This paper investigates the linear stability of the 2D Boussinesq equations under vertical or full dissipation near Couette flow. We establish exponential decay estimates for vorticity and temperature perturbations as well as their y-derivatives, along with hypoelliptic regularization results in the x-variable. This result reveals that all nontrivial horizontal Fourier modes decay exponentially. Furthermore, we extend the decay exponents presented in Tao et al. (J. Differ. Equ., 267: 1731-1747, 2019) to more general cases.
We construct adaptive Matérn radial basis function (MA-RBF) modifications of the classical two-step Adams–Bashforth and one-step Adams–Moulton formulas for first-order initial value problems (IVPs). The proposed schemes are based on a C4-smooth Matérn kernel and recover the corresponding classical Adams formulas in the flat limit, when the shape parameter tends to zero. Taylor expansions of the local truncation errors yield explicit adaptive values, used in place of the squared shape parameter, that raise the order of both formulas from two to three. The resulting MA-RBF Adams–Bashforth formula and Adams–Bashforth–Moulton predictor–corrector scheme are tested on two nonlinear benchmark IVPs. The numerical results confirm the predicted third-order convergence and show accuracy gains over the corresponding classical Adams schemes.
We construct a Bogovskii-operator for a heterogeneous domain consisting of two bulk regions separated by a thin perforated layer with thickness and periodicity of order ɛ. In particular, we consider various types of boundary conditions at the outer boundary of the domain and determine the explicit dependence of the operator’s norm on ɛ. Finally, we apply this result to establish uniform a priori bounds for the pressure and then derive the effective problem for the corresponding microscopic Stokes model.
General higher-order rogue wave solutions are derived for the parity-time-symmetric (2+1)-dimensional nonlocal nonlinear Schrödinger equation. By combining the Hirota’s bilinear method with the Kadomtsev–Petviashvili hierarchy reduction technique, explicit rogue wave solutions are constructed in terms of Schur polynomials. Their dynamical behaviors are systematically investigated. Furthermore, the formation mechanism of rogue wave patterns with fixed y=y0 under internal parameters is explored. The outer region of the generated wave patterns consists of multiple fundamental rogue waves, whose configurations are governed by the parameters α2m+1 and β2m+1. While lower-order rogue waves tend to emerge in the central region.
We prove a nonrelativistic limit theorem for an electrostatic Klein–Gordon–Poisson system in three spatial dimensions. Within a finite-energy weak-solution framework, we show that the second-order time correction vanishes as the relativistic parameter tends to zero, and that the limiting dynamics is governed by the Schrödinger–Poisson system. The main contribution of this note is a combined hyperbolic–elliptic energy structure, which provides simultaneous control of the Klein–Gordon component and the self-consistent Poisson field. This energy mechanism yields the compactness needed to pass to the weak limit in the nonlocal coupling and justifies the nonrelativistic limit without using an explicit Newtonian potential representation.
Reconstructing multipolar acoustic sources from far-field data is a highly ill-posed inverse problem when the count, locations, types, and intensities are all unknown. We propose a physics-guided deep learning framework that fully reconstructs the sources at a fixed frequency, using four-channel Direct Sampling Method (DSM) indicator functions as physics-informed inputs within a divide-and-conquer pipeline. Numerical experiments demonstrate accuracy, robustness to noise, and flexibility.
In this study, the generalized finite difference method (GFDM) is applied for the first time to anti-plane magneto-electro-elastic (MEE) problems involving inclusions. In the GFDM framework, partial derivatives at each node are approximated by linear combinations of the variable values at surrounding scattered nodes within a local influence domain. This meshless approach, which uses only nodal values for derivative approximation, simplifies the numerical treatment of the considered multi-field coupling problems. To handle the discontinuity in material properties across the interfaces, a domain decomposition technique (DDT) is employed for problems involving inclusions. The accuracy and effectiveness of the proposed method are verified through numerical examples. The results show that the GFDM yields accurate solutions for the problems under consideration and remains robust even in extreme cases, such as when the material parameters of the matrix and inclusions differ by a factor of 100.
This paper is concerned with the existence of infinitely many normalized solutions to the following Schrödinger equation −Δu+V(x)u+λu=ulogu2,x∈RN,∫RN|u|2dx=a,where the potential V(x)∈C(RN,R) and lim|x|→∞V(x)=+∞. The main difficulties come from the fact that the corresponding energy functional is not of class C1 on H1(RN). We overcome the difficulties by constructing a new, well-behaved function to optimize the nonlinear term. And we consider problems that can be set in more general space, which does not require the following assumption ∫RNu2|logu2|dx<∞.To the best of our knowledge, this paper is the first to establish the existence of infinitely many normalized solutions to the logarithmic nonlinear Schrödinger equation in this more general space.
A novel mixed spectral-Galerkin method based on generalized ball polynomials is proposed for solving the biharmonic equation on the unit ball, with the fourth-order problem decoupled into a system of second-order equations. The corresponding discrete scheme yields a strictly diagonal stiffness matrix, which significantly enhances the computational efficiency. Rigorous a priori error estimates are established in both the L2- and H1-norms. A numerical experiment confirms the high efficiency and accuracy of the proposed scheme.
We study the low Mach number limit of the 3D axisymmetric isentropic compressible MHD system with viscosity but without magnetic diffusion in R3. The initial data are large and axisymmetric: the velocity has only radial and axial components, while the magnetic field is purely azimuthal. Using a cylindrical-coordinate decomposition, high-order energy estimates and the global strong solution of the corresponding axisymmetric incompressible non-resistive MHD system (Lei, 2015), we prove that for any finite T>0 and sufficiently small Mach number ϵ (with threshold depending on T), the compressible strong solution exists on [0,T] and converges to the incompressible one with a convergence rate of ϵ1+β in the sense of squared energy norms. A key difficulty is the coordinate singularity (1/r2 terms), which is resolved by desingularized perturbation variables.
In this paper, we prove the uniform boundedness and global existence of solutions to an HIV model with CTL immune response in the case of weak chemotaxis. The proof relies on a combination of energy estimates and the self-mapping approach.
This paper develops a balanced-norm supercloseness analysis for the first-order system Petrov–Galerkin finite element discretization of singularly perturbed reaction–diffusion problems on Bakhvalov-type meshes. By introducing auxiliary variables and a curl constraint, the original problem is reformulated as a first-order system, leading to a variational formulation that is coercive and continuous in the balanced norm. Although Bakhvalov-type meshes feature smoother grading in layer regions, their non-uniform transition points introduce fundamental difficulties to the theoretical analysis. To address this crucial challenge, the main contribution of this work is to construct a novel interpolation operator tailored to the structural features of boundary and corner layers. Based on this operator, we establish the robust optimal-order balanced-norm supercloseness result that is uniformly valid with respect to the singular perturbation parameter.
This paper presents a temporal semi-discrete scheme for a nonstationary Stokes hemivariational inequality. To approximate the time derivative, a mid-point discretization is used in the first time step, and the BDF2 for the other time steps. Under appropriate solution regularity assumptions, the scheme is shown to be of second order with respect to the time step-size.
