Multiscale analysis for diffusion-driven neutrally stable states

The Turing instabilities for reaction–diffusion systems are studied from the Fourier normal modes which appear by searching the solution obtained from linearization of the reaction–diffusion system at the spatially homogeneous steady state. The linear stability analysis is only appropriate when the temporal eigenvalues associated to every given spatial eigenvalue have non-zero real part. If the real part of the temporal eigenvalue in a normal mode is equal to zero there is no enough information coming from the linearized system. Given an arbitrary spatial eigenvalue, by equating to zero the real part of the corresponding temporal eigenvalue will lead to a neutral stability manifold in the parameter space. If for a given spatial eigenvalue the other parameters in the reaction–diffusion process drive the system to the neutral manifold, then neither stability nor instability can be warranted by the usual linear analysis. In order to give a sketch of the nonlinear analysis we use a multiple scales method. As an application, we analyze the behavior of solutions to the Schnakenberg trimolecular reaction kinetics in the presence of diffusion.     

Two-level Galerkin–Lagrange multipliers method for the stationary Navier–Stokes equations

In this paper, we combine the Galerkin–Lagrange multiplier (GLM) method with the two-level method to solve the stationary Navier–Stokes equations in order to avoid the time-consuming process and the construction of zero-divergence elements. Different quadrilateral partitions are used for approximating the velocity and the pressure. Then some error estimates are obtained and some numerical results of the GLM method and the two-level GLM method are given. The results show that the two-level method based on the GLM method is more efficient than the GLM method under the convergence rate of same order.     

Quasistationary Approximation in the Rotating Ring Problem

We consider the planar rotation-symmetric motion by inertia of a viscous incompressible fluid in a ring with free boundary. We reduce the corresponding initial-boundary value problem for the Navier–Stokes equations to some problem for a coupled system of one parabolic equation and two ordinary differential equations. We suppose that the coefficient of the derivatives of the sought functions with respect to time (the quasistationary parameter) is small; so the system is singularly perturbed. In this article we construct an asymptotic expansion for a solution to the rotating ring problem in a small quasistationary parameter and obtain a smallness estimate for the difference between the exact and approximate solutions.     

Conservation laws for Camassa–Holm equation, Dullin–Gottwald–Holm equation and generalized Dullin–Gottwald–Holm equation

In this paper we construct the conservation laws for the Camassa–Holm equation, the Dullin–Gottwald–Holm equation (DGH) and the generalized Dullin–Gottwald–Holm equation (generalized DGH). The variational derivative approach is used to derive the conservation laws. Only first order multipliers are considered. Two multipliers are obtained for the Camassa–Holm equation. For the DGH and generalized DGH equations the variational derivative approach yields two multipliers; thus two conserved vectors are obtained.     

Nonlinear Potentials for Hamilton–Jacobi–Bellman Equations. II

A formal method of constructing the viscosity solutions for abstract nonlinear equations of Hamilton–Jacobi–Bellman (HJB) type was developed in the previous work of the author. A new advantage of this method (which was called an nonlinear potentials method) is that it gives a possibility to choose at the first step an expected regularity of the solution and then – to construct this solution. This makes the whole procedure more simple because an analysis of regularity of viscosity solutions is usually the most complicated step.Nonlinear potentials method is a generalization of Krylov's approach to study HJB equations.In this article nonlinear potentials method is applied to elliptic degenerate HJB equations in R^d with variable coefficients.     

Steady flows of Jeffrey–Hamel type from the half-plane into an infinite channel. 1. Linearization on an antisymmetric solution

In this paper our objective is to provide physically reasonable solutions for the stationary Navier–Stokes equations in a two-dimensional domain with two outlets to infinity, a semi-strip Π₋ and a half-plane K. The same problem in an aperture domain, i.e. in a domain with two half-plane outlets to infinity, has been studied but only under symmetry restrictions on the data. Here, we assume that the main asymptotic term of the solution takes an antisymmetric form in K and apply the technique of weighted spaces with detached asymptotics, i.e. we use spaces where the functions have prescribed asymptotic forms in the outlets.After first showing that the corresponding Stokes problem admits a unique solution if and only if certain compatibility conditions are satisfied, we write the Navier–Stokes equations as a perturbation of the Stokes problem and the crucial compatibility condition as an algebraic equation by which the flux becomes determined. Assuming that the coefficient of the main (antisymmetric) asymptotic term of the solution in K does not vanish and that the data are sufficiently small, we use a contraction principle to solve the Navier–Stokes system coupled with the algebraic equation.Finally, we discuss the ill-posedness of the Navier–Stokes problem with prescribed flux.     

Modified Implicit–Explicit BDF Methods for Nonlinear Parabolic Equations

Implicit–explicit multistep methods for nonlinear parabolic equations were recently analyzed. If the implicit scheme is one of the backward differentiation formulae (BDF) of order up to six, then the corresponding implicit–explicit method of the same order is stable provided the stability constant is less than a specific scheme-dependent constant. Based on BDF, implicit methods are constructed such that the corresponding implicit–explicit scheme of the same order exhibits improved stability properties.     

Structure of the set of stationary solutions of phase-lock equations in superconductivity

In previous article [M. Zhan, Phase-lock equations and its connections to Ginzburg–Landau equations of superconductivity, J. Nonlinear Anal. 42 (2000) 1063–1075], we introduced a system of equations (phase-lock equations) to model the superconductivity phenomena. We investigated its connection to Ginzburg–Landau equations and proved the existence and uniqueness of both weak and strong solutions. In this article, we study the steady-state problem associated with the phase-lock equations. We prove that the steady-state problem has multiple solutions and show that the solution set enjoys some structural properties as proved by Foias and Teman for the Navier–Stokes equations in [C. Foias, R. Teman, Structure of the set of stationary solutions of the Navier–Stokes equations, Commun. Pure Appl. Math. XXX (1977) 149–164].     

Large time behavior of solutions of Hamilton–Jacobi equations with periodic boundary data

We study the large time behavior of viscosity solutions of Hamilton–Jacobi equations with periodic boundary data on bounded domains. We establish a result on convergence of viscosity solutions to state constraint asymptotic solutions or periodic asymptotic solutions depending on the sign of critical value as time goes to infinity.     

Yang–Mills Theory over Compact Symplectic Manifolds

In this paper, Yang–Mills theory over a compactKähler manifold is naturally extended to a compactsymplectic manifold. The relation betweenthe Yang–Mills equation and symplecticstructure is explicitly clarified, and the moduli spaceof Yang–Mills connections over a compactsymplectic manifold is constructed. Furthermore, theabsolute minima of the Yang–Mills functional arecharacterized, and finite dimensionality ofthe moduli space of the minimizers of the Yang–Millsfunctional is shown.     

A two-grid method based on Newton iteration for the Navier–Stokes equations

In this paper, we consider a two-grid method for resolving the nonlinearity in finite element approximations of the equilibrium Navier–Stokes equations. We prove the convergence rate of the approximation obtained by this method. The two-grid method involves solving one small, nonlinear coarse mesh system and two linear problems on the fine mesh which have the same stiffness matrix with only different right-hand side. The algorithm we study produces an approximate solution with the optimal asymptotic in h and accuracy for any Reynolds number. Numerical example is given to show the convergence of the method.     

Asymptotic solutions of a system of equations of second order which are concentrated in a neighborhood of a ray

By means of the ray method with a complex eikonal, formal asymptotic solutions are constructed of a system of equations of second order on a smooth Riemannian manifold which are concentrated in a neighborhood of a fixed ray.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 104, pp. 170–179, 1981.The author wishes to thank V. M. Babich and V. S. Buldyrev for their attention to the work.     

An a Posteriori Error Estimate for a Semi-Lagrangian Scheme for Hamilton–Jacobi Equations

We present an a posteriori estimate for a first order semi-Lagrangian method for Hamilton–Jacobi equations. The result requires piecewise C ^1,1 regularity of the viscosity solution and is stated for the Bellman equation related to the infinite horizon problem, although it can be applied to more general Hamilton–Jacobi equations with convex Hamiltonians. This estimate suggests different numerical indicators that can be used to construct an adaptive algorithm for the approximation of the viscosity solution.     

On the Asymptotic Solution of Non-Hamiltonian Systems Exhibiting Sustained Resonance

With the exception of some special examples, much of the literature on the formal construction of asymptotic solutions of systems exhibiting sustained resonance concerns Hamiltonian problems, for which the reduced problem is of order two when a single resonance is present. In the Hamiltonian case, the resonance manifold is a curve that is explicitly defined by the governing equations and is independent of the actual sustained resonance solution. When the basic standard form system is non-Hamiltonian, with M slow and N fast variables, the corresponding reduced problem is of order M + 1; in general it involves all of the slow variables, P₁,…, P_M, plus the resonant phase Q. In this paper, the solution of a general non-Hamiltonian system in standard form is formally constructed for the case of a single sustained resonance. First, a well-known example is reviewed, for which the projection of the solutions on the resonance manifold can be derived a priori, independent of the evolution of Q. Then, the general case is solved, using a generalization of the multiple scale method of Kuzmak-Luke, where knowledge of the asymptotic solution for Q (as well as higher-order terms) is needed to define the projection of the solution on the resonance manifold. The results simplify significantly when initial conditions are chosen exactly on the resonance manifold; the modifications necessary for arbitrary initial conditions are also given. Two examples are discussed in detail to illustrate the procedure. The asymptotic results are confirmed for several test cases by comparison with numerical integrations of the exact equations.     

Construction of Runge–Kutta methods of Crouch–Grossman type of high order

An approach is described to the numerical solution of order conditions for Runge–Kutta methods whose solutions evolve on a given manifold. This approach is based on least squares minimization using the Levenberg–Marquardt algorithm. Methods of order four and five are constructed and numerical experiments are presented which confirm that the derived methods have the expected order of accuracy.     

Existence and global bounds for a fluid model of plasma display technology

This article considers the fluid model for the discharge of plasma particle species in display technology. The fluid equations are coupled with Poisson's equation, which describes the effect of the charged particles on the electric field. The diffusion and mobility coefficients for the positive ion particles depend on the electric field, while those for the electrons depend on the electron mean energy. The reaction rates are proportional to the products of the densities of the reacting particles involved in the particular ionization, conversion or recombination reactions. Moreover, the ionization coefficients are dependent on the electric field, which varies spatially and temporally. The main ionization and discharge reactions are described by an initial-boundary value problem for a system of coupled parabolic–elliptic partial differential equations. The system is first analyzed by upper–lower solution method. By means of the a priori bounds obtained for an arbitrary time, the existence of solution for the initial-boundary value problem is proved in an appropriate Hölder space.     

Heat Flow for the Yang–Mills–Higgs Field and the Hermitian Yang–Mills–Higgs Metric

For a parameter > 0, we study a type of vortex equations, which generalize the well-known Hermitian–Einstein equation, for a connection A and a section of a holomorphic vector bundle E over a Kähler manifold X. We establish a global existence of smooth solutions to heat flow for a self-dual Yang–Mills–Higgs field on E. Assuming the -stability of (E, ), we prove the existence of the Hermitian Yang–Mills–Higgs metric on the holomorphic bundle E by studying the limiting behaviour of the gauge flow.     

Hypersurfaces with Prescribed Affine Gauss–Kronecker Curvature

We extend a procedure for solving particular fourth order PDEs by splitting them into two linked second order Monge–Ampère equations. We use this for the global study of Blaschke hypersurfaces with prescribed Gauss–Kronecker curvature.     

Variable-Order ESIRK Methods for Stiff Differential Equations

ESIRK methods (Effective order Singly-Implicit Runge–Kutta methods) have been shown to be efficient for the numerical solution of stiff differential equations. In this paper, we consider a new implementation of these methods with a variable order strategy. We show that the efficiency of the ESIRK method for stiff problems is improved by using the proposed variable order schemes.     

Iterative Solution of SPARK Methods Applied to DAEs

In this article a broad class of systems of implicit differential–algebraic equations (DAEs) is considered, including the equations of mechanical systems with holonomic and nonholonomic constraints. Solutions to these DAEs can be approximated numerically by applying a class of super partitioned additive Runge–Kutta (SPARK) methods. Several properties of the SPARK coefficients, satisfied by the family of Lobatto IIIA-B-C-C^*-D coefficients, are crucial to deal properly with the presence of constraints and algebraic variables. A main difficulty for an efficient implementation of these methods lies in the numerical solution of the resulting systems of nonlinear equations. Inexact modified Newton iterations can be used to solve these systems. Linear systems of the modified Newton method can be solved approximately with a preconditioned linear iterative method. Preconditioners can be obtained after certain transformations to the systems of nonlinear and linear equations. These transformations rely heavily on specific properties of the SPARK coefficients. A new truly parallelizable preconditioner is presented.