Seminar: Nonlinear Problems of Mathematical Physics

 

• Date    January  13    2023.                                                                                                                  Speaker:  : A.M. Tuffaha, Department  of Mathematics and Statistics , American University of Sharjah                                                                                                                                                         Title: On the well-posedness of an inviscid fluid-structure 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 Euler-Bernoulli beam equation. The fluid-structure 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 a-priori estimates and construct local-in-time 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 Taylor-Rayleigh 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 Ablowitz-Kaup- Newell-Segur (AKNS) nonlinear evolution equations are obtained by applying the Lax hierarchy of the generalized Zakharov-Shabat (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 N-soliton solutions for the integro-differential 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 models

  Abstract: In this talk, we consider the degenerate semi-linear 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 bell-shaped 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, Harrop-Griffits 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 convolution-type 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 Navier-Stokes Equations.
      Abstract: I’ll discuss recent regularity results for axisymmetric solutions to the Navier-Stokes equations. Among them are slightly supercritical assumptions providing smoothness, which, for example, exclude Type I blowups of axisymmetric solutions.

 

• • • Date: May 13 , 2022
      Speaker: Engin Basakoglu, Department of Mathematics, Bogaziçi University, Istanbul
      Title: The Hirota-Satsuma system on the torus.
      Abstract: In this talk, we consider the Hirota-Satsuma 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 Hirota-Satsuma 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 Hirota-Satsuma system, once having the analogous smoothing estimates, we discuss the existence and smoothness of a global attractor in the energy space.

 

• • • 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 Kelvin-Voigt (Navier-Stokes-Voigt) equations, which describes the motion of a viscous incompressible non-Newtonian fluid. The direct problems for the system of Kelvin-Voigt (Navier-Stokes-Voigt) 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 well-known 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 Kelvin-Voigt equations, which consist to determine a coefficient of the right-hand side, by the given additional condition.

 

• • • Date: April 1 , 2022
      Speaker: Brian Straughan, Durham University, UK
      Title: Stability problems for Kelvin – Voigt fluids.
      Abstract: We introduce the Kelvin-Voigt 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 Kelvin-Voigt fluid are outlined and we then discuss several continuous dependence and structural stability issues.

 

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

 

• • • 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 well-known 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 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 finite-dimensional one, that is unstable, and an infinite-dimensional one, that is stable. We shall stabilize the finite-dimensional component by a simple, linear, finite-dimensional 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 7, 2022
      Speaker: Varga Kalantarov, Koc University, Istanbul
      Title: On the convective Brinkman-Forchheimer equations
      Abstract: : The convective Brinkman-Forchheimer (CBF) equations describe the motion of incompressible fluid flows in a saturated porous medium. One can consider CBF equations as a damped Navier-Stokes equations with damping \alpha u+\beta |u|^{r-1}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 Leray-Hopf 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 Minty-Browder technique in the proofs. Finally, we discuss the global in time strong solutions to CBF equations.

 

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

 

• • • 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 5, 2021 at 17:30 (Istanbul time)
      Speaker: Hasan Inci, Koc University, Istanbul
      Title: On the local well posedness of the two component Camassa-Holm equation
      Abstract: : The Camassa-Holm equation models the motion of waves in shallow water. In this talk we consider a two component generalization of the Camassa-Holm equation. local well posedness in the sense of Hadamard means local existence, uniqueness and stability of solutions. the two component camassa-holm 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: 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 blow-up for the shallow water equations, in particular, for Camassa-Holm(C-H) and the related equations and systems is an interesting question which remains unclear for special cases and awaits a detailed investigation. Blow-up 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 blow-up mechanism for solutions of Cauchy problem for C-H with certain initial profiles. It should be noted that blow-up 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 local-in-space blow-up 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 blow-up conditions for some generalizations of the C-H equation.

 

• • • 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 4, 2021 at 17:00 (Istanbul time)
      Speaker: Andrey Shishkov, Mathematical Institute of RUDN University , Moscow
      Title: Large and very singular solutions of semi-linear elliptic equations
      Abstract: There will be done a review of results about nonnegative solutions of semi-linear equations of diffusion-nonlinear 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: 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 (non-local or Wentzel type) will be also discussed.

 

• • • Date: May 21, 2021 at 17:00 (Istanbul time)
      Speaker: Jerry Bona, University of Illinois at Chicago
      Title: Initial- and Boundary-Value 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 initial-boundary-value 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 initial-boundary-value 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 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 non-divergence 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 non-divergence 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: 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 well-posedness 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 well-posedness 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 16, 2021
      Speaker: Tekin Dereli, Koç University, Istanbul
      Abstract: Vacuum Polarization Effects in Bertotti-Robinson Spacetimes
      Abstract: Bertotti-Robinson 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 Einstein-Maxwell solutions.We show here that the Bertotti-Robinson metrics together with Wu-Yang magnetic pole potentials give a class of static solutions of a
      system of non-minimally coupled  Einstein-Yang-Mills equations,  that may be relevant for investigating vacuum polarization effects in a first order perturbative approach to quantum fields.

 

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

 

• • • Date: April 2, 2021
      Speaker: Hasan Inci, Koç University, Istanbul
      Titile:: Euler equations and cassical mechanics.
      Abstract:: Vacuum Polarization Effects in Bertotti-Robinson Spacetimes
      Abstract: Bertotti-Robinson 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 Einstein-Maxwell solutions.We show here that the Bertotti-Robinson metrics together with Wu-Yang magnetic pole potentials give a class of static solutions of a
      system of non-minimally coupled  Einstein-Yang-Mills equations,  that may be relevant for investigating vacuum polarization effects in a first order perturbative approach to quantum fields.

 

• • • Date: April March 16, 2021
      Speaker: Albert Erkip, Sabanci University, Istanbul
      Titile:: On Klein-Gordon equations.
      Abstract:: In this talk we will go through some well-known results on the Klein-Gordon equation. We start with a discussion of the well-posedness 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: March 19, 2021
      Speaker: Sergey Zelik, University of Surrey, Guilford, UK
      Titile:: Inertial Manifolds for Dispersive PDE’s.

 

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