A NN-uniform quantitative Tanaka's theorem for the conservative Kac's NN-particle system with Maxwell molecules

This paper considers the space homogenous Boltzmann equation with Maxwell molecules and arbitrary angular distribution. Following Kac's program, emphasis is laid on the the associated conservative Kac's stochastic $N$-particle system, a Markov process with binary collisions conserving energy and total momentum. An explicit Markov coupling (a probabilistic, Markovian coupling of two copies of the process) is constructed, using simultaneous collisions, and parallel coupling of each binary random collision on the sphere of collisional directions. The euclidean distance between the two coupled systems is almost surely decreasing with respect to time, and the associated quadratic coupling creation (the time variation of the averaged squared coupling distance) is computed explicitly. Then, a family (indexed by $\delta > 0$) of $N$-uniform ''weak'' coupling / coupling creation inequalities are proven, that leads to a $N$-uniform power law trend to equilibrium of order ${\sim}_{ t \to + \infty} t^{-\delta}$, with constants depending on moments of the velocity distributions strictly greater than $2(1 + \delta)$. The case of order $4$ moment is treated explicitly, achieving Kac's program without any chaos propagation analysis. Finally, two counter-examples are suggested indicating that the method: (i) requires the dependance on $>2$-moments, and (ii) cannot provide contractivity in quadratic Wasserstein distance in any case.Comment: arXiv admin note: text overlap with arXiv:1312.225

Scalable and Quasi-Contractive Markov Coupling of Maxwell Collision

This paper considers space homogenous Boltzmann kinetic equations in dimension $d$ with Maxwell collisions (and without Grad's cut-off). An explicit Markov coupling of the associated conservative (Nanbu) stochastic $N$-particle system is constructed, using plain parallel coupling of isotropic random walks on the sphere of two-body collisional directions. The resulting coupling is almost surely decreasing, and the $L_2$-coupling creation is computed explicitly. Some quasi-contractive and uniform in $N$ coupling / coupling creation inequalities are then proved, relying on $2+\alpha$-moments ($\alpha >0$) of velocity distributions; upon $N$-uniform propagation of moments of the particle system, it yields a $N$-scalable $\alpha$-power law trend to equilibrium. The latter are based on an original sharp inequality, which bounds from above the coupling distance of two centered and normalized random variables $(U,V)$ in $\R^d$, with the average square parallelogram area spanned by $(U-U_\ast,V-V_\ast)$, $(U_\ast,V_\ast)$ denoting an independent copy. Two counter-examples proving the necessity of the dependance on $>2$-moments and the impossibility of strict contractivity are provided. The paper, (mostly) self-contained, does not require any propagation of chaos property and uses only elementary tools.Comment: 29 page

Geometric optics and boundary layers for Nonlinear Schrodinger equations

We justify supercritical geometric optics in small time for the defocusing semiclassical Nonlinear Schrodinger Equation for a large class of non-necessarily homogeneous nonlinearities. The case of a half-space with Neumann boundary condition is also studied.Comment: 44 page

A simple criterion of transverse linear instability for solitary waves

We prove an abstract instability result for an eigenvalue problem with parameter. We apply this criterion to show the transverse linear instability of solitary waves on various examples from mathematical physics.Comment: The main result has been improved and its proof simplifie

Stability and instability of the KdV solitary wave under the KP-I flow

We consider the KP-I and gKP-I equations in $\mathbb{R}\times (\mathbb{R}/2\pi \mathbb{Z})$. We prove that the KdV soliton with subcritical speed $0<c<c^*$ is orbitally stable under the global KP-I flow constructed by Ionescu and Kenig \cite{IK}. For supercritical speeds $c>c^*$, in the spirit of the work by Duyckaerts and Merle \cite{DM}, we sharpen our previous instability result and construct a global solution which is different from the solitary wave and its translates and which converges to the solitary wave as time goes to infinity. This last result also holds for the gKP-I equation