Local solvability of a nonstationary leakage problem for an ideal incompressible fluid, 3

In this paper we prove the existence and uniqueness of solutions of the leakage problem for the Euler equations in bounded domain Ω C R³ with corners π/n, n = 2, 3… We consider the case where the tangent components of the vorticity vector are given on the part S₁ of the boundary where the fluid enters the domain. We prove the existence of an unique solution in the Sobolev space W_p^l(Ω), for arbitrary natural l and p > 1. The proof is divided on three parts: (1) the existence of solutions of the elliptic problem in the domain with corners where v – velocity vector, ω – vorticity vector and n is an unit outward vector normal to the boundary, (2) the existence of solutions of the following evolution problem for given velocity vector (3) the method of successive approximations, using solvability of problems (1) and (2).     

Upper and lower bounds for higher order eigenvalues of some semilinear elliptic equations

We prove upper and lower bounds on the eigenvalues (as the norm of the eigenfunction tends to zero) in bifurcation problems for a class of semilinear elliptic equations in bounded domains of R^N. It is shown that these bounds are computable in terms of the eigenvalues of the associated linear equation.     

High resolution of 2D natural convection in a cavity by the DQ method

In this paper, the differential quadrature method (DQ) is applied to solve the benchmark problem of 2D natural convection in a cavity by utilizing the velocity–vorticity form of the Navier–Stokes equations, which is governed by the velocity Poisson equation, continuity equation and vorticity transport equation as well as energy equation. The coupled equations are simultaneously solved by imposing the vorticity definition at boundary without any iterative procedure. The present model is properly utilized to obtain results in the range of Rayleigh number (10³${10}^3$–10⁷${10}^7$) and H/L$H/L$ aspect ratios varying from 1 to 3. Nusselt numbers computed for 10³?Ra?10⁷${10}^3?\text{<italic>Ra</italic>}?{10}^7$ in a cavity show excellent agreement with the results available in the literature. Additionally, the detailed features of flow phenomena such as velocity, temperature, vorticity, and streamline plots are also delineated in this work. Thus, it is convinced that the DQ method is capable of solving coupled differential equations efficiently and accurately.     

Approximation of operators with reproducing nonlinearities

We consider an equation of the form Au+N(u)=u in a Hilbert space and assume that the nonlinearity N is reproducing relative to a known sequence of vectors. Under this assumption the Rayleigh-Ritz-Galerkin approximations lead to a simple class of nonlinear algebraic eigenvalue problems.In a general variational case we show that Lusternik-Schnirelmann critical values of Rayleigh-Ritz-Galerkin problems provide upper bounds to those of the original problem. Lower bounds are constructed in the case N(u)=B^*(Bu)³.The author would like to thank the European Research Office for their assistance in this research.     

Asymptotic behavior of the solutions of certain parabolic equations

Some laws in physics describe the change of a flux and are represented by parabolic equations of the form (*) \documentclass{article}\pagestyle{empty}\begin{document}$$\frac{{\partial u}}{{\partial t}}=\frac{\partial}{{\partial x_j }}(\eta \frac{{\partial u}}{{ax_j}}-vju),$$\end{document} j≤m, where η and v_j are functions of both space and time. We show under quite general assumptions that the solutions of equation (*) with homogeneous Dirichlet boundary conditions and initial condition u(x, 0) = u_o(x) satisfy The decay rate d > 0 only depends on bounds for η, v and G § R^m the spatial domain, while the constant c depends additionally on which norm is considered. For the solutions of equation (*) with homogeneous Neumann boundary conditions and initial condition u₀(x) ≥ 0 we derive bounds d₁u₁ ≤ u(x, t) ≤ d₂u₂, Where d_i, i = 1, 2, depend on bounds for η, v and G, and the u_i are bounds on the initial condition u₀.     

Componentwise error analysis for FFTs with applications to fast Helmholtz solvers

We analyze the stability of the Cooley-Tukey algorithm for the Fast Fourier Transform of ordern=2 ^k and of its inverse by using componentwise error analysis.We prove that the components of the roundoff errors are linearly related to the result in exact arithmetic. We describe the structure of the error matrix and we give optimal bounds for the total error in infinity norm and inL ₂ norm.The theoretical upper bounds are based on a worst case analysis where all the rounding errors work in the same direction. We show by means of a statistical error analysis that in realistic cases the max-norm error grows asymptotically like the logarithm of the sequence length by machine precision.Finally, we use the previous results for introducing tight upper bounds on the algorithmic error for some of the classical fast Helmholtz equation solvers based on the Faster Fourier Transform and for some algorithms used in the study of turbulence.     

Riesz transform,Gaussian bounds and the method of wave equation

For an abstract self-adjoint operator L and a local operator A we study the boundedness of the Riesz transform AL^– on L^p for some > 0. A very simple proof of the obtained result is based on the finite speed propagation property for the solution of the corresponding wave equation. We also discuss the relation between the Gaussian bounds and the finite speed propagation property. Using the wave equation methods we obtain a new natural form of the Gaussian bounds for the heat kernels for a large class of the generating operators. We describe a surprisingly elementary proof of the finite speed propagation property in a more general setting than it is usually considered in the literature.As an application of the obtained results we prove boundedness of the Riesz transform on L^p for all p (1,2] for Schrödinger operators with positive potentials and electromagnetic fields. In another application we discuss the Gaussian bounds for the Hodge Laplacian and boundedness of the Riesz transform on L^p of the Laplace-Beltrami operator on Riemannian manifolds for p > 2.Mathematics Subject Classification (1991): 42B20The author was partially supported by Summer Research Award from New Mexico State University.in final form: 8 June 2003     

Stabilization of solutions of initial-boundary problems for quasilinear parabolic equations

Lower bounds are obtained for solutions of the initial-boundary Dirichlet problem for high order equations. Sharp bounds are also obtained for ess sup¦u(x, t)¦ of the Neumann initial-boundary problem for a second- order equation in D=x(t >0), where (Rⁿ, n 2 is a domain with noncompact convex boundary.Translated from Ukrainskii Maternaticheskii Zhurnal, Vol. 44, No. 10, pp. 1441–1450, October, 1992.     

Existence theorem and blow-up criterion of the strong solutions to the magneto-micropolar fluid equations

In this paper we study the magneto-micropolar fluid equations in ℝ³, prove the existence of the strong solution with initial data in H^s(ℝ³) for , and set up its blow-up criterion. The tool we mainly use is Littlewood–Paley decomposition, by which we obtain a Beale–Kato–Majda-type blow-up criterion for smooth solution (u, ω, b) that relies on the vorticity of velocity ∇ × u only. Copyright © 2007 John Wiley & Sons, Ltd.     

Exponential growth of the vorticity gradient for the Euler equation on the torus

We prove that there are solutions to the Euler equation on the torus with C^1,α$C^{1,\alpha }$ vorticity and smooth except at one point such that the vorticity gradient grows in L^∞$L^{\infty }$ at least exponentially as t→∞$t\to \infty$. The same result is shown to hold for the vorticity Hessian and smooth solutions. Our proofs use a version of a recent result by Kiselev and Šverák [6].     

Propagation of oscillations to 2D incompressible Euler equations

The asymptotic expansions are studied for the vorticity to 2D incompressible Euler equations with-initial vorticity , where ϕ₀(x) satisfies |d ϕ₀(x)|≠0 on the support of and is sufficiently smooth and with compact support in ℝ² (resp. ℝ²×T) The limit,v(t,x), of the corresponding velocity fields {v ^ɛ(t,x)} is obtained, which is the unique solution of (E) with initial vorticity ω₀(x). Moreover, (ℤ²)) for all 1≽p∞, where and ϕ(t,x) satisfy some modulation equation and eikonal equation, respectively.     

Low‐curvature image simplifiers: Global regularity of smooth solutions and Laplacian limiting schemes

We consider a class of fourth‐order nonlinear diffusion equations motivated by Tumblin and Turk's “low‐curvature image simplifiers” for image denoising and segmentation. The PDE for the image intensity u is of the form where g(s) = k²/(k² + s²) is a “curvature” threshold and λ denotes a fidelity‐matching parameter. We derive a priori bounds for Δu that allow us to prove global regularity of smooth solutions in one space dimension, and a geometric constraint for finite‐time singularities from smooth initial data in two space dimensions. This is in sharp contrast to the second‐order Perona‐Malik equation (an ill‐posed problem), on which the original LCIS method is modeled. The estimates also allow us to design a finite difference scheme that satisfies discrete versions of the estimates, in particular, a priori bounds on the smoothness estimator in both one and two space dimensions. We present computational results that show the effectiveness of such algorithms. Our results are connected to recent results for fourth‐order lubrication‐type equations and the design of positivity‐preserving schemes for such equations. This connection also has relevance for other related fourth‐order imaging equations. © 2004 Wiley Periodicals, Inc.     

Vortex energy and 360° Néel walls in thin‐film micromagnetics

We study the vortex pattern in ultrathin ferromagnetic films of circular crosssection. The model is based on the following energy functional: for in‐plane magnetizations m: B² → S¹ in the unit disc . The avoidance of volume charges ? · m ≠ 0 in B² and surface charges m · ν ≠ 0 on δB² leads to the formation of a vortex in the limit ε → 0. At the level ε > 0 the vortex is regularized by the formation of a 360° Néel wall (a one‐dimensional transition layer with core of scale ε) concentrated along a radius of B². We derive the limiting energy of the vortex by matching upper and lower bounds. Our analysis on the lower bound is based on a dynamical system argument and an interpolation inequality with sharp leading‐order constant, while the upper bound uses the leading‐order energy for 360° Néel walls. © 2010 Wiley Periodicals, Inc.     

Some bounds for the deviation and interpolation points in Chebyshev approximation

Some bounds are given for the deviation and interpolation points of a functionf based on bounds forf ⁽ⁿ⁺²⁾/f ⁽ⁿ⁺¹⁾. These bounds are in terms of the deviation and interpolation points ofe ^vx , –1 x 1, wherev is a parameter. The behavior of these points asv is also discussed.The results of this paper are contained in the second author's Master's thesis submitted to the University of Wyoming in May 1969.     

Lower Bounds for n-Term Approximations of Plane Convex Sets and Related Topics

In this paper, we establish lower bounds for n-term approximations in the metric of L ²⁽I ² ) of characteristic functions of plane convex subsets of the square I ² with respect to arbitrary orthogonal systems. It is shown that, as n, these bounds cannot decrease more rapidly than .     

Cones and error bounds for linear iterations

Consider an ordered Banach space with a cone of positive elementsK and a norm ∥·∥. Let [a,b] denote an order-interval; under mild conditions, ifx*∈[a,b] then $$||x * - \tfrac{1}{2}(a + b)|| \leqslant \tfrac{1}{2}||b - a||.$$ This inequality is used to generate error bounds in norm, which provide on-line exit criteria, for iterations of the type $$x_r = Ax_{r - 1} + a,A = A^ + + A^ - ,$$ whereA ⁺ andA ^? are bounded linear operators, withA ⁺ K ?K andA ^? K ? ?K. Under certain conditions, the error bounds have the form $$\begin{gathered} ||x * - x_r || \leqslant ||y_r ||,y_r = (A^ + - A^ - )y_{r - 1} , \hfill \\ ||x * - x_r || \leqslant \alpha ||\nabla x_r ||, \hfill \\ ||x * - \tfrac{1}{2}(x_r + x_{r - 1} )|| \leqslant \tfrac{1}{2}||\nabla x_r ||. \hfill \\ \end{gathered} $$ These bounds can be used on iterative methods which result from proper splittings of rectangular matrices. Specific applications with respect to certain polyhedral cones are given to the classical Jacobi and Gauss-Seidel splittings.     

Improved upper bounds for the critical probability of oriented percolation in two dimensions

We refine the method of our previous paper [2] which gave upper bounds for the critical probability in two-dimensional oriented percolation. We use our refinement to show that © 1994 John Wiley & Sons, Inc.     

A priori estimates for superlinear and subcritical elliptic equations: the Neumann boundary condition case

We consider here solutions of a nonlinear Neumann elliptic equation Δu +?f (x, u) =?0 in Ω, ?u/?ν =?0 on ?Ω, where Ω is a bounded open smooth domain in ${\mathbb{R}^N, N\geq2}$ and f satisfies super-linear and subcritical growth conditions. We prove that L ^∞?bounds on solutions are equivalent to bounds on their Morse indices.     

Critical Points of Real Polynomials and Topology of Real Algebraic T-Surfaces

The paper is devoted to a special class of real polynomials, so-called T-polynomials, which arise in the combinatorial version of the Viro theorem. We study the relation between the numbers of real critical points of a given index of a T-polynomial and the combinatorics of lattice triangulations of Newton polytopes. We obtain upper bounds for the numbers of extrema and saddles of generic T-polynomials of a given degree in three variables, and derive from them upper bounds for Betti numbers of real algebraic surfaces in defined by T-polynomials. The latter upper bounds are stronger than the known upper bounds for arbitrary real algebraic surfaces in . Another result is the existence of an asymptotically maximal family of real polynomials of degree min three variables with 31m ³/36 + O(m ²) saddle points.     

Eigenvalue bounds for elliptic operators

Lower bounds for the real parts of the points in the spectrum of elliptic equations are derived. These bounds, depending only on the diameter L of the domain G and on the maximum norm M of the coefficients a, b, are optimal. They are always positive and thus the spectrum is bounded away from the imaginary axis. This result is then used to prove an “anti-dynamo theorem” for magnetic fields with plane symmetry in the case of a compressible steady flow surrounded by a perfect conductor.