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 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. This picture should already hint at the great delicacy in studying boundary layers (source internet):

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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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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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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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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}. 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).

The major open problem is to understand the large time behavior of solutions to the 2D Euler equations. This is notoriously difficult, being time-reversible Hamiltonian system and having conserved energy and many invariant Casimir’s

\displaystyle \mathcal{E}[u] = \int_\Omega \frac{|u|^2}{2} \;dx , \qquad \mathcal{C}[\omega] = \int_\Omega \Phi(\omega) \; dx

that are conserved for all times, for any reasonable function {\Phi(\cdot)}. A great reference that discusses in depth this topics and other questions in fluid dynamics is this lecture notes by V. Sverak.

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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 provide 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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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. The result is similar to that obtained by Bedrossian ’17 for the Vlasov–Poisson–Fokker–Planck equation and by Masmoudi-Zhao ’19 for Navier-Stokes equations near Couette. However, unlike these works, the linearized operator cannot be inverted explicitly due to the complexity of the Landau collision operator. A framework follows, which is purely energy method, combining Guo’s weighted energy method with the hypocoercive energy method and the Klainerman’s vector field method. In this blog post, I give a flavor of our proof.

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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 simple 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 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

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I am sadden to learn that Bob Glassey 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.