Studies in Applied Mathematics最新文献

The paper is devoted to the dynamical stability of pullback attractors for non-autonomous Lagrangian-averaged Navier–Stokes equations with delays. First, using the Ascoli–Arzelà theorem, we prove the pullback asymptotic compactness of solution operators to establish the existence of pullback attractors. Second, we prove the pointwise upper semicontinuity of pullback attractors as the delay time approaches zero. Eventually, we further investigate their uniform upper semicontinuity when the delay time converges to zero. Combining these two results, the latter strengthens the former. To the best of our knowledge, this paper appears to be the first to address both types of upper semicontinuity for pullback attractors, particularly the interesting uniform case.

We study wave propagation through a one-dimensional array of subwavelength resonators with periodically time-modulated material parameters. Focusing on a high-contrast regime, we use a scattering framework based on Fourier expansions and scattering matrix techniques to capture the interactions between an incident wave and the temporally varying system. This way, we derive a formulation of the total energy flux corresponding to time-dependent systems of resonators. We show that the total energy flux is composed of the transmitted and reflected energy fluxes and derive an optical theorem which characterizes the energy balance of the system. We provide a number of numerical experiments to investigate the impact of the time-dependency, the operating frequency, and the number of resonators on the maximal attainable energy gain and energy loss. Moreover, we show the existence of lasing points, at which the total energy diverges. Our results lay the foundation for the design of energy dissipative or energy amplifying systems.

In this article, we investigate the two-dimensional pressureless Euler equations with three constant Riemann initial data. Our primary focus is on the wave interactions involving contact discontinuities and delta shocks. A distinguishing feature of the solution is the emergence of a delta shock wave which is characterized by a Dirac delta function appearing in both the density and internal energy variables. By exploiting generalized characteristic analysis, nine topologically distinct solution patterns are derived. Some of these configurations exhibit features similar to Mach-reflection and in certain cases, vacuum regions may also develop. To validate the theoretical results, numerical simulations are carried out using a semidiscrete central upwind scheme. The comparison between analytical and numerical results demonstrates excellent agreement, providing deeper insights into the complex dynamics of wave interactions in the pressureless Euler system.

This paper is concerned with the propagation dynamics for a spatial discrete waterborne pathogen model with temporal periodicity. We firstly establish the spreading speed of the model. Then, applying the asymptotic fixed-point theorem via a pair of time-periodic upper and lower solutions, we show that the model admits time-periodic traveling waves when the basic reproduction number is larger than one and the wave speed is greater than a threshold speed. In addition, when the basic reproduction number is either smaller than one; or larger than one but the wave speeds are less than the threshold speed, we further prove the non-existence of time-periodic traveling waves by using the methods of comparison principle. From our result, one can see the minimum wave speed for the time-periodic traveling waves is exactly the same with the spreading speed of the model.

In this paper, we concern with wave interaction for the Chaplygin gas dynamics with a Dirac measure source term, in which the delta standing wave, delta shock wave, contact discontinuity and stationary contact discontinuity are involved. A general framework is provided to investigate the wave interaction problem. Under this framework, we analyze the possible wave patterns in each case, and obtain all of exact solutions which rigorously show the evolution process of various waves. Intriguingly, it is observed that when a contact discontinuity collides with a stationary contact discontinuity, it then passes through the stationary contact discontinuity and evolves a delta shock wave. It is also noticed that when a delta shock wave interacts with a stationary contact discontinuity, it then penetrates the stationary contact discontinuity and bifurcates multiple delta contact discontinuities. What is more, after the interaction between a contact discontinuity and a delta standing wave, they evolve either a delta shock wave followed by a stationary contact discontinuity, or multiple delta contact discontinuities followed by a stationary contact discontinuity. Finally, our technique and results can contribute to the studies on wave interactions for quasilinear hyperbolic systems with a general singular source term.

We study orthogonal polynomials on a fully symmetric planar domain $�\Omega$ that is generated by a certain triangle in the first quadrant. For a family of weight functions on $�\Omega$, we show that orthogonal polynomials that are even in the second variable on $�\Omega$ can be identified with orthogonal polynomials on the unit disk composed with a quadratic map, and the same phenomenon can be extended to the domain generated by the rotation of $�\Omega$ in higher dimensions. The connection allows an immediate deduction of results for approximation and Fourier orthogonal expansions on these fully symmetric domains. It applies, for example, to analysis on a sector of a disk in two variables and the conic domain derived from the rotation of a sector in high dimensions.