We are concerned with deriving sharp exponential decay estimates (i.e. with maximum rate and minimum multiplicative constant) for linear, hypocoercive evolution equations. Using a modal decomposition of the model allows to assemble a Lyapunov functional using Lyapunov matrix inequalities for each Fourier mode.
We shall illustrate the approach on the 1D Goldstein-Taylor model, a2-velocity transport-relaxation equation. On the torus the lowest Fourier modes determine the spectral gap of the whole equation in $L^{2}$. By contrast, on the whole real line the Goldstein-Taylor model does not have a spectral gap, since the decay rate of the Fourier modes approaches zero in the small mode limit. Hence, the decay is reduced to algebraic.
In the final part of the talk we consider the Goldstein-Taylor model with non-constant relaxation rate, which is hence not amenable to a modal decomposition. In this case we construct a Lyapunov functional of pseudodifferential nature, one that is motivated by the modal analysis in the constant case.The robustness of this approach is illustrated on a multi-velocity GoldsteinTaylor model, yielding explicit rates of convergence to the equilibrium.
This is joint work with J. Dolbeault, A. Einav, C. Schmeiser, B. Signorello, and T. Wöhrer.[-]

Quantized vortices have been experimentally observed in type-II superconductors, superfluids, nonlinear optics, etc. In this talk, I will review different mathematical equations for modeling quantized vortices in superfluidity and superconductivity, including the nonlinear Schrödinger/Gross-Pitaevskii equation, Ginzburg-Landau equation, nonlinear wave equation, etc. Asymptotic approximations on single quantized vortex state and the reduced dynamic laws for quantized vortex interaction are reviewed and solved approximately in several cases. Collective dynamics of quantized vortex interaction based on the reduced dynamic laws are presented. Extension to bounded domains with different boundary conditions are discussed.[-]

This talk is devoted to the quasi linear approximation for solutions of the Vlasov equation a very popular tool in Plasma Physic cf. [4] which proposes, for the quantity:
(1)
$$
q(t,\ v)=\int_{\mathbb{R}_{v}^{d}}f(x,\ v,\ t)dx)\ ,
$$
the solution of a parabolic, linear or non linear evolution equation
(2)
$$
\partial_{t}q(t,\ v)-\nabla_{v}(D(q,\ t;v)\nabla_{v}q)=0
$$
Since the Vlasov equation is an hamiltonian reversible dynamic while (2) is not reversible whenever $D(q,\ t,\ v)\# 0$ the problem is subtle. Hence I did the following things :

1. Give some sufficient conditions, in particular in relation with the Landau damping that would imply $D(q,\ t,\ v)\simeq 0$. a situation where the equation (2) with $D(q,\ t;v)=0$ does not provides a meaning full approximation.

2. Building on contributions of [7] and coworkers show the validity of the approximation (2) for large time and for a family of convenient randomized solutions. This is justified by the fact that the assumed randomness law is in agreement which what is observed by numerical or experimental observations (cf. [1]).

3. In the spirit of a Chapman Enskog approximation formalize the very classical physicist approach (cf. [6] pages 514-532) one can show [3] that under analyticity assumptions this approximation is valid for short time. As in [6] one of the main ingredient of this construction is based on the spectral analysis of the linearized equation and as such it makes a link with a classical analysis of instabilities in plasma physic.

Remarks

In some sense the two approaches are complementary The short time is purely deterministic and the stochastic is based on the intuition that over longer time the randomness will take over of course the transition remains from the first regime to the second remains a challenging open problem. The similarity with the transition to turbulence in fluid mechanic is striking It is underlined by the fact that the tensor
$$
\lim_{\epsilon\rightarrow 0}\mathbb{D}^{\epsilon}(t,\ v)=\lim_{\epsilon\rightarrow 0}\int dx\int_{0}^{\frac{t}{\epsilon^{2}}}d\sigma E^{\epsilon}(t,\ x+\sigma v)\otimes E^{\epsilon}(t-\epsilon^{2}\sigma,\ x)
$$
which involves the electric fields here plays the role of the Reynolds stress tensor.

2 Obtaining, for some macroscopic description, a space homogenous equation for the velocity distribution is a very natural goal. Here the Vlasov equation is used as an intermediate step in the derivation. And more generally it appears as an example of weak turbulence. In particular defining what would be the physical natural probability seems related to the derivation of $\mathrm{e}$ of the Lenard-Balescu equation as done in [5].[-]

This lecture is devoted to the characterization of convergence rates in some simple equations with mean field nonlinear couplings, like the Keller-Segel and Nernst-Planck systems, Cucker-Smale type models, and the Vlasov-Poisson-Fokker-Planck equation. The key point is the use of Lyapunov functionals adapted to the nonlinear version of the model to produce a functional framework adapted to the asymptotic regime and the corresponding spectral analysis.[-]

The dynamical low-rank approximation is a low-rank factorization updating technique. It leads to differential equations for factors in a decomposition of the solution, which need to be solved numerically. The dynamical low-rank method seems particularly suitable for solving kinetic equations, because in many relevant cases the effective dynamics takes place on a lower-dimensional manifold and thus the solution has low rank. In this way, the 5-dimensional (3 space, 2 angle) radiation transport problem is reduced, both in computational cost as well as in memory footprint. We show several numerical examples.[-]

Numerical approximation of the Boltzmann equation is a challenging problem due to its high-dimensional, nonlocal, and nonlinear collision integral. Over the past decade, the Fourier-Galerkin spectral method has become a popular deterministic method for solving the Boltzmann equation, manifested by its high accuracy and potential of being further accelerated by the fast Fourier transform. Albeit its practical success, the stability of the method is only recently proved by Filbet, F. & Mouhot, C. in [Trans.Amer.Math.Soc. 363, no. 4 (2011): 1947-1980.] by utilizing the”spreading” property of the collision operator. In this work, we provide anew proof based on a careful L2 estimate of the negative part of the solution. We also discuss the applicability of the result to various initial data, including both continuous and discontinuous functions. This is joint work with Kunlun Qi and Tong Yang.[-]

We propose a modulated free energy which combines of the method previously developed by the speaker together with the modulated energy introduced by S. Serfaty. This modulated free energy may be understood as introducing appropriate weights in the relative entropy to cancel the more singular terms involving the divergence of the flow. This modulated free energy allows to treat singular interactions of gradient-flow type and allows potentials with large smooth part, small attractive singular part and large repulsive singular part. As an example, a full rigorous derivation (with quantitative estimates) of some chemotaxis models, such as Patlak-Keller Segel system in the subcritical regimes, is obtained. This is joint work with D. Bresch and Z. Wang.[-]

The aim of this talk is the rigorous derivation of crossdiffusion systems from stochastic, moderately interacting many-particle systems for multiple species. Applications include animal populations and neuronal ensembles. The mean-field limit leads to nonlocal cross-diffusion systems, while the limit of vanishing interaction radius gives local cross-diffusion equations. This allows for the derivation of fluid-type models that can be found in neuronal networks and of Shigesada-Kawasaki-Teramoto population models. The derivation uses the techniques of Oehlschläger. The entropy structure of the limiting models is discussed and some numerical experiments are presented.[-]