On the Free Boundary to the Incompressible Euler Equations

In this paper, we consider an inviscid, incompressible, irrotational fluid in a region of R^3 with free boundary. Motivated by [1], we find that in this particular case, we do not need the complicated energy functional in [1], instead we can use a simpler replacement and get the a priori energy estimate for a positive time, which depends only on the initial data.     

Convergence of the Vlasov-Poisson-Boltzmann System to the Incompressible Euler Equations

In this paper we establish the convergence of the Vlasov-Poisson-Boltzmann system to the incompressible Euler equations in the so-called quasi-neutral regime. The convergence is rigorously proved for time intervals on which the smooth solution of the Euler equations of the incompressible fluid exists. The proof relies on the relative-entropy method.     

Convergence of the Marker-and-Cell Scheme for the Incompressible Navier–Stokes Equations on Non-uniform Grids

We prove in this paper the convergence of the Marker-and-Cell scheme for the discretization of the steady-state and time-dependent incompressible Navier–Stokes equations in primitive variables, on non-uniform Cartesian grids, without any regularity assumption on the solution. A priori estimates on solutions to the scheme are proven; they yield the existence of discrete solutions and the compactness of sequences of solutions obtained with family of meshes the space step and, for the time-dependent case, the time step of which tend to zero. We then establish that the limit is a weak solution to the continuous problem.     

The Convergence of the Navier–Stokes–Poisson System to the Incompressible Euler Equations

ABSTRACT

The combining quasineutral and inviscid limit of the Navier–Stokes–Poisson system in the torus 𝕋 ^d , d ≥ 1 is studied. The convergence of the Navier–Stokes–Poisson system to the incompressible Euler equations is proven for the global weak solution and for the case of general initial data.     

A IPN×IPN Spectral Element Projection Method for the Unsteady Incompressible Navier-Stokes Equations

In this paper, we present a PN×PN spectral element method and a detailed comparison with existing methods for the unsteady incompressible Navier-Stokes equations. The main purpose of this work consists of: (i) detailed comparison and discussion of some recent developments of the temporal discretizations in the frame of spectral element approaches in space; (ii) construction of a stable PN×PN method together with a PN→PN-2 post-filtering. The link of different methods will be clarified. The key feature of our method lies in that only one grid is needed for both velocity and pressure variables, which differs from most well-known solvers for the Navier-Stokes equations. Although not yet proven by rigorous theoretical analysis, the stability and accuracy of this one-grid spectral method are demonstrated by a series of numerical experiments.     

A IP_N×IP_N Spectral Element Projection Method for the Unsteady Incompressible Navier-Stokes Equations

In this paper,we present a IP_N×IP_N spectral element method and a detailed comparison with existing methods for the unsteady incompressible Navier-Stokes equa- tions.The main purpose of this work consists of:(i) detailed comparison and discussion of some recent developments of the temporal discretizations in the frame of spectral el- ement approaches in space;(ii) construction of a stable IP_N×IP_N method together with a IP_N→IP_(N-2) post-filtering.The link of different methods will be clarified.The key feature of our method lies in that only one grid is needed for both velocity and pressure variables,which differs from most well-known solvers for the Navier-Stokes equations. Although not yet proven by rigorous theoretical analysis,the stability and accuracy of this one-grid spectral method are demonstrated by a series of numerical experiments.     

Dispersion Analysis of Multi-symplectic Scheme for the Nonlinear Schrodinger Equations

In this paper,we study the dispersive properties of multi-symplectic discretizations for the nonlinear Schrodinger equations.The numerical dispersion relation and group velocity are investigated.It is found that the numerical dispersion relation is relevant when resolving the nonlinear Schrodinger equations.     

The 3D Incompressible Euler Equations with a Passive Scalar: A Road to Blow-Up?

The three-dimensional incompressible Euler equations with a passive scalar θ are considered in a smooth domain $\varOmega\subset \mathbb{R}^{3}$ with no-normal-flow boundary conditions $\boldsymbol{u}\cdot\hat{\boldsymbol{n}}|_{\partial\varOmega} = 0$ . It is shown that smooth solutions blow up in a finite time if a null (zero) point develops in the vector B=?q×?θ, provided B has no null points initially: $\boldsymbol{\omega} = \operatorname{curl}\boldsymbol {u}$ is the vorticity and q=ω??θ is a potential vorticity. The presence of the passive scalar concentration θ is an essential component of this criterion in detecting the formation of a singularity. The problem is discussed in the light of a kinematic result by Graham and Henyey (Phys. Fluids 12:744–746, 2000) on the non-existence of Clebsch potentials in the neighbourhood of null points.     

Weak and Strong Solutions for the Stokes Approximation of Non-homogeneous Incompressible Navier-Stokes Equations

In this paper,the Dirichlet problem of Stokes approximate of non-homogeneous incompressibleNavier-Stokes equations is studied.It is shown that there exist global weak solutions as well as global andunique strong solution for this problem,under the assumption that initial density ρ_0(x)is bounded away from0 and other appropriate assumptions(see Theorem 1 and Theorem 2).The semi-Galerkin method is applied toconstruct the approximate solutions and a prior estimates are made to elaborate upon the compactness of theapproximate solutions.     

A Note on the Rate of Convergence of the Euler–Maruyama Method for Stochastic Differential Equations

Stochastic processes are approximated by wavelet operators. For the stochastic process and the scale function satisfying some conditions, a new method is given to estimate the degree of approximation.     

Symmetries,Lagrangian and Conservation Laws for the Maxwell Equations

The evolution equations of Maxwell’s equations has a Lagrangian written in terms of the electric E and magnetic H fields, but admit neither Lorentz nor conformal transformations. The additional equations ∇⋅E=0, ∇⋅H=0 guarantee the Lorentz and conformal invariance, but the resulting system is overdetermined, and hence does not have a Lagrangian. The aim of the present paper is to attain a harmony between these two contradictory properties and provide a correspondence between symmetries and conservation laws using the Lagrangian for the evolutionary part of Maxwell’s equations.     

Energy Error Estimates for the Projection-Difference Method with the Crank–Nicolson Scheme for Parabolic Equations

We solve an abstract parabolic problem in a separable Hilbert space, using the projection-difference method. The spatial discretization is carried out by the Galerkin method and the time discretization, by the Crank–Nicolson scheme. On assuming weak solvability of the exact problem, we establish effective energy estimates for the error of approximate solutions. These estimates enable us to obtain the rate of convergence of approximate solutions to the exact solution in time up to the second order. Moreover, these estimates involve the approximation properties of the projection subspaces, which is illustrated by subspaces of the finite element type.     

Backward Euler Scheme,Singular Hölder Norms,and Maximal Regularity for Parabolic Difference Equations

We show finite difference analogues of maximal regularity results for discretizations of abstract linear parabolic problems. The involved spaces are discrete versions of spaces of Hölder continuous functions, which can be singular in 0. The main tools are real interpolation and Da Prato–Grisvard's theory of the sum of linear operators.     

Free Evolution in the Lax–Phillips Scheme for Second-Order Operator Differential Equations

Necessary and sufficient conditions are obtained for an operator differential equation to describe free evolution in the Lax–Phillips scattering scheme.     

Justification of Asymptotic Two-dimensional Model for Steady Navier-Stokes Equations for Incompressible Flow

We study the asymptotic behavior of solutions to steady Navier-Stokes equations for incompressible flow in thin three-dimensional deformed cylinders. We prove that a sequence of the solutions converges strongly to a solution of a corresponding two-dimensional asymptotic model if the thickness of the cylinders converges to zero.     

A Regularity Criterion for the Navier–Stokes Equations

We prove that a weak solution u = (u ¹, u ², u ³) to the Navier–Stokes equations is strong, if any two components of u satisfy Prodi–Ohyama–Serrin's criterion. As a local regularity criterion, we prove u is bounded locally if any two components of the velocity lie in L ^{6, ∞}.     

Inviscid Incompressible Limits Under Mild Stratification: A Rigorous Derivation of the Euler–Boussinesq System

We consider the full Navier–Stokes–Fourier system in the singular regime of small Mach and large Reynolds and Péclet numbers, with ill prepared initial data on an unbounded domain \(\Omega \subset R^3\) with a compact boundary. We perform the singular limit in the framework of weak solutions and identify the Euler–Boussinesq system as the target problem.     

Euler–Lagrange Equations for Lagrangians Containing Complex-order Fractional Derivatives

Two variational problems of finding the Euler–Lagrange equations corresponding to Lagrangians containing fractional derivatives of real- and complex-order are considered. The first one is the unconstrained variational problem, while the second one is the fractional optimal control problem. The expansion formula for fractional derivatives of complex-order is derived in order to approximate the fractional derivative appearing in the Lagrangian. As a consequence, a sequence of approximated Euler–Lagrange equations is obtained. It is shown that the sequence of approximated Euler–Lagrange equations converges to the original one in the weak sense as well as that the sequence of the minimal values of approximated action integrals tends to the minimal value of the original one.