Existence and Decay of Solutions of a Nonlinear Viscoelastic Problem with a Mixed Nonhomogeneous Condition

We study the initial-boundary value problem for a nonlinear wave equation given by u_{tt}-u_{xx}+\int_{0}^{t}k(t-s)u_{xx}(s)ds+ u_{t}^{q-2}u_{t}=f(x,t,u) , 0 < x < 1, 0 < t < T, u_{x}(0,t)=u(0,t), u_{x}(1,t)+\eta u(1,t)=g(t), u(x,0)=\^u_{0}(x), u_{t}(x,0)={\^u}_{1}(x), where \eta \geq 0, q\geq 2 are given constants {\^u}_{0}, {\^u}_{1}, g, k, f are given functions. In part I under a certain local Lipschitzian condition on f, a global existence and uniqueness theorem is proved. The proof is based on the paper [10] associated to a contraction mapping theorem and standard arguments of density. In Part} 2, under more restrictive conditions it is proved that the solution u(t) and its derivative u_{x}(t) decay exponentially to 0 as t tends to infinity.Comment: 26 page

Optimal Control of the Strong-Field Laser Ionization of Clusters in Helium Droplets

The strong-field ionization dynamics of Ag and Xe clusters are studied using fs pulse shaping. By tailoring the temporal shapes of the laser pulses, the coupling of the energy into the Ag clusters can be controlled, leading to a maximum yield of highly charged atomic ions and an enhancement of the highest atomic charge states. For Xe, fitness scans of the laser parameters show that a two-step ionization scheme gives rise to an extreme charging of the clusters. Three-pulse trains obtained in an optimization experiment are even more effective and result in maximum yields of different chosen charge states

Stability for the Boussinesq system on real hyperbolic Manifolds and application

In this paper we study the global existence and stability of mild solution for the Boussinesq system on the real hyperbolic manifold $\mathbb{H}^d(\mathbb{R})$ ($d \geqslant 2$). We will consider a couple of Ebin-Marsden's Laplace and Laplace-Beltrami operators associated with the corresponding linear system which provides a vectorial heat semigoup. First, we prove the existence and the uniqueness of the bounded mild solution for the linear system by using certain dispersive and smoothing estimates of the vectorial heat semigroup. Next, using the fixed point arguments, we can pass from the linear system to the semilinear system to establish the existence of the bounded mild solution. We will prove the exponential stability of such solution by using the cone inequality. Finally, we give an application of stability to the existence of periodic mild solution for the Boussinesq system.Comment: 23 pages. arXiv admin note: substantial text overlap with arXiv:2209.0780

On asymptotically almost periodic solutions to the Navier-Stokes equations in hyperbolic manifolds

In this paper we extend a recent work \cite{HuyXuan2020} to study the forward asymptotically almost periodic (AAP-) mild solution of Navier-Stokes equation on the real hyperbolic manifold $\mathbb{H}^d(\mathbb{R})$ with dimension $d \geq 2$. Using the dispertive and smoothing estimates for Stokes equation \cite{Pi} we invoke the Massera-type principle to prove the existence and uniqueness of the AAP- mild solution for the Stokes equation in $L^p(\Gamma(TM)))$ space with $p>d$. We then establish the existence and uniqueness of the small AAP- mild solutions of the Navier-Stokes equation by using the fixed point argument. The asymptotic behaviour (exponential decay and stability) of these small solutions are also related. Our results extend also \cite{FaTa2013} for the forward asymptotic mild solution of the Navier-Stokes equation on the curved background.Comment: 21 page

Selection of Exercises to Improve the Effectiveness of Physical Education under the Project of Elective Sports Cockpitching for Female Students K2023 Hanoi Metropolitan University

The interview results have selected 36 exercises belonging to 5 exercise groups to improve the effectiveness of physical education according to the project for female Hanoi Metropolitan University students studying the optional sport of shuttlecock, including Group 1: Exercises to develop strength quickly has 7 exercises; Group 2: Strength development exercises have 6 exercises; Group 3: Exercises to develop endurance has 5 exercises; Group 4: Exercises to develop flexibility and coordination has 5 exercises; Group 5: Exercises with the bridge has 13 exercises with an agreement level of &gt; 70% or more and an average score of &gt; 3.41 - 4.20 points. &nbsp

Large time behavior of differential equations with drifted periodic coefficients modeling Carbon storage in soil

This paper is concerned with the linear ODE in the form $y'(t)=\lambda\rho(t)y(t)+b(t)$, $\lambda <0$ which represents a simplified storage model of the carbon in the soil. In the first part, we show that, for a periodic function $\rho(t)$, a linear drift in the coefficient $b(t)$ involves a linear drift for the solution of this ODE. In the second part, we extend the previous results to a classical heat non-homogeneous equation. The connection with an analytic semi-group associated to the ODE equation is considered in the third part. Numerical examples are given.Comment: 18 page

Stabilization for equal-order polygonal finite element method for high fluid velocity and pressure gradient

This paper presents an adapted stabilisation method for the equal-order mixed scheme of finite elements on convex polygonal meshes to analyse the high velocity and pressure gradient of incompressible fluid flows that are governed by Stokes equations system. This technique is constructed by a local pressure projection which is extremely simple, yet effective, to eliminate the poor or even non-convergence as well as the instability of equal-order mixed polygonal technique. In this research, some numerical examples of incompressible Stokes fluid flow that is coded and programmed by MATLAB will be presented to examine the effectiveness of the proposed stabilised method

On attractor's dimensions of the modified Leray-alpha equation

The primary objective of this paper is to investigate the modified Leray-alpha equation on the two-dimensional sphere $\mathbb{S}^2$, the square torus $\mathbb{T}^2$ and the three-torus $\mathbb{T}^3$. In the strategy, we prove the existence and the uniqueness of the weak solutions and also the existence of the global attractor for the equation. Then we establish the upper and lower bounds of the Hausdorff and fractal dimensions of the global attractor on both $\mathbb{S}^2$ and $\mathbb{T}^2$. Our method is based on the estimates for the vorticity scalar equations and the stationary solutions around the invariant manifold that are constructed by using the Kolmogorov flows. Finally, we will use the results on $\mathbb{T}^2$ to study the lower bound for attractor's dimensions on the case of $\mathbb{T}^3$.Comment: 24 page