HomeArticles posted by ttn12

Landau damping for the borderline Vlasov-Dirac-Benney system

October 27, 2025October 27, 2025 ttn12 Kinetic theory, New papers, Plasma Physics Analytic spaces, Borderline analyticity, Landau damping, Vlasov-Dirac-Benney

The Vlasov-Dirac-Benney system refers to the Vlasov theory for charged particles when the interaction potential is a Dirac delta function, namely

$\displaystyle \partial_t f + v\cdot \nabla_x f + E \cdot \nabla_v f = 0 \ \ \ \ \ (1)$

where the self-consistent electric field is computed through

$\displaystyle E = -\nabla_x \rho[f] \ \ \ \ \ (2)$

with density $\rho[f] =\int f(t,x,v)\; dv - n_{0}$ (compared with the classical Vlasov-Poisson system $E = -\nabla_x (-\Delta_x)^{-1}\rho[f]$ ). The name was coined by C. Bardos due to a close link to a model for water waves derived by Benney. The system is in fact ill-posed for general initial data in any Sobolev and Gevrey spaces, and the question of Landau damping for the borderline analytic data has always been open and of great interest. I shall briefly discuss how such a Vlasov theory arises, and mention my recent work that resolves this very question. This work is dedicated to my teacher and friend Dang Duc Trong in occasion of his 60th birthday.

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Stability of shear flows near a boundary

June 3, 2025August 1, 2025 ttn12 Books, boundary layers, fluid dynamics, Instabilities, Lecture notes

I’ve just released this book titled Stability of shear flows near a boundary, written jointly with E. Grenier, on the arXiv. This book is devoted to the study of the linear and nonlinear stability of shear flows and boundary layers for Navier Stokes equations for incompressible fluids with Dirichlet boundary conditions in the case of small viscosity. The aim of this book is to provide a comprehensive presentation to recent advances on boundary layers stability. It targets graduate students and researchers in mathematical fluid dynamics and only assumes that the readers have a basic knowledge on ordinary differential equations and complex analysis. No prerequisites are required in fluid mechanics, excepted a basic knowledge on Navier Stokes and Euler equations, including Leray’s theorem. This book consists of three parts. Part I is devoted to the presentation of classical results and methods: Green functions techniques, resolvent techniques, analytic functions. Part II focuses on the linear analysis, first of Rayleigh equations, then of Orr Sommerfeld equations. This enables the construction of Green functions for Orr Sommerfeld, and then the construction of the resolvent of linearized Navier Stokes equations. Part III details the construction of approximate solutions for the complete nonlinear problem and nonlinear instability results. We had plan to complete a few additional chapters near the end of the book, but ultimately decided to release the full unpublished version that we had back in August 2020.

Remarks on Landau damping

May 16, 2025May 16, 2025 ttn12 Kinetic theory, New papers, Plasma Physics Analyticity, Landau damping, Phase mixing, Plasma echoes, Vlasov-Poisson

I’ve just posted on arxiv this short paper “Remarks on Landau damping”, which gives a quick overview of phase mixing, Landau damping, plasma echoes, and in particular, provides a self-contained proof, for non specialists, of the celebrated result by Mouhot-Villani in the analytic case. This was written in honor of Dang Duc Trong, a kind mentor to many Vietnamese mathematicians, on the occasion of his 60th birthday. Specifically, consider the following classical Vlasov-Poisson system

$\displaystyle \partial_t f + v\cdot \nabla_x f + E \cdot \nabla_v f = 0 \ \ \ \ \ (1)$

$\displaystyle E = -\nabla_x \phi, \qquad -\Delta_x \phi = \rho \ \ \ \ \ (2)$

modeling the dynamics of excited electrons confined on a torus ${\mathbb{T}_x^3\times \mathbb{R}^3}$ or in the whole space ${\mathbb{R}^3_x\times \mathbb{R}^3_v}$ , in which ${\rho(t,x) = \int_{\mathbb{R}^3} f(t,x,v)\; dv-n_{\mathrm{ion}}}$ denotes the charged density, and ${ n_{\mathrm{ion}}}$ is a non-negative constant representing the uniform ions background. The Cauchy problem is rather classical, going back to the works by Lions-Perthame, Pfaffelmoser, and Schaeffer in the early 90s, which assert that smooth initial data ${f(0,x,v)}$ with finite moments give rise to global-in-time smooth solutions. However, their large time behavior is largely open due to the presence of plasma echoes and rich underlying physics, which we shall now discuss.

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The Vlasov theory for relativistic plasmas near vacuum

April 29, 2025April 30, 2025 ttn12 Kinetic theory, Plasma Physics Phase mixing, Relativistic plasmas, Scattering theory, Transport dispersion, Vlasov-Maxwell

In plasma physics, the dynamics of collisionless charged particles can often be modeled by the relativistic Vlasov-Maxwell system, namely the transport equation

$\displaystyle \partial_t f + \hat v \cdot \nabla_x f + \mathcal K\cdot \nabla_v f = 0 \ \ \ \ \ (1)$

for the unknown one-particle density distribution ${f(t,x,v)}$ in the phase space ${\mathbb{R}_x^3 \times \mathbb{R}^3_v}$ , coupled with the Lorentz force ${\mathcal K = E + \hat v \times B}$ , whose electromagnetic fields are computed though the classical Maxwell equations

$\displaystyle \begin{aligned} \partial_t B + \nabla_x \times E = 0, \qquad \nabla_x \cdot E &= \rho[f], \\ - \partial_t E + \nabla_x \times B = {\bf j}[f], \qquad \nabla_x \cdot B & =0. \end{aligned} \ \ \ \ \ (2)$

The particles travel with relativistic velocities ${\hat v = v/\langle v\rangle}$ with ${\langle v\rangle =\sqrt{1+|v|^2}}$ (normalizing the speed of light to be one), and the charge and current densities are defined by ${\rho[f] = \int f(t,x,v) dv - n_{\mathrm{ion}}}$ and ${{\bf j}[f] = \int \hat v f(t,x,v)\; dv - {\bf j}_{\mathrm{ion}}}$ for some fixed constant background densities ${n_{\mathrm{ion}}}$ and ${{\bf j}_{\mathrm{ion}}}$ (i.e. for sake of presentation, we simply focus on the dynamics of excited electrons, fixing a uniform ion background).

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Landau damping for relativistic plasmas

January 26, 2025March 4, 2025 ttn12 Kinetic theory, Plasma Physics Landau damping, Langmuir waves, Relativistic plasmas, Vlasov-Maxwell

Of great interest is to resolve the Final State Conjecture of charged particles in the kinetic theory of plasma physics (borrowing words from general relativity, though amusingly lesser mathematical progress is made for the case of plasmas!), asserting the scattering to neutrality in the far field plus a superposition of coherent states or trapped particles, and identifying relaxation mechanism plays a fundamental role in resolving the conjecture. Despite an extensive list of works in the mathematical literature towards the understanding of the large-time dynamics of non-relativistic plasmas (already encoding extremely rich underlying physics and challenging mathematics – see Survival Threshold), very little is known in the case of relativistic plasmas, including the basic question of whether stable equilibria exist, not to mention the apparent lack of study on relaxation mechanism. In this blog article, I shall highlight my recent joint work with D. Han-Kwan (CNRS, Nantes) and F. Rousset (Paris-Sud, Orsay) which resolved the linear stability theory of relativistic plasmas near radial spatially homogenous equilibria, the first such a result, paving the waves for many possible future advances on the subject.

Figure: Depicted is a description of phase mixing on torus (in the whole space, particles simply scatter away), one of the key relaxation mechanisms in plasma physics.

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Survival threshold for plasma oscillations

January 19, 2025February 4, 2025 ttn12 Kinetic theory, Plasma Physics Landau damping, Phase mixing, Plasma oscillation, Survival threshold, Vlasov-Poisson

In plasma physics, plasma oscillations, also known as Langmuir’s oscillatory waves, refer to the oscillatory behavior of excited electrons in a non-trivial non-equilibrium state of a plasma. In this article, I shall describe how plasma oscillations arise via the Vlasov’s collisionless kinetic theory with long-range Coulomb’s pair interaction between charged particles, namely through the Vlasov-Poisson system. I shall then introduce the so-called survival threshold of spatial frequencies (namely, the inverse of wavelengths) that characterizes the dynamics of excited electrons near spatially homogenous steady states:

Phase mixing above survival threshold
Plasma oscillations below survival threshold

in which above / below refers to elementary waves with wavenumbers larger / smaller than the threshold. Phase mixing is a pure transport damping mechanism which yields rapid decay for the electric field, while plasma oscillations are not damped, but disperse like a Klein-Gordon dispersive wave in the whole space (i.e. the survival of oscillations below threshold, noting these oscillatory modes may occur on a large torus as well). The classical Landau damping then occurs due to resonant interaction between the two regimes at survival threshold, which we shall detail below. Eventually, the following figure captures the dynamics of the electric field whose dispersion relation, say $\lambda_\pm(k)$ , obeys (1) Klein-Gordon’s pure oscillations below survival threshold (i.e. no damping $\Re \lambda_\pm(k)=0$ ), (2) Landau damping at the threshold (i.e. onset of damping $\Re \lambda_\pm(k)<0$ ), and (3) phase mixing above the threshold (i.e. exponential damping $\Re \lambda_\pm(k)\lesssim-|k|$ ):

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Two Mini-courses on Wave and Kinetic Theory

May 18, 2024January 27, 2025 ttn12 Conferences

Prior to the Summer School in Math Physics in Hue (August 5-10) this summer, there will be two mini-courses on the wave and kinetic theory in the week of July 16 – 19, 2024, held at the VIASM, Hanoi.

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Summer School in Mathematical Physics in Vietnam

February 20, 2024January 27, 2025 ttn12 Conferences Math Physics, Vietnam

Phan Thanh Nam (LMU Munich) and I have been organizing a Summer School series in Mathematical Physics in Vietnam, hosted by the Vietnam Institute for Advanced Study in Mathematics (VIASM) and local universities in Vietnam, first in Hanoi (August, 2022 ), then Quy Nhon (August, 2023), and the next one is in Hue (August 5-10, 2024). The goal of the summer school series is to bring together leading international experts, young researchers, and students from the diverse areas of mathematical physics and partial differential equations to disseminate the recent developments and to train the next generation scientists in the field.

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The inviscid limit problem for Navier-Stokes equations

August 18, 2022August 26, 2022 ttn12 boundary layers, fluid dynamics Boundary layers, Boundary vorticity, Inviscid limit, Navier-Stokes

A longstanding open problem is to establish the inviscid limit of classical solutions to the incompressible Navier-Stokes equations for smooth initial data on a domain with boundaries. The question is of great physical and mathematical interest, and it deeply links to the transition to turbulence in fluids that may possibly take place faster than expected due to the presence of a boundary. In this article, I shall give a quick overview of this subject, and then highlight some recent works with my former student, Trinh T. Nguyen, (currently a Van Vleck Assistant Professor at University of Wisconsin, Madison), whose main results establish the inviscid limit for smooth data that are only required to be analytic locally near the boundary. This may be the best possible type of positive results that one can hope for, given the known violent instabilities at the boundary, which I shall discuss below. Before getting on, this picture should already hint at the great delicacy in studying boundary layers (source internet):

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Dispersion for Schrödinger equations

August 3, 2022August 3, 2022 ttn12 Lecture notes, Schrodinger Dispersion, Schrodinger

The Schrödinger equation forms the basic principles of quantum mechanics (like that of Newton’s second law in classical mechanics). It also plays an important role in describing waves at an appropriate regime in classical fluid dynamics (e.g., water waves) and plasma physics (e.g., Langmuir’s waves or oscillations in a plasma!). In this quick note, I shall present a few basic properties and classical results for the Schrödinger equations, focusing mainly on the defocusing cubic nonlinear equations

$\displaystyle i\partial_t u + \Delta u = |u|^2 u \ \ \ \ \ (1)$

on ${\mathbb{R}_+ \times \mathbb{R}^d}$ , ${d\ge 1}$ (also known as the Gross-Pitaevskii equation). These notes are rather introductory and classical (e.g., Tao’s lecture notes), which I’m using as part of my lectures at the summer school that P. T. Nam and I are running this week on “the Mathematics of interacting Bose gases” at VIASM, Hanoi, Vietnam (August 1-5, 2022)!

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Two special issues in memory of Bob Glassey

July 22, 2022July 22, 2022 ttn12 Uncategorized Bob Glassey

Yan Guo, Walter Strauss, and I organized two special issues on Nonlinear Waves and Kinetic Theory dedicated to the memory of Bob Glassey, who sadly passed away recently (I wrote an eulogy of his passing on this blog). The special issues are now published on the Kinetic and Related Models journal, Issue 1 and Issue 2.

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A roadmap to nonuniqueness of L^p weak solutions to Euler

January 21, 2022March 3, 2022 ttn12 fluid dynamics, Instabilities

In this post, I discuss a rather classical roadmap to obtain the non-uniqueness of ${L^p}$ weak solutions to the classical incompressible Euler equations; namely, focusing on the two-dimensional case, which reads in the vorticity formulation for vorticity function ${\omega}$ :

$\displaystyle \partial_t \omega + v \cdot \nabla \omega = 0, \qquad v = \nabla^\perp \Delta^{-1}\omega \ \ \ \ \ (1)$

on ${\mathbb{R}^2}$ , with initial vorticity in ${L^p}$ (hence, vorticity remains in ${L^p}$ for all times). It’s known, going far back to Yudovich ’63, that weak solutions with bounded vorticity are unique, leaving open the question of uniqueness of solutions whose vorticity is only in ${L^p}$ for ${p\in [1,\infty)}$ . This blog post is to discuss the possible quick roadmap to proving nonuniqueness arising from the instability nature of fluid models, focusing on the Euler equations (1).

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Notes on the large time of Euler equations and inviscid damping

June 13, 2021June 29, 2021 ttn12 fluid dynamics Arnold stability, Euler equations, inviscid damping

Consider the classical incompressible Euler equations in two dimension; namely written in the vorticity formulation, the transport equation for the unknown scalar vorticity ${\omega = \omega(t,x,y)}$ ,

$\displaystyle \partial_t \omega + (u \cdot \nabla) \omega =0$

posed on a spatial domain ${\Omega \subset \mathbb{R}^2}$ , where the velocity field ${u\in \mathbb{R}^2}$ is obtained through the Biot-Savart law ${u = \nabla^\perp \Delta^{-1} \omega}$ (with convention $(a_1,a_2)^\perp = (a_2,-a_1)$ ). By construction, the velocity field ${u}$ is incompressible: ${\nabla \cdot u=0}$ . When dealing with domains with a boundary, ${\Delta^{-1} }$ is defined together with a Dirichlet boundary condition that corresponds to the no-penetration condition of fluids ${u\cdot n =0}$ on the boundary.

The global well-posedness theory of 2D Euler is classical: (1) smooth initial data give rise to solutions that remain smooth for all times (e.g. the Beale-Kato-Majda criterium holds, as vorticity is uniformly bounded for all times; in fact, being transported along the volume-preserving flow, all ${L^p}$ norms of vorticity are conserved), and (2) weak solutions with bounded vorticity are unique (see Yudovich ’63).

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Generator functions and their applications

May 31, 2021June 17, 2021 ttn12 fluid dynamics, Kinetic theory, New papers Analyticity, Boundary layers, Cauchy-Kovalevskaya, Generator functions, Landau damping

The Cauchy-Kovalevskaya theorem is a classical convenient tool to construct analytic solutions to partial differential equations, which allows one to view and treat them as if they are ordinary differential equations:

$\displaystyle u_t = F(t,x,u,u_x)$

for unknown functions ${u(t,x)}$ in ${t\ge 0}$ and ${x\in \mathbb{R}^d}$ (or some spatial domain). Roughly, if ${F(t,x,u,w)}$ is locally analytic near a point ${(0,x_0,u_0,w_0)}$ then the PDE has a unique solution ${u(t,x)}$ which is analytic near ${(0,x_0)}$ , as established by Cauchy (1842) and generalized by Kovalevskaya as part of her dissertation (1875). The theorem has found many applications such as in fluid dynamics and kinetic theory where it is used to provide existence of analytic solutions including those that are obtained at certain asymptotic limits (e.g., inviscid limit, Boltzmann-to-fluid limit, Landau damping, inviscid damping,…). There are modern formulations of the abstract theorem: see, for instance, Asano, Baouendi and Goulaouic, Caflisch, Nirenberg, and Safonov. In this blog post, I present generator functions, as an alternative approach to the use of the Cauchy-Kovalevskaya theorem, recently introduced in my joint work with E. Grenier (ENS Lyon), and discuss the versatility and simplicity of their use to applications. In addition to providing existence of analytic solutions, I will also mention another use of generator functions to capture some physics that would be otherwise missed for analytic data (namely, an analyticity framework to capture physical phenomena that are not seen for analytic data!).

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Landau damping and extra dissipation for plasmas in the weakly collisional regime

May 23, 2021May 28, 2021 ttn12 Kinetic theory, Plasma Physics Enhanced dissipation, Landau damping, Landau equations, Vector field method, Vlasov-Poisson

Sanchit Chaturvedi (Stanford), Jonathan Luk (Stanford), and I just submitted the paper “The Vlasov–Poisson–Landau system in the weakly collisional regime”, where we prove Landau damping and extra dissipation for plasmas modeled by the physical Vlasov-Poisson-Landau system in the weakly collisional regime ${\nu\ll1}$ , where ${\nu}$ is the collisional parameter. The results are obtained for Sobolev data that are ${\nu^{1/3}}$ -close to global Maxwellians on the torus ${\mathbb{T}_x^3\times \mathbb{R}_v^3}$ . While Landau damping is a classical subject in plasma physics that predicts mixing and relaxation without dissipation of the electric field in a plasma, extra dissipation arises due to the interplay between phase mixing and entropic relaxation, or between transport and diffusion, which enhances decay to a faster rate than the usual diffusion rate. In this blog post, I give a flavor of the proof of our results, where we develop a purely energy method which combines Guo’s weighted energy method with the hypocoercive energy method and the Klainerman’s vector field method.

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Plasma echoes near stable Penrose data

June 22, 2020June 24, 2022 ttn12 Kinetic theory, Plasma Physics Landau damping, Plasma echoes, Vlasov-Poisson

Echoes in a plasma are the excitement of new waves due to nonlinear interaction. The excitement may happen at an arbitrarily large time, which is the main source of difficulties in understanding Landau damping. For analytic data, the echoes are suppressed as the electric field is exponentially localized in time, and the nonlinear Landau damping holds for such data, as was first obtained by Mouhot and Villani in their celebrated work (Acta Math 2011; see also the extension to include Gevrey data). The nonlinear Landau damping remains largely elusive for less regular data (e.g., data with Sobolev regularity).

Recently, in a collaboration with E. Grenier (ENS Lyon) and I. Rodnianski (Princeton), we give an elementary proof of the known Landau damping results, which I also blogged it here, that were seen as a perturbation of the free transport dynamics, whose damping is direct (that is, the phase mixing). In the companion paper with E. Grenier and I. Rodnianski, we construct a class of echo solutions, which are arbitrarily large in any Sobolev spaces (in particular, they do not belong to the analytic or Gevrey classes studied by Mouhot and Villani), but nonetheless, the nonlinear Landau damping holds. In this blog post, I shall briefly discuss the plasma echo mechanism and our new results.

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Landau damping for analytic and Gevrey data

June 6, 2020June 8, 2020 ttn12 Kinetic theory, Plasma Physics Landau damping, Vlasov-Poisson

Landau damping is a classical subject in Plasma Physics, which studies decay of the electric field in a collisionless plasma in the large time. The damping was discovered and fully understood by Landau in the 40s for the linearized evolution near Maxwellians, and later extended by O. Penrose in the 60s for general spatially homogenous equilibria. The first mathematical proof of the nonlinear Landau damping was given by Mouhot and Villani for analytic data in their celebrated work (Acta Math, 2011). Their proof was then simplified, and the result was extended by Bedrossian, Masmoudi, and Mouhot to include data in certain Gevrey classes (Annals of PDEs, 2016).

Recently, in a collaboration with E. Grenier (ENS Lyon) and I. Rodnianski (Princeton), we give an elementary proof of these same results, which I shall give a sketch of it in this blog post. To avoid some tedious algebra, I mainly focus on the analytic case, which is precisely the case originally studied by Mouhot and Villani, leaving some remarks to the Gevrey cases at the very end of the post, where you’ll also find the slides of my recent lectures over Zoom on this topics.

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Bob Glassey

February 17, 2020February 17, 2020 ttn12 Uncategorized

I am sadden to learn that Bob Glassey, a Professor Emeritus at Indiana, passed away this weekend after a long illness. Bob was a pioneer in the mathematical study of kinetic theory and nonlinear wave equations. He, together with Walter Strauss, was the first to initiate the mathematical study of Vlasov-Maxwell systems that describe the dynamics of a collisionless plasma. One of his fundamental theorems, known as Glassey-Strauss’ theorem (ARMA 1986), is to assert that solutions to the relativistic Vlasov-Maxwell system in the three dimensional space do not develop singularities as long as the velocity support remains bounded. The latter condition was subsequently verified by him and his former PhD student Jack Schaeffer for the case of low dimensions; namely, when particles are limited to one or two spatial domains. Their work has inspired several attempts from the mathematical community to tackle the full three dimensional case, which remains an outstanding open problem in the field.

Together with J. Schaeffer, Bob was also one of the first to initiate the mathematical study of Landau damping for Vlasov-Poisson systems in the presence of low frequency (or unconfined spatial domain). More precisely, for confined plasma (say, plasma on a torus), it was discovered and fully understood by Landau in the 40s that at the linearized level near a Gaussian, the electric field decays exponentially or polynomially depending on the regularity of initial data in the large time. The linear Landau damping remains to hold for more general spatially homogenous equilibria, known as Penrose stable equilibria. Later, Mouhot and Villani (Acta Math, 2011) verified this damping for data with analyticity for the nonlinear equations. In the unconfined case, Glassey and Schaeffer proved that the linear damping holds and is optimal at a much slower rate, which is surprisingly worse for Gaussians, due to the failure of the Penrose stability condition that holds in the confined case.

Bob also made fundamental and beautiful studies on the blowup issue for semilinear Heat, Wave, and Schr\”odinger (e.g., the Glassey’s trick), among other things. His book “The Cauchy problem in kinetic theory (SIAM 1996)” remains a fundamental textbook in the field.

Although Bob was already retired when I came to Indiana for my graduate study, he kindly participated and generously offered valuable guidances in a working seminar that I ran on the DiPerna-Lions theory for Boltzmann equations in the summer of 2008.

Landau damping for screened Vlasov-Poisson on the whole space

September 9, 2019September 15, 2019 ttn12 Kinetic theory Landau damping, Vlasov-Poisson

In a recent joint work with Daniel Han-Kwan (CMLS, Ecole polytechnique) and Frédéric Rousset (Paris-Sud University), we give an alternative proof of the Landau damping for screened Vlasov-Poisson system near stable homogenous equilibria on the whole space, a result that was first established by Bedrossian, Masmoudi and Mouhot, for data with finite Sobolev regularity (they remarked that 36 derivatives were sufficient).

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Dafermos and Rodnianski’s r^p-weighted approach to decay for wave equations

November 14, 2018 ttn12 General Relativity, Uncategorized r^p approach, Wave equations

Dafermos and Rodnianski introduced an ${r^p}$ -weighted vector field method to obtain boundedness and decay of solutions to wave equations on a Lorentzian background, avoiding to use global vector field multipliers and commutators with weights in ${t}$ and thus proving to be more robust in dealing with black holes such as in Schwarzschild and Kerr background metrics. The approach has found numerous applications; see, for instance, Dafermos-Rodnianski, Moschidis, or Keir for many insightful discussions. In this blog post, I will give some basic details of this approach, based on lecture notes of S. Klainerman, focusing mostly on the flat Minkowski spacetime.

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