Research group: Nonlinear Analysis and PDE
Here are the main themes our group is currently working on (the updated report for the 2017 HCERES evaluation committee is this file (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";
- Discretization and numerical analysis for reaction-convection-diffusion equations and systems.
- 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";
- 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";
- 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";
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.
Control of Partial Differential Equations
- This theme is associated to the GDRE CONEDP.
- Control of hyperbolic equations on networks.
- Control of entropy solutions of conservation laws.
- Finite time stabilization associated to ANR project Finite4SoS.
Applied Problems, Interdisciplinary Interactions
- These activities are mainly connected to the CASCIMODOT 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 and LMA UMR CNRS 7348.
- 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.