Benjamin Texier

From Newton to Boltzmann: Hard Spheres and Short-range Potentials

We provide a rigorous derivation of the Boltzmann equation as the mesoscopic limit of systems of hard spheres, or Newtonian particles interacting via a short-range potential, as the number of particles

A Stability Criterion for High-Frequency Oscillations

N

On Nonlinear Stabilization of Linearly Unstable Maps

goes to infinity and the characteristic length of interaction

Transition to Longitudinal Instability of Detonation Waves is Generically Associated with Hopf Bifurcation to Time-Periodic Galloping Solutions

\e

Relative Poincaré-Hopf bifurcation and Galloping Instability of Traveling Waves

simultaneously goes to

Hopf Bifurcation of Viscous Shock Waves in Compressible Gas Dynamics and MHD

0,

From Newton to Boltzmann: the case of short-range potentials

in the Boltzmann-Grad scaling

The onset of instability in first-order systems

N \e^{d-1} \equiv 1.

Nash-Moser iteration and singular perturbations

The time of validity of the convergence is a fraction of the average time of first collision, due to a limitation of the time on which one can prove uniform estimates for the BBGKY and Boltzmann hierarchies. Our proof relies on the fundamental ideas of Lanford, and the important contributions of King, Cercignani, Illner and Pulvirenti, and Cercignani, Gerasimenko and Petrina. The main novelty here is the detailed study of pathological trajectories involving recollisions, which proves the term-by-term convergence for the correlation series expansion.

Existence of quasilinear relaxation shock profiles in systems with characteristic velocities

We show that a simple Levi compatibility condition determines stability of WKB solutions to semilinear hyperbolic initial-value problems issued from highly-oscillating initial data with large amplitudes. The compatibility condition involves the hyperbolic operator, the fundamental phase associated with the initial oscillation, and the semilinear source term; it states roughly that hyperbolicity is preserved around resonances. If the compatibility condition is satisfied, the solutions are defined over time intervals independent of the wavelength, and the associated WKB solutions are stable under a large class of initial perturbations. If the compatibility condition is not satisfied, resonances are exponentially amplified, and arbitrarily small initial perturbations can destabilize the WKB solutions in small time. The amplification mechanism is based on the observation that in frequency space, resonances correspond to points of weak hyperbolicity. At such points, the behavior of the system depends on the lower order terms through the compatibility condition. The analysis relies, in the unstable case, on a short-time Duhamel representation formula for solutions of zeroth-order pseudo-differential equations. Our examples include coupled Klein-Gordon systems, and systems describing Raman and Brillouin instabilities.