Frontiers in Fluid and Kinetic Partial Differential Equations

Abstract: I will present an abstract result showing that, for a quasilinear evolution problem, the continuity of the data-to-solution map follows automatically from the estimates that are usually established in the proof of existence of solutions. This is joint work with N. Burq, M. Ifrim, D. Tataru, and C. Zuily.

Abstract: In 2014, Jia and Sverak constructed self-similar solutions evolving from arbitrarily large scaling-invariant initial data in 3D and conjectured that they go unstable at high Reynolds numbers and thereby generate non-unique solutions. In 2D, Leray-Hopf solutions are unique, but this picture may still hold in the infinite-energy class. We construct self-similar solutions evolving from arbitrarily large scaling-invariant initial data in 2D and present numerical evidence of non-uniqueness. Joint work with Julien Guillod (Sorbonne Universite and ENS), Mikhail Korobkov, and Xiao Ren (Fudan University).

Abstract: [Abstract To Be Announced]

Abstract: We will be interested in the solutions to the Vlasov–Maxwell system arising from sufficiently regular  initial data, with a small distribution function. In particular, we will compare their asymptotic behavior  with that of the solutions to the linearised system. While the electromagnetic field can be approximated by  a linear solution, the distribution function exhibits a modified scattering dynamic: due to the long–range  effects of the Lorentz force, it converges along linear characteristics corrected by a logarithmic term. A  key step in defining these modified characteristics is to identify an effective Lorentz force that governs  the asymptotic behavior of the force field.

Abstract: In 2003, Bressan proposed a conjecture on the mixing efficiency of incompressible flows, which remains open.  This talk surveys progress toward resolving Bressan’s mixing conjecture and presents a new result confirming  its asymptotic validity for time-periodic velocity fields. We accomplish this by adapting dynamical systems  tools to the non-smooth framework of DiPerna-Lions flows. Furthermore, we discuss links to bounds on metric  entropy and extensions of the Ruelle inequality.

Abstract: We present recent works with Zaher Hani and Xiao Ma, in which we derive the Boltzmann equation from the  hard sphere dynamics in the Boltzmann-Grad limit, for the full time range in which the (strong) solution to  the Boltzmann equation exists. This is done in the Euclidean setting in any dimension d ≥ 2, and in the  periodic setting in dimensions d ∈ {2,3}. As a corollary, we also derive the corresponding fluid  equations from the the hard sphere dynamics. This executes the original program, proposed in Hilbert's Sixth  Problem in 1900, pertaining to the derivation of hydrodynamic equations from colliding particle systems, via  the Boltzmann equation as the intermediate step.

Abstract: We will discuss some results concerning the long-time behavior of solutions to the two-dimensional  incompressible Euler and Navier-Stokes equations. One at zero viscosity and long times, the other at long  time and subsequently zero viscosity.

Abstract: In the first part of the talk, we survey recent results on the nonlinear Boltzmann equation for kinetic  shear flow. We discuss the issue in two cases either on the finite interval with finite energy or on the  infinite interval with infinite energy at infinite time. In the second part, we focus on a recent study of  the diffusive limit of the time evolutionary Boltzmann equation in the half space T² × R⁺ for a small  Knudsen number ε > 0. For boundary conditions in the normal direction, it involves diffuse reflection  moving with a tangent velocity proportional to ε on the wall, whereas the far field is described by a global  Maxwellian with zero bulk velocity. The incompressible Navier-Stokes equations, as the corresponding formal  fluid dynamic limit, admit a specific time-dependent shearing solution known as the Rayleigh profile, which  accounts for the effect of the tangentially moving boundary on the flow at rest in the far field. Using the  Hilbert expansion method, for well-prepared initial data we construct the Boltzmann solution around the  Rayleigh profile without initial singularity over any finite time interval.

Abstract: It is well-established that shear flows are linearly unstable provided the viscosity is small enough, when  the horizontal Fourier wave number lies in some interval, between the so-called lower and upper marginally  stable curves. In this article, we prove that, under a natural spectral assumption, shear flows undergo a  Hopf bifurcation near their upper marginally stable curve. In particular, close to this curve, there exists  space periodic traveling waves solutions of the full incompressible Navier-Stokes equations. For the  linearized operator, the occurrence of an essential spectrum containing the entire negative real axis causes  certain difficulties which must be overcome. Moreover, if this Hopf bifurcation is super-critical, these  time and space periodic solutions are linearly and nonlinearly asymptotically stable. This is a joint work  with D. Bian and G. Iooss.

Abstract: In this talk, I will present recent advances in the study of vortex dynamics for the two-dimensional Euler  equations. I will discuss results on the desingularization of time-periodic point vortex configurations,  both in rigid and non-rigid frameworks. The focus will be on the rigorous construction of a leapfrogging  motion associated with Love’s four-vortex configuration, obtained through a combination of KAM theory and  the Nash–Moser iterative scheme.

Abstract: I will talk about some recent work on the problem of establishing a wave turbulence theory for water waves  systems. This is a classical problem in Mathematical Physics, going back to pioneering work of Hasselmann.  To address it we propose a new mechanism, based on a combination of two main ingredients: (1) deterministic  energy estimates for all solutions that are small in L∞-based norms, and (2) probabilistic  arguments aimed at understanding propagation of randomness on long time intervals. This is joint work with  Yu Deng and Fabio Pusateri.

Abstract: We introduce a new approach to derive mean-field limits for first- and second-order particle systems with  singular interactions. It is based on a duality approach combined with the analysis of linearized dual  correlations, and it allows to cover for the first time arbitrary square-integrable interaction forces at  possibly vanishing temperature. The approach also provides convergence rates, and some statistical form of  Central Limit Theorem at the limit. This corresponds to joint works with D. Bresch, M. Duerinckx, and  N. Khoury.

Abstract: Classical variational approach of maximizing the kinetic energy under constraints provides nonlinear  stability of the maximizing vortex configuration in various settings, but this approach fails to handle the  situations where the vorticity is concentrated at multiple points in the fluid domain. This is simply  because such configurations are not even local kinetic energy maximizers, even when we restrict the  admissible class using all known coercive conserved quantities. We present results on nonlinear stability of  superpositions of several Lamb dipoles, obtained by combining classical variational principle with  dynamical bootstrapping schemes. This is based on several joint works with Ken Abe, Kyudong Choi, and  Yao Yao.

Abstract: This talk studies the non-cutoff Boltzmann equation for hard potentials in a perturbative setting. We first  establish a sharp short-time estimate on the radius of analyticity and Gevrey regularity of mild solutions.  Furthermore, we obtain a global-in-time radius estimate in Gevrey space. The proof combines hypoelliptic  estimates with the macro-micro decomposition.

Abstract: The BGK waves are the steady states for the 1d Vlasov-Poisson system. We consider their linear stability and derive a simple criterion. This is joint work with D. Bian, E. Grenier and W. Huang.

Abstract: A central theme in this body of work is the rigorous derivation of mean-field limits for systems of  particles with singular interactions—notably Coulomb and Riesz types. These systems are governed by  gradient flows, conservative flows, and may include stochastic (noisy) effects. The modulated energy method  is introduced as a tool to quantify convergence from a discrete particle system to a continuum PDE limit. At  the heart of this approach lies a commutator-type functional inequality, which has seen significant recent  progress. Global-in-time convergence is also addressed.

Abstract: We discuss recent results showing that the standard Fisher information is monotone in time for the space homogeneous Boltzmann and Landau equations. This new Lyapunov functional allows us to establish the existence of global smooth solutions in all cases that remained open. To prove this monotonicity, we introduce a novel doubling-variables technique and reduce the problem to an inequality in the family of the log-Sobolev inequalities.

Abstract: We will report recent progress on the 2-D immersed boundary problem with the Navier-Stokes equation, which  models coupled motion of a 1-D closed elastic string and ambient fluid in the entire plane. This is based on  joint works with Dongyi Wei.

Abstract: We consider the 2D, incompressible Navier-Stokes equations near the Couette flow, ω^(NS) = 1 + ε ω,  set on the channel T × [-1, 1], supplemented with Navier boundary conditions on the perturbation,  ω|_{y = ±1} = 0. We are simultaneously interested in two asymptotic regimes that are classical in  hydrodynamic stability: the long time, t → ∞, stability of background shear flows, and the inviscid  limit, ν → 0 in the presence of boundaries. Given small (ε ≪ 1, but independent of ν) Gevrey 2-datum,  ω₀^(ν)(x, y), that is supported away from the boundaries y = ±1. This is the first nonlinear  asymptotic stability result of its type, which combines three important physical phenomena at the nonlinear  level: inviscid damping, enhanced dissipation, and long-time inviscid limit in the presence of boundaries.

Abstract: When the steady flows are away from stagnation, the associated Euler equations can be locally reduced to a  semilinear equation. On the other hand, stagnation of flows is not only an interesting phenomenon in fluid  mechanics, but also plays a significant role in understanding many important properties of fluid equations.  It also induces many challenging problems in analysis. First, we discuss the scenario when the Euler  equations can be reduced to a single semilinear equation in terms of stream function. Second, we give a  classification of incompressible Euler flows via the set of flow angles. Finally, the applications for  vanishing viscosity limit of fluid via these classifications will be addressed.

Abstract: We derive the Φ⁴₃ measure on the torus as a rigorous limit of the quantum Gibbs state of an  interacting Bose gas, where the limiting classical measure describes the critical behavior of the Bose gas  just above the Bose–Einstein phase transition. Since the quantum problem is typically formulated using a  nonlocal interaction potential, a key challenge is to approximate the local Φ⁴₃ theory by a Hartree  measure with a nonlocal interaction. This requires uniform estimates on the Hartree measure, which are  achieved using techniques from recent development on stochastic quantization and paracontrolled calculus.  The connection to the quantum problem is then established by applying the variational approach and deriving  a quantitative convergence of the quantum correlation functions to those of the Hartree classical field.

Abstract: In this talk, I will present some recent stability results on 2D and 3D Stokes-transport equations around the  Couette with non-homogeneous density background. This is based on joint works with Daniel Sinambela and  Weiren Zhao.