A particle method for the homogeneous Landau equation

We propose a novel deterministic particle method to numerically approximate the Landau equation for plasmas. Based on a new variational formulation in terms of gradient flows of the Landau equation, we regularize the collision operator to make sense of the particle solutions. These particle solutions solve a large coupled ODE system that retains all the important properties of the Landau operator, namely the conservation of mass, momentum and energy, and the decay of entropy. We illustrate our new method by showing its performance in several test cases including the physically relevant case of the Coulomb interaction. The comparison to the exact solution and the spectral method is strikingly good maintaining 2nd order accuracy. Moreover, an efficient implementation of the method via the treecode is explored. This gives a proof of concept for the practical use of our method when coupled with the classical PIC method for the Vlasov equation.Comment: 27 pages, 14 figures, debloated some figures, improved explanations in sections 2, 3, and

Decaying Turbulence in Generalised Burgers Equation

We consider the generalised Burgers equation $$\frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} - \nu \frac{\partial^2 u}{\partial x^2}=0,\ t \geq 0,\ x \in S^1,$$ where $f$ is strongly convex and $\nu$ is small and positive. We obtain sharp estimates for Sobolev norms of $u$ (upper and lower bounds differ only by a multiplicative constant). Then, we obtain sharp estimates for small-scale quantities which characterise the decaying Burgers turbulence, i.e. the dissipation length scale, the structure functions and the energy spectrum. The proof uses a quantitative version of an argument by Aurell, Frisch, Lutsko and Vergassola \cite{AFLV92}. Note that we are dealing with \textit{decaying}, as opposed to stationary turbulence. Thus, our estimates are not uniform in time. However, they hold on a time interval $[T_1, T_2]$, where $T_1$ and $T_2$ depend only on $f$ and the initial condition, and do not depend on the viscosity. These results give a rigorous explanation of the one-dimensional Burgers turbulence in the spirit of Kolmogorov's 1941 theory. In particular, we obtain two results which hold in the inertial range. On one hand, we explain the bifractal behaviour of the moments of increments, or structure functions. On the other hand, we obtain an energy spectrum of the form $k^{-2}$. These results remain valid in the inviscid limit.Comment: arXiv admin note: substantial text overlap with arXiv:1201.5567, arXiv:1107.486

Domains of analyticity of Lindstedt expansions of KAM tori in dissipative perturbations of Hamiltonian systems

Many problems in Physics are described by dynamical systems that are conformally symplectic (e.g., mechanical systems with a friction proportional to the velocity, variational problems with a small discount or thermostated systems). Conformally symplectic systems are characterized by the property that they transform a symplectic form into a multiple of itself. The limit of small dissipation, which is the object of the present study, is particularly interesting. We provide all details for maps, but we present also the modifications needed to obtain a direct proof for the case of differential equations. We consider a family of conformally symplectic maps $f_{\mu, \epsilon}$ defined on a $2d$-dimensional symplectic manifold $\mathcal M$ with exact symplectic form $\Omega$; we assume that $f_{\mu,\epsilon}$ satisfies $f_{\mu,\epsilon}^*\Omega=\lambda(\epsilon) \Omega$. We assume that the family depends on a $d$-dimensional parameter $\mu$ (called drift) and also on a small scalar parameter $\epsilon$. Furthermore, we assume that the conformal factor $\lambda$ depends on $\epsilon$, in such a way that for $\epsilon=0$ we have $\lambda(0)=1$ (the symplectic case). We study the domains of analyticity in $\epsilon$ near $\epsilon=0$ of perturbative expansions (Lindstedt series) of the parameterization of the quasi--periodic orbits of frequency $\omega$ (assumed to be Diophantine) and of the parameter $\mu$. Notice that this is a singular perturbation, since any friction (no matter how small) reduces the set of quasi-periodic solutions in the system. We prove that the Lindstedt series are analytic in a domain in the complex $\epsilon$ plane, which is obtained by taking from a ball centered at zero a sequence of smaller balls with center along smooth lines going through the origin. The radii of the excluded balls decrease faster than any power of the distance of the center to the origin

Passing to the Limit in a Wasserstein Gradient Flow: From Diffusion to Reaction

We study a singular-limit problem arising in the modelling of chemical reactions. At finite {\epsilon} > 0, the system is described by a Fokker-Planck convection-diffusion equation with a double-well convection potential. This potential is scaled by 1/{\epsilon}, and in the limit {\epsilon} -> 0, the solution concentrates onto the two wells, resulting into a limiting system that is a pair of ordinary differential equations for the density at the two wells. This convergence has been proved in Peletier, Savar\'e, and Veneroni, SIAM Journal on Mathematical Analysis, 42(4):1805-1825, 2010, using the linear structure of the equation. In this paper we re-prove the result by using solely the Wasserstein gradient-flow structure of the system. In particular we make no use of the linearity, nor of the fact that it is a second-order system. The first key step in this approach is a reformulation of the equation as the minimization of an action functional that captures the property of being a curve of maximal slope in an integrated form. The second important step is a rescaling of space. Using only the Wasserstein gradient-flow structure, we prove that the sequence of rescaled solutions is pre-compact in an appropriate topology. We then prove a Gamma-convergence result for the functional in this topology, and we identify the limiting functional and the differential equation that it represents. A consequence of these results is that solutions of the {\epsilon}-problem converge to a solution of the limiting problem.Comment: Added two sections, corrected minor typos, updated reference

Well-posedness and long-time behavior for a class of doubly nonlinear equations

This paper addresses a doubly nonlinear parabolic inclusion of the form $A(u_t)+B(u)\ni f$. Existence of a solution is proved under suitable monotonicity, coercivity, and structure assumptions on the operators $A$ and $B$, which in particular are both supposed to be subdifferentials of functionals on $L^2(\Omega)$. Moreover, under additional hypotheses on $B$, uniqueness of the solution is proved. Finally, a characterization of $\omega$-limit sets of solutions is given and we investigate the convergence of trajectories to limit points