Studies in Applied Mathematics最新文献

Based on the development in dealing with nonlocal boundary conditions, we propose a seamless local-nonlocal coupling diffusion model in this paper. In our model, a finite constant interaction horizon is equipped in the nonlocal part and transmission conditions are imposed on a co-dimension one interface. To achieve a seamless coupling, we introduce an auxiliary function to merge the nonlocal model with the local part and design a proper coupling transmission condition to ensure the stability and convergence. In addition, by introducing bilinear form, well-posedness of the proposed model can be proved and convergence to a standard elliptic transmission model with first order in $�H^1$ norm can be derived.

Subwavelength resonance is a vital acoustic phenomenon in contrasting media. The narrow bandgap width of single-layered resonator has prompted the exploration of multi-layered metamaterials as an effective alternative, which consist of alternating nests of high-contrast materials, called “resonators”, and a background media. In this paper, we develop a general mathematical framework for studying acoustics within multi-layered high-contrast structures. First, by using layer potential techniques, we establish the representation formula in terms of a matrix type operator with a block tridiagonal form for multi-layered structures within general geometry. Then, we prove the existence of subwavelength resonances via the Gohberg–Sigal theory, which generalizes the celebrated Minnaert resonances in single-layered structures. Intriguingly, we find that the primary contribution to mode splitting lies in the fact that as the number of nested resonators increases, the degree of the corresponding characteristic polynomial also increases, while the type of resonance (consists solely of monopolar resonances) remains unchanged. Furthermore, we derive original formulas for the subwavelength resonance frequencies of concentric dual-resonator. Numerical results for different nested resonators are presented to corroborate the theoretical findings.

We propose two novel multiagent systems characterized by discontinuous vector fields, designed to track a given moving or stationary target while ensuring collision avoidance. Unlike traditional approaches that employ continuous vector fields, we adopt Filippov's differential inclusion framework to model the dynamics at discontinuity points. For each model, we present an admissible set of initial conditions, system parameters, kernel functions, and a target configuration that guarantees both qualitative and quantitative asymptotic tracking of a prescribed target while avoiding collisions.

We investigate the dynamics of the modified Korteweg-de Vries (mKdV) equation from the perspective of the weak Kolmogorov-Arnold-Moser (KAM) theory, and obtain that the associated Hamiltonian system admits a weak KAM solution. This solution satisfies the Hamilton–Jacobi equation in the sense of the minimal measure, which allows the system to be reduced to a class of integrable Hamiltonian systems. The minimal measure is closely related to the Aubry–Mather theory. Consequently, we establish the existence of a weak solution of the perturbed mKdV equation, despite the presence of system resonance. Finally, we extend this method to quasi-periodic perturbed mKdV equations and coupled mKdV equations, proving the existence of weak KAM solutions.

This work studies the global well-posedness of the Oldroyd-B model with anisotropic viscosity. While global existence and uniqueness of strong solutions for the fully dissipative Oldroyd-B model were established in Constantin–Kliegl [Archive for Rational Mechanics and Analysis, 206 (2012): 725–740], under $��H^2��\left(�R^2�\right)��$ initial data, the horizontally viscous counterpart—where dissipation of velocity acts only along the horizontal direction—remains unexplored. We establish the global existence and uniqueness of strong solutions to the initial-boundary value problem for the horizontally viscous Oldroyd-B model with arbitrary large initial data in the vertical strip domain $��[�0�,�1�]�\times �R�$ and the periodic channel $��[�0�,�1�]�\times �T�$, where $�T$ represents the one-dimensional torus. To the best of our knowledge, this work provides the first rigorous proof of global-in-time existence and uniqueness of strong solutions for the partially dissipative Oldroyd-B model. In addition, we investigate the long-time asymptotic behavior of solutions for small initial data in the periodic channel $��[�0�,�1�]�\times �T�$. Our results extend the current analytical understanding of visco