Projet ANR : Stab

Most physical, biological and economic phenomena can be described by evolution equations.Such equations can be linear or more often non-linear, reversible or non-reversible, deterministic or stochastic. It is interesting to know certain qualitative properties of their solutions, in particular their regularity and long time behavior. These properties can be studied by analytic, probabilistic or approximation methods. The reversible setting, in which the underlying operator of the partial differential equation is symmetric, or equivalently, in which the law of the stochastic process is unchanged by time reversal, is well understood. There the long time behavior and regularity properties are mainly studied by means of functional inequalities such as the Poincaré and logarithmic Sobolev inequalities. These inequalities give bounds on Liapunov functionals of the evolution (entropies); they can be obtained for instance by Bakry-Emery methods of using Liapunov conditions, which stem from MCMC methods. In certain models the dynamics can be better understood by introducing particle methods (hence, large systems of coupled differential equations) or, following F. Otto, by interpreting the evolution as a gradient flow of an entropy with respect to a Wasserstein distance, hence providing a natural time discretization.

In the present proposal we wish to go beyond this academic setting and study the stability and stabilization for both more realistic and more complex evolution equations, and in particular non-reversible or/and degenerate : instances are kinetic Fokker-Planck type hypoelliptic equations, Navier-Stokes type equations, for polymers, mean field equations and systems of reaction-diffusion equations modeling chemical reactions. For such models the techniques used in the reversible setting must be refined and extended, but overall new analytic, probabilistic and numerical techniques must be developed : in particular we intend to stress on the numerical study of such models. The stability of numerical schemes is a fundamental issue whose study is based on discrete functional inequalities and on associated (jump) stochastic processes. Estimating the constants in these inequalities is a main question which we intend to study using tools such as the curvature of discrete spaces, extremal functions, or Liapunov functions as developed by members of the present proposal in other settings. The issues under consideration have analytic, probabilistic and numerical aspects : this is why the present proposal gathers researchers in the fields of PDEs, discrete and continuous Markov processes and numerical analysis.

The project is divided into three interconnected main parts.

1. Nonlinear stability of numerical schemes and particle systems : asymptotic preserving schemes, semi-lagrangian methods, splitting techniques, mean field particle systems.
2. Stabilization for non-reversible models including hypocoercivity, systems of PDEs, jump processes, discrete time dynamics and acceleration of convergence.
3. Perturbation and stability of functional inequalities, including the computation of explicit bounds, qualitative properties of optimal functions like symmetry breaking, boundary conditions, non-local operators.