• Abstract

The paper investigates the well-posedness of global solutions and the existence of global attractors for weakly damped FDS nonlinear wave equations. It establishes the well-posedness of weak solutions using Galerkin approximation and a priori estimate. Subsequently, a dynamical system is constructed based on the well-posedness of the solution. The existence of a bounded absorbing set for the equations and the smooth properties of the operator semigroup are presented, leading to the existence of a global attractor.

• General Summary

• Abstract

We develop a locally mass-conservative enriched Petrov-Galerkin (EPG) method without any penalty term for the simulation of Darcy flow in fractured porous media. The discrete fracture model is applied to model the fractures as the lower dimensional fracture interfaces. The new method enriches the approximation trial space of the conforming continuous Galerkin (CG) method with bubble functions and enriches the approximation test space of the CG method with piecewise constant functions in the fractures and the surrounding porous media. We propose a framework for constructing the bubble functions and consider a decoupled algorithm for the EPG method. The solution of the pressure can be decoupled into two steps with a standard CG method and a post-processing correction. The post-processing correction based on the bubble functions in the matrix and the fractures can be solved separately, which is useful for parallel computing. We derive a priori and a posteriori error estimates for the problem. Numerical examples are presented to illustrate the performance of the proposed method.

• General Summary

• Abstract

This paper is concerned with the time decay estimates of  the fourth order Schrodinger  operator $H=\Delta^{2}+V(x)$ in dimension three, where $V(x)$ is  a real valued decaying potential.   Assume that zero is  a regular point or the first kind resonance of $H$,  and $H$ has no positive eigenvalues, we established the following  time optimal decay estimates of  $e^{-itH}$ with a regular term $H^{\alpha/4}$:

 $$\|H^{\alpha/4}e^{-itH}P_{ac}(H)\|_{L^1-L^\infty}\lesssim |t|^{-\frac{3+\alpha}{4}}, \quad   0 \leq \alpha \leq 3.$$

When zero is the second or  third kind  resonance of $H$,   their decay will be significantly changed. We remark that such improved  time decay estimates with the extra regular term $H^{\alpha/4}$  will be interesting in the well-posedness and scattering of nonlinear fourth order Schrodinger equations with potentials.

• General Summary