Anne de Bouard (CNRS, École Polytechnique)
Title:Introduction to stochastic PDE.
Abstract: PDEs with stochastic (white in time) perturbations are widely used to model phenomena with small scale fluctuations. After having given a few examples of such models, I will introduce basic tools for the analysis of the well posedness of stochastic PDEs (stochastic analysis in infinite dimension). The typical examples that I will keep in mind are the additive Navier-Stokes equation in 2D, and the nonlinear Schrödinger equation, with additive or multiplicative noise.
The course will then concentrate on the large time behavior of solutions in the dissipative case (invariant measures, convergence to equilibrium). Finally, I will explain how the tools developed for diffusion approximation in finite dimension (that is for stochastic differential equations) may be generalized to PDEs, in order to justify the use of white noise in time.
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Fabrice Béthuel (Sorbonne Université)
Title:Asymptotics for two-dimensional elliptic Allen-Cahn systems.
Abstract:The formation of codimension-one interfaces for multi-well gradient-driven problems is well-known and established in the scalar case, where the equation is often referred to as the Allen-Cahn equation. The vectorial case in contrast is quite open. This lack of results and insight is to a large extent related to the absence of known monotonicity formula. After a presentation of the thory in the scalar case, I will focus on the elliptic case in two dimensions, and presents some results which extend to the vectorial case in two dimensions most of the results obtained for the scalar case. I will also emphasize some specific features of the vectorial case.
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Léo Bigorgne (CNRS, Université de Rennes 1)
Title: Modified scattering for small data solutions to the Vlasov-Maxwell system.
Abstract: We will capture the asymptotic behavior of the solutions to the Vlasov-Maxwell system arising from sufficiently small and regular data. Our analysis is based on vector fields methods, allowing us to exploit the null structure of the equations, and the Glassey-Strauss decomposition of the electromagnetic field. In particular, we will see that the electromagnetic field approaches, for large time, a solution to the vacuum Maxwell equations. Due to the long-range effects of the Lorentz force, the Vlasov field converges along logarithmic corrections of the linear characteristics.
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Emeric Bouin (Université Paris-Dauphine)
Title: Hydrodynamic limits and hypocoercivity for kinetic equations with heavy tails.
Abstract:In this talk, I will review some results about long time behaviour of linear kinetic equations for which the microscopic equilibrium (that is, the kernel of the reorientation operator) is typically a density with polynomial decay. There will be no space confinement and the reorientation operator could be of scattering, Fokker-Planck or Levy-Fokker-Planck types. I will first present a spectral approach a la Ellis and Pinsky that yields to a unified treatment of the macroscopic limits for this kind of equations and then focus on re-shaping the Dolbeault-Mouhot-Schmeiser L2 hypocoercivity method to get explicit rates of decay to zero in suitable weighted norms. It time allows, I will describe extensions and related works. This comes mainly from joint works with Dolbeault, Lafleche and Mouhot.
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Marie Doumic (INRIA de Paris)
Title: Analysis and calibration of depolymerising systems.
Abstract: Shrinkage 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. In many applications, the dynamic nature of the experiments, as well as their nanoscale, makes it challenging to estimate their features. In this talk, we review some inverse problems linked to the estimation of the initial size-distribution and of the fragmentation characteristics.
Departing from a model of discrete depolymerisation, we first evaluate the impact of using continuous approximations to solve the initial-state estimation problem. At second order, the asymptotic model becomes an advection-diffusion equation, where the diffusion is a corrective term. This approximation is much more accurate, but we face a classical accuracy versus stability trade-off: the inverse reconstruction reveals to be severely ill-posed. Thanks to Carleman inequalities and to 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 by H. Rezaei.
To estimate the fragmentation kernel in experiments of polymer breakage, we propose several approaches based on the continuous fragmentation equation, studying and making use either of the long-term, the transient or the short-term dynamics. Error estimates in Bounded Lipshitz norm are obtained. This is a joint work with M. Escobedo and M. Tournus, based on biological questions and experiments of W.F. Xue.
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David Gérard-Varet (Université Paris Cité)
Title: Mixing in active suspensions.
Abstract: We consider a model introduced by D. Saintillan and M. Shelley to describe active suspensions of elongated particles. This model, which generalizes the classical Doi model for passive suspensions, couples a Stokes equation for the fluid substrate and a transport equation for the density distribution of particles in space and orientation. We investigate mixing properties of this model (damping and enhanced dissipation). The main new feature of the analysis is that the usual velocity variable of the euclidean space is replaced by an orientation variable on the sphere, which is responsible for strong qualitative changes and new mathematical difficulties. This is joint work with M. Coti Zelati and H. Dietert.
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Thomas Giletti (Université de Lorraine)
Title: Propagating fronts and terraces in multistable reaction-diffusion equations.
Abstract:This talk will be devoted to propagation phenomena for a general scalar reaction-diffusion equation, i.e. when it may admit an arbitrarily large number of stationary states. Large time propagation can no longer be described by a single front, but by a family of several stacked fronts (or `propagating terrace') involving intermediate transient equilibria. I will review several methods, differing in their degree of generality (homo- or heterogeneous, one- or multi-dimensional, semi- or non-linear equations...), to handle such dynamics.
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Megan Griffin-Pickering (University College London)
Title: Recent results on the quasi-neutral limit for the ionic Vlasov-Poisson system.
Abstract: Vlasov-Poisson type systems are well known kinetic models for plasma. The precise structure of the model differs according to which species of particle it describes, with the classical version of the system describing electrons. The model for ions, however, includes an additional exponential nonlinearity in the equation for the electrostatic potential, which creates several additional mathematical difficulties.
The Debye length is a characteristic length scale of a plasma describing the scale of electrostatic interaction. In real plasmas this length is typically very small, and in physics applications frequently assumed to be very close to zero. This motivates the study of the limiting behaviour of Vlasov-Poisson type systems as the Debye length tends to zero relative to the observation scale—known as the ‘quasi-neutral’ limit. In the case of the ionic model, the formal limit is the kinetic isothermal Euler system; however, this limit is highly non-trivial to justify rigorously and known to be false in some cases without very strong regularity conditions and/or structural conditions.
I will present a recent work, a collaboration with Mikaela Iacobelli, in which we prove the quasi-neutral limit for the ionic Vlasov-Poisson system for a certain class of rough (L^\infty) data: that is, data that may be expressed as perturbations of an analytic function, small in the sense of Wasserstein distances. The smallness of the perturbation that we require is much less restrictive than in the previously known results.
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Jacek Jendrej (CNRS, Université Sorbonne Paris Nord)
Title: Dynamics of kink clusters for scalar fields in dimension 1+1
Abstract: We consider classical scalar fields in dimension 1+1 with a self-interaction potential being a symmetric double-well. Such a model admits non-trivial static solutions called kinks and antikinks. A kink cluster is a solution approaching, for large positive times, a superposition of alternating kinks and antikinks whose velocities converge to 0 and mutual distances grow to infinity. Our main result is a determination of the asymptotic behaviour of any kink cluster at the leading order.
Our results are partially inspired by the notion of "parabolic motions" in the Newtonian n-body problem. I will present this analogy and mention its limitations. If time allows, I will explain the role of kink clusters as universal profiles for formation of multi-kink configurations.
This is a joint work with Andrew Lawrie from MIT.
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Ivàn Moyano (Université Côte-d'Azur)
Title: Uncertainty principles in control theory of PDEs.
Abstract: In this talk we review some classical and recent results relating the uncertainty principles for the Laplacian with the controllability and stabilisation of some linear PDEs. The uncertainty principles for the Fourier transforms state that a square integrable function cannot be both localised in frequency and space without being zero, and this can be further quantified resulting in unique continuation inequalities in the phase spaces. Applying these ideas to the spectrum of the Laplacian on a compact Riemannian manifold, Lebeau and Robbaino obtained their celebrated result on the exact controllability of the heat equation in arbitrarily small time. The relevant quantitative uncertainty principles known as spectral inequalities in the literature can be adapted to a number of different operators, including the Laplace-Beltami operator associated to C^1 metrics or some Schödinger operators with long-range potentials, as we have shown in recent results in collaboration with Gilles Lebeau (Nice) and Nicolas Burq (Orsay), with a significant relaxation on the localisation in space. As a consequence, we obtain a number of corollaries on the decay rate of damped waves with rough dampings, the simultaneous controllability of heat equations with different boundary conditions and the controllability of the heat equation with rough controls.
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Toan Nguyen (Penn State University)
Title: Landau damping and the survival threshold.
Abstract: The talk is to precise the classical notion of Landau damping and to provide the survival threshold of spatial frequencies that dictates the transition from purely oscillatory modes, known as Langmuir waves, to the free dynamics of electrons near spatially homogeneous backgrounds, classically modeled by the Vlasov-Poisson system in plasma physics or the Hartree-Coulomb equations in quantum mechanics. The transition occurs due to the exact resonant interaction between excited electrons and the oscillatory waves, namely the classical Landau damping.
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Tristan Robert (Université de Lorraine)
Title: On path-wise regularization by noise for some nonlinear dispersive PDEs.
Abstract: In this talk, we will consider nonlinear dispersive PDEs with a rough (distributional) time coefficient in front of the dispersion. We will review the nonlinear Young integral approach to this kind of problem and show that, for a wide class of strongly non-resonant dispersive PDEs, one can greatly improve on the well-posedness theory provided that the time coefficient is irregular enough. We will also discuss the case of strongly non-resonant equations perturbed by completely resonant non-linearities.
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Laura Spinolo (Istituto di Matematica Applicata e Tecnologie Informatiche, Pavie)
Title: The singular local limit of nonlocal conservation laws.
Abstract: Consider a so-called nonlocal conservation law, that is a continuity equation where the velocity field depends on the solution through the convolution with a given kernel. In the singular limit where the convolution kernel is replaced by a Dirac delta, one formally recovers a scalar conservation law. In this talk I will address the following question: can we rigorously justify this formal limit? Recent counter-examples in general rule out convergence to the entropy admissible solution of the limit conservation law.
However, in the specific framework of traffic models (with anisotropic convolution kernels) convergence holds, under suitable assumptions. My presentation will be based on joint works with Maria Colombo, Gianluca Crippa and Elio Marconi.
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Lorenzo Zambotti (Sorbonne Université)
Title: The revolution of pathwise stochastic analysis.
Abstract: The last 25 years have seen the introduction of a new approach to stochastic
analysis, first with the theory of rough paths in the case of stochastic differential equations (T. Lyons, M. Gubinelli) and then with the theory of regularity structures or stochastic paracontrolled distributions in the case of stochastic partial differential equations (M. Hairer, M. Gubinelli). This new framework has a very strong analytic flavor and is based on a new approach to the classical question of how to multiply two distributions. In this talk I will try to present some of the main ideas of this theory.
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