#### International Video-workshop on Infinite Dimensional Dynamical Systems

2020-08-28

Sergey Zelik is a professor of University of Surrey, UK. He got the PhD in mathematics from Moscow State University. His research field is infinitedimensional dissipative dynamical systems generated by partial differential equations of mathematical physics.

We discuss the dynamics generated by weakly damped wave equations in bounded 3D domains where the damping exponent depends explicitly on time and may change sign. It is shown that in the case when the non-linearity is superlinear, the considered equation remains dissipative if the weighted mean value of the dissipation rate remains positive.Two principally different cases are considered. In the case when this mean is uniform (which corresponds to deterministic dissipation rate), it is shown that the considered system possesses smooth uniform attractors as well as non-autonomous exponential attractors. In the case where the mean is not uniform (which corresponds to the random dissipation rate), the tempered random attractor is constructed. In contrast to the usual situation, this random attractor is expected to have infinite dimension.

Alain Miranville is a professor of Universiy of Poitiers, France. His research field is qualitative properties of parabolic PDE's.

Our aim in this talk is to discuss mathematical models for glial cells and energy metabolism in the brain. In particular we discuss the existence of global in times solutions for a Cahn-Hilliard type model.

Morgan Pierre is a professor of University of Poitiers, France.  His research field is models of phase transition and phase separation, such as Allen-Cahn and Cahn-Hilliard type equations.

We consider a space semidiscretization of the Allen-Cahn equation by P1 finite elements. We build a family of exponential attractors associated to the discretized equations which is robust as the mesh parameterhtends to 0. As a corollary, we obtain an upper bound on the fractal dimension of the global attractor which is independent of h. Our proof is adapted from the result of Efendiev, Miranville and Zelik concerning the continuity of exponential attractors under perturbation of the underlying semigroup. We will also discuss the case of a time discretization and some perspectives.

Xiaoying Han ia a professor of Auburn University, USA. He got the PhD in mathematics from State University of New York at Buffalo. His research field is modeling, analysis, and simulation of dynamical systems in biology, ecology, chemical engineering, etc.

In this talk I will introduce a few interesting lattice dynamical systems arising from neural network applications. Their long term dynamics will be studied by theory and techniques of dynamical systems. The talk will focus on the models and methodology applied to study these models, without including details of technical proofs.

Maurizio Grasselli is a professor of Politecnico di Milano, Italy. His research field is infinitedimensional dissipative dynamical systems.

I intend to present some recent results on a model of phase separation in (incompressible) liquids proposed by M.-H. Giga, A. Kirshtein, and C. Liu in 2018. This model consists of the Navier-Stokes system coupled with the conserved Allen-Cahn equation with Flory-Huggins  potential. Density and viscosity may depend on the phase field. The inviscid case will also be discussed.

Tomas Caraballo is a Professor of University of Sevilla, Spain. His research field is stochastic partial differential equations and partial functional differential equations.

This talk is first devoted to the local and global existence of mild solutions for a class of fractional impulsive stochastic differential equations with infinite delay driven by both $\mathbb{K}$-valued Q-cylindrical Brownian motion and fractional Brownian motion with Hurst parameter $H\in(1/2,1)$. A general framework which provides an effective way to prove the continuous dependence of mild solutions on initial value is established under some appropriate assumptions. Furthermore, it is also proved the exponential decay to zero of solutions to fractional stochastic impulsive differential equations with infinite delay. Finally, some comments and remarks will be mentioned concerning the existence of attractings sets.

Roberto Triggiani is a Professor of University of Memphis, USA. His research field is control theory of partial differential equations.

The study of uniform stabilization of Navier-Stokes equations by feedback controls was initiated about 20 years ago. The following problem remained open: can the localized, boundary-based, stabilizing controls be asserted to be finite dimensional also for d=3? Prior results (2015) required the additional assumption that the Initial Conditions be compactly supported. We shall provide an affirmative solution of this problem. It will require a drastic change of the functional setting from the Sobolev-Hilbert based setting of past literature to a Besov space setting with tight indeces. Moreover, a novel procedure will be given. It will require establishing maximal regularity of the linearized, boundary feedback uniformly stable problem to handle the non-linear analysis. This is joint work with Irena Lasiecka and Buddhika Priyasad.

Irena Lasiecka is a Professor of University of Memphis, USA. She got the PhD in mathematics in University of Warsaw.  His research field is control theory of partial differential equations.

An appearance of a flutter in oscillating structures is an endemic phenomenon. Most common causes are vibrations induced by the moving flow of a gas (air, liquid) which is interacting with the structure. Typical examples include: turbulent jets, vibrating bridges [Tacoma bridge], oscillating facial palate at the onset of apnea. In the case of an aircraft it may compromise its safety. The intensity of the flutter depends heavily on the speed of the flow (subsonic, transonic or supersonic regimes). Thus, reduction or attenuation of flutter is one of the key problems in aeroelasticity with applications to a variety of fields including aerospace engineering, structural engineering, medicine and life sciences.