Talks (titles and abstracts)
Kazuo Aoki: Boundary conditions for the Boltzmann equation based on a kinetic model of gas-surface interaction
Boundary conditions for the Boltzmann equation are investigated on the basis of a kinetic model of gas-surface interaction. The model takes into account gas and physisorbed molecules interacting with a surface potential and colliding with surface and bulk phonons. The interaction layer is assumed to be thinner than the mean free path of the gas molecules, and the phonons are assumed to be at equilibrium. The asymptotic kinetic equation for the inner physisorbate layer, which forms a steady half-space problem, is derived and used to investigate boundary conditions for the Boltzmann equation valid outside the physisorbate layer. To be more specific, new models of the boundary condition are proposed on the basis of iterative solutions of the half-space problem and are assessed by the direct numerical analysis of the problem. In addition, some rigorous mathematical results for the half-space problem are presented.
This is a joint work with Vincent Giovangigli, François Golse (Ecole Polytechnique) and Shingo Kosuge (Kyoto University).
Claude Bardos: Boundary effects in the zero viscosity limit of solutions of Navier Stokes equations with no slip boundary condition
In this talk I consider the \(\nu\rightarrow\) limit of solutions of the \(2d\) Navier-Stokes equation with no slip boundary condition and will elaborate on two complementary problems:
- The convergence to the solution of the Euler equations under strong analyticity hypothesis during a short time \(0<t<T\) to emphasize the role of the curvature of the boundary on this time \(T\) of validity in connection with the size of Görtler vortices.
- To prove that the Onsager \(\frac13\) on the solution of the \(u(x,t)\) of the Euler equation implies the same regularity for the pressure and then use this remark to prove that in the zero viscosity limit of solutions \(u_\nu\) bounded in \(L^\infty((0,T, C^{0,\alpha})\) with \(\alpha>\frac13\) there is no anomalous energy dissipation.
These observations are part of a programme initiated with E. Titi around 2007 and continuing with the contribution other colleagues in particular presently To Nguyen, Tr. Nguyen and D. Boutros.
Yann Brenier: Gravitational models related to fluid mechanics and optimal transport
I will review some recent results linking these topics, through joint works with with Philippe Anjolras, Luigi Ambrosio and Aymeric Baradat, as well as Ivan Moyano.
Emanuele Caglioti: Random matching in 2d: some recent results
I will consider the 2-dimensional random matching problem in two dimensional sets.
In a challenging paper, Caracciolo et. al., on the basis of a subtle linearization of the Monge Ampere equation, conjectured that the expected value of the square of the Wasserstein distance, with exponent 2, between two samples of N uniformly distributed points in the unit square is logN / 2πN plus corrections.This and other related conjectures has been proved by Ambrosio et al. in a series of challenging papers.
In the talk I will review the results cited above and some extensions.
Also, I will consider the case for densities defined in all the plane. In particular, in the case of the Gaussian distribution it is possible to determine the leading behavior of the expected cost (joint work with Francesca Pieroni).
Marie Doumic: Analysis and estimation for fragmentation systems
Breakage of large particles, either through depolymerisation (i.e. progressive shortening) or through fragmentation (breakage into smaller pieces) may be modelled by discrete equations, of Becker-Döring type, or by continuous ones, like fragmentation equation or Lifshitsz-Slyozov system. In this talk, we study the time dynamics of such systems, in the perspective of estimating the functional parameters of the equation through partial observation of the solution - either fragmentation rate and kernel for the fragmentation equation, or the initial condition for the depolymerisation problem.
Departing from a model of discrete depolymerisation, we first evaluate the impact of using continuous approximations to solve the initial-state estimation problem. The second order approximation reveals more accurate, but we face an accuracy versus stability trade-off: the inverse reconstruction reveals to be severely ill-posed. Thanks to Carleman inequalities and log-convexity estimates, we prove observability results and error estimates for a Tikhonov regularisation. This is a joint work with P. Moireau, inspired by experiments carried out by H. Rezaei's team.
As concerns the estimation of the fragmentation kernel, we proposed several approaches based on the continuous fragmentation equation, studying and using either the long-term, the transient or the short-term dynamics - this last approach revealing more fruitful. Error estimates in Bounded Lipshitz norm are obtained, the time window of observation playing the role of a regularisation parameter. This is a joint work with M. Escobedo and M. Tournus, based on biological questions and experiments carried out by W.F. Xue's team.
Isabelle Gallagher: Convergence of strong solutions from the Boltzmann equation to the Navier-Stokes equations
It is well-known that the Boltzmann equation converges to the Navier-Stokes equations, in various settings: renormalized solutions towards Leray solutions, or strong solutions (unique, for a short time) towards Fukita-Kato type solutions. In a work with Isabelle Tristani in 2020 we revisited this question in the case of strong solutions, with the aim of using what is known on the life span of the Navier-Stokes equations to obtain similar results (typically global existence in two space dimensions, for a small enough Knudsen number) for the Boltzmann equations including in the case of "ill-prepared data". This was achieved for smooth initial data (in space and velocities, and exponentially small in velocities). We will describe this work, and also a work in progress with Kleber Carrapatoso and Isabelle Tristani, where we seek initial data as close as possible to the requirements of the Navier-Stokes equations.
Irene M. Gamba: Weak turbulence modeled by a quasilinear diffusion system for electrostatic and magnetized plasma systems
After a quick derivation of the quasilinear diffusion models for electrostatic and highly magnetized plasma systems arises for high energy small perturbation from bulk states, in electrostatic and highly magnetized mean field regimes. This is weak turbulence regime for periodic flows in a tori admitting non-Maxwellian stationary states.
The model is obtain by a reduction of the Vlasov-Maxwell (or Vlasov) systems in nonequilibrium regimes showing that perturbation of moving Gaussians in momenta exhibit instabilities that derail the approximation of Maxwellian limiting states for long time.
In particular, we show a rigorous proof for existence of weak solutions in the electrostatic case model, for the time dynamics describing the evolution of a lower dimensional system in momenta-spectral energy density, which deviate from statistical equilibrium, contrary to Landau damping effect. The proof is based on extended methods from porous media flows models with non-linear gradient forms and source terms. A first manuscript is work in collaboration, and includes William Porteous for recent work in progress.
In addition, numerical approximations to non-equilibrium statistical states have been obtained in both the electrostatic and three-dimensional flows with Kun Huang, Michael Abdelmalik and Boris Briezman.
Cyril Imbert: Partial regularity in time for the space-homogeneous Boltzmann equation with very soft potentials
We consider the space-homogeneous Boltzmann equation with very soft potentials. We are interested in the regularity of the weak solutions constructed as in Villani (1998). We estimate how big or small is the set of singular times. More precisely, we give an upper bound for the Hausdorff measure of this singular set in terms of the 2 parameters appearing in Boltzmann collision operator in the non cut-off case. This is a joint work with F. Golse and L. Silvestre.
Pierre-Emmanuel Jabin: A new duality approach to the mean-field limit
We introduce a new duality method to obtain the mean-field limit for large systems of interacting particles. The approach is based on a new notion of dual cumulants, which are defined from the solutions to the dual Liouville or backward Kolmogorov equation. At the limit, the rescaled dual cumulants solve a simple hierarchy for which we can prove uniqueness. This strategy allows to derive both the mean-field limit and propagation of chaos with very few assumptions: it applies to both first and second order systems, with or without diffusion, and it only requires the interaction kernel to be in L^2 with potential extensions in L^p.
This is a joint work with D. Bresch and M. Duerinckx.
Shi Jin: Quantum Computation of partial differential equations and linear algebra problems
Quantum computers have the potential to gain algebraic and even up to exponential speed up compared with its classical counterparts, and can lead to technology revolution in the 21st century. Since quantum computers are designed based on quantum mechanics principle, they are most suitable to solve the Schrodinger equation, and linear PDEs (and ODEs) evolved by unitary operators. The most efficient quantum PDE solver is quantum simulation based on solving the Schrodinger equation. It will be interesting to explore what other problems in scientific computing, such as ODEs, PDEs, linear algebra and optimization problems, can be handled by quantum simulation. became challenging for general PDEs, more so for nonlinear ones, and
We will first give a short “mathematician’s survival kit” on quantum computing, then discuss three topics:
1) We introduce the “warped phase transform” to map general linear PDEs and ODEs to Schrodinger equation or with unitary evolution operators in higher dimension so they are suitable for quantum simulation. This method also allows us to do quantum simulation for iterative methods in linear algebra.
2) For (nonlinear) Hamilton-Jacobi equation and scalar nonlinear hyperbolic equations we use the level set method to map them—exactly—to phase space linear PDEs so they can be implemented with quantum algorithms and we gain quantum advantages for various physical and numerical parameters.
3) For PDEs with uncertain coefficients, we introduce a transformation so the uncertainty only appears in the initial data, allowing us to compute ensemble averages with multiple initial data with just one run, instead of multiple runs as in Monte-Carlo or stochastic collocation type sampling algorithms.
Finally we will give some open problems.
Dave Levermore: Global Dynamics for the Kompaneets Equation
The Kompaneets equation governs the evolution of a photon energy spectrum due to Compton scattering in a spatially homogeneous plasma. We prove some results concerning the long-time convergence of solutions to Bose--Einstein equilibria and the failure of photon conservation. In particular, we show the total photon number can decrease with time via an outflux of photons at the zero-energy boundary. The ensuing accumulation of photons at zero energy is analogous to Bose--Einstein condensation. We provide two conditions that guarantee a photon loss occurs, and show that once a loss is initiated then it persists forever. We prove that every solution has a large-time limit that is a Bose-Einstein density that can be characterized in terms of the total photon loss. Additionally, we provide some results concerning the behavior of the solution near the zero-energy boundary, an Oleinik inequality, a comparison principle, and show that the solution operator is an L^1 contraction. None of these results impose a boundary condition at the zero-energy boundary.
Pierre-Louis Lions: Large Random Matrices and PDEs
Tai-Ping Liu: Boltzmann Equation and Fluid Dynamics: Singular Layers
The Boltzmann equation in the kinetic theory for gases is related to the fluid dynamics as a consequence of the H-Theorem. The H-Theorem says that the gases have the tendency to approach thermal equilibrium. Near the thermal equilibrium, the Boltzmann equation can be accurately approximated by fluid dynamics models. It has been recognized that there are singular layers for the Boltzmann equation where the gases can be far from thermal equilibrium. The study of these singular layers is key to the study of the relationship between the Boltzmann equation and the fluid dynamics. There are boundary, initial and shock layers. Natural physical setups often involve the coupling of different types of singular layer. We will present examples to illustrate the physical phenomena and the Green's function approach for solving some of these problems.
Frank Merle: On the Soliton Resolution for Energy Critical Wave Equation in the radial case
Clément Mouhot: tba
Thierry Paul: Mean field limit of multi-agent systems, graph-limit and dynamical systems associated to PDEs
We consider the large number limit of agents - namely distinguishable particles - systems. We show that the so-called graph limit of such dynamical systems is the hydrodynamic Euler equation inherited from the obtained kinetic mean field equation. As a by-product, we show that a large class of PDEs are the graph limit of multi-agent systems that we explicitly construct.
Nataša Pavlović: On the effective dynamics of Bose-Fermi mixtures
Investigating degenerate mixtures of bosons and fermions is an extremely active area of research in experimental physics for constructing and understanding novel quantum bound states such as those in superconductors, superfluids, and supersolids. These ultra-cold Bose-Fermi mixtures are fundamentally different from degenerate gases with only bosons or fermions. They not only show an enriched phase diagram, but also a fundamental instability due to energetic considerations coming from the Pauli exclusion principle. Inspired by this activity in the physics community, recently we started exploring the mathematical theory of Bose-Fermi mixtures. One of the main challenges is understanding the physical scales of the system that allow for suitable analysis. We will describe how we overcame this challenge by identifying a novel scaling regime in which the fermion distribution behaves semi-clasically, but the boson field remains quantum-mechanical. In this regime, the bosons are much lighter and more numerous than the fermions. The talk is based on the joint work with Esteban Cárdenas and Joseph Miller.
Peter Pickl: Microscopic derivation of Vlasov-Dirac-Benney equation with short-range pair potentials
Deriving the dynamics of many-body-systems analytically or numerically is often a very challenging task. Thus simplified effective descriptions of such systems which give good approximations are sought for. One important example for such a system is a plasma, which consists of positively charged ions as well as electrons. Due to the mass ratio between ions and electrons, the electrons compensate the long-range part of the ion-ion interaction and the system is effectively described by ions with delta-like interactions. The effective description one gets for such a system is the so-called Vlasov-Dirac-Benney.
In the talk I will present a recent proof for propagation of chaos of an N-particle system in three dimensions with pair potentials of the form \(N^{3\beta-1} \phi(N^{\beta}x)\) for \(\beta\in\left[0,\frac{1}{7}\right]\) and \(\phi\in L^{\infty}(\mathbb{R}^3)\cap L^1(\mathbb{R}^3)\). In particular, for typical initial data, we show convergence of the Newtonian trajectories to the characteristics of the Vlasov-Dirac-Benney system. The result is joint work with Manuela Feistl.
Mario Pulvirenti: On the validity of the Boltzmann equation for quantum particle systems
I this talk I discuss the problem of a rigorous derivation of the Boltzmann equation for Bosons and Fermions in the weak-coupling limit.
Sylvia Serfaty: The attractive log gas: propagation of chaos, stability and uniqueness questions
We consider the dynamics of a system of particles with logarithmic attractive interaction, on the torus, at inverse temperature beta. We show phase transitions on the stability and uniqueness of the uniform distribution. Investigating the mean-field convergence of the system by the modulated free energy method, we deduce that uniform-in-time convergence is not always true. This is joint work with Antonin Chodron de Courcel and Matthew Rosenzweig.
Alexis Vasseur: Inviscid limit from Navier-stokes to small BV solutions of Euler
In 2004, Bianchini and Bressan proved the inviscid limit to small BV solutions of 1D conservation laws using so-called artificial viscosities. However, the question remained open as to whether this inviscid limit could be achieved with physical viscosities. In this presentation, we establish that small BV solutions to the isentropic Euler equations can indeed be obtained as inviscid limits from the compressible barotropic Navier-Stokes equations, thus resolving the problem for the Isentropic Euler. This is a joint work with Geng Chen and Moon-Jin Kang.