Stability and error analysis for a spectral Galerkin method for the Navier‐Stokes equations with H2 or H1 initial data

In this article we consider a spectral Galerkin method with a semi‐implicit Euler scheme for the two‐dimensional Navier‐Stokes equations with H² or H¹ initial data. The H²‐stability analysis of this spectral Galerkin method shows that for the smooth initial data the semi‐implicit Euler scheme admits a large time step. The L²‐error analysis of the spectral Galerkin method shows that for the smoother initial data the numerical solution u exhibits faster convergence on the time interval [0, 1] and retains the same convergence rate on the time interval [1, ∞). © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005.     

A shallow‐water approximation to the full water wave problem

We demonstrate that the system of the Green‐Naghdi equations as a two‐directional, nonlinearly dispersive wave model is a close approximation to the two‐dimensional full water wave problem. Based on the energy estimates and the proof of the well‐posedness for the Green‐Naghdi equations and the water wave problem, we compare solutions of the two systems, showing that without restrictions on the wave amplitude, any two solutions of the two systems remain close, at least in some finite time within the shallow‐water regime, provided that their initial data are close in the Banach space H^s × H^s+1 for some s > . As a consequence, we show that if the depth of the water compared with the wavelength is sufficiently small, the two solutions exist for the same finite time using the uniformly bounded energies defined in the paper. © 2006 Wiley Periodicals, Inc.     

Local well‐posedness and zero‐α limit for the Euler‐α equations

In this paper, we prove the local‐in‐time existence and a blow‐up criterion of solutions in the Besov spaces for the Euler‐α equations of inviscid incompressible fluid flows in . We also establish the convergence rate of the solutions of the Euler‐α equations to the corresponding solutions of the Euler equations as the regularization parameter α approaches 0 in . Copyright © 2016 John Wiley & Sons, Ltd.     

Unconditional superconvergence analysis of an H1‐galerkin mixed finite element method for nonlinear Sobolev equations

An efficient H¹‐Galerkin mixed finite element method (MFEM) is presented with and zero order Raviart‐Thomas elements for the nonlinear Sobolev equations. On one hand, the existence and uniqueness of the solutions of the semidiscrete approximation scheme are proved and the super close results of order for the original variable u in a broken H¹ norm and the auxiliary variable in norm are deduced without the boundedness of the numerical solution in ‐norm. Conversely, a linearized Crank‐Nicolson fully discrete scheme with the unconditional super close property is also developed through a new approach, while previous literature always require certain time step conditions (see the references below). Finally, a numerical experiment is included to illustrate the feasibility of the proposed method. Here h is the subdivision parameter and τ is the time step.     

CONVERGENCE RATES FOR THE COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH GENERAL FORCES

For the viscous and heat-conductive fluids governed by the compressible Navier- Stokes equations with external force of general form in R^3, there exist nontrivial stationary solutions provided the external forces are small in suitable norms, which was studied in article [15], and there we also proved the global in time stability of the stationary solutions with respect to initial data in H^3-framework. In this article, the authors investigate the rates of convergence of nonstationary solutions to the corresponding stationary solutions when the initial data are small in H^3 and bounded in L6/5.     

Optimal decay rate of the bipolar Euler–Poisson system with damping in dimension three

By rewriting a bipolar Euler–Poisson equations with damping into a Euler equation with damping coupled with a Euler–Poisson equation with damping and using a new spectral analysis, we obtain the optimal decay results of the solutions in L² norm. More precisely, the velocities u₁ and u₂ decay at the L²?rate , which is faster than the normal L²‐rate for the heat equation and the Navier–Stokes equations. In addition, the decay rates of the disparities of two densities ρ₁?ρ₂ and the disparity of two velocities u₁?u₂ could reach to and in L² norm, respectively. Copyright © 2015 John Wiley & Sons, Ltd.     

Global attractors for 2‐D wave equations with displacement‐dependent damping

In this paper the long‐time behaviour of the solutions of 2‐D wave equation with a damping coefficient depending on the displacement is studied. It is shown that the semigroup generated by this equation possesses a global attractor in H(Ω) × L₂(Ω) and H²(Ω)∩H(Ω) × H(Ω). Copyright © 2009 John Wiley & Sons, Ltd.     

On the decay exponent of isotropic turbulence

It has long been observed that after a short initial transient period of time the decay of the velocity .uctuations of high Reynolds number homogeneous isotropic turbulence follows closely the algebraic law ∼ t^–n. From experiments and DNS data it is noticed that the numerical value for the exponent is n ≈ 1.2. In the last decade values close to the latter one have been adopted to calibrate turbulence models such as the k‐ε‐model. It will be shown that if such an algebraic decay exists it corresponds to the invariant (similarity) solution of the equations of fluid dynamics under the group of scaling. Most important it is shown that if such solutions exist the decay exponents are fixed due to certain invariants. For the Navier‐Stokes equation the invariant is a constant Reynolds number and it follows n = 1. For the Euler equation the value is prescribed to owing to the conservation of energy with finite initial energy. It is interesting to note that the experimental and DNS data indicate towards a decay induced by the Euler equations rather than the one conformal to the Navier‐Stokes equations.     

Decay of a model system of radiating gas

This paper concerns the Cauchy problem of a model system to the radiating gas in . Large time behaviors of classical solutions to the Cauchy problem are studied without needing the smallness assumption of initial perturbation in L¹‐norm. We obtain the optimal H^N‐norm time‐decay rates of the solutions in with 1 ≤ n ≤ 4 by applying the Fourier splitting method introduced by Schonbek (1980) with a slight modification and an energy method. Furthermore, when initial perturbation is bounded in L^p‐norm (p ∈ (1,2]), optimal L^p‐ L² decay estimates of the derivatives of solutions are shown. Copyright © 2013 John Wiley & Sons, Ltd.     

A Petrov‐Galerkin method with quadrature for semi‐linear second‐order hyperbolic problems

We propose and analyze a Crank–Nicolson quadrature Petrov–Galerkin (CNQPG) ‐spline method for solving semi‐linear second‐order hyperbolic initial‐boundary value problems. We prove second‐order convergence in time and optimal order H² norm convergence in space for the CNQPG scheme that requires only linear algebraic solvers. We demonstrate numerically optimal order H^k, k = 0,1,2, norm convergence of the scheme for some test problems with smooth and nonsmooth nonlinearities. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

On the Global Behavior of Solutions of a Coupled System of Nonlinear Schrödinger Equation

We mainly study a system of two coupled nonlinear Schrödinger equations where one equation includes gain and the other one includes losses. This model constitutes a generalization of the model of pulse propagation in birefringent optical fibers. We aim in this study at partially answering a question of some authors in [1]: “Is the H¹‐norm of the solution globally bounded in the Manakov case, when ?” We found that in the Manakov case, and when , the solution stays in , and also that the H¹‐norm of the solution cannot blow up in finite time. In the Manakov case, an estimate of the total energy is provided, which is different from that has been given in [1]. These results are corroborated by numerical results that have been obtained with a finite element solver well adapted for that purpose.     

Unstable Manifolds of Euler Equations

We consider a steady state v₀ of the Euler equation in a fixed bounded domain in ?ⁿ. Suppose the linearized Euler equation has an exponential dichotomy of unstable and center‐stable subspaces. By rewriting the Euler equation as an ODE on an infinite‐dimensional manifold of volume‐preserving maps in W^{k, q} the unstable (and stable) manifolds of v₀ are constructed under a certain spectral gap condition that is satisfied for both two‐dimensional and three‐dimensional examples. In particular, when the unstable subspace is finite dimensional, this implies the nonlinear instability of v₀ in the sense that arbitrarily small W^{k, q} perturbations can lead to L² growth of the nonlinear solutions. © 2013 Wiley Periodicals, Inc.     

The fictitious domain method with L2‐penalty for the Stokes problem with the Dirichlet boundary condition

We consider the fictitious domain method with L²‐penalty for the Stokes problem with the Dirichlet boundary condition. First, we investigate the error estimates for the penalty method at the continuous level. We obtain the convergence of order in H¹‐norm for the velocity and in L²‐norm for the pressure, where is the penalty parameter. The L²‐norm error estimate for the velocity is upgraded to . Moreover, we derive the a priori estimates depending on for the solution of the penalty problem. Next, we apply the finite element approximation to the penalty problem using the P1/P1 element with stabilization. For the discrete penalty problem, we prove the error estimate in H¹‐norm for the velocity and in L²‐norm for the pressure, where h denotes the discretization parameter. For the velocity in L²‐norm, the convergence rate is improved to . The theoretical results are verified by the numerical experiments.     

Uniform stability of spectral nonlinear Galerkin methods

This article provides a stability analysis for the backward Euler schemes of time discretization applied to the spatially discrete spectral standard and nonlinear Galerkin approximations of the nonstationary Navier‐Stokes equations with some appropriate assumption of the data (λ, u₀, f). If the backward Euler scheme with the semi‐implicit nonlinear terms is used, the spectral standard and nonlinear Galerkin methods are uniform stable under the time step constraint Δt ≤ (2/λλ₁). Moreover, if the backward Euler scheme with the explicit nonlinear terms is used, the spectral standard and nonlinear Galerkin methods are uniform stable under the time step constraints Δt = O(λ) and Δt = O(λ), respectively, where λ ≤ λ, which shows that the restriction on the time step of the spectral nonlinear Galerkin method is less than that of the spectral standard Galerkin method. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

Nonstationary Stokes system in weighted Sobolev spaces

We consider an initial‐boundary value problem for nonstationary Stokes system in a bounded domain Omega??³ with slip boundary conditions. We assume that Ω is crossed by an axis L. Let us introduce the following weighted Sobolev spaces with finite norms: and where ?(x) = dist{x, L}. We proved the result. Given the external force f∈L_{2, ?µ}(Ω^T), initial velocity v₀∈H(Ω), µ∈?₊\? there exist velocity v∈H(Ω^T) and the pressure p, ?p∈L_{2, ?µ}(Ω^T) and a constant c, independent of v, p, f, such that As we consider the Stokes system in weighted Sobolev spaces the following two things must be used:

• 1. the slip boundary condition and
• 2. the Helmholtz–Weyl decomposition.

Copyright © 2010 John Wiley & Sons, Ltd.     

On the blow‐up criterion of the periodic incompressible fluids

Kato, Ponce, Beale and Majda prove the existence and uniqueness of maximal solution of Euler and Navier–Stokes equations and some blow‐up criterion. In the periodic case, we establish that if the maximum time T* is finite, then the growth of is at least of the order of (T* ? t)^{?2m / 5}. Copyright © 2012 John Wiley & Sons, Ltd.     

On analyticity of the ‐Stokes semigroup for some non‐Helmholtz domains

Consider the Stokes equations in a sector‐like C ³ domain . It is shown that the Stokes operator generates an analytic semigroup in for . This includes domains where the ‐Helmholtz decomposition fails to hold. To show our result we interpolate results of the Stokes semigroup in and L ² by constructing a suitable non‐Helmholtz projection to solenoidal spaces.     

The Heat Equation Shrinks Ising Droplets to Points

Let be a bounded, smooth enough domain of ?². For L > 0 consider the continuous‐time, zero‐temperature heat bath stochastic dynamics for the nearest‐neighbor Ising model on (?/L)² (the square lattice with lattice spacing 1/L) with initial condition such that σ_x =?1 if x ? and σ_x = + 1 otherwise. We prove the following classical conjecture due to H. Spohn: In the diffusive limit where time is rescaled by L² and L → ∞, the boundary of the droplet of “‐” spins follows a deterministic anisotropic curve‐shortening flow such that the normal velocity is given by the local curvature times an explicit function of the local slope. Locally, in a suitable reference frame, the evolution of the droplet boundary follows the one‐dimensional heat equation. To our knowledge, this is the first proof of mean‐curvature‐type droplet shrinking for a lattice model with genuine microscopic dynamics. An important ingredient is in our forthcoming work, where the case of convex was solved. The other crucial point in the proof is obtaining precise regularity estimates on the deterministic curve‐shortening flow. This builds on geometric and analytic ideas of Grayson, Gage and Hamilton, Gage and Li, Chou and Zhu, and others.© 2015 Wiley Periodicals, Inc.     

An asymptotic expansion for the semi‐infinite sum of Dirac‐δ functions

In this paper, we derive an asymptotic expansion for the semi‐infinite sum of Dirac‐δ functions centered at discrete equidistant points defined by the set . The method relies on the Laplace transform of the semi‐infinite sum of Dirac‐δ functions. The derived series distribution takes the form of the Euler‐Maclaurin summation when the distributions are defined for complex or real‐valued continuous functions over the interval . For n=1, the series expansion contributes with a term equal to δ(x)/2, which survives in the limit when a→0⁺. This term represents a correction term, which is in general omitted in calculations of the density of states of quantum confined systems by finite‐size effects.     

Asymptotics of solutions for sub critical non‐convective type equations

We study the Cauchy problem for non‐linear dissipative evolution equations (1) where ?? is the linear pseudodifferential operator and the non‐linearity is a quadratic pseudodifferential operator (2) û ≡ ?_x→ξ u is the Fourier transformation. We consider non‐convective type non‐linearity, that is we suppose that a(t,0,y) ≠ 0. Let the initial data , are sufficiently small and have a non‐zero total mass , where is the weighted Sobolev space. Then we give the main term of the large time asymptotics of solutions in the sub critical case. Copyright © 2004 John Wiley & Sons, Ltd.