 8. (joint work with Abba Rmadan and Atanas G. Stefanov)
 Date December 9 2022 . Speaker: Albert Erkip, Sabancı University, Istanbul Title: A comparison of solutions of convolutiontype unidirectional wave equations
Abstract : In this talk, we prove a comparison result for a general class of nonlinear dispersive unidirectional wave equations where dispersion is caused by a convolution in space. A typical example of such a model is the Benjamin Bona Mahony (BBM) equation. The natural question is to determine how solutions depend on the convolution kernel; in particular whether solutions stay close whenever the kernels are close in some suitable sense. To simplify the question we will look at the zero dispersion limit, namely when the kernel approaches the Dirac measure.

 Date: June 21, 2022
Speaker: Gregory Seregin, Oxford University, UK
Title: Axisymmetric Solutions to the NavierStokes Equations.
Abstract: I’ll discuss recent regularity results for axisymmetric solutions to the NavierStokes equations. Among them are slightly supercritical assumptions providing smoothness, which, for example, exclude Type I blowups of axisymmetric solutions.
 Date: June 21, 2022


 Date: May 13 , 2022
Speaker: Engin Basakoglu, Department of Mathematics, Bogaziçi University, Istanbul
Title: The HirotaSatsuma system on the torus.
Abstract: In this talk, we consider the HirotaSatsuma system, a coupled Korteweg de Vries system (CKdV), with periodic boundary conditions. The model explains two long waves’ interactions with separate dispersion relations. CKdV systems have a wide range implementations in different fields of chemistry, biology, hydrodynamics, mechanics, plasma physics, etc. The talk concerns with the dynamics of solutions to the HirotaSatsuma system. In particular, given 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 of the system. To cope with the resonances stemming from interactions of various frequencies, dependent on the coupling parameter, we invoke the Diophantine approximation of real numbers. Moreover, we address the forced and damped version of the HirotaSatsuma system, once having the analogous smoothing estimates, we discuss the existence and smoothness of a global attractor in the energy space.
 Date: May 13 , 2022



 Date: April 15 , 2022
Speaker: Khompysh Khonatbek, Al–Farabi Kazakh National University, Almaty, Kazakhstan
Title: Stability problems for Kelvin – Voigt fluids.
Abstract: One of the mathematical model of hydrodynamics is the system of KelvinVoigt (NavierStokesVoigt) equations, which describes the motion of a viscous incompressible nonNewtonian fluid. The direct problems for the system of KelvinVoigt (NavierStokesVoigt) equations consist of defining a velocity and a pressure of fluid. There are many works on the direct problems, one can say, that the most wellknown of them are works of Oskolkov. The inverse problems of hydrodynamics arise in the mathematical modeling of processes that in addition to solutions of the direct problem (velocity, pressure), it is necessary to determine some physical parameters, which are unknown or impossible to direct measure. In this talk, we discuss the unique solvability of some inverse problems for KelvinVoigt equations, which consist to determine a coefficient of the righthand side, by the given additional condition.
 Date: April 15 , 2022



 Date: April 1 , 2022
Speaker: Brian Straughan, Durham University, UK
Title: Stability problems for Kelvin – Voigt fluids.
Abstract: We introduce the KelvinVoigt equations of variable order for thermal convection studies. A hierarchy of viscoelastic fluids is introduced via the names of Stokes, Maxwell, Oldroyd, Kelvin and Voigt. The basic problems of thermal convection in a KelvinVoigt fluid are outlined and we then discuss several continuous dependence and structural stability issues.
 Date: April 1 , 2022



 Date: March 25 , 2022
Speaker: Turker Ozsari, Bilkent University, Ankara
Title: Noncontrollability of the Schroedinger equation on the half space.
Abstract: We discuss the lack of exact boundary controllability for the Schrödinger equation on the half space in low dimensions (half line and half plane). We also extend this result to the biharmonic Schrödinger equation. Our method relies on recent developments in local wellposedness theory for initial boundary value problems of these equations. In particular, we use temporal estimates of spatial traces in fractional Sobolev spaces. The noncontrollability results obtained here contrast with the dynamics in the case of bounded domains where controllability property holds true.
 Date: March 25 , 2022



 Date: February 25, 2022
Speaker: Sevdzhan Hakkaev, Department Mathematics, Trakya University, Edirne
Title: On the stability fo periodic waves.
 Date: February 25, 2022



 Date: January 28, 2022
Speaker: Ilker Koçyigit, Koç University, Istanbul
Title: Improved reconstruction conditions for array imaging.
Abstract: In this talk, we first give some examples from the field of inverse problems and some of their applications. Then, we will talk about array imaging which has various applications such as radar/sonar imaging and medical imaging. We will discuss its formulation as a minimization problem and talk about some known sufficient conditions for exact recovery. Then, we will present an improved recovery condition based on coherence conditions. We compare this method to the wellknown restricted isometry property and coherence conditions. We will discuss resolution estimates using it when the exact recovery of the image is not possible.
 Date: January 28, 2022



 Date: January 14, 2022 at 17:30 (Istanbul time)
Speaker: Ionut Munteanu, Department of Mathematics, Alexandru Ion Cuza University of Iasi
Title: Boundary stabilization of parabolic type equations by proportional type feedback forms
Abstract: Here we shall present ome results on boundary exponential stabilization to trajectories of semilinear heat equations. We shall use the method of splitting the system into two parts: a finitedimensional one, that is unstable, and an infinitedimensional one, that is stable. We shall stabilize the finitedimensional component by a simple, linear, finitedimensional boundary controller. We shall compare this method to the backstepping control design technique. We shall highlight the differences and the similarities. Then we shall try to merge them. Some applications will be provided.
 Date: January 14, 2022 at 17:30 (Istanbul time)



 Date: January 7, 2022
Speaker: Varga Kalantarov, Koc University, Istanbul
Title: On the convective BrinkmanForchheimer equations
Abstract: : The convective BrinkmanForchheimer (CBF) equations describe the motion of incompressible fluid flows in a saturated porous medium. One can consider CBF equations as a damped NavierStokes equations with damping \alpha u+\beta u^{r1}u, where u is the velocity field and r>=1 is the absorption exponent. In this talk, we discuss the existence and uniqueness of a global in time weak solution in the LerayHopf sense satisfying the energy equality to critical and supercritical CBF equations (r>=3) in two and three dimensional bounded as well as periodic domains. We exploit the monotonicity as well as the demicontinuity properties of the linear and nonlinear operators and the MintyBrowder technique in the proofs. Finally, we discuss the global in time strong solutions to CBF equations.
 Date: January 7, 2022



 Date: December 17, 2021
Speaker: Attila Askar, Koç University, Istanbul
Title: NavierStokes ve Schrödinger denklemleri: Matematik ve Fizikte iki uç ile Kuvantum olaylarda sönüm.
 Date: December 17, 2021

 Zaman: Nov 26, 2021 05:30 PM
 Konuşmacı: Alp Eden
 Konuşma başlığı: “Örnek bir uygulamalı matematikçi: William Prager (19031980)”
 Özet. William Prager Alman bilim anlayışının iyi bir temsilcisidir. Göttingen Üniversitesinde Ludwig Prandtl ’ın yanında Mühendis; İstanbul Üniversitesinde Richard de Mises ’in yanında Arf ’ın deyimiyle uygulamalı matematikçi; Brown Üniversitesinde birçok ilke imzasını koymuş bir ekol kurucu olarak hem bir uygulamalı matematikçi (SIAM) hem bir fizikçi (AIP) hem bir deneysel mühendis (ASME) hem de teorik mühendis (SES) olabilmiş nadir kişilerden biridir. Konuşmamda onun hayatından bazı kesitler sunacağım. Özellikle Türkiye’de uzun süren etkisini göz önüne sermeye çalışacağım.


 Date: November 12, 2021
Speaker: Yu. Alkhutov, Vladimir State University, Russia
Title: Elliptic and parabolic equations of the second order with nonstandard growth condition .
 Date: November 12, 2021



 Date: November 5, 2021 at 17:30 (Istanbul time)
Speaker: Hasan Inci, Koc University, Istanbul
Title: On the local well posedness of the two component CamassaHolm equation
Abstract: : The CamassaHolm equation models the motion of waves in shallow water. In this talk we consider a two component generalization of the CamassaHolm equation. local well posedness in the sense of Hadamard means local existence, uniqueness and stability of solutions. the two component camassaholm equation is locally well posed in the sobolev spaces $H^s$ for $s > 3/2$ in the sense of hadamard. a natural question is how the solutions depend on the initial data. we will show that the dependence on the initial data is nowhere locally Lipschitz. This shows that the two component Camassa – Holm equation is very sensitive to perturbations in the initial data.
 Date: November 5, 2021 at 17:30 (Istanbul time)



 Date: June 11, 2021 at 16:30 (Istanbul time)
Speaker: Emil Novruz, Gebze Technical University, Gebze
Title: On blow up phenomena for shallow water equations
Abstract: The precise mode of blowup for the shallow water equations, in particular, for CamassaHolm(CH) and the related equations and systems is an interesting question which remains unclear for special cases and awaits a detailed investigation. Blowup techniques are quite specific to each type of equation; there is no general method. In the present talk we will describe the details of the blowup mechanism for solutions of Cauchy problem for CH with certain initial profiles. It should be noted that blowup criteria for such type equations systematically involve the computation of some global quantities or other global conditions like antisymmetry assumptions or sign conditions on the associated potential. In this context, the subject of special interest is a localinspace blowup criterion which only involves the values of initial function and its derivative in a single point of the real line. In the talk we also will try to focus on such type blowup conditions for some generalizations of the CH equation.
 Date: June 11, 2021 at 16:30 (Istanbul time)



 Date: June 11, 2021 at 17:00 (Istanbul time)
Speaker: Mustafa Polat, Yeditepe University, Istanbul
Title: Blow up of solutions of fourth order nonlinear pseudoparabolic equations
Abstract: The problem of blow up in a finite time of solutions of initial boundary value problems for fourth order pseudoparabolic equations and lower bound of blow up of solutions of these kind of equations will be discussed.
 Date: June 11, 2021 at 17:00 (Istanbul time)



 Date: June 4, 2021 at 17:00 (Istanbul time)
Speaker: Andrey Shishkov, Mathematical Institute of RUDN University , Moscow
Title: Large and very singular solutions of semilinear elliptic equations
Abstract: There will be done a review of results about nonnegative solutions of semilinear equations of diffusionnonlinear degenerate absorption, which take infinite value on some subsets or all boundary of the domain under consideration: so called very singular and large solutions. Some classical and new sharp necessary and sufficient conditions of existence and uniqueness of such solutions will be discussed.
 Date: June 4, 2021 at 17:00 (Istanbul time)



 Date: May 28, 2021 at 17:00 (Istanbul time)
Speaker: Mansur Ismailov, Gebze Technical University, Gebze
Title: Inverse problem for finding of time – dependent coefficent of the diffusion equation
Abstract: The inverse problems problem of finding the time dependent coefficients and source terms will be considered. Under some regularity conditions on the data it will be shown that the connsidered inverse problems are well posed. Difficulties arising in the study of inverse problems with boundary conditions different from the classical boundary conditions (nonlocal or Wentzel type) will be also discussed.
 Date: May 28, 2021 at 17:00 (Istanbul time)



 Date: May 21, 2021 at 17:00 (Istanbul time)
Speaker: Jerry Bona, University of Illinois at Chicago
Title: Initial and BoundaryValue Problems for Nonlinear, Dispersive Wave Equations
Abstract: While posing nonlinear, dispersive wave equations with initial data defined in all of space or in a periodic setting leads to the most elegant mathematical theory, in applications it is seldom the case that we know the initial value everywhere in space at a single instance of time. This observation has led to the development of theory for initialboundaryvalue problems for such equations. In laboratory or field situations, the initial data is often unknown and so taken to be zero and the solutions develop by way of being forced from the boundary. Boundary forcing can sometimes be measured reasonably accurately as a function of time. A similar remark holds true for numerical methods for the approximation of solutions. In the early stages of the development of nonlinear, dispersive equations, numerical schemes were invariably set in either a periodic domain or with homogeneous Dirichlet boundary conditions. That changed later in the 20th century when such models began to be used in real world situations.
It is the goal of this lecture to introduce some particular nonlinear, dispersive models and discuss their initialboundaryvalue formulations. The presentation will start with some relatively simple, classical style theory and branch into more modern methods using detailed Fourier methods and inverse scattering theory.
 Date: May 21, 2021 at 17:00 (Istanbul time)



 Date: May 7, 2021
Speaker: Darya Apushkinskaya, Peoples’ Friendship University of Russia ,Moscow,
Saarland University, Saarbrüken
Title : When Hopf’s lemma remains valid?
Abstract: The Hopf lemma, known also as the“boundary point principle”, is one of the important tools in qualitative analysis of partial differential equations. This lemma states that a nonconstant supersolution of a partial differential equation with a minimum value at a boundary point, must increase linearly Aaway from its boundary minimum provided the boundary is smooth enougn.
For general operators of nondivergence type with bounded measurable coefficients this result was established in elliptic case independently by E. Hopf and O. Oleinik (1952) and in parabolic case by L. Nirenberg (1953). The first result for elliptic equations with divergence structure was proved by R. Finn and D. Gilbarg (1957). Later the efforts of many mathematicians were aimed at the extension of the classes of admissible opeartors and at the reduction of the boundary smoothness.
We present several versions of the Hopf lemma for general elliptic equations in divergence and nondivergence forms under the sharp requirements on the coefficients of equations and on the boundary of a domain. Also we provide a new sharp counterexample.
The talk is based on results obtained in collaboration with Alexander Nazarov.
 Date: May 7, 2021



 Date: April 30, 202
Title: Periodic nonlinear Schrödinger problems (Doğrusal olmayan periyodik Schrödinger problemleri)
Speaker: Burak Gurel, Boğaziçi University, Istanbul
Abstract: We discuss several types of NLS Cauchy problems on torus in the form of a literature survey. We first explain the role of Strichartz estimates in proving wellposedness results in unbounded domains and lack of their existence in the periodic setting until Bourgain’s monumental GAFA papers in 1993. We then present Bourgain’s wellposedness results for NLS with various nonlinearity exponents on torus in various dimensions. Some more recents results especially for critical nonlinearities will also be mentioned. Finally, we briefly talk about the proofs of periodic Strichartz estimates with an emphasis on how analytic number theoretical tools become essential in counting integer points that solve certain Diophantine equations arising from the linear solution operator of NLS.
 Date: April 30, 202



 Date: April 16, 2021
Speaker: Tekin Dereli, Koç University, Istanbul
Abstract: Vacuum Polarization Effects in BertottiRobinson Spacetimes
Abstract: BertottiRobinson spacetimes are topologically $AdS_2 \times S^2$ and described by a conformally flat metric. Together with the Coulomb electric potential, they provide a class of static, geodetically complete EinsteinMaxwell solutions.We show here that the BertottiRobinson metrics together with WuYang magnetic pole potentials give a class of static solutions of a
system of nonminimally coupled EinsteinYangMills equations, that may be relevant for investigating vacuum polarization effects in a first order perturbative approach to quantum fields.
 Date: April 16, 2021



 Date: April 9, 2021
Speaker: Alp Eden, Bogaziçi University, Istanbul
Titile:: Hilbert’s 16. problem.
 Date: April 9, 2021



 Date: April 2, 2021
Speaker: Hasan Inci, Koç University, Istanbul
Titile:: Euler equations and cassical mechanics.
Abstract:: Vacuum Polarization Effects in BertottiRobinson Spacetimes
Abstract: BertottiRobinson spacetimes are topologically $AdS_2 \times S^2$ and described by a conformally flat metric. Together with the Coulomb electric potential, they provide a class of static, geodetically complete EinsteinMaxwell solutions.We show here that the BertottiRobinson metrics together with WuYang magnetic pole potentials give a class of static solutions of a
system of nonminimally coupled EinsteinYangMills equations, that may be relevant for investigating vacuum polarization effects in a first order perturbative approach to quantum fields.
 Date: April 2, 2021



 Date: April March 16, 2021
Speaker: Albert Erkip, Sabanci University, Istanbul
Titile:: On KleinGordon equations.
Abstract:: In this talk we will go through some wellknown results on the KleinGordon equation. We start with a discussion of the wellposedness of the Cauchy problem and possible “correct” choices of function spaces. We then proceed to the question of the time span of the solution; namely look at conditions ensuring global existence versus finite time blow up. Finally we consider the potential well method (the classical 1975 paper of Payne and Sattinger Saddle Points and Instability of Nonlinear Hyperbolic Equations, Israel J. of Math. 22) and see how the threshold between global existence and blow up can be determined..
 Date: April March 16, 2021



 Date: March 19, 2021
Speaker: Sergey Zelik, University of Surrey, Guilford, UK
Titile:: Inertial Manifolds for Dispersive PDE’s.
 Date: March 19, 2021



 Date: March 12, 2021
Speaker: Turker Ozsari, Bilkent University, Ankara
Title:: Dispersion estimates for solutions of the forth order Schrödinger equation on the half line.
 Date: March 12, 2021


 Date: March 5, 2021
Speaker: Varga Kalantarov, Koç University, Istanbul
Title:: About some nonlinear PDE’s and the program of the seminar.
 Date: March 5, 2021