An official website of the United States government
Except in the trivial case of spatially uniform flow, the advection–diffusion operator of a passive scalar tracer is linear and non-self-adjoint. In this study, we exploit the linearity of the governing equation and present an analytical eigenfunction approach for computing solutions to the advection–diffusion equation in two dimensions given arbitrary initial conditions, and when the advecting flow field at any given time is a plane parallel shear flow. Our analysis illuminates the specific role that the non-self-adjointness of the linear operator plays in the solution behaviour, and highlights the multiscale nature of the scalar mixing problem given the explicit dependence of the eigenvalue–eigenfunction pairs on a multiscale parameter$$q=2{\rm i}k\,{\textit {Pe}}$$, where$$k$$is the non-dimensional wavenumber of the tracer in the streamwise direction, and$${\textit {Pe}}$$is the Péclet number. We complement our theoretical discussion on the spectra of the operator by computing solutions and analysing the effect of shear flow width on the scale-dependent scalar decay of tracer variance, and characterize the distinct self-similar dispersive processes that arise from the shear flow dispersion of an arbitrarily compact tracer concentration. Finally, we discuss limitations of the present approach and future directions.
more »
« less
Taylor dispersion and phase mixing in the non‐cutoff Boltzmann equation on the whole space
Abstract 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 (that is, infinite Knudsen number ). 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 and its interplay with the singular collision operator. For ‐wavenumbers with , one sees anenhanced dissipationeffect wherein the characteristic decay time‐scale is accelerated to , where is the singularity of the kernel ( being the Landau collision operator, which is also included in our analysis); for , one seesTaylor dispersion, wherein the decay time‐scale is accelerated to . Additionally, we prove almost uniform phase mixing estimates. For macroscopic quantities such as the density , these bounds imply almost uniform‐in‐ decay of in due to phase mixing and dispersive decay.
more »
« less
- Award ID(s):
- 2348453
- PAR ID:
- 10532050
- Publisher / Repository:
- Wiley
- Date Published:
- Journal Name:
- Proceedings of the London Mathematical Society
- Volume:
- 129
- Issue:
- 1
- ISSN:
- 0024-6115
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
We establish dispersive estimates and local decay estimates for the time evolution of non-self-adjoint matrix Schrödinger operators with threshold resonances in one space dimension. In particular, we show that the decay rates in the weighted setting are the same as in the regular case after subtracting a finite rank operator corresponding to the threshold resonances. Such matrix Schrödinger operators naturally arise from linearizing a focusing nonlinear Schrödinger equation around a solitary wave. It is known that the linearized operator for the 1D focusing cubic NLS equation exhibits a threshold resonance. We also include an observation of a favorable structure in the quadratic nonlinearity of the evolution equation for perturbations of solitary waves of the 1D focusing cubic NLS equation.more » « less
-
Abstract While it is well known that constant rotation induces linear dispersive effects in various fluid models, we study here its effect on long time nonlinear dynamics in the inviscid setting. More precisely, we investigate stability in the 3d rotating Euler equations in with afixedspeed of rotation. We show that for any , axisymmetric initial data of sufficiently small size ε lead to solutions that exist for a long time at least and disperse. This is a manifestation of the stabilizing effect of rotation, regardless of its speed. To achieve this we develop an anisotropic framework that naturally builds on the available symmetries. This allows for a precise quantification and control of the geometry of nonlinear interactions, while at the same time giving enough information to obtain dispersive decay via adapted linear dispersive estimates.more » « less
-
Abstract We investigate the unbiased model for money exchanges: agents give at random time a dollar to one another (if they have one). Surprisingly, this dynamics eventually leads to a geometric distribution of wealth (shown empirically by Dragulescu and Yakovenko, and rigorously by several follow-up papers). We prove a uniform-in-time propagation of chaos result as the number of agents goes to infinity, which links the stochastic dynamics to a deterministic infinite system of ordinary differential equations. This deterministic description is then analyzed by taking advantage of several entropy–entropy dissipation inequalities and we provide a quantitative almost-exponential rate of convergence toward the equilibrium (geometric distribution) in relative entropy.more » « less
-
PurposeTo develop a physics‐guided deep learning (PG‐DL) reconstruction strategy based on a signal intensity informed multi‐coil (SIIM) encoding operator for highly‐accelerated simultaneous multislice (SMS) myocardial perfusion cardiac MRI (CMR). MethodsFirst‐pass perfusion CMR acquires highly‐accelerated images with dynamically varying signal intensity/SNR following the administration of a gadolinium‐based contrast agent. Thus, using PG‐DL reconstruction with a conventional multi‐coil encoding operator leads to analogous signal intensity variations across different time‐frames at the network output, creating difficulties in generalization for varying SNR levels. We propose to use a SIIM encoding operator to capture the signal intensity/SNR variations across time‐frames in a reformulated encoding operator. This leads to a more uniform/flat contrast at the output of the PG‐DL network, facilitating generalizability across time‐frames. PG‐DL reconstruction with the proposed SIIM encoding operator is compared to PG‐DL with conventional encoding operator, split slice‐GRAPPA, locally low‐rank (LLR) regularized reconstruction, low‐rank plus sparse (L + S) reconstruction, and regularized ROCK‐SPIRiT. ResultsResults on highly accelerated free‐breathing first pass myocardial perfusion CMR at three‐fold SMS and four‐fold in‐plane acceleration show that the proposed method improves upon the reconstruction methods use for comparison. Substantial noise reduction is achieved compared to split slice‐GRAPPA, and aliasing artifacts reduction compared to LLR regularized reconstruction, L + S reconstruction and PG‐DL with conventional encoding. Furthermore, a qualitative reader study indicated that proposed method outperformed all methods. ConclusionPG‐DL reconstruction with the proposed SIIM encoding operator improves generalization across different time‐frames /SNRs in highly accelerated perfusion CMR.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1112/plms.12616
