Augmented Lagrangian preconditioners for the Oseen–Frank model of nematic and cholesteric liquid crystals

BIT Numerical Mathematics - We propose a robust and efficient augmented Lagrangian-type preconditioner for solving linearizations of the Oseen–Frank model arising in nematic and cholesteric...     

An Oseen scheme for the conduction–convection equations based on a stabilized nonconforming method

An Oseen iterative scheme for the stationary conduction–convection equations based on a stabilized nonconforming finite element method is given. The stability and error estimates are analyzed, which show that the presented method is stable and has good precision. Numerical results are shown to support the developed theory analysis and demonstrate the good effectiveness of the given method.     

Two-level defect-correction Oseen iterative stabilized finite element methods for the stationary Navier–Stokes equations

Two-level defect-correction Oseen iterative stabilized finite element methods for the stationary Navier–Stokes equations based on local Gauss integration are considered in this paper. The methods combine the defect-correction method and the two-level strategy with the locally stabilized method. Moreover, the stability and convergence of the presented methods are deduced. Finally, numerical tests confirm the theoretical results of the presented methods.     

Well-posedness for the system of the Hamilton–Jacobi and the continuity equations

Let H (t, x, p) be a Hamiltonian function that is convex in p. Let the associated Lagrangian satisfy the nonstandard minorization condition where m > 0, ω > 0, and C ≥ 0 are constants. Under some additional conditions, we prove that the associated value function is the unique viscosity solution of S _t + H(t, x, ∇S) = 0 in , without any conditions at infinity on the solution. Here ωT < π/2. To the Hamilton–Jacobi equation corresponding to the classical action integrand in mechanics, we adjoin the continuity equation and establish the existence and uniqueness of a viscosity–measure solution (S, ρ) of

Liouville theorems for the Navier–Stokes equations and applications

We study bounded ancient solutions of the Navier–Stokes equations. These are solutions with bounded velocity defined in R ⁿ × (−1, 0). In two space dimensions we prove that such solutions are either constant or of the form u(x, t) = b(t), depending on the exact definition of admissible solutions. The general 3-dimensional problem seems to be out of reach of existing techniques, but partial results can be obtained in the case of axisymmetric solutions. We apply these results to some scenarios of potential singularity formation for axi-symmetric solutions, and obtain extensions of results in a recent paper by Chen, Strain, Tsai and Yau [4].     

Convergence and continuous dependence for the Brinkman–Forchheimer equations

This paper investigates the flow of fluid in a porous medium which is described in the Brinkman–Forchheimer equations and obtains the structural stability results for the coefficients.     

Relationship between the Udwadia–Kalaba equations and the generalized Lagrange and Maggi equations

In their paper “A New Perspective on Constrained Motion,” F. E. Udwadia and R. E. Kalaba propose a new form of matrix equations of motion for nonholonomic systems subject to linear nonholonomic second-order constraints. These equations contain all of the generalized coordinates of the mechanical system in question and, at the same time, they do not involve the forces of constraint. The equations under study have been shown to follow naturally from the generalized Lagrange and Maggi equations; they can be also obtained using the contravariant form of the motion equations of a mechanical system subjected to nonholonomic linear constraints of second order. It has been noted that a similar method of eliminating the forces of constraint from differential equations is usually useful for practical purposes in the study of motion of mechanical systems subjected to holonomic or classical nonholonomic constraints of first order. As a result, one obtains motion equations that involve only generalized coordinates of a mechanical system, which corresponds to the equations in the Udwadia–Kalaba form.     

Perfectly matched layers for the heat and advection–diffusion equations

We design a perfectly matched layer for the advection–diffusion equation. We show that the reflection coefficient is exponentially small with respect to the damping parameter and the width of the PML and independently of the advection and of the viscosity parameters. Numerical tests assess the efficiency of the approach.     

Numerical solution of the Cauchy problem for the Painlevé I and II equations

A numerical method for solving the Cauchy problem for the first and second Painlevé differential equations is proposed. The presence of movable poles of the solution is allowed. The positions of the poles are not a priori known and are determined in the process of solving the equation. The proposed method is based on the transition to an auxiliary system of differential equations in a neighborhood of a pole. The equations in this system and its solution have no singularities in either the pole or its neighborhood. Numerical results confirming the efficiency of this method are presented.     

ℋ-matrix techniques for approximating large covariance matrices and estimating its parameters

In this work the task is to use the available measurements to estimate unknown hyper-parameters (variance, smoothness parameter and covariance length) of the covariance function. We do it by maximizing the joint log-likelihood function. This is a non-convex and non-linear problem. To overcome cubic complexity in linear algebra, we approximate the discretised covariance function in the hierarchical (ℋ-) matrix format. The ℋ-matrix format has a log-linear computational cost and storage O(knlogn), where rank k is a small integer. On each iteration step of the optimization procedure the covariance matrix itself, its determinant and its Cholesky decomposition are recomputed within ℋ-matrix format. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Control problems for parabolic and hyperbolic equations via the theory ofG and Γ convergence

Summary A sequence of optimal control problems for systems governed by PDE'sis considered. The parameter (index of an element of the sequence) appears in the cost functionals which have integral form, as well as in the state equations which are of parabolic or hyperbolic type. It is proved that, under some -convergence of the cost functionals and some convergence of the indicator functions of sets of admissible solutions, the optimal solutions exist and converge to an optimal solution of the limit problem.     

Carleman estimates and unique continuation property for the Kadomtsev–Petviashvili equations

We study the unique continuation property for the generalized Kadomtsev–Petviashvili (KP) equations and its regularized version. We use Carleman estimates to prove that if the solution of the KP equations vanishes in an open subset, then this solution is identically equal to zero in the horizontal component of the open subset.     

RBSDE''''s with jumps and the related obstacle problems for integral-partial differential equations

The author proves, when the noise is driven by a Brownian motion and an independent Poisson random measure, the one-dimensional reflected backward stochastic differential equation with a stopping time terminal has a unique solution. And in a Markovian framework, the solution can provide a probabilistic interpretation for the obstacle problem for the integral-partial differential equation.     

Numerical method for the KdV–Kawahara and Benney–Lin equations

We are concerned with the initial-boundary-value problem associated to the Korteweg – de Vries – Kawahara (KdVK) equation and Benney – Lin (BL) equation, which are transport equations perturbed by dispersive terms of 3rd and 5th order and a term of 4th order in the case of (BL) equation. These equations appear in several fluid dynamics problems. We obtain local smoothing effects that are uniform with respect to the size of the interval. We also propose a simple finite-difference scheme for the problem and prove its stability. Finally, we give some numerical examples. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global existence and uniform decay for the coupled Klein-Gordon-Schr?dinger equations

This paper is concerned to the existence, uniqueness and uniform decay for the solutions of the coupled Klein-Gordon-Schr?dinger damped equations where ω is a bounded domain of R ⁿ , n≤ 3, F : R ²→R is a C ¹-function; γ, β; θ are constants such that γ, β > 0 and 1 ≤ 2θ≤ 2. Received January 1999 – Accepted October 1999     

Nonlocal problems for the equations of Kelvin-Voight fluids and their ε-approximations

In this paper, we study some nonlocal problems for the Kelvin-Voight equations (1) and the penalized Kelvin-Voight equations (2): the first and second initial boundary-value problems and the first and second time periodic boundary problems. We prove that these problems have global smooth solutions of the classW _∞ ¹ (?⁺;W ₂ ^2+k (Ω)),k=1,2,...;Ω??³. Bibliography: 25 titles.     

On scaling invariance and type-Ⅰ singularities for the compressible Navier-Stokes equations

We find a new scaling invariance of the barotropic compressible Navier-Stokes equations. Then it is shown that type-Ⅰ singularities of solutions with■ can never happen at time T for all adiabatic number γ 1. Here κ 0 does not depend on the initial data.This is achieved by proving the regularity of solutions under■ This new scaling invariance also motivates us to construct an explicit type-Ⅱ blowup solution for γ 1.