In the paper, we consider the penalty finite element methods (FEMs) for the stationary Smagorinsky model. Firstly, a one-grid penalty FEM is proposed and analyzed. Since this method is nonlinear, a novel linearized iteration scheme is derived for solving it. We also derived the stability and convergence of numerical solutions for this iteration scheme. Furthermore, a two-grid penalty FEM is developed for Smagorinsky model. Under $\varepsilon<

In this paper, we study the stabilization of a coupled wave system formed by one localized non-regular fractional viscoelastic damping of Kelvin-Voigt type and localized non-smooth coefficients. Our main aim is to prove that the C0-semigroup associated with this model is strong stability and decays polynomially at a rate of t−1. By introducing a new system to deal with fractional Kelvin-Voigt damping, we obtain a new equivalent augmented system, so as to show the well-posedness of the system based on Lumer-Phillips theorem. We achieve the strong stability for the C0-semigroup associated with this new model by using a general criteria of Arendt-Batty, and then turn out a polynomial energy decay rate of order t−1 with the help of a frequency domain approach.

In this paper, we give a new higher dimensional Hermite-Hadamard inequality for a function $f:\prod\limits_{i=1}^n[a_i,b_i]\subset\mathbb{R}^n\rightarrow\mathbb{R}$ which is semiconvex of rate $(k_1,k_2,…,k_n)$ on the co-ordinates. This generalizes some existing results on Hermite-Hadamard inequalities of S.S. Dragomir. In addition, we explain the Hermite-Hadamard inequality from the point of view of optimal mass transportation with cost function $c(x,y):=f(y-x)+\sum_{i=1}^n\frac{k_i}{2}\big|x_i-y_i\big|^2$, where $f(\cdot):\prod\limits_{i=1}^n[a_i,b_i]\rightarrow[0,\infty)$ is semiconvex of rate $(k_1,k_2,…,k_n)$ on the co-ordinates and $x=(x_1,x_2,…,x_n)$, $y=(y_1,y_2,…,y_n)\in\prod\limits_{i=1}^n[a_i,b_i]$. Furthermore, by using the higher dimensional Hermite-Hadamard inequality, we compare the transport cost in different transport models on the sphere $\mathbb{S}^2$.

In this paper, we prove the weighted Lorentz and weighted Orlicz estimates for the weak solutions to the higher order parabolic systems with the leading coefficients satisfying a small BMO norm condition. As a byproduct, we obtain the weighted estimates for the higher order elliptic systems.

The motion of plane curves specified by hyperbolic inverse mean curvature with a constant force term is considered. We proved that this flow remains the convexity for any forced term. Furthermore, we give an example to understand how the constant forced term $c$ affects this hyperbolic inverse mean flow. Particularly, the asymptotic behavior of the flow under different initial conditions is discussed.

In this paper, we propose an algorithm for solving a two-point boundary-value problem for a linear differential equation with constant delay subject to a nonlinear boundary condition. We derive sufficient conditions for the convergence of the algorithm and for the existence of an isolated solution to the problem under study. A numerical example is provided.

A general approach is developed for discriminating strong and hereditary symmetric operators. The recursion operator of the Blaszak-Marciniak (BM) equation hierarchy is proved to be strong and hereditary symmetric. As an example of discrete soliton equations related to 3×3 matrix spectral problems, the τ-symmetries and Lie algebra structure of the BM equation are built firstly.

Consider the nonautonomous semilinear evolution equation of type: $(\star) \; u’(t)=A(t)u(t)+f(t,u(t)), \; t \in \mathbb{R},$ where $ A(t), \ t\in \mathbb{R} $ is a family of closed linear operators on a Banach space $X$, the nonlinear term $f$, acting on some real interpolation spaces, is assumed to be almost periodic only in a weak sense (i.e. in Stepanov sense) with respect to $t$ and Lipschitzian in bounded sets with respect to the second variable. We prove the existence and uniqueness of positive almost periodic solutions in the strong sense (Bohr sense) for equation $ (\star) $ using the exponential dichotomy approach. Then, we establish a new composition result of Stepanov almost periodic functions by assuming only the continuity of $f$ in the second variable. Moreover, we provide an application to a nonautonomous system of reaction–diffusion equations describing a Lotka–Volterra predator–prey model with diffusion and time–dependent parameters in a generalized almost periodic environment.

The subject of this research is the time-domain scattering problem for a three-dimensional layered elastic shell submerged in a two-layered fluid separated by an unbounded rough surface. The essence of the problem is to model the scattering interaction between the given elastic shell and some incident wave in a two-layered medium. Using the exact transparent boundary condition (TBC), we reformulate the unbounded scattering problem into an equivalent initial-boundary value problem. The well-posedness is proved for the problem by the Laplace transform and variational method in the s-domain. Moreover, we show that the reduced problem has a unique weak solution by the energy method. The priori estimates with explicit dependence on the time are derived for the acoustic pressure and the elastic displacement in the time domain.

This paper studies the global regularity problem for the 2D micropolar Rayleigh-B e ̵́ nard convection system with velocity zero dissipation, micro-rotation velocity Laplace dissipation and temperature critical dissipation. By introducing a combined quantity and using the technique of Littlewood-Paley decomposition, we establish the global regularity result of solutions to this system.

This paper is concerned with the asymptotic stability and instability of solutions to a variable coefficient logarithmic wave equation with nonlinear damping and memory term. This model describes wave travelling through nonhomogeneous viscoelastic materials. By choosing appropriate multiplier and using weighted energy method, we prove the exponential decay of the energy. Besides, we also obtain the instability at the infinity of the solutions in the presence of the nonlinear damping.

In this paper we propose a numerical method for American multi-asset options under jump-diffusion model based on the combination of the exponential time differencing (ETD) technique for the differential operator and Gauss-Hermite quadrature for the integral term. In order to simplify the computational stencil and improve characteristics of the ETD-scheme mixed derivative eliminating transformation is applied. The results are compared with recently proposed methods.

We consider a new class of semiparametric spatio-temporal models with unknown and banded autoregressive coefficient matrices. The setting represents a type of sparse structure in order to include as many panels as possible. We apply the local linear method and least squares method for Yule-Walker equation to estimate trend function and spatio-temporal autoregressive coefficient matrices respectively. We also balance the over-determined and under-determined phenomena in part by adjusting the order of extracting sample information. Both the asymptotic normality and convergence rates of the proposed estimators are established. The proposed methods are further illustrated using both simulation and case studies, the results also show that our estimator is stable among different sample size, and it performs better than the traditional method with known spatial weight matrices.

Recently, an infinitesimal approach for finding reciprocal transformations has been proposed. The method uses the group analysis approach and consists of similar steps as for finding an equivalence group of transformations. The new method provides a systematic tool for finding classes of reciprocal transformations (group of reciprocal transformations). Similar to the classical group analysis this approach can be also applied for finding all reciprocal transformations (not only composing a group) of the equations under study. The present paper provides this algorithm. As an illustration, the method is applied to the two-dimensional stationary gas dynamics equations. Equivalence group, group of reciprocal transformations and completeness of all discrete reciprocal transformations are presented in the paper. The results are stated in form of a theorem.

Abstract. In this manuscript we consider the internal control problem for the fifth order KdV type equation, commonly called the Kawahara equation, on unbounded domains. Precisely, under certain hypotheses over the initial and boundary data, we are able to prove that there exists an internal control input such that solutions of the Kawahara equation satisfies an integral overdetermination condition. This condition is satisfied when the domain of the Kawahara equation is posed in the real line, left half-line and right half-line. Moreover, we are also able to prove that there exists a minimal time in which the integral overdetermination condition is satisfied. Finally, we show a type of exact controllability associated with the “mass” of the Kawahara equation posed in the half-line.

In this work we the study of the asymptotic behavior in the whole line of a thermoelastic structure with interfacial slip and second sound. We prove several polynomial decay estimates depending on the smoothness of initial data. The proof is based on the semigroup approach, the energy method by introducing a Lyapunov functional and Fourier transform.

A Kalman reduced form is obtained for linear systems over Hermite rings. This reduced form gives information of the set of assignable polynomials to a given linear system.

In this paper, we study the long-time dynamical behavior of the non-autonomous velocity-vorticity-Voigt model of the 3D Navier-Stokes equations with damping and memory. We first investigate the existence and uniqueness of weak solutions to the initial boundary value problem for above-mentioned model. Next, we prove the existence of uniform attractor of this problem, where the time-dependent forcing term $f \in L^2_b(\mathbb{R}; H^{-1}(\Omega))$ is only translation bounded instead of translation compact. The results in this paper will extend and improve some results in Yue, Wang (Comput. Math. Appl., 2020) in the case of non-autonomous and contain memory kernels which have not been studied before.