Uniform time of existence of the smooth solution for 3D Euler-α equations with periodic boundary conditions

2018 ◽  
Vol 28 (10) ◽  
pp. 1881-1897
Author(s):  
Aibin Zang
Keyword(s):  
Euler Equations ◽  
Smooth Solution ◽  
Initial Velocity ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Time Interval ◽  
Periodic Domain ◽  
Unique Classical Solution ◽  
Euler Formula ◽  
Incompressible Euler

After reformulating the incompressible Euler-[Formula: see text] equations in 3D periodic domain, one obtains that there exists a unique classical solution of Euler-[Formula: see text] equations in uniform time interval independent of [Formula: see text]. It is shown that the solutions of the Euler-[Formula: see text] converge to the corresponding solutions of Euler equation in [Formula: see text] in space, uniformly in time. It also follows that the [Formula: see text] [Formula: see text] solutions of Euler-[Formula: see text] equations exist in any fixed sub-interval of the maximum existing interval for the Euler equations provided that initial velocity is regular enough and [Formula: see text] is sufficiently small.

The Zero-Mach Limit of Compressible Euler Equations

2013 ◽  
Vol 275-277 ◽  
pp. 518-521
Author(s):  
Shu Wang
Keyword(s):  
Mach Number ◽  
Euler Equations ◽  
Initial Velocity ◽  
Compressible Flows ◽  
Incompressible Flows ◽  
Compressible Euler Equations ◽  
Time Interval ◽  
Compressible Euler ◽  
Three Space ◽  
Incompressible Euler

The aim of this paper is to prove that compressible Euler equations in two and three space dimensions converge to incompressible Euler equations in the limit as the Mach number tends to zero. No smallness restrictions are imposed on the initial velocity, or the time interval. We assume instead that the incompressible flows exists and is reasonably smooth on a given time interval, and prove that compressible flows converge uniformly on that time interval.

One-Dimensional Unsteady Periodic Flow Model with Boundary Conditions Constrained by Differential Equations

2009 ◽  
Vol 131 (6) ◽  
Author(s):  
Nhan T. Nguyen
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Euler Equations ◽  
Periodic Boundary ◽  
Closed Loop ◽  
Fluid Transport ◽  
Periodic Boundary Conditions ◽  
Transport Systems ◽  
Variable Approach ◽  
Upwind Method

This paper describes a modeling method for closed-loop unsteady fluid transport systems based on 1D unsteady Euler equations with nonlinear forced periodic boundary conditions. A significant feature of this model is the incorporation of dynamic constraints on the variables that control the transport process at the system boundaries as they often exist in many transport systems. These constraints result in a coupling of the Euler equations with a system of ordinary differential equations that model the dynamics of auxiliary processes connected to the transport system. Another important feature of the transport model is the use of a quasilinear form instead of the flux-conserved form. This form lends itself to modeling with measurable conserved fluid transport variables and represents an intermediate model between the primitive variable approach and the conserved variable approach. A wave-splitting finite-difference upwind method is presented as a numerical solution of the model. An iterative procedure is implemented to solve the nonlinear forced periodic boundary conditions prior to the time-marching procedure for the upwind method. A shock fitting method to handle transonic flow for the quasilinear form of the Euler equations is presented. A closed-loop wind tunnel is used for demonstration of the accuracy of this modeling method.

scholarly journals Limits of the Stokes and Navier–Stokes equations in a punctured periodic domain

2019 ◽  
Vol 18 (02) ◽  
pp. 211-235
Author(s):  
Michel Chipot ◽  
Jérôme Droniou ◽  
Gabriela Planas ◽  
James C. Robinson ◽  
Wei Xue
Keyword(s):  
Boundary Conditions ◽  
Periodic Boundary ◽  
Stokes Equations ◽  
Periodic Boundary Conditions ◽  
Dirichlet Boundary Conditions ◽  
Navier Stokes ◽  
Periodic Domain ◽  
Navier Stokes Equations ◽  
Forcing Function ◽  
Connected Set

We treat three problems on a two-dimensional “punctured periodic domain”: we take [Formula: see text], where [Formula: see text] and [Formula: see text] is the closure of an open connected set that is star-shaped with respect to [Formula: see text] and has a [Formula: see text] boundary. We impose periodic boundary conditions on the boundary of [Formula: see text], and Dirichlet boundary conditions on [Formula: see text]. In this setting we consider the Poisson equation, the Stokes equations, and the time-dependent Navier–Stokes equations, all with a fixed forcing function [Formula: see text], and examine the behavior of solutions as [Formula: see text]. In all three cases we show convergence of the solutions to those of the limiting problem, i.e. the problem posed on all of [Formula: see text] with periodic boundary conditions.

Molecular Dynamics Using Non-Variational Polarizable Force Fields: Theory, Periodic Boundary Conditions Implementation and Application to the Bond Capacity Model

2019 ◽  
Author(s):  
Pier Paolo Poier ◽  
Louis Lagardere ◽  
Jean-Philip Piquemal ◽  
Frank Jensen
Keyword(s):  
Boundary Conditions ◽  
Periodic Boundary ◽  
Force Fields ◽  
Periodic Boundary Conditions ◽  
Computational Effort ◽  
Polarizable Force Fields ◽  
Energy Models ◽  
Capacity Model ◽  
Energy Gradient ◽  
Polarization Models

<div> <div> <div> <p>We extend the framework for polarizable force fields to include the case where the electrostatic multipoles are not determined by a variational minimization of the electrostatic energy. Such models formally require that the polarization response is calculated for all possible geometrical perturbations in order to obtain the energy gradient required for performing molecular dynamics simulations. </p><div> <div> <div> <p>By making use of a Lagrange formalism, however, this computational demanding task can be re- placed by solving a single equation similar to that for determining the electrostatic variables themselves. Using the recently proposed bond capacity model that describes molecular polarization at the charge-only level, we show that the energy gradient for non-variational energy models with periodic boundary conditions can be calculated with a computational effort similar to that for variational polarization models. The possibility of separating the equation for calculating the electrostatic variables from the energy expression depending on these variables without a large computational penalty provides flexibility in the design of new force fields. </p><div><div><div> </div> </div> </div> <p> </p><div> <div> <div> <p>variables themselves. Using the recently proposed bond capacity model that describes molecular polarization at the charge-only level, we show that the energy gradient for non-variational energy models with periodic boundary conditions can be calculated with a computational effort similar to that for variational polarization models. The possibility of separating the equation for calculating the electrostatic variables from the energy expression depending on these variables without a large computational penalty provides flexibility in the design of new force fields. </p> </div> </div> </div> </div> </div> </div> </div> </div> </div>

scholarly journals Global classical solutions to the elastodynamic equations with damping

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mengmeng Liu ◽  
Xueyun Lin
Keyword(s):  
Initial Data ◽  
Periodic Boundary ◽  
Energy Estimate ◽  
Periodic Boundary Conditions ◽  
Deformation Tensor ◽  
Classical Solutions ◽  
Small Initial Data ◽  
Weighted Energy ◽  
Global Classical Solutions ◽  
Weighted Energy Estimate

AbstractIn this paper, we show the global existence of classical solutions to the incompressible elastodynamics equations with a damping mechanism on the stress tensor in dimension three for sufficiently small initial data on periodic boxes, that is, with periodic boundary conditions. The approach is based on a time-weighted energy estimate, under the assumptions that the initial deformation tensor is a small perturbation around an equilibrium state and the initial data have some symmetry.

scholarly journals Normal periodic solutions for the fractional abstract Cauchy problem

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jennifer Bravo ◽  
Carlos Lizama
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Periodic Solution ◽  
Periodic Boundary ◽  
Abstract Cauchy Problem ◽  
Periodic Boundary Conditions ◽  
Sectorial Operator ◽  
Closed Linear Operator ◽  
Resolvent Set ◽  
Spectral Angle

AbstractWe show that if A is a closed linear operator defined in a Banach space X and there exist $t_{0} \geq 0$ t 0 ≥ 0 and $M>0$ M > 0 such that $\{(im)^{\alpha }\}_{|m|> t_{0}} \subset \rho (A)$ { ( i m ) α } | m | > t 0 ⊂ ρ ( A ) , the resolvent set of A, and $$ \bigl\Vert (im)^{\alpha }\bigl(A+(im)^{\alpha }I \bigr)^{-1} \bigr\Vert \leq M \quad \text{ for all } \vert m \vert > t_{0}, m \in \mathbb{Z}, $$ ∥ ( i m ) α ( A + ( i m ) α I ) − 1 ∥ ≤ M  for all  | m | > t 0 , m ∈ Z , then, for each $\frac{1}{p}<\alpha \leq \frac{2}{p}$ 1 p < α ≤ 2 p and $1< p < 2$ 1 < p < 2 , the abstract Cauchy problem with periodic boundary conditions $$ \textstyle\begin{cases} _{GL}D^{\alpha }_{t} u(t) + Au(t) = f(t), & t \in (0,2\pi ); \\ u(0)=u(2\pi ), \end{cases} $$ { D t α G L u ( t ) + A u ( t ) = f ( t ) , t ∈ ( 0 , 2 π ) ; u ( 0 ) = u ( 2 π ) , where $_{GL}D^{\alpha }$ D α G L denotes the Grünwald–Letnikov derivative, admits a normal 2π-periodic solution for each $f\in L^{p}_{2\pi }(\mathbb{R}, X)$ f ∈ L 2 π p ( R , X ) that satisfies appropriate conditions. In particular, this happens if A is a sectorial operator with spectral angle $\phi _{A} \in (0, \alpha \pi /2)$ ϕ A ∈ ( 0 , α π / 2 ) and $\int _{0}^{2\pi } f(t)\,dt \in \operatorname{Ran}(A)$ ∫ 0 2 π f ( t ) d t ∈ Ran ( A ) .

scholarly journals Sharp Lyapunov-type inequalities for second-order half-linear difference equations with different kinds of boundary conditions

2021 ◽  
Vol 115 (3) ◽  
Author(s):  
Robert Stegliński
Keyword(s):  
Difference Equation ◽  
Boundary Conditions ◽  
Difference Equations ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Second Order ◽  
Linear Difference Equations ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation

AbstractIn this work, we establish optimal Lyapunov-type inequalities for the second-order difference equation with p-Laplacian $$\begin{aligned} \Delta (\left| \Delta u(k-1)\right| ^{p-2}\Delta u(k-1))+a(k)\left| u(k)\right| ^{p-2}u(k)=0 \end{aligned}$$ Δ ( Δ u ( k - 1 ) p - 2 Δ u ( k - 1 ) ) + a ( k ) u ( k ) p - 2 u ( k ) = 0 with Dirichlet, Neumann, mixed, periodic and anti-periodic boundary conditions.

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