*Here are the main themes our group is currently working on (the updated report for the 2017 HCERES evaluation committee is this file [1] (in french)).*

## Nonlinear Elliptic and Parabolic Partial Differential Equations, Nonlocal and Integro-Differential Equations, and their Numerical Analysis

- Existence, uniqueness and general properties of solutions of nonlinear elliptic and parabolic pdes (classical solutions, solutions in Sobolev spaces, viscosity solutions);
- Nonlinear potential theory, boundary trace, measure data. Singularities;
- Reaction-diffusion equations and systems; bifurcations; travelling waves; connections with biological problems.
- Nonlocal equations : boundary conditions, regularity and large time behavior of solutions for evolution equations. This theme is partially associated to the ANR project "Weak Kam beyond Hamilton-Jacobi" [2];
- Discretization and numerical analysis for reaction-convection-diffusion equations and systems.

## Dispersive Equations

- Well-posedness (existence, uniqueness, continuity with respect to initial data) of dispersive equations in spaces of functions with low regularity. This theme is partially associated to the ANR project "GEO-DISP" [3];
- Large time behavior of solutions to dispersive hamiltonian equations (orbital and asymptotic stability of solitary waves, scattering operator);
- Finite time blow-up with or without dissipative effects.

## Hamilton-Jacobi Equations and Deterministic Optimal Control Problems

- Nonstandard deterministic control problems, problems with discontinuity, stratified domains, uniqueness of viscosity solutions for the associated Hamilton-Jacobi Equation. This theme is associated to the ANR project "Hamilton-Jacobi equations on heterogeneous structures and networks" [4];
- Large time behavior of solutions of Hamilton-Jacobi Equations : problems with boundary conditions, ergodic problems, regularity... This theme is associated to the ANR project "Weak Kam beyond Hamilton-Jacobi" [2];

## Analysis and Approximation of Conservation Laws

- Well-posedness of interface and boundary-value problems for scalar hyperbolic conservation laws, numerical approximation, and applications;
- Analysis of constrained and non-local models involving hyperbolic conservation laws;
- Degenerate parabolic problems, porous media models and their approximation, associated to ANR Project GEOPOR [5].

## Control of Partial Differential Equations

- This theme is associated to the GDRE CONEDP [6].
- Control of hyperbolic equations on networks.
- Control of entropy solutions of conservation laws.
- Finite time stabilization associated to ANR project Finite4SoS [7].

## Applied Problems, Interdisciplinary Interactions

- These activities are mainly connected to the CASCIMODOT [8] research project (Scientific Computing and Modeling in the Universities of Orléans and Tours) whose objective is to promote encounters and collaborations between different actors of modeling or scientific computing.
- TempoGnRH Project in Neuroscience. Aim : propose and analyze models to understand the pulsatile secretion of GnRH neurons. Collaboration with PRC - UMR CNRS 7247 - UMR INRA 0085 [9] and LMA UMR CNRS 7348 [10].
- Macroscopic modeling of pedestrian and vehicle dynamics, discretization and analysis. - PHC Polonium No.31460NC "Nonlocal nonlinear hyperbolic conservation laws: modeling, analysis, approximations".
- Spatial biological control of insects invasion, collaboration with IRBI - UMR CNRS 7261 [11].

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DossierEvaluationEquipeAnalyseFinal.pdf [1] | 202.85 Ko |