Toan Nguyen

Pennsylvania State University | Penn State · Department of Mathematics

We revisit the proof of Landau damping near stable homogenous equilibria of Vlasov–Poisson systems with screened interactions in the whole space \(\mathbb {R}^d\) (for \(d\ge 3\)) that was first established by Bedrossian, Masmoudi and Mouhot in [5]. Our proof follows a Lagrangian approach and relies on precise pointwise in time dispersive estimates...

We had introduced so called generators functions to precisely follow the regularity of analytic solutions of Navier-Stokes equations earlier (see Grenier and Nguyen [Ann. PDE 5 (2019)]. In this short note, we give a short presentation of these generator functions and use them to construct analytic solutions to classical evolution equations, which p...

Consider the Vlasov–Poisson–Landau system with Coulomb potential in the weakly collisional regime on a 3 3 -torus, i.e. ∂ t F ( t , x , v ) + v i ∂ x i F ( t , x , v ) + E i ( t , x ) ∂ v i F ( t , x , v ) = ν Q ( F , F ) ( t , x , v ) , E ( t , x ) = ∇ Δ − 1 ( ∫ R 3 F ( t , x , v ) d v − ∫ − T 3 ∫ R 3 F ( t , x , v ) d v d x ) , \begin{align*} \pa...

In this paper, we derive sharp bounds on the semigroup of the linearized incompressible Navier-Stokes equations near a stationary shear layer in the half space (R+2 or R+3), with Dirichlet boundary conditions, assuming that this shear layer in spectrally unstable for Euler equations. In the inviscid limit, due to the prescribed no-slip boundary con...

We study the nonlocal vectorial transport equation \(\partial _ty+ ({\mathbb {P}}y \cdot \nabla ) y=0\) on bounded domains of \({\mathbb {R}}^d\) where \({\mathbb {P}}\) denotes the Leray projector. This equation was introduced to obtain the unique optimal rearrangement of a given map \(y_0\) as the infinite time limit of the solution with initial...

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 target...

We study the linearized Vlasov-Poisson system around suitably stable homogeneous equilibria on $\mathbb{R}^d\times \mathbb{R}^d$ (for any $d \geq 1$) and establish dispersive $L^\infty$ decay estimates in the physical space.

In this paper, we give an elementary proof of the nonlinear Landau damping for the Vlasov-Poisson system near Penrose stable equilibria on the torus $\mathbb{T}^d \times \mathbb{R}^d$ that was first obtained by Mouhot and Villani in \cite{MV} for analytic data and subsequently extended by Bedrossian, Masmoudi, and Mouhot \cite{BMM} for Gevrey-$\gam...

In this paper we construct particular solutions to the classical Vlasov-Poisson system near stable Penrose initial data on $\mathbb{T} \times \mathbb{R}$ that are a combination of elementary waves with arbitrarily high frequencies. These waves mutually interact giving birth, eventually, to an infinite cascade of echoes of smaller and smaller amplit...

In [Grenier-Nguyen], we introduced so called {\em generators} functions to precisely follow the regularity of analytic solutions of Navier Stokes equations. In this short note, we give a presentation of these generator functions and use them to give existence results of analytic solutions to some classical equations, namely to hyperbolic equations,...

This paper is the continuation of a program, initiated in Grenier-Nguyen [8,9], to derive pointwise estimates on the Green function of Orr Sommerfeld equations. In this paper we focus on long wavelength perturbations, more precisely horizontal wavenumbers $\alpha$ of order $\nu^{1/4}$, which correspond to the lower boundary of the instability area...

We revisit the proof of Landau damping near stable homogenous equilibria of Vlasov-Poisson systems with screened interactions in the whole space $\mathbb{R}^d$ (for $d\geq3$) that was first established by Bedrossian, Masmoudi and Mouhot. Our proof follows a Lagrangian approach and relies on precise pointwise in time dispersive estimates in the phys...

In this paper we give a proof of an Onsager type conjecture on conservation of energy and entropies of weak solutions to the relativistic Vlasov--Maxwell equations. As concerns the regularity of weak solutions, say in Sobolev spaces $W^{\alpha,p}$, we determine Onsager type exponents $\alpha$ that guarantee the conservation of all entropies. In par...

We establish the inviscid limit of the incompressible Navier-Stokes equations on the whole plane $\mathbb{R}^2$ for initial data having vorticity as a superposition of point vortices and a regular component. In particular, this rigorously justifies the vortex-wave system from the physical Navier-Stokes flows in the vanishing viscosity limit, a mode...

We study the nonlocal vectorial transport equation $\partial_ty+ (\mathbb{P} y \cdot \nabla) y=0$ on bounded domains of $\mathbb{R}^d$ where $\mathbb{P}$ denotes the Leray projector. This equation was introduced to obtain the unique optimal rearrangement of the initial map $y_0$ as its steady states (\cite{AHT, Macthesis, Brenier09}). We rigorously...

We consider the quantum Boltzmann equation, which describes the growth of the condensate, or in other words, models the interaction between excited atoms and a condensate. In this work, the full form of Bogoliubov dispersion law is considered, which leads to a detailed study of surface integrals inside the collision operator on energy manifolds. We...

The classical Orr-Sommerfeld equations are the resolvent equations of the linearized Navier-Stokes equations around a stationary shear layer profile in the half plane. In this paper, we derive pointwise bounds on the Green function of the Orr-Sommerfeld problem away from its critical layers.

We establish the inviscid limit of the incompressible Navier--Stokes equations on the whole plane R^2 for initial data having vorticity as a superposition of point vortices and a regular component. In particular, this rigorously justifies the vortex-wave system from the physical Navier--Stokes flows in the vanishing viscosity limit, a model that wa...

In their classical work, Sammartino and Caflisch (Commun Math Phys 192(2):433–461, 1998a; Commun Math Phys 192(2):463–491, 1998b) proved the inviscid limit of the incompressible Navier–Stokes equations for well-prepared data with analytic regularity in the half-space. Their proof is based on the detailed construction of Prandtl’s boundary layer asy...

In this paper, we study the Vlasov-Maxwell system in the non-relativistic limit, that is in the regime where the speed of light is a very large parameter. We consider data lying in the vicinity of homogeneous equilibria that are stable in the sense of Penrose (for the Vlasov-Poisson system), and prove Sobolev stability estimates that are valid for...

The aim of this paper is to investigate the stability of Prandtl boundary layers in the vanishing viscosity limit: $\nu \to 0$. In \cite{Grenier}, one of the authors proved that there exists no asymptotic expansion involving one Prandtl's boundary layer with thickness of order $\sqrt\nu$, which describes the inviscid limit of Navier-Stokes equation...

We study the large time behavior of solutions to two-dimensional Euler and Navier-Stokes equations linearized about shear flows of the mixing layer type in the unbounded channel $\mathbb{T} \times \mathbb{R}$. Under a simple spectral stability assumption on a self-adjoint operator, we prove a local form of the linear inviscid damping that is unifor...

In $1904$, Prandtl introduced his famous boundary layer in order to describe the behavior of solutions of incompressible Navier Stokes equations near a boundary as the viscosity goes to $0$. His Ansatz was that the solution of Navier Stokes equations can be described as a solution of Euler equations, plus a boundary layer corrector, plus a vanishin...

In this paper, we prove the nonlinear stability under localized perturbations of spectrally stable time-periodic source defects of reaction-diffusion systems. Consisting of a core that emits periodic wave trains to each side, source defects are important as organizing centers of more complicated flows. Our analysis uses spatial dynamics combined wi...

The wave turbulence equation is an effective kinetic equation that describes the dynamics of wave spectra in weakly nonlinear and dispersive media. Such a kinetic model was derived by physicists in the 1960s, though the well-posedness theory remains open due to the complexity of resonant interaction kernels. In this paper, we provide a global uniqu...

In their classical work, Caflisch and Sammartino proved the inviscid limit of the incompressible Navier-Stokes equations for well-prepared data with analytic regularity in the half-space. Their proof is based on the detailed construction of Prandtl's boundary layer asymptotic expansions. In this paper, we give a direct proof of the inviscid limit f...

In this paper, we study Prandtl's boundary layer asymptotic expansion for incompressible fluids on the half-space in the inviscid limit. In \cite{Gr1}, E. Grenier proved that Prandtl's Ansatz is false for data with Sobolev regularity near Rayleigh's unstable shear flows. In this paper, we show that this Ansatz is also false for Rayleigh's stable sh...

This is a continuation and completion of the program (initiated in \cite{GrN1,GrN2}) to derive pointwise estimates on the Green function and sharp bounds on the semigroup of linearized Navier-Stokes around a generic stationary boundary layer profile. This is done via a spectral analysis approach and a careful study of the Orr-Sommerfeld equations,...

This paper concerns the validity of the Prandtl boundary layer theory in the inviscid limit for steady incompressible Navier–Stokes flows. The stationary flows, with small viscosity, are considered on [0,L]×R+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepa...

In this paper, we derive sharp bounds on the semigroup of the linearized Navier-Stokes equations near a stationary boundary layer on the half space. The bounds are obtained uniformly in the inviscid limit. Delicate boundary layer norms are introduced in order to capture the true boundary layer behavior of vorticity near the boundary. As an immediat...

In this paper, we construct the Green function for the classical Orr-Sommerfeld equations, which are the linearized Navier-Stokes equations around an unstable shear layer. As an immediate application, we derive uniform sharp bounds on the semigroup of the linearized Navier-Stokes problem in the vanishing viscosity limit.

Although many studies have been carried on to understand the Hasselmann-Zakharov weak turbulence equation for capillary waves since its derivation in the 60's, the question about the existence and uniqueness of solutions to the equation still remains unanswered, due to the complexity of the equation. This work provides a solution to the problem.

We study the high Reynolds number limit of a viscous fluid in the presence of a rough boundary. We consider the two-dimensional incompressible Navier-Stokes equations with Navier slip boundary condition, in a domain whose boundaries exhibit fast oscillations in the form $x_2 = \varepsilon^{1+\alpha} \eta(x_1/\varepsilon)$, $\alpha > 0$. Under suita...

In this paper, we develop an abstract framework to establish ill-posedness in the sense of Hadamard for some nonlocal PDEs displaying unbounded unstable spectra. We apply it to prove the ill-posedness for the hydrostatic Euler equations as well as for the kinetic incompressible Euler equations and the Vlasov-Dirac-Benney system.

The aim of this paper is to provide a justification of the Maxwell-Boltzmann approximation of electron density from kinetic models. First, under reasonable regularity assumption, we rigorously derive a reduced kinetic model for the dynamics of ions, while electrons satisfy the Maxwell-Boltzmann relation. Second, we prove that equilibria of the elec...

Consider a system of N particles interacting through Newton’s second law with Coulomb interaction potential in one spatial dimension or a C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt...

We study the instability of solutions to the relativistic Vlasov-Maxwell systems in two limiting regimes: the classical limit when the speed of light tends to infinity and the quasineutral limit when the Debye length tends to zero. First, in the classical limit $\varepsilon \to 0$, with $\varepsilon$ being the inverse of the speed of light, we cons...

We study the Cauchy problem for first-order quasi-linear systems of partial differential equations. When the spectrum of the initial principal symbol is not included in the real line, i.e., when hyperbolicity is violated at initial time, then the Cauchy problem is strongly unstable, in the sense of Hadamard. This phenomenon, which extends the linea...

We establish various criteria, which are known in the incompressible case, for the validity of the inviscid limit for the compressible Navier-Stokes flows considered in a general domain $\Omega$ in $\mathbb{R}^n$ with or without a boundary. In the presence of a boundary, a generalized Navier boundary condition for velocity is assumed, which in part...

We establish the global-in-time existence and uniqueness of classical solutions to the "one and one-half" dimensional relativistic Vlasov--Maxwell systems in a bounded interval, subject to an external magnetic field which is infinitely large at the spatial boundary. We prove that the large external magnetic field confines the particles to a compact...

In this paper we show how the stability of Prandtl boundary layers is linked to the stability of shear flows in the incompressible Navier Stokes equations. We then recall classical physical instability results, and give a short educational presentation of the construction of unstable modes for Orr Sommerfeld equations. We end the paper with a conje...

In this paper, we construct growing modes of the linearized Navier-Stokes equations about generic stationary shear flows of the boundary layer type in a regime of sufficiently large Reynolds number: $R \to \infty$. Notably, the shear profiles are allowed to be linearly stable at the infinite Reynolds number limit, and so the instability presented i...

This paper concerns spectral instability of shear flows in the incompressible Navier-Stokes equations with sufficiently large Reynolds number: $R\to \infty$. It is well-documented in the physical literature, going back to Heisenberg, C.C. Lin, Tollmien, Drazin and Reid, that generic plane shear profiles other than the linear Couette flow are linear...

This short note is to announce our recent results [2,3] which provide a complete mathematical proof of the viscous destabilization phenomenon, pointed out by Heisenberg (1924), C.C. Lin and Tollmien (1940s), among other prominent physicists. Precisely, we construct growing modes of the linearized Navier-Stokes equations about general stationary she...

This paper is a first step toward understanding the effect of toroidal geometry on the rigorous stability theory of plasmas. We consider a collisionless plasma inside a torus, modeled by the relativistic Vlasov-Maxwell system. The surface of the torus is perfectly conducting and it reflects the particles specularly. We provide sharp criteria for th...

In an appropriate moving coordinate frame, source defects are time-periodic solutions to reaction-diffusion equations that are spatially asymptotic to spatially periodic wave trains whose group velocities point away from the core of the defect. In this paper, we rigorously establish nonlinear stability of spectrally stable source defects in the com...

We investigate the influence of the topography on the lake equations which describe the two-dimensional horizontal velocity of a three-dimensional incompressible flow. We show that the lake equations are structurally stable under Hausdorff approximations of the fluid domain and $L^p$ perturbations of the depth. As a byproduct, we obtain the existen...

In this paper we provide sharp criteria for linear stability or instability of equilibria of collisionless plasmas in the presence of boundaries. Specifically, we consider the relativistic Vlasov-Maxwell system with specular reflection at the boundary for the particles and with the perfectly conducting boundary condition for the electromagnetic fie...

This note concerns a nonlinear ill-posedness of the Prandtl equation and an invalidity of asymptotic boundary-layer expansions of incompressible fluid flows near a solid boundary. Our analysis is built upon recent remarkable linear ill-posedness results established by G\'erard-Varet and Dormy [2], and an analysis in Guo and Tice [5]. We show that t...

We study the inviscid limit problem of the incompressible flows in the presence of both impermeable regular boundaries and a hypersurface transversal to the boundary across which the inviscid flow has a discontinuity jump. In the former case, boundary layers have been introduced by Prandtl as correctors near the boundary between the inviscid and vi...

Coherent structures are solutions to reaction-diffusion systems that are time-periodic in an appropriate moving frame and spatially asymptotic at $x=\pm\infty$ to spatially periodic travelling waves. This paper is concerned with sources which are coherent structures for which the group velocities in the far field point away from the core. Sources a...

We establish long-time stability of multi-dimensional noncharacteristic boundary layers of a class of hyperbolic–parabolic systems including the compressible Navier–Stokes equations with inflow [outflow] boundary conditions, under the assumption of strong spectral, or uniform Evans, stability. Evans stability has been verified for small-amplitude l...

In the lines of a recent paper by Gerard-Varet and Dormy, we establish various ill-posedness results for the Prandtl equation. By considering perturbations of stationary shear flows, we show that for some linearizations of the Prandtl equation and some $C^\infty$ initial data, local in time $C^\infty$ solutions do not exist. At the nonlinear level,...

We study nonlinear time-asymptotic stability of small--amplitude planar Lax shocks in a model consisting of a system of multi--dimensional conservation laws coupled with an elliptic system. Such a model can be found in context of dynamics of a gas in presence of radiation. Our main result asserts that the standard uniform Evans stability condition...

Extending investigations of Yarahmadian and Zumbrun in the strictly parabolic case, we study time-asymptotic stability of arbitrary (possibly large) amplitude noncharacteristic boundary layers of a class of hyperbolic–parabolic systems including the Navier–Stokes equations of compressible gas, and magnetohydrodynamics with inflow or outflow boundar...

We establish long-time stability of multidimensional viscous shocks of a general class of symmetric hyperbolic-parabolic systems with variable multiplicities, notably including the equations of compressible magnetohydrodynamics (MHD) in dimensions $d\ge 2$ . This extends the existing result established by Zumbrun for systems with characteristics of...

Extending previous work with Lattanzio and Mascia on the scalar (in fluid-dynamical variables) Hamer model for a radiative gas, we show nonlinear orbital asymptotic stability of small-amplitude shock profiles of general systems of coupled hyperbolic--eliptic equations of the type modeling a radiative gas, that is, systems of conservation laws coupl...

This work establishes nonlinear orbital asymptotic stability of scalar radiative shock profiles, namely, traveling wave solutions to the simplified model system of radiating gas \cite{Hm}, consisting of a scalar conservation law coupled with an elliptic equation for the radiation flux. The method is based on the derivation of pointwise Green functi...

We extend our recent work with K. Zumbrun on long-time stability of multi-dimensional noncharacteristic viscous boundary layers of a class of symmetrizable hyperbolic-parabolic systems. Our main improvements are (i) to establish the stability for a larger class of systems in dimensions $d\ge 2$, yielding the result for certain magnetohydrodynamics...

Building on work of Barker, Humpherys, Lafitte, Rudd, and Zumbrun in the shock wave case, we study stability of compressive, or "shock-like", boundary layers of the isentropic compressible Navier-Stokes equations with gamma-law pressure by a combination of asymptotic ODE estimates and numerical Evans function computations. Our results indicate stab...

A class of cross diffusion parabolic systems given on bounded domains of IR n, with arbitrary n, is investigated. We show that there is a global attractor with finite Hausdorff dimension which attracts all solutions. The result will be applied to the generalized Shigesada, Kawasaki and Teramoto (SKT) model with Lotka-Volterra reactions. In addition...

Boundedness and Hölder regularity of solutions to a class of strongly coupled elliptic systems are investigated. The Hölder estimates for the gradients of solutions are also established. Finally, the fixed point theory is applied to prove existence of positive solution(s) for general cross diffusion elliptic systems.

A class of strongly coupled degenerate parabolic system is considered. Sufficient conditions will be given to show that bounded weak solutions are Hölder continuous everywhere. The general theory will be applied to a generalized porous media type Shigesada-Kawasaki-Teramoto model in population dynamics.

The purpose of this paper is to investigate the dynamics of a class of triangular parabolic systems given on bounded domains of arbitrary dimension. In particular, the existence of global attractors and the persistence property will be established.

A class of triangular parabolic systems given on bounded domains of R n with arbitrary n is investigated. Sufficient conditions on the structure of the systems are found to assure that weak solutions exist globally.

We investigate the existence of a global attractor for a class of triangular cross diffusion systems in domains of any dimension. These systems includes the Shigesada-Kawasaki-Teramoto (SKT) model, which arises in population dynamics and has been studied in two dimensional domains. Our results apply to the (SKT) system when the dimension of the dom...

Computer printout. Thesis (M.S.)--University of Texas at San Antonio, 2006. Includes vita. Includes bibliographical references (leaves 105-109)