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Mini-courses

A. Ben Abdessalem, Paris, France
3h mini-course : Geometry and PDEs Abstract

Dans ce cours, et après quelques rappels de géométrie, nous montrerons comment certains problèmes géométriques ont conduit
à des célèbres équations et des problèmes d'EDP sur les variétés (en particulier, au problème de Yamabe).
On fera une ébauche des différentes méthodes utilisées pour montrer l'existence de solutions.
A. Shirikyan, Cergy Pontoise, France
3h mini-course: Control and stabilisation for viscous conservation laws Abstract

The main goal of this course is to present various results on controllability for the viscous Burgers equation. The problem is studied on a finite interval with the Dirichlet or periodic boundary conditions, assuming that the control is localised in space. We begin with a result on non-controllability showing that the phase space of the problem contains large areas that cannot be reached even in infinite time. We next turn to the question of controllability and stabilisation to trajectories. Two types of result will be presented: global stabilisation and local exact controllability. Some generalisations to multidimensional PDEs and open problems will also be discussed.
Ph. Souplet, Paris Nord, France.
3h mini-course: Diffusive Hamilton-Jacobi equations and their singularities Abstract

This mini-course is an introduction to the diffusive Hamilton-Jacobi equation
(DHJ) u_t-\Delta u=|Du|^p,
which plays an important role in optimal stochastic control.
The qualitative analysis of (DHJ) has undergone numerous developments in the past 15 years and reveals a rich variety of phenomena. We will focus on questions of gradient blowup (GBU) for the initial-Dirichlet problem, which occurs in the superquadratic case $p>2$, and treat the following questions:
- Local and global existence: classical and viscosity solutions
- Sufficient conditions for GBU or for global classical existence
- Bernstein type estimates and localization of singularities on the boundary
- Single-point GBU on the boundary and final space profiles
- Time rates of GBU
- Post GBU behavior for viscosity solutions: loss and recovery of boundary conditions, multiple time singularities.

Invited talks

A. Benaissa, Sidi Bel Abbes, Algeria
Title: On the stability of some degenerate evolution problems of hyperbolic type Abstract

In this talk, we are interested in the study of a one dimensional setting of some degenerate evolution problems under boundary feedbacks. In particular, we study stabilization of a degenerate wave equation and a system of degenerate wave equations coupled via zero order terms. We prove optimal energy decay rate, by using a spectral approach.

The talk is a part of the following papers:
References
[1] H.Zerkouk, C. Aichi, A. Benaissa, On the stability of a degenerate wave equation under fractional feedbacks acting on the degenerate boundary , J. Dyn. Control Syst. 28 (2022)-3, 601-633.
[2] Indirect boundary stabilization of weakly coupled degenerate wave equations , Paper in preparation.
A. Duca, Nancy, France
Title: Bilinear controls for non-linear Schrödinger equations Abstract

We consider the nonlinear Schrödinger equation (NLS) in the presence of an external potential field whose time-dependent amplitude plays the role of control. In the first part of the talk, we assume the NLS equation on a torus of arbitrary dimension and we ensure the approximate controllability between any pair of eigenstates with respect to the L^2−norm. The result is valid in arbitrarily small time and when the potential satisfies a specific saturation property. The proof is obtained by a multiplicative version of a geometric control approach introduced by Agrachev and Sarychev. In the second part, we consider the NLS equation on an interval and in the presence of Dirichlet type boundary conditions. We ensure the local exact controllability with respect to the H^3 -norm. We use a bilinear control that acts on the equation through four directions: three Fourier modes and a generic one. The Fourier modes are chosen so that they satisfy the saturation property introduced in the first part of the talk. These modes approximately control the linearised equation. The generic direction instead allows to prove its exact controllability. The inverse function theorem finally yields the local exact controllability of the NLS equation.

Project developped in collaboration with Vahagn Nersesyan.
T. Duyckaerts, Paris, France
Title: Dynamics of nonlinear wave equations Abstract

In this talk I will present several results on the global dynamics of the wave equation with a focusing nonlinearity, highlighting in particular the strong influence of the power of the nonlinearity on the behaviour of global solutions.
R. Dziri, University Tunis el Manar, Tunisia
Title: Shape derivative for Navier-Stokes flow with Navier boundary condition Abstract

The talk deals with a shape derivative result for the time-dependent NavierStokes equations with Navier Boundary condition. For that purpose, we make use of a weak version of the implicit function theorem and obtain the existence of weak shape derivative for the considered system. An application to the drag shape minimization will also be presented.
T.-E. Ghoul, Abu Dhabi, Emirates
Title: Stable self-similar magnetic reconnection. Abstract

Magnetic reconnection is known in plasma physics as a singularity formation on the current density, or on the lines of the magnetic field. Physicists except that 3D incompressible Hall MHD is a good model for the fast magnetic reconnection. We will present a new result that we obtained on 3D incompressible viscous Hall MHD without resistivity.
We prove that the singularity is of self-similar nature and is stable in the analytic topology.
This is the first result of smooth blowup on 3D incompressible Navier Stokes even if the blow up mechanism is due to the Hall effect on the magnetic field.
R. Hadiji, Créteil, France
Title: Review of some problems related to Ginzburg-Landau energy Abstract

The complex Ginzburg-Landau equation describes a vast variety of phenomena as second-order phase transitions, superconductivity, super- fluidity, Bose-Einstein condensation and liquid crystals.
After a short introduction to the standard complex Ginzburg-Landau equation, I will review some of its important variants such the presence of a weight. I will describe also a theoretical problem linked to some particular potentials for the Ginzburg-Landau energy.
T. Hmidi, Abu Dhabi, Emirates
Title: On time quasi-periodic solutions for singular models in geophysical flows Abstract

In this talk I will discuss the emergence of time periodic and quasi-periodic solutions in two-dimensional turbulent flows. We explore their existence around stationary radial profiles and particular attention will be devoted to the disc and the annulus. sWe use tools from bifurcation theory, KAM techniques and Nash-Moser scheme.
S. Ibrahim, Victoria, Canada
Title: The Relativistic Vlasov-Maxwell system in the context of plasmas of fusion Abstract

This talk is devoted to the Relativistic Vlasov-Maxwell (RVM) system in space dimension three. After reviewing the main known results about local and global well-posedness of weak and strong solutions, I will present a new representation formula of the momentum increment. As a consequence, I will show how the domain of influence in momentum is controlled by mild information on the initial data. In particular, this allows us to construct smooth solutions to the RVM system in the regime of dense, hot and strongly magnetized plasmas, like in fusion reactors.

This is a joint work with C. Cheverry.
N. Nouaili, Paris, France
Title: Singularities in the complex Ginzburg Landau equation Abstract

The equation of complex Ginzburg Landau (CGL) have a long history in physics. The CGL-equation describes a lot of phenomena including nonlinear waves, second-order phase transitions, and superconductivity The study of collapse, chaotic or blow-up solutions of equation appears in many works; in the description of an unstable plane Poiseuille flow or in the context of binary mixtures.

The talk will present constructive examples of finite-time blow-up solution to the CGL equation.
A. Rhandi, Salerno, Italy
Title: Approximate controllability of networks systems Abstract

In this talk, the rich feedback theory of regular linear systems in the Salamon-Weiss sense as well as some advanced tools in semigroup theory are used to formulate and solve control problems for network systems. More precisely, we derive necessary and sufficient conditions for approximate control- lability of such systems. These criteria, in some particular cases, are given by the well-known Kalman’s controllability rank condition.

Work in collaboration with Y. El gantouh and S. Hadd.

Short talks

Y. Abdelbadie, Tlemcen, Algeria
Title: On a fractional KPZ type equation including nonlocal gradient terms: the stationary case. Abstract

M. Antabli, University Tunis el Manar
Title: Homoclinic solutions in R^2 for singular hamiltonian systems with weak force potentials Abstract

S. Benmansour, Higher School of Management -Tlemcen, Algeria
Title: Multiplicity result for nonlocal elliptic problem involving critical growth Abstract

In this work we investigate the existence of two distinct solutions for a nonlocal Kirchhoff type problem with critical Sobolev exponent. The approach is essentially based on variational methods.
W. Boughamda, University of Carthage, Tunisia
Title: Stability of strings with variable coefficients under joint damping Abstract

This study is devoted to the stabilization problem of a star-shaped network of strings with variable physical coefficients, under a feedback control at the common node and an extreme node, and subject to Dirichlet or Neumann boundary conditions on the two others nodes. We show that the system has a sequence of (generalized) eigenfunctions, which forms a Riesz basis with parentheses for the state Hilbert space. The spectrum-determined growth condition fulfills. We prove the exponential stability of the system under some conditions on the physical coefficients. Our approach is based on the spectrum mapping theorem, and our method not only leads to an improved result, it is also simpler than the method used by Guo and Xu (Glasgow Math J 53:481–499 2011) in easier situation. At the end, we prove that a phenomenon of lack of uniform stability occurs when a point mass is concentrated in the common node.
H. Boughazi, High School of Management of Tlemcen, Algeria
Title: Nodal Solutions for a Paneitz-Branzon type equation Abstract
H. Bouzidi University of Monastir, Tunisia
Title: Exact boundary controllability of the linear biharmonic Schrödinger equation with variable coefficients Abstract

The Talk deals with the exact boundary controllability of the linear fourth-order Schrödinger equation with space-dependent coefficients and clamped boundary conditions on a bounded interval. The control acts on the first spatial derivative at the right endpoint. We prove that this control system is exactly controllable at any time T>0.
It is done using a spectral analysis and the nonharmonic Fourier series.
A. Chaabani, University Tunis el Manar, Tunisia
Title: New regularity criterion for 3D Navier Stokes equations Abstract
A. Dieb, Tieret, Algeria
Title: Uniqueness and nondegeneracy for Dirichlet fractional problems in bounded domains via asymptotic methods Abstract

We consider positive solutions of a fractional Lane-Emden-type problem in a bounded do- main with Dirichlet conditions. We show that uniqueness and nondegeneracy hold for the asymptotically linear problem in general domains. Furthermore, we also prove that all the known uniqueness and nondegeneracy results in the local case extend to the nonlocal regime when the fractional parameter s is sufficiently close to 1.
K. Ghabi, University Tunis el Manar, Tunisia
Title: Classification des solutions d'indices de Morse finis des problèmes avec condition de Robin sur le bord Abstract
R. Ghanmi, University of Jendouba, Tunisia
Title: Inhomogeneous coupled non-linear Schrödinger systems Abstract

A. Hannachi, University of Batna 2
Title: On the Global Well-Posedness of the Axisymmetric Viscous Boussinesq System in Critical Lebesque spaces Abstract

The current communication will be focused on the global well-posedness of the axisymmetric viscous Boussinesq system when the initial data in critical Lebesgue spaces . More clearly, if the initial vorticity and the initial density are in L1(Ω) × L1(R3), then the velocity vector field is not squart summable. We aim at deriving analogous results for the classical two-dimensional and three-dimensional axisymmetric Navier-Stokes equations recently obtained by Th. Gallay and V. Sverák.
N. Irzi, University Abderrahmane Mira, Bejaia, Algeria
Title: Existence of three weak solutions for fourth-order Leray-Lions problem with indefinite weights Abstract

In this work, we establish the existence of at least three weak solutions for the fourth-order problem with indefinite weights involving the Leray–Lions operator with nonstandard growth conditions. The proof of our main result uses variational methods and the critical theorem of Bonanno and Marano [On the structure of the critical set of nondifferentiable functions with a weak compactness condition].
A. Matallah, Higher School of Management -Tlemcen, Algeria
Title: Existence of solutions for a Kirchhoff type problem with critical exponent Abstract

In this work, we use variational methods to prove the existence of multiple solutions for a nonhomogeneous critical elliptic problem of Kirchhoff type.
N. Souayeh, University of Gafsa, Tunisia
Title: Indirect stability of coupled wave equations with local Kelvin-Voigt damping. Abstract

We consider a coupled wave system with local Kelvin-Voigt damping on one wave only in the interval (-1,1), and with different propagation speeds. We investigate the effectiveness of the indirect control and prove that the stability and the regularity of the solutions depend on the smoothness of the damping coefficient.
M. Zerguine, University of Betna 2, Algeria
Title: Rigidity aspects of singular patches in stratified flows Abstract

The current presentation establishes the local well-posedness issue for the 2d inviscid Boussinesq system when the vorticity is given by a singular patch. We give a signi􏰾cant improvement of Hassainia-Hmidi (J. Math. Anal. Appl. 430:2 (2015), 777􏰼 809) by replacing their compatibility assumption on the density with a constraint on its platitude degree on the singular set. The second main contribution focuses on the same issue for the partial viscous Boussinesq system. We demonstrate a uniformly well-posedness theory with respect to the vanishing conductivity. This issue is much harder than the inviscid case and one should carefully deal with various di􏱀culties related to some aspects of the transport-di􏰽usion equation. One of them appears at the level of weak a priori estimates where the re􏰾ned analysis for a transport-di􏰽usion equation subject to a logarithmic singular potential is required. The Further di􏱀culty comes from some commutators arising in the control of the co-normal regularity that we counterbalance in part by the maximal smoothing e􏰽ects of the heat semi-􏰿ow. Notice that there is a frequency logarithmic loss induced by the low regularity of the velocity that we handle using Hölder type estimates with time loss of regularity.