$H^2$-Stabilization of the Isothermal Euler Equations: a Lyapunov Function Approach

The authors consider the problem of boundary feedback stabilization of the 1D Euler gas dynamics locally around stationary states and prove the exponential stability with respect to the H2-norm.To this...     

Some results for the primitive equations with physical boundary conditions

In this paper, we consider the (simplified) 3-dimensional primitive equations with physical boundary conditions. We show that the equations with constant forcing have a bounded absorbing ball in the H ¹-norm and that a solution to the unforced equations has its H ¹-norm decay to 0. From this, we argue that there exists an invariant measure (on H ¹) for the equations under random kick-forcing.     

A priori error estimates of finite volume method for nonlinear optimal control problem

In this paper, we study a priori error estimates for the finite volume element approximation of nonlinear optimal control problem. The schemes use discretizations based on a finite volume method. For the variational inequality, we use the method of the variational discretization concept to obtain the control. Under some reasonable assumptions, we obtain some optimal order error estimates. The approximate order for the state, costate and control variables is O(h ²) or \(O\left( {{h^2}\sqrt {\left| {\ln h} \right|} } \right)\) in the sense of L ²-norm or L ^∞-norm. A numerical experiment is presented to test the theoretical results. Finally, we give some conclusions and future works.     

The resolution of the bounded L2 curvature conjecture in general relativity

This paper reports on the recent proof of the bounded L² curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the L²-norm of the curvature and a lower bound of the volume radius of the corresponding initial data set.     

Jacobi Spectral Methods for Volterra-Urysohn Integral Equations of Second Kind with Weakly Singular Kernels

Abstract

In this article, we discuss Jacobi spectral Galerkin and iterated Jacobi spectral Galerkin methods for Volterra-Urysohn integral equations with weakly singular kernels and obtain the convergence results in both the infinity and weighted L²-norm. We show that the order of convergence in iterated Jacobi spectral Galerkin method improves over Jacobi spectral Galerkin method. We obtain the convergence results in two cases when the exact solution is sufficiently smooth and non-smooth. For finding the improved convergence results, we also discuss Jacobi spectral multi-Galerkin and iterated Jacobi spectral multi-Galerkin method and obtain the convergence results in weighted L²-norm. In fact, we prove that the iterated Jacobi spectral multi-Galerkin method improves over iterated Jacobi spectral Galerkin method. We provide numerical results to verify the theoretical results.     

The convergence of non-Newtonian fluids to Navier-Stokes equations

In this paper, the convergence of solutions for incompressible dipolar viscous non-Newtonian fluids is investigated. We obtain the conclusion that the solutions of non-Newtonian fluids converge to the solutions of Navier-Stokes equations in the sense of L²-norm (resp. H¹-norm), as the viscosities tend to zero and the initial data belong to H¹(Ω) (resp. H²(Ω)). Moreover, we obtain L^∞-norm convergence of solutions if the initial data belong to H²(Ω).     

Annihilating sets for the short time Fourier transform

We obtain a class of subsets of R2d such that the support of the short time Fourier transform (STFT) of a signal f∈L²(Rd) with respect to a window g∈L²(Rd) cannot belong to this class unless f or g is identically zero. Moreover we prove that the L²-norm of the STFT is essentially concentrated in the complement of such a set. A generalization to other Hilbert spaces of functions or distributions is also provided. To this aim we obtain some results on compactness of localization operators acting on weighted modulation Hilbert spaces.     

Bregman iterative model using the G-norm

In this paper, we analyze the Bregman iterative model using the G-norm. Firstly, we show the convergence of the iterative model. Secondly, using the source condition and the symmetric Bregman distance, we consider the error estimations between the iterates and the exact image both in the case of clean and noisy data. The results show that the Bregman iterative model using the G-norm has the similar good properties as the Bregman iterative model using the L2-norm.     

Pointwise estimates of solutions for the multi-dimensional bipolar Euler–Poisson system

In the paper, we consider a multi-dimensional bipolar hydrodynamic model from semiconductor devices and plasmas. This system takes the form of Euler–Poisson with electric field and frictional damping added to the momentum equations. By making a new analysis on Green’s functions for the Euler system with damping and the Euler–Poisson system with damping, we obtain the pointwise estimates of the solution for the multi-dimensions bipolar Euler–Poisson system. As a by-product, we extend decay rates of the densities \({\rho_i(i=1,2)}\) in the usual L²-norm to the L^p-norm with \({p\geq1}\) and the time-decay rates of the momentums m_i(i = 1,2) in the L²-norm to the L^p-norm with p > 1 and all of the decay rates here are optimal.     

An h-p Petrov-Galerkin finite element method for linear Volterra integro-differential equations

We analyze an h-p version Petrov-Galerkin finite element method for linear Volterra integrodifferential equations. We prove optimal a priori error bounds in the L ²- and H ¹-norm that are explicit in the time steps, the approximation orders and in the regularity of the exact solution. Numerical experiments confirm the theoretical results. Moreover, we observe that the numerical scheme superconverges at the nodal points of the time partition.