Corrigendum to “Discrepancy principle for the dynamical systems method” [Communications in Nonlinear Science and Numerical Simulation 10 (2005) 95–101]

Consider an operator equation B(u) − f = 0 in a real Hilbert space. Let us call this equation ill-posed if the operator B′(u) is not boundedly invertible, and well-posed otherwise. The dynamical systems method (DSM) for solving this equation consists of a construction of a Cauchy problem, which has the following properties: (1) it has a global solution for an arbitrary initial data, (2) this solution tends to a limit as time tends to infinity, (3) the limit is the minimal-norm solution to the equation B(u) = f. A global convergence theorem is proved for DSM for equation B(u) − f = 0 with monotone operators B.     

Asymptotic behavior of solutions to anisotropic conservation laws in two-dimensional space

In this paper, we investigate the asymptotic behavior of solutions for anisotropic conservation laws in two-dimensional space, provided with step-like initial conditions that approach the constant states u_± (u₋<u₊) as x→±∞, respectively. It shows that there is a global classical solution that converges toward the rarefaction wave, ie, the unique entropy solution of the Riemann problem for the nonviscous Burgers' equation in one-dimensional space.     

Two-Scale Convergence Method and Scattering Problems

In this paper, we deal with the steady-state acoustic wave equation in the space ℝ³ diffracted by an obstacle made by an inhomogeneous medium and located in a bounded domain. The inhomogeneity of the medium depends on a parameter ε > 0. If the solution u _ε converges to a solution u ₀ of the limit problem as ε → 0, as in the homogenization process, then we can use the two-scale convergence method to study the convergence of the gradient.__________Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 10, Suzdal Conference-4, 2003.     

An analysis for a high‐order difference scheme for numerical solution to uxx = F(x,t, u,ut, ux)

This article is concerned with a high‐order difference scheme presented by Jain, Jain, and Mohanty for the nonlinear parabolic equation u_xx = F(x, t, u, u_t, u_x) with Dirichlet boundary conditions. The solvability of the difference scheme is proved by Brower's fixed point theorem and the uniqueness of the difference solution is obtained by showing that the coefficient matrix is strictly column‐wise diagonal dominant. The boundedness and convergence of the difference scheme are obtained. The convergence order is 4 in space and 2 in time in L_∞‐norm. A numerical example is provided to illustrate the validity of the theoretical results. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq , 2006     

Dynamical Systems Method of gradient type for solving nonlinear equations with monotone operators

A version of the Dynamical Systems Method (DSM) of gradient type for solving equation F(u)=f where F:H→H is a monotone Fréchet differentiable operator in a Hilbert space H is studied in this paper. A discrepancy principle is proposed and the convergence to the minimal-norm solution is justified. Based on the DSM an iterative scheme is formulated and the convergence of this scheme to the minimal-norm solution is proved.     

Solving the generalized regularized long wave equation on the basis of a reproducing kernel space

On the basis of a reproducing kernel space, an iterative algorithm for solving the generalized regularized long wave equation is presented. The analytical solution in the reproducing kernel space is shown in a series form and the approximate solution u_n is constructed by truncating the series to n terms. The convergence of u_n to the analytical solution is also proved. Results obtained by the proposed method imply that it can be considered as a simple and accurate method for solving such evolution equations.     

New bounds for the Kuramoto‐Sivashinsky equation

We show that every L‐periodic mean‐zero solution u of the Kuramoto‐Sivashinsky equation is on average o(L) for L ? 1, in the sense that for any T > 0 the space average of | u(t) | is bounded by for any t > T and any L sufficiently large. For this we argue that on large spatial scales, the solution behaves like an entropy solution of the inviscid Burgers equation. The analysis of this non‐standard perturbation of the Burgers equation is based on a “div‐curl” argument. © 2004 Wiley Periodicals, Inc.     

Homogenization of Hamilton‐Jacobi‐Bellman equations with respect to time‐space shifts in a stationary ergodic medium

We consider a family {u_? (t, x, ω)}, ? < 0, of solutions to the equation ?u_?/?t + ?Δu_?/2 + H (t/?, x/?, ?u_?, ω) = 0 with the terminal data u_?(T, x, ω) = U(x). Assuming that the dependence of the Hamiltonian H(t, x, p, ω) on time and space is realized through shifts in a stationary ergodic random medium, and that H is convex in p and satisfies certain growth and regularity conditions, we show the almost sure locally uniform convergence, in time and space, of u_?(t, x, ω) as ? → 0 to the solution u(t, x) of a deterministic averaged equation ?u/?t + H?(?u) = 0, u(T, x) = U(x). The “effective” Hamiltonian H? is given by a variational formula. © 2007 Wiley Periodicals, Inc.     

Unique solvability of the inverse problem of determination of the leading coefficient in a parabolic equation

We study existence and uniqueness of the solution for the inverse problem of determination of the unknown coefficient ϱ(t) multiplying u _t in a nondivergence parabolic equation. As additional information, the integral of the solution over the domain of space variables with some given weight function is specified. The coefficients of the equation depend both on time and on the space variables.     

Mathematical and numerical study of a system of conservation laws

The system of equations (f (u))_t − (a(u)v + b(u))_x = 0 and u_t − (c(u)v + d(u))_x = 0, where the unknowns u and v are functions depending on , arises within the study of some physical model of the flow of miscible fluids in a porous medium. We give a definition for a weak entropy solution (u, v), inspired by the Liu condition for admissible shocks and by Krushkov entropy pairs. We then prove, in the case of a natural generalization of the Riemann problem, the existence of a weak entropy solution only depending on x/t. This property results from the proof of the existence, by passing to the limit on some approximations, of a function g such that u is the classical entropy solution of u_t − ((cg + d)(u))_x = 0 and simultaneously w = f (u) is the entropy solution of w_t − ((ag + b)(f⁽⁻¹⁾(w)))_x = 0. We then take v = g(u), and the proof that (u, v) is a weak entropy solution of the coupled problem follows from a linear combination of the weak entropy inequalities satisfied by u and f (u). We then show the existence of an entropy weak solution for a general class of data, thanks to the convergence proof of a coupled finite volume scheme. The principle of this scheme is to compute the Godunov numerical flux with some interface functions ensuring the symmetry of the finite volume scheme with respect to both conservation equations.     

A Model Equation Illustrating Subcritical Instability to Long Waves in Shear Flows

A model equation somewhat more general than Burger's equation has been employed by Herron [1] to gain insight into the stability characteristics of parallel shear flows. This equation, namely, u_t + uu_y = u_xx + u_yy, has an exact solution U(y) = ?2tanh y. It was shown in [1] that this solution is linearly stable, and more recently, Galdi and Herron [3] have proved conditional stability to finite perturbations of sufficiently small initial amplitude using energy methods. The present study utilizes multiple-scaling methods to derive a nonlinear evolution equation for a long-wave perturbation whose amplitude varies slowly in space and time. A transformation to the heat-conduction equation has been found which enables this amplitude equation to be solved exactly. Although all disturbances ultimately decay due to diffusion, it is found that subcritical instability is possible in that realistic disturbances of finite initial amplitude can amplify substantially before finally decaying. This behavior is probably typical of perturbations to shear flows of practical interest, and the results illustrate deficiencies of the energy method.     

Stability and convergence of the spectral Galerkin method for the Cahn‐Hilliard equation

A spectral Galerkin method in the spatial discretization is analyzed to solve the Cahn‐Hilliard equation. Existence, uniqueness, and stabilities for both the exact solution and the approximate solution are given. Using the theory and technique of a priori estimate for the partial differential equation, we obtained the convergence of the spectral Galerkin method and the error estimate between the approximate solution u_N(t) and the exact solution u(t). © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

A Dimensional Splitting Method for Quasilinear Hyperbolic Equations with Variable Coefficients

A numerical method is presented for the variable coefficient, nonlinear hyperbolic equation u _t + _i=1 ^d V _i(x, t)f _i(u) _{x i} = 0 in arbitrary space dimension for bounded velocities that are Lipschitz continuous in the x variable. The method is based on dimensional splitting and uses a recent front tracking method to solve the resulting one-dimensional non-conservative equations. The method is unconditionally stable, and it produces a subsequence that converges to the entropy solution as the discretization of time and space tends to zero. Four numerical examples are presented; numerical error mechanisms are illustrated for two linear equations, the efficiency of the method compared with a high-resolution TVD method is discussed for a nonlinear problem, and finally, applications to reservoir simulation are presented.     

Two inverse problems of finding a coefficient in a parabolic equation

For a parabolic equation, we consider inverse problems of reconstructing a coefficient that depends on the space variables alone. The first problem is to find a lower-order coefficient c(x) multiplying u(x, t), and the second problem is to find the coefficient a(x) multiplying Δu. As additional information, the integral of the solution with respect to time with some weight function is given. The coefficients of the equation depend both on time and on the space variables. We obtain sufficient conditions for the existence of generalized solutions of our problems; moreover, for the first problem, we also prove uniqueness and construct an iterative sequence that converges to the desired coefficient almost everywhere in the domain. We present examples of input data of these problems for which the assumptions of our theorems are necessarily true.     

Global existence to a reaction–diffusion equation through a self‐similar‐like upper solution

A demonstration method is presented, which will ensure the existence of positive global solutions in time to the reaction–diffusion equation ?u_t+Δu+u^p=0 in ?ⁿ×[0, ∞), for exponents p?3 and space dimensions n?3. This method does not require the initial value to have a specific uniform smallness condition, but rather to satisfy a bell‐like form. The method is based on a specific upper solution, which models the diffusion process of the heat equation. The upper solution is not self‐similar, but does have a self‐similar‐like form. After transforming the reaction–diffusion problem into an equivalent one, whose initial value is uniformly very small, a local solution is obtained in the time interval [0, 1] by the use of this upper solution. This local solution is then extended to [0, ∞) through an infinite sequence of extensions. At each step, an appropriate change of variables will transform the extension into a problem nearly identical to the local problem in [0, 1]. These transformations exploit the diffusive and self‐similar‐like nature of the upper solution. Copyright © 2009 John Wiley & Sons, Ltd.     

An O(k2 + kh2 + h4) arithmetic average discretization for the solution of 1‐D nonlinear parabolic equations

This article develops a new two‐level three‐point implicit finite difference scheme of order 2 in time and 4 in space based on arithmetic average discretization for the solution of nonlinear parabolic equation ε u_xx = f(x, t, u, u_x, u_t), 0 < x < 1, t > 0 subject to appropriate initial and Dirichlet boundary conditions, where ε > 0 is a small positive constant. We also propose a new explicit difference scheme of order 2 in time and 4 in space for the estimates of (?u/?x). The main objective is the proposed formulas are directly applicable to both singular and nonsingular problems. We do not require any fictitious points outside the solution region and any special technique to handle the singular problems. Stability analysis of a model problem is discussed. Numerical results are provided to validate the usefulness of the proposed formulas. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Nondeterminism of Linear Operators and Lower Entropy Estimates

Let u be a (bounded, linear) operator from a Hilbert space ℋ into the Banach space C(T), the space of continuous functions on the compact metric space T. We introduce and investigate numbers τ _n (u), n≥1, measuring the degree of determinism of the operator u. The slower τ _n (u) decreases, the less determined are functions in the range of u by their values on a certain set of points. It is shown that n ^−1/2 τ _n (u)≤2e _n (u), where e _n (u) are the (dyadic) entropy numbers of u. Furthermore, we transform the notion of strong local nondeterminism from the language of stochastic processes into that of linear operators. This property, together with a lower entropy estimate for the compact space T, leads to a lower estimate for τ _n (u), hence also for e _n (u). These results are used to prove sharp lower entropy estimates for some integral operators, among them, Riemann–Liouville operators with values in C(T) for some fractal set T. Some multi-dimensional extensions are treated as well.      

Monte-Carlo approximations for 2d Navier-Stokes equations with measure initial data

We are interested in proving Monte-Carlo approximations for 2d Navier-Stokes equations with initial data u ₀ belonging to the Lorentz space L ^2,∞ and such that curl u ₀ is a finite measure. Giga, Miyakawaand Osada [7] proved that a solution u exists and that u=K* curl u, where K is the Biot-Savartkernel and v = curl u is solution of a nonlinear equation in dimension one, called the vortex equation. In this paper, we approximate a solution v of this vortex equationby a stochastic interacting particlesystem and deduce a Monte-Carlo approximation for a solution of the Navier-Stokesequation. That gives in this case a pathwise proofof the vortex algorithm introducedby Chorin and consequently generalizes the works ofMarchioro-Pulvirenti [12] and Méléardv [15] obtained in the case of a vortex equation with bounded density initial data. Received: 6 October 1999 / Revised version: 15 September 2000 / Published online: 9 October 2001     

Stokes and Navier–Stokes equations with nonhomogeneous boundary conditions

In this paper, we study the existence and regularity of solutions to the Stokes and Oseen equations with nonhomogeneous Dirichlet boundary conditions with low regularity. We consider boundary conditions for which the normal component is not equal to zero. We rewrite the Stokes and the Oseen equations in the form of a system of two equations. The first one is an evolution equation satisfied by Pu, the projection of the solution on the Stokes space – the space of divergence free vector fields with a normal trace equal to zero – and the second one is a quasi-stationary elliptic equation satisfied by (I−P)u, the projection of the solution on the orthogonal complement of the Stokes space. We establish optimal regularity results for Pu and (I−P)u. We also study the existence of weak solutions to the three-dimensional instationary Navier–Stokes equations for more regular data, but without any smallness assumption on the initial and boundary conditions.     

On the Transfer of Generalized Functions by an Evolution Flow

We investigate properties of a solution of a stochastic differential equation with interaction and their dependence on a space variable. It is shown that x(u, t) − u belongs to S under certain conditions imposed on the coefficients, and, furthermore, it depends continuously on the initial measure as an element of S. We also study the problem of the existence of a solution of the equation governed by a generalized function. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 8, pp. 1020 – 1029, August, 2005.