Attractors for partly dissipative lattice dynamic systems in l2×l2

In this paper, we study the asymptotic behavior of solutions for the partly dissipative lattice dynamical systems in $l^2\times l^2$. We prove the asymptotic compactness of the solutions and then establish the existence of the global attractor in $l^2\times l^2$. The upper semicontinuity of the global attractor is also considered by finite-dimensional approximations of attractors for the lattice systems.     

Convergence rates of Neumann problems for Stokes systems

In quantitative homogenization of the Neumann problems for Stokes systems with rapidly oscillating periodic coefficients, this paper studies the convergence rates of the velocity in $L^2$ and $H^1$ as well as those of the pressure term in $L^2$, without any smoothness assumptions on the coefficients.     

Notes on the convergence order of gradient schemes for time fractional differential equations

We apply the GDM (Gradient Discretization Method), developed recently, as discretization in space to time-fractional diffusion and diffusion-wave equations with a fractional derivative of Caputo type in any space dimension.In the case of time-fractional diffusion equations, we establish an implicit scheme, and we prove an $L^{\infty }(L^2)$-error estimate. A similar result in a discrete $L^{\infty }(H_0^1)$–norm is also stated.To construct the numerical scheme for the time-fractional diffusion-wave equation, we write the equation in the form of a system of two low-order equations. We state an a prior estimate result that helps us to derive error estimates in discrete semi-norms of $L^{\infty }(H^1)$ and $H^1(L^2)$. The convergence is unconditional. Another gradient scheme is also suggested. We state its convergence results, which improve the convergence order proved recently for a SUSHI scheme.These results hold then for all the schemes within the framework of GDM: conforming and nonconforming finite element, mixed finite element, hybrid mixed mimetic family, some Multi-Point Flux approximation finite volume schemes, and some discontinuous Galerkin schemes.     

Weak version of restriction estimates for spheres and paraboloids in finite fields

We study $L^p-L^r$ restriction estimates for algebraic varieties in d-dimensional vector spaces over finite fields. Unlike the Euclidean case, if the dimension d is even, then it is conjectured that the $L^{(2d+2)/(d+3)}-L^2$ Stein–Tomas restriction result can be improved to the $L^{(2d+4)/(d+4)}-L^2$ estimate for both spheres and paraboloids in finite fields. In this paper we show that the conjectured $L^p-L^2$ restriction estimate holds in the specific case when test functions under consideration are restricted to d-coordinate functions or homogeneous functions of degree zero. To deduce our result, we use the connection between the restriction phenomena for our varieties in d dimensions and those for homogeneous varieties in $(d+1)$ dimensions.     

The topological structure of complete noncompact submanifolds in spheres

In this paper, we first show that δ-super stable complete noncompact minimal submanifolds in $S^{m+n}$ or $R^{m+n}$ with $\delta >{(\frac{m?1}{m})}^2$ admit no nontrivial $L^2$ harmonic 1-forms and have only one nonparabolic end, which generalizes Cao–Shen–Zhu's result in [2] on stable minimal hypersurface in $R^{m+1}$ and Lin's result in [13] on $\frac{m?1}{m}$-super stable minimal submanifolds in $R^{m+n}$. Second, we prove that the dimension of the space of $L^2$ harmonic p-forms on $M^m$ is zero or finite and there is only one nonparabolic end or finitely many nonparabolic ends of M under the assumptions on the Schrödinger operators involving the squared norm of the traceless second fundamental form.