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Hydrodynamic Cucker--Smale Model with Normalized Communication Weights and Time Delay
Choi, Young-Pil,Haskovec, Jan Society for Industrial and Applied Mathematics 2019 SIAM journal on mathematical analysis Vol.51 No.3
<P>We study a hydrodynamic Cucker--Smale-type model with time delay in communication and information processing, in which agents interact with each other through normalized communication weights. The model consists of a pressureless Euler system with time-delayed nonlocal alignment forces. We resort to its Lagrangian formulation and prove the existence of its global-in-time classical solutions. Moreover, we derive a sufficient condition for the asymptotic flocking behavior of the solutions. Finally, we show the presence of a critical phenomenon for the Eulerian system posed in the spatially one-/two-dimensional setting.</P>
The Fokker--Planck Equation with Absorbing Boundary Conditions in Bounded Domains
Hwang, Hyung Ju,Jang, Juhi,Jung, Jaewoo Society for Industrial and Applied Mathematics 2018 SIAM journal on mathematical analysis Vol.50 No.2
<P>In this paper, we study the initial-boundary value problem of the Fokker--Planck equation with absorbing boundary conditions in multidimensional bounded domains. First, we establish a well-posedness theory by constructing a solution for the regularized equation and passing to the limit by uniform <TEX>$L^{1}$</TEX> and <TEX>$L^{\infty }$</TEX> estimates. Second, for a general multidimensional domain with a smooth boundary, we show that the solution is smooth far from the singular set and locally Hölder continuous up to the singular set.</P>
Bloch Waves in Bubbly Crystal Near the First Band Gap: A High-Frequency Homogenization Approach
Ammari, Habib,Lee, Hyundae,Zhang, Hai Society for Industrial and Applied Mathematics 2019 SIAM journal on mathematical analysis Vol.51 No.1
<P>This paper is concerned with the high-frequency homogenization of bubbly phononic crystals. It is a follow-up of the work [H. Ammari et al., <italic toggle='yes'>J. Differential Equations</I>, 263 (2017), pp. 5610--5629], which shows the existence of a subwavelength band gap. This phenomena can be explained by the periodic inference of cell resonance which is due to the high contrast in both the density and bulk modulus between the bubbles and the surrounding medium. In this paper, we prove that the first Bloch eigenvalue achieves its maximum at the corner of the Brillouin zone. Moreover, by computing the asymptotic of the Bloch eigenfunctions in the periodic structure near that critical frequency, we demonstrate that these eigenfunctions can be decomposed into two parts: one part is slowly varying and satisfies a homogenized equation, while the other is periodic across each elementary crystal cell and is varying. They rigorously justify, in the nondilute case, the observed superfocusing of acoustic waves in bubbly crystals near and below the maximum of the first Bloch eigenvalue and confirm the band gap opening near and above this critical frequency.</P>