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Jacob Bedrossian

Publications and source records attributed to Jacob Bedrossian.

At least 37 records · Page 2Linked to original sources
arXiv2023

Non-invariance of Gaussian Measures under the 2D Euler Flow

In this article we consider the two-dimensional incompressible Euler equations and give a sufficient condition on Gaussian measures of jointly independent Fourier coefficients supported on $H^σ(\mathbb{T}^2)$ ($σ>3$) such that these measures are not invariant (in vorticity form). We show that this condition holds on an open and dense set in suitable topologies (and so is generic in a Baire category sense) and give some explicit examples of Gaussian measures which are not invariant. We also pose a few related conjectures which we believe to be approachable.

math.AP↗
arXiv2022

A brief introduction to the mathematics of Landau damping

In these short, rather informal, expository notes I review the current state of the field regarding the mathematics of Landau damping, based on lectures given at the CIRM Research School on Kinetic Theory, November 14--18, 2022. These notes are mainly on Vlasov-Poisson in $(x,v) \in \mathbb T^d \times \mathbb R^d$ however a brief discussion of the important case of $(x,v) \in \mathbb R^d \times \mathbb R^d$ is included at the end. The focus will be nonlinear and these notes include a proof of Landau damping on $(x,v) \in \mathbb T^d \times \mathbb R^d$ in the Vlasov--Poisson equations meant for graduate students, post-docs, and others to learn the basic ideas of the methods involved. The focus is also on the mathematical side, and so most references are from the mathematical literature with only a small number of the many important physics references included. A few open problems are included at the end. These notes are not currently meant for publication so they may not be perfectly proof-read and the reference list might not be complete. If there is an error or you have some references which you think should be included, feel free to send me an email and I will correct it when I get a chance.

math.AP↗
arXiv2022

Taylor dispersion and phase mixing in the non-cutoff Boltzmann equation on the whole space

In this paper, we describe the long-time behavior of the non-cutoff Boltzmann equation with soft potentials near a global Maxwellian background on the whole space in the weakly collisional limit (i.e. infinite Knudsen number $1/ν\to \infty$). Specifically, we prove that for initial data sufficiently small (independent of the Knudsen number), the solution displays several dynamics caused by the phase mixing/dispersive effects of the transport operator $v \cdot \nabla_x$ and its interplay with the singular collision operator. For $x$-wavenumbers $k$ with $|k|\ggν$, one sees an enhanced dissipation effect wherein the characteristic decay time-scale is accelerated to $O(1/ν^{\frac{1}{1+2s}} |k|^{\frac{2s}{1+2s}})$, where $s \in (0,1]$ is the singularity of the kernel ($s=1$ being the Landau collision operator, which is also included in our analysis); for $|k|\ll ν$, one sees Taylor dispersion, wherein the decay is accelerated to $O(ν/|k|^2)$. Additionally, we prove almost-uniform phase mixing estimates. For macroscopic quantities as the density $ρ$, these bounds imply almost-uniform-in-$ν$ decay of $(t\nabla_x)^βρ$ in $L^\infty_x$ due to Landau damping and dispersive decay.

math.AP↗
arXiv2022

The Vlasov-Poisson and Vlasov-Poisson-Fokker-Planck systems in stochastic electromagnetic fields: local well-posedness

In this paper, we construct unique, local-in-time strong solutions to the Vlasov-Poisson (VP) and Vlasov-Poisson-Fokker-Planck (VPFP) systems subjected to external, spatially regular, white-in-time electromagnetic fields in $\mathbb T^d \times \mathbb R^d$. Initial conditions are taken $H^σ$ with $σ> d/2 + 1$ (in addition to polynomial velocity weights). We additionally show that solutions to the VPFP are instantly $C^\infty_{x,v}$ due to hypoelliptic regularization if the external force fields are smooth. The external forcing arises in the kinetic equation as a stochastic transport in velocity, which means, together with the anisotropy between $x$ and $v$ in the nonlinearity, that the local theory is a little more complicated than comparable fluid mechanics equations subjected to either additive stochastic forcing or stochastic transport. Although stochastic electromagnetic fields are often discussed in the plasma physics literature, to our knowledge, this is the first mathematical study of strong solutions to nonlinear stochastic kinetic equations.

math.AP↗
arXiv2022

Chaos in stochastic 2d Galerkin-Navier-Stokes

We prove that all Galerkin truncations of the 2d stochastic Navier-Stokes equations in vorticity form on any rectangular torus subjected to hypoelliptic, additive stochastic forcing are chaotic at sufficiently small viscosity, provided the frequency truncation satisfies $N\geq 392$. By "chaotic" we mean having a strictly positive Lyapunov exponent, i.e. almost-sure asymptotic exponential growth of the derivative with respect to generic initial conditions. A sufficient condition for such results was derived in previous joint work with Alex Blumenthal which reduces the question to the non-degeneracy of a matrix Lie algebra implying Hörmander's condition for the Markov process lifted to the sphere bundle (projective hypoellipticity). The purpose of this work is to reformulate this condition to be more amenable for Galerkin truncations of PDEs and then to verify this condition using a) a reduction to genericity properties of a diagonal sub-algebra inspired by the root space decomposition of semi-simple Lie algebras and b) computational algebraic geometry executed by Maple in exact rational arithmetic. Note that even though we use a computer assisted proof, the result is valid for all aspect ratios and all sufficiently high dimensional truncations; in fact, certain steps simplify in the formal infinite dimensional limit.

math.PR↗
arXiv2022

A regularity method for lower bounds on the Lyapunov exponent for stochastic differential equations

We put forward a new method for obtaining quantitative lower bounds on the top Lyapunov exponent of stochastic differential equations (SDEs). Our method combines (i) an (apparently new) identity connecting the top Lyapunov exponent to a Fisher information-like functional of the stationary density of the Markov process tracking tangent directions with (ii) a novel, quantitative version of Hörmander's hypoelliptic regularity theory in an $L^1$ framework which estimates this (degenerate) Fisher information from below by a $W^{s,1}_{\mathrm{loc}}$ Sobolev norm. This method is applicable to a wide range of systems beyond the reach of currently existing mathematically rigorous methods. As an initial application, we prove the positivity of the top Lyapunov exponent for a class of weakly-dissipative, weakly forced SDE; in this paper we prove that this class includes the Lorenz 96 model in any dimension, provided the additive stochastic driving is applied to any consecutive pair of modes.

math.DS↗
arXiv2022

Stationary measures for stochastic differential equations with degenerate damping

A variety of physical phenomena involve the nonlinear transfer of energy from weakly damped modes subjected to external forcing to other modes which are more heavily damped. In this work we explore this in (finite-dimensional) stochastic differential equations in $\mathbb R^n$ with a quadratic, conservative nonlinearity $B(x,x)$ and a linear damping term $-Ax$ which is degenerate in the sense that $\mathrm{ker} A \neq \emptyset$. We investigate sufficient conditions to deduce the existence of a stationary measure for the associated Markov semigroups. Existence of such measures is straightforward if $A$ is full rank, but otherwise, energy could potentially accumulate in $\mathrm{ker} A$ and lead to almost-surely unbounded trajectories, making the existence of stationary measures impossible. We give a relatively simple and general sufficient condition based on time-averaged coercivity estimates along trajectories in neighborhoods of $\mathrm{ker} A$ and many examples where such estimates can be made.

math.PR↗
arXiv2022

Vortex filament solutions of the Navier-Stokes equations

We consider solutions of the Navier-Stokes equations in $3d$ with vortex filament initial data of arbitrary circulation, that is, initial vorticity given by a divergence-free vector-valued measure of arbitrary mass supported on a smooth curve. First, we prove global well-posedness for perturbations of the Oseen vortex column in scaling-critical spaces. Second, we prove local well-posedness (in a sense to be made precise) when the filament is a smooth, closed, non-self-intersecting curve. Besides their physical interest, these results are the first to give well-posedness in a neighborhood of large self-similar solutions of $3d$ Navier-Stokes, as well as solutions which are locally approximately self-similar.

math.AP↗
arXiv2021

Encoded Prior Sliced Wasserstein AutoEncoder for learning latent manifold representations

While variational autoencoders have been successful in several tasks, the use of conventional priors are limited in their ability to encode the underlying structure of input data. We introduce an Encoded Prior Sliced Wasserstein AutoEncoder wherein an additional prior-encoder network learns an embedding of the data manifold which preserves topological and geometric properties of the data, thus improving the structure of latent space. The autoencoder and prior-encoder networks are iteratively trained using the Sliced Wasserstein distance. The effectiveness of the learned manifold encoding is explored by traversing latent space through interpolations along geodesics which generate samples that lie on the data manifold and hence are more realistic compared to Euclidean interpolation. To this end, we introduce a graph-based algorithm for exploring the data manifold and interpolating along network-geodesics in latent space by maximizing the density of samples along the path while minimizing total energy. We use the 3D-spiral data to show that the prior encodes the geometry underlying the data unlike conventional autoencoders, and to demonstrate the exploration of the embedded data manifold through the network algorithm. We apply our framework to benchmarked image datasets to demonstrate the advantages of learning data representations in outlier generation, latent structure, and geodesic interpolation.

cs.LG↗
arXiv2021

Nonlinear inviscid damping and shear-buoyancy instability in the two-dimensional Boussinesq equations

We investigate the long-time properties of the two-dimensional inviscid Boussinesq equations near a stably stratified Couette flow, for an initial Gevrey perturbation of size $\varepsilon$. Under the classical Miles-Howard stability condition on the Richardson number, we prove that the system experiences a shear-buoyancy instability: the density variation and velocity undergo an $O(t^{-1/2})$ inviscid damping while the vorticity and density gradient grow as $O(t^{1/2})$. The result holds at least until the natural, nonlinear timescale $t \approx \varepsilon^{-2}$. Notice that the density behaves very differently from a passive scalar, as can be seen from the inviscid damping and slower gradient growth. The proof relies on several ingredients: (A) a suitable symmetrization that makes the linear terms amenable to energy methods and takes into account the classical Miles-Howard spectral stability condition; (B) a variation of the Fourier time-dependent energy method introduced for the inviscid, homogeneous Couette flow problem developed on a toy model adapted to the Boussinesq equations, i.e. tracking the potential nonlinear echo chains in the symmetrized variables despite the vorticity growth.

math.AP↗
arXiv2020

The Batchelor spectrum of passive scalar turbulence in stochastic fluid mechanics at fixed Reynolds number

In 1959, Batchelor predicted that the stationary statistics of passive scalars advected in fluids with small diffusivity $κ$ should display a $|k|^{-1}$ power spectrum along an inertial range contained in the viscous-convective range of the fluid model. This prediction has been extensively tested, both experimentally and numerically, and is a core prediction of passive scalar turbulence. In this article we provide a rigorous proof of a version of Batchelor's prediction in the $κ\to 0$ limit when the scalar is subjected to a spatially-smooth, white-in-time stochastic source and is advected by the 2D Navier-Stokes equations or 3D hyperviscous Navier-Stokes equations in $\mathbb{T}^d$ forced by sufficiently regular, nondegenerate stochastic forcing. Although our results hold for fluids at arbitrary Reynolds number, this value is fixed throughout. Our results rely on the quantitative understanding of Lagrangian chaos and passive scalar mixing established in our recent works. Additionally, in the $κ\to 0$ limit, we obtain statistically stationary, weak solutions in $H^{-ε}$ to the stochastically-forced advection problem without diffusivity. These solutions are almost-surely not locally integrable distributions with non-vanishing average anomalous flux and satisfy the Batchelor spectrum at all sufficiently small scales. We also prove an Onsager-type criticality result which shows that no such dissipative, weak solutions with a little more regularity can exist.

math.AP↗
arXiv2020

Uniqueness Criteria for the Oseen Vortex in the 3d Navier-Stokes Equations

In this paper, we consider the uniqueness of solutions to the 3d Navier-Stokes equations with initial vorticity given by $ω_0 = αe_z δ_{x = y = 0}$, where $δ_{x=y= 0}$ is the one dimensional Hausdorff measure of an infinite, vertical line and $α\in \mathbb R$ is an arbitrary circulation. This initial data corresponds to an idealized, infinite vortex filament. One smooth, mild solution is given by the self-similar Oseen vortex column, which coincides with the heat evolution. Previous work by Germain, Harrop-Griffiths, and the first author implies that this solution is unique within a class of mild solutions that converge to the Oseen vortex in suitable self-similar weighted spaces. In this paper, the uniqueness class of the Oseen vortex is expanded to include any solution that converges to the initial data in a sufficiently strong sense. This gives further evidence in support of the expectation that the Oseen vortex is the only possible mild solution that is identifiable as a vortex filament. The proof is a 3d variation of a 2d compactness/rigidity argument in $t \searrow 0$ originally due to Gallagher and Gallay.

math.AP↗
arXiv2020

Quantitative spectral gaps and uniform lower bounds in the small noise limit for Markov semigroups generated by hypoelliptic stochastic differential equations

We study the convergence rate to equilibrium for a family of Markov semigroups $\{\mathcal{P}_t^ε\}_{ε> 0}$ generated by a class of hypoelliptic stochastic differential equations on $\mathbb{R}^d$, including Galerkin truncations of the incompressible Navier-Stokes equations, Lorenz-96, and the shell model SABRA. In the regime of vanishing, balanced noise and dissipation, we obtain a sharp (in terms of scaling) quantitative estimate on the exponential convergence in terms of the small parameter $ε$. By scaling, this regime implies corresponding optimal results both for fixed dissipation and large noise limits or fixed noise and vanishing dissipation limits. As part of the proof, and of independent interest, we obtain uniform-in-$ε$ upper and lower bounds on the density of the stationary measure. Upper bounds are obtained by a hypoelliptic Moser iteration, the lower bounds by a de Giorgi-type iteration (both uniform in $ε$). The spectral gap estimate on the semigroup is obtained by a weak Poincaré inequality argument combined with quantitative hypoelliptic regularization of the time-dependent problem.

math.AP↗
arXiv2020

Suppression of plasma echoes and Landau damping in Sobolev spaces by weak collisions in a Vlasov-Fokker-Planck equation

In this paper, we study Landau damping in the weakly collisional limit of a Vlasov-Fokker-Planck equation with nonlinear collisions in the phase-space $(x,v) \in \mathbb T_x^n \times \mathbb R^n_v$. The goal is four-fold: (A) to understand how collisions suppress plasma echoes and enable Landau damping in agreement with linearized theory in Sobolev spaces, (B) to understand how phase mixing accelerates collisional relaxation, (C) to understand better how the plasma returns to global equilibrium during Landau damping, and (D) to rule out that collision-driven nonlinear instabilities dominate. We give an estimate for the scaling law between Knudsen number and the maximal size of the perturbation necessary for linear theory to be accurate in Sobolev regularity. We conjecture this scaling to be sharp (up to logarithmic corrections) due to potential nonlinear echoes in the collisionless model.

math.AP↗
arXiv2020

Linearized wave-damping structure of Vlasov-Poisson in $\mathbb R^3$

In this paper we study the linearized Vlasov-Poisson equation for localized disturbances of an infinite, homogeneous Maxwellian background distribution in $\mathbb R^3_x \times \mathbb R^3_v$. In contrast with the confined case $\mathbb T^d _x \times \mathbb R_v ^d$, or the unconfined case $\mathbb R^d_x \times \mathbb R^d_v$ with screening, the dynamics of the disturbance are not scattering towards free transport as $t \to \pm \infty$: we show that the electric field decomposes into a very weakly-damped Klein-Gordon-type evolution for long waves and a Landau-damped evolution. The Klein-Gordon-type waves solve, to leading order, the compressible Euler-Poisson equations linearized about a constant density state, despite the fact that our model is collisionless, i.e. there is no trend to local or global thermalization of the distribution function in strong topologies. We prove dispersive estimates on the Klein-Gordon part of the dynamics. The Landau damping part of the electric field decays faster than free transport at low frequencies and damps as in the confined case at high frequencies; in fact, it decays at the same rate as in the screened case. As such, neither contribution to the electric field behaves as in the vacuum case.

math.AP↗
arXiv2019

Almost-sure enhanced dissipation and uniform-in-diffusivity exponential mixing for advection-diffusion by stochastic Navier-Stokes

We study the mixing and dissipation properties of the advection-diffusion equation with diffusivity $0 < κ\ll 1$ and advection by a class of random velocity fields on $\mathbb T^d$, $d=\{2,3\}$, including solutions of the 2D Navier-Stokes equations forced by sufficiently regular-in-space, non-degenerate white-in-time noise. We prove that the solution almost surely mixes exponentially fast uniformly in the diffusivity $κ$. Namely, that there is a deterministic, exponential rate (independent of $κ$) such that all mean-zero $H^1$ initial data decays exponentially fast in $H^{-1}$ at this rate with probability one. This implies almost-sure enhanced dissipation in $L^2$. Specifically that there is a deterministic, uniform-in-$κ$, exponential decay in $L^2$ after time $t \gtrsim |\log κ|$. Both the $O(|\log κ|)$ time-scale and the uniform-in-$κ$ exponential mixing are optimal for Lipschitz velocity fields and, to our knowledge, are the first rigorous examples of velocity fields satisfying these properties (deterministic or stochastic). This work is also a major step in our program on scalar mixing and Lagrangian chaos necessary for a rigorous proof of the Batchelor power spectrum of passive scalar turbulence.

math.AP↗
arXiv2019

Inviscid damping and enhanced dissipation of the boundary layer for 2D Navier-Stokes linearized around Couette flow in a channel

We study the 2D Navier-Stokes equations linearized around the Couette flow $(y,0)^t$ in the periodic channel $\mathbb T \times [-1,1]$ with no-slip boundary conditions in the vanishing viscosity $ν\to 0$ limit. We split the vorticity evolution into the free evolution (without a boundary) and a boundary corrector that is exponentially localized to at most an $O(ν^{1/3})$ boundary layer. If the initial vorticity perturbation is supported away from the boundary, we show inviscid damping of both the velocity and the vorticity associated to the boundary layer. For example, our $L^2_t L^1_y$ estimate of the boundary layer vorticity is independent of $ν$, provided the initial data is $H^1$. For $L^2$ data, the loss is only logarithmic in $ν$. Note both such estimates are false for the vorticity in the interior. To the authors' knowledge, this inviscid decay of the boundary layer vorticity seems to be a new observation not previously isolated in the literature. Both velocity and vorticity satisfy the expected $O(\exp(-δν^{1/3}α^{2/3}t))$ enhanced dissipation in addition to the inviscid damping. Similar, but slightly weaker, results are obtained also for $H^1$ data that is against the boundary initially. For $L^2$ data against the boundary, we at least obtain the boundary layer localization and enhanced dissipation.

math.AP↗