An official website of the United States government
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
On the stabilizing effect of rotation in the 3d Euler equations
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
- Award ID(s):
- 2106650
- PAR ID:
- 10442134
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Communications on Pure and Applied Mathematics
- Volume:
- 76
- Issue:
- 12
- ISSN:
- 0010-3640
- Format(s):
- Medium: X Size: p. 3553-3641
- Size(s):
- p. 3553-3641
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
This research focuses on studying the scattering phenomenon. Scattering electromagnetic waves from a rotating conducting cylinder is investigated when the material of the conducting cylinder is linear, homogeneous, isotropic, and dispersive. This study is an extension of a previous work that investigated the effect of the rotating conducting cylinder on the scattered phase and amplitude, when the material of the conducting cylinder is linear, homogeneous, isotropic, and nondispersive. One of the important result of the previous work is that the Franklin transformation is a proper and more accurate method to calculate the effect of the rotation, and gives more accurate results than Galilean transformation. In this research, the Franklin transformation will be used to investigate the effect of the rotation of the object on the scattered phase and magnitude of the incident waves. The two types of incident waves (E-wave and H-wave) will be considered herein. The simulation results will clearly display the behavior of the scattered phase and magnitude with changes to the incident frequency, the speed of rotation, and the radius of the very good conducting cylinder. Moreover, this result is compared with the result of the previous work (non- dispersive material) to show the behavior of the scattered phase and magnitude when the incident frequency, speed of the rotation and radius of the very good conducting cylinder is changed.more » « less
-
Abstract We construct a class of global, dynamical solutions to the 3 d Euler equations near the stationary state given by uniform “rigid body” rotation. These solutions are axisymmetric, of Sobolev regularity, have non-vanishing swirl and scatter linearly, thanks to the dispersive effect induced by the rotation. To establish this, we introduce a framework that builds on the symmetries of the problem and precisely captures the anisotropic, dispersive mechanism due to rotation. This enables a fine analysis of the geometry of nonlinear interactions and allows us to propagate sharp decay bounds, which is crucial for the construction of global Euler flows.more » « less
-
Abstract Long time dynamics of the smoothed step initial value problem or dispersive Riemann problem for the Benjamin‐Bona‐Mahony (BBM) equationare studied using asymptotic methods and numerical simulations. The catalog of solutions of the dispersive Riemann problem for the BBM equation is much richer than for the related, integrable, Korteweg‐de Vries equation. The transition width of the initial smoothed step is found to significantly impact the dynamics. Narrow width gives rise to rarefaction and dispersive shock wave (DSW) solutions that are accompanied by the generation of two‐phase linear wavetrains, solitary wave shedding, and expansion shocks. Both narrow and broad initial widths give rise to two‐phase nonlinear wavetrains or DSW implosion and a new kind of dispersive Lax shock for symmetric data. The dispersive Lax shock is described by an approximate self‐similar solution of the BBM equation whose limit asis a stationary, discontinuous weak solution. By introducing a slight asymmetry in the data for the dispersive Lax shock, the generation of an incoherent solitary wavetrain is observed. Further asymmetry leads to the DSW implosion regime that is effectively described by a pair of coupled nonlinear Schrödinger equations. The complex interplay between nonlocality, nonlinearity, and dispersion in the BBM equation underlies the rich variety of nonclassical dispersive hydrodynamic solutions to the dispersive Riemann problem.more » « less
-
Abstract The nonlinear mechanical responses of rocks and soils to seismic waves play an important role in earthquake physics, influencing ground motion from source to site. Continuous geophysical monitoring, such as ambient noise interferometry, has revealed co‐seismic wave speed reductions extending tens of kilometers from earthquake sources. However, the mechanisms governing these changes remain challenging to model, especially at regional scales. Using a nonlinear damage model constrained by laboratory experiments, we develop and apply an open‐source 3D discontinuous Galerkin method to simulate regional co‐seismic wave speed changes during the 2015 Mw7.8 Gorkha earthquake. We find pronounced spatial variations of co‐seismic wave speed reduction, ranging from <0.01% to >50%, particularly close to the source and within the Kathmandu Basin, while disagreement with observations remains. The most significant reduction occurs within the sedimentary basin and varies with basin depths, whereas wave speed reductions correlate with the fault slip distribution near the source. By comparing ground motions from simulations with elastic, viscoelastic, elastoplastic, and nonlinear damage rheologies, we demonstrate that the nonlinear damage model effectively captures low‐frequency ground motion amplification due to strain‐dependent wave speed reductions in soft sediments. We verify the accuracy of our approach through comparisons with analytical solutions and assess its scalability on high‐performance computing systems. The model shows near‐linear strong and weak scaling up to 2,048 nodes, enabling efficient large‐scale simulations. Our findings provide a physics‐based framework to quantify nonlinear earthquake effects and emphasize the importance of damage‐induced wave speed variations for seismic hazard assessment and ground motion predictions.more » « less
