We study the (2+1)-dimensional continuum model for the evolution of stepped epitaxial surface under long-range elastic interaction proposed by Xu and Xiang (SIAM J. Appl. Math. 69 (2009), 1393–1414). The long-range interaction term and the two length scales in this model makes PDE analysis challenging. Moreover, unlike in the (1+1)-dimensional case, there is a nonconvexity contribution in the total energy in the (2+1)-dimensional case, and it is not easy to prove that the solution is always in the well-posed regime during the evolution. In this paper, we propose a modified (2+1)-dimensional continuum model based on the underlying physics. This modification fixes the problem of possible ill-posedness due to the nonconvexity of the energy functional. We prove the existence and uniqueness of both the static and dynamic solutions and derive a minimum energy scaling law for them. We show that the minimum energy surface profile is mainly attained by surfaces with step meandering instability. This is essentially different from the energy scaling law for the (1+1)-dimensional epitaxial surfaces under elastic effects attained by step bunching surface profiles. We also discuss the transition from the step bunching instability to the step meandering instability in (2+1)-dimensions.

We investigate whether the inhibition phenomenon of the Rayleigh-Taylor (RT) instability by a horizontal magnetic field can be mathematically verified for a non-resistive viscous magnetohydrodynamic (MHD) fluid in a two-dimensional (2D) horizontal slab domain. This phenomenon was mathematically analyzed by Wang (J. Math. Phys., 53:073701, 2012) for stratified MHD fluids in the linearized case. To our best knowledge, the mathematical verification of this inhibition phenomenon in the nonlinear case still remains open. In this paper, we prove such inhibition phenomenon for the (nonlinear) inhomogeneous, incompressible, viscous case with Navier (slip) boundary condition. More precisely, we show that there is a critical number of the field strength $m_C,$ such that if the strength $|m|$ of a horizontal magnetic field is bigger than $m_C,$ then the small perturbation solution around the magnetic RT equilibrium state is algebraically stable in time. Moreover, we also provide a nonlinear instability result when $|m|∈[0,m_C).$

The core-shell structure design is an important subject in science and engineering, which also plays a key role in wave scattering and target reconstructions. This work aims to develop a novel boundary integral equation method for solving the acoustic scattering from a 3D core-shell structure in a two-layered lossy medium. The boundary integral equation contains continuous and weakly singular kernels. The well-posedness of the scattering problem is established by combining the integral equation, variational, and operator theory techniques. The study lays the groundwork for future numerical methods for layered obstacles and rough surfaces composite scattering and inverse scattering problems.

Minimax optimization problems are an important class of optimization problems arising from both modern machine learning and from traditional research areas. We focus on the stability of constrained minimax optimization problems based on the notion of local minimax point by Dai and Zhang (2020). Firstly, we extend the classical Jacobian uniqueness conditions of nonlinear programming to the constrained minimax problem and prove that this set of properties is stable with respect to small $\mathcal{C}^2$ perturbation. Secondly, we provide a set of conditions, called Property A, which does not require the strict complementarity condition for the upper level constraints. Finally, we prove that Property A is a sufficient condition for the strong regularity of the Kurash-Kuhn-Tucker (KKT) system at the KKT point, and it is also a sufficient condition for the local Lipschitzian homeomorphism of the Kojima mapping near the KKT point.

The aim of this paper is to analyze the robust convergence of a class of parareal algorithms for solving parabolic problems. The coarse propagator is fixed to the backward Euler method and the fine propagator is a high-order single step integrator. Under some conditions on the fine propagator, we show that there exists some critical $J_∗$ such that the parareal solver converges linearly with a convergence rate near 0.3, provided that the ratio between the coarse time step and fine time step named $J$ satisfies $J ≥ J_∗.$ The convergence is robust even if the problem data is nonsmooth and incompatible with boundary conditions. The qualified methods include all absolutely stable single step methods, whose stability function satisfies $|r(−∞)|<1,$ and hence the fine propagator could be arbitrarily high-order. Moreover, we examine some popular high-order single step methods, e.g., two-, three- and four-stage Lobatto IIIC methods, and verify that the corresponding parareal algorithms converge linearly with a factor 0.31 and the threshold for these cases is $J_∗ = 2.$ Intensive numerical examples are presented to support and complete our theoretical predictions.

We studied linear triblock copolymer and homopolymer mixtures and constructed a new Nakazawa-Ohta-type model to describe the phase separation of the system. For this high-order, nonlocal and multicomponent phase field system, we construct two second-order, linear and energy-stable schemes. We also proved the energy dissipation and mass conservation of the schemes. Finally, numerical simulations were presented in the 3D case, which verified the theoretical analysis. In addition to those reported phase structures, we found some new ones, which can give some guidance to experimenters.

Compared with the real Laplacian eigenvalues of undirected networks, the ones of asymmetrical directed networks might be complex, which is able to trigger additional collective dynamics, including the oscillatory behaviors. However, the high dimensionality of the reaction-diffusion systems defined on directed networks makes it difficult to do in-depth dynamic analysis. In this paper, we strictly derive the Hopf normal form of the general two-species reaction-diffusion systems defined on directed networks, with revealing some noteworthy differences in the derivation process from the corresponding on undirected networks. Applying the obtained theoretical framework, we conduct a rigorous Hopf bifurcation analysis for an SI reaction-diffusion system defined on directed networks, where numerical simulations are well consistent with theoretical analysis. Undoubtedly, our work will provide an important way to study the oscillations in directed networks.