Methods and Applications of Analysis Articles (Project Euclid)

Lyapunov's Functions and Existence of Integral Manifolds for Impulsive Differental Systems with Time-varying Delay

Thu, 05 Aug 2010 15:41 EDT

Gani Stamov

Source: Methods Appl. Anal., Volume 16, Number 3, 291--298.

Abstract:
In this paper the existence of integral manifolds for impulsive differential systems with time-varying delay and with impulsive effect at fixed moments are investigated. The main results are obtained by using of piecewise continuous Lyapunov’s functions and Razumikhin’s technique.

Completeness of Eigenfunctions of Sturm-Liouville Problems with Transmission Conditions

Thu, 05 Aug 2010 15:41 EDT

Aiping Wang, Jiong Sun, Xiaoling Hao, Siqin Yao

Source: Methods Appl. Anal., Volume 16, Number 3, 299--312.

Abstract:
In this paper, we investigate a class of Sturm-Liouville problems with eigenparameter-dependent boundary conditions and transmission conditions at an interior point. A self-adjoint linear operator $A$ is defined in a suitable Hilbert space $H $such that the eigenvalues of such a problem coincide with those of $A$. We show that the operator $A$ has only point spectrum, the eigenvalues of the problem are algebraically simple, and the eigenfunctions of $A$ are complete in $H$.

Local Time Decay for a Quasilinear Schrodinger Equation

Thu, 05 Aug 2010 15:41 EDT

J. E. Lin

Source: Methods Appl. Anal., Volume 16, Number 3, 313--320.

Abstract:
We study the solutions of a quasilinear Schrödinger equation which has been derived in many areas of physical modeling. Using the Morawetz Radial Identity, we show that the local energy of a solution is integrable in time and the local $L^2$ norm of the solution approaches zero as time approaches the infinity.

Concentration in Lotka-Volterra Parabolic or Integral Equations: A General Convergence Result

Thu, 05 Aug 2010 15:41 EDT

Guy Barles, Sepideh Mirrahimi, Benoît Perthame

Source: Methods Appl. Anal., Volume 16, Number 3, 321--340.

Abstract:
We study two equations of Lotka-Volterra type that describe the Darwinian evolution of a population density. In the first model a Laplace term represents the mutations. In the second one we model the mutations by an integral kernel. In both cases, we use a nonlinear birth-death term that corresponds to the competition between the traits leading to selection. In the limit of rare or small mutations, we prove that the solution converges to a sum of moving Dirac masses. This limit is described by a constrained Hamilton-Jacobi equation. This was already proved in Dirac concentrations in "Lotka-Volterra parabolic PDEs", Indiana Univ. Math. J. , 57:7 (2008), pp. 3275–3301, for the case with a Laplace term. Here we generalize the assumptions on the initial data and prove the same result for the integro-differential equation.

A Paradigm for Time-periodic Sound Wave Propagation in the Compressible Euler Equations

Thu, 05 Aug 2010 15:41 EDT

Blake Temple, Robin Young

Source: Methods Appl. Anal., Volume 16, Number 3, 341--364.

Abstract:
We formally derive the simplest possible periodic wave structure consistent with time-periodic sound wave propagation in the $3 × 3$ nonlinear compressible Euler equations. The construction is based on identifying the simplest periodic pattern with the property that compression is counter-balanced by rarefaction along every characteristic. Our derivation leads to an explicit description of shock-free waves that propagate through an oscillating entropy field without breaking or dissipating, indicating a new mechanism for dissipation free transmission of sound waves in a nonlinear problem. The waves propagate at a new speed, (different from a shock or sound speed), and sound waves move through periods at speeds that can be commensurate or incommensurate with the period. The period determines the speed of the wave crests, (a sort of observable group velocity), but the sound waves move at a faster speed, the usual speed of sound, and this is like a phase velocity. It has been unknown since the time of Euler whether or not time-periodic solutions of the compressible Euler equations, which propagate like sound waves, are physically possible, due mainly to the ubiquitous formation of shock waves. A complete mathematical proof that waves with the structure derived here actually solve the Euler equations exactly, would resolve this long standing open problem.

Parabolic and Elliptic Systems with VMO Coefficients

Thu, 05 Aug 2010 15:41 EDT

Hongjie Dong, Doyoon Kim

Source: Methods Appl. Anal., Volume 16, Number 3, 365--388.

Abstract:
We consider second order parabolic and elliptic systems with leading coefficients having the property of vanishing mean oscillation (VMO) in the spatial variables. An $L_q − L_p$ theory is established for systems both in divergence and non-divergence form.

Convergence Rate to the Nonlinear Waves for Viscous Conservation Laws on the Half Line

Thu, 05 Aug 2010 15:41 EDT

Itsuko Hashimoto, Yoshihiro Ueda, Shuichi Kawashima

Source: Methods Appl. Anal., Volume 16, Number 3, 389--402.

On the Convergence of Fully-discrete High-Resolution Schemes with van Leer's Flux Limiter for Conservation Laws

Thu, 05 Aug 2010 15:41 EDT

Nan Jiang

Source: Methods Appl. Anal., Volume 16, Number 3, 403--422.

Abstract:
A class of fully-discrete high-resolution schemes using flux limiters was constructed by P. K. Sweby, which amounted to add a limited anti-diffusive flux to a first order scheme. This technique has been very successful in obtaining high-resolution, second order, oscillation free, explicit difference schemes. However, the entropy convergence of such schemes has been open. For the scalar convex conservation laws, we use one of Yang’s convergence criteria to show the entropy convergence of the schemes with van Leer’s flux limiter when the building block of the schemes is the Godunov or the Engquish-Osher. The entropy convergence of the corresponding problems in semi-discrete case, for convex conservation laws with or without a source term, has been settled by Jiang and Yang.

Cascade of Phase Shifts and Creation of Nonlinear Focal Points for Supercritical Semiclassical Hartree Equation

Tue, 12 Oct 2010 09:43 EDT

Satoshi Masaki

Source: Methods Appl. Anal., Volume 16, Number 4, 403--458.

Abstract:
We consider the semiclassical limit of the Hartree equation with a data causing a focusing at a point. We study the asymptotic behavior of phase function associated with the WKB approximation near the caustic when a nonlinearity is supercritical. In this case, it is known that a phase shift occurs in a neighborhood of focusing time in the case of focusing cubic nonlinear Schrödinger equation. Thanks to the smoothness of the nonlocal nonlinearities, we justify the WKB-type approximation of the solution for a data which is larger than in the previous results and is not necessarily well-prepared. We also show by an analysis of the limit hydrodynamical equaiton that, however, this WKB-type approximation breaks down before reaching the focal point: Nonlinear effects lead to the formation of singularity of the leading term of the phase function.

Asymptotic Behavior of Oscillating Radial Solutions to Certain Nonlinear Equations, Part II

Tue, 12 Oct 2010 09:43 EDT

Changfeng Gui, Xue Luo, Feng Zhou

Source: Methods Appl. Anal., Volume 16, Number 4, 459--468.

Secant-like Method for Solving Generalized Equations

Tue, 12 Oct 2010 09:43 EDT

Ioannis K. Argyros, Saïd Hilout

Source: Methods Appl. Anal., Volume 16, Number 4, 469--478.

Abstract:
In A Kantorovich–type analysis for a fast iterative method for solving nonlinear equations , and Convergence and applications of Newton–type iterations , Argyros introduced a new derivative–free quadratically convergent method for solving a nonlinear equation in Banach space. In this paper, we extend this method to generalized equations in order to approximate a locally unique solution. The method uses only divided differences operators of order one. Under some Lipschitz–type conditions on the first and second order divided differences operators and Lipschitz–like property of set–valued maps, an existence–convergence theorem and a radius of convergence are obtained. Our method has the following advantages: we extend the applicability of this method than all the previous ones, and we do not need to evaluate any Fréchet derivative. We provide also an improvement on the radius of convergence for our algorithm, under some center–condition and less computational cost. Numerical examples are also provided.

On Breakdown of Solutions to the Full Compressible Navier-Stokes Equations

Tue, 12 Oct 2010 09:43 EDT

Xiangdi Huang, Jing Li

Source: Methods Appl. Anal., Volume 16, Number 4, 479--490.

A Blowup Criterion for the Full Compressible Navier-Stokes Equations

Tue, 12 Oct 2010 09:43 EDT

Xiangdi Huang

Source: Methods Appl. Anal., Volume 16, Number 4, 491--506.

Abstract:
In this paper, we establish a blow up criterion for strong solutions of the full compressible Navier-Stokes equations just in terms of the gradient of the velocity. It shows that the gradient of the velocity alone dominates the global existence of strong solutions.

Bounds on Front Speeds for Inviscid and Viscous G-equations

Tue, 12 Oct 2010 09:43 EDT

James Nolen, Jack Xin, Yifeng Yu

Source: Methods Appl. Anal., Volume 16, Number 4, 507--520.

Spectrum and Functions of Operators on Direct Families of Banach Spaces

Tue, 12 Oct 2010 09:43 EDT

M. I. Gil

Source: Methods Appl. Anal., Volume 16, Number 4, 521--534.

Singular Support of the Scattering Kernel for the Rayleigh Wave in Perturbed Half-Spaces

Mon, 06 Dec 2010 09:10 EST

Mishio Kawashita, Hideo Soga

Source: Methods Appl. Anal., Volume 17, Number 1, 1--48.

Abstract:
This paper deals with the Rayleigh wave scattering on perturbed half-spaces in the framework of the Lax-Phillips type. Singular parts of the scattering kernel for this scattering are closely connected with singularities of the Rayleigh wave passing through the perturbation on the boundary. This can be described by estimating the singular support of the scattering kernel on the Rayleigh wave channel. The proof is based on a representation formula of the scattering kernel that was obtained in the previous work. However, the formula does not suit the situation of the Rayleigh wave, even though it is a natural extension of Majda’s formula for the usual wave equation. Hence, the formula needs to be reformed, and the problem needs to be reduced to a pseudo-differential equation on the boundary governing the Rayleigh wave. Key methods for the reduced problem are construction of an approximate solution for the Rayleigh wave and analysis of an oscillatory integral distilled by using the solution. The phase function of the oscillatory integral is always degenerate along the characteristic curve of the Rayleigh wave. This degeneracy is handled by introducing a certain criterion for the regularity of the distribution defined by the oscillatory integral.

Dynamical Laws of the Coupled Gross-Pitaevskii Equations for Spin-1 Bose-Einstein Condensates

Mon, 06 Dec 2010 09:10 EST

Weizhu Bao, Yanzhi Zhang

Source: Methods Appl. Anal., Volume 17, Number 1, 49--80.

Abstract:
In this paper, we derive analytically the dynamical laws of the coupled Gross- Pitaevskii equations (CGPEs) without/with an angular momentum rotation term and an external magnetic field for modelling nonrotating/rotating spin-1 Bose-Eintein condensates. We prove the conservation of the angular momentum expectation when the external trapping potential is radially symmetric in two dimensions and cylindrically symmetric in three dimensions; obtain a system of first order ordinary differential equations (ODEs) governing the dynamics of the density of each component and solve the ODEs analytically in a few cases; derive a second order ODE for the dynamics of the condensate width and show that it is a periodic function without/with a perturbation; construct the analytical solution of the CGPEs when the initial data is chosen as a stationary state with its center- of-mass shifted away from the external trap center. Finally, these dynamical laws are confirmed by the direct numerical simulation results of the CGPEs.

Smoothness Criterion for Navier-Stokes Equations in Terms of Regularity along the Streamlines

Mon, 06 Dec 2010 09:10 EST

Chi Hin Chan

Source: Methods Appl. Anal., Volume 17, Number 1, 81--104.

On Bellman's Equations with VMO Coefficients

Mon, 06 Dec 2010 09:10 EST

Nicolai V. Krylov

Source: Methods Appl. Anal., Volume 17, Number 1, 105--122.

On the Differential Operators of the Generalized Fifth-order Korteweg-de Vries Equation

Mon, 06 Dec 2010 09:10 EST

Chun-Te Lee

Source: Methods Appl. Anal., Volume 17, Number 1, 123--136.

Abstract:
In this paper, we present the differential operators of the generalized fifth-order KdV equation. We give formal proofs on the Hamiltonian property including the skew-adjoint property and Jacobi identity by the use of prolongation method. Our results show that there are five 3-order Hamiltonian operators, which can be used to construct the Hamiltonians, and no 5-order operators are shown to pass the Hamiltonian test, although there are infinite number of them, and are skew-adjoint.

Semi-classical Analysis of a Conjoint Crossing of Three Symmetric Modes

Mon, 21 Feb 2011 09:22 EST

Clotilde Fermanian Kammerer, Vidian Rousse

Source: Methods Appl. Anal., Volume 17, Number 2, 137--164.

Abstract:
In this article we focus on a semiclassical Schrödinger equation with matrix-valued potential presenting a symmetric conjoint crossing of three eigenvalues. The potential we consider is well-known in the chemical literature as a pseudo Jahn-Teller potential. We analyze the energy transfers which occur between the three modes in terms of Wigner measures.

Wave Interactions for the Pressure Gradient Equations

Mon, 21 Feb 2011 09:22 EST

T. Raja Sekhar, V. D. Sharma

Source: Methods Appl. Anal., Volume 17, Number 2, 165--178.

Abstract:
In this paper, we solve the Riemann problem for a coupled hyperbolic system of conservation laws, which arises as an intermediate model in the flux splitting method for the computation of Euler equations in gasdynamics. We study the properties of solutions involving shock and rarefaction waves, and establish their existence and uniqueness. We present numerical examples for different initial data, and finally discuss all possible elementary wave interactions; it is noticed that in certain cases the resulting wave pattern after interaction is substantially different from that which arises in isentropic gasdynamics.

Spectral Asymptotics and Quasiclassical Analysis of Schrödinger Type Operators

Mon, 21 Feb 2011 09:22 EST

Andrea Ziggioto

Source: Methods Appl. Anal., Volume 17, Number 2, 179--190.

Global Smooth Solutions of the Three-dimensional Modified Phase Field Crystal Equation

Mon, 21 Feb 2011 09:22 EST

Cheng Wang, Steven M. Wise

Source: Methods Appl. Anal., Volume 17, Number 2, 191--212.

Nonexistence Results for Hessian Inequality

Mon, 21 Feb 2011 09:22 EST

Qianzhong Ou

Source: Methods Appl. Anal., Volume 17, Number 2, 213--224.

Abstract:
In this paper , the author proves a Liouville type theorem for some Hessian entire inequality with sub-lower-critical exponent, via suitable choices of test functions and the argument of integration by parts .

A Liapunov-Schmidt Reduction for Time-Periodic Solutions of the Compressible Euler Equations

Mon, 23 May 2011 16:26 EDT

Blake Temple, Robin Young

Source: Methods Appl. Anal., Volume 17, Number 23, 225--262.

Abstract:
Following the authors’ earlier work in "A paradigm for time-periodic sound wave propagation in the compressible Euler equations," and "Time-periodic linearized solutions of the compressible Euler equations and a problem of small divisors," we show that the nonlinear eigenvalue problem introduced in "Time-periodic linearized solutions of the compressible Euler equations and a problem of small divisors," can be recast in the language of bifurcation theory as a perturbation of a linearized eigenvalue problem. Solutions of this nonlinear eigenvalue problem correspond to time periodic solutions of the compressible Euler equations that exhibit the simplest possible periodic structure identified in "A paradigm for time-periodic sound wave propagation in the compressible Euler equations." By a Liapunov-Schmidt reduction we establish and refine the statement of a new infinite dimensional KAM type small divisor problem in bifurcation theory, whose solution will imply the existence of exact time-periodic solutions of the compressible Euler equations. We then show that solutions exist to within an arbitrarily high Fourier mode cutoff. The results introduce a new small divisor problem of quasilinear type, and lend further strong support for the claim that the time-periodic wave pattern described at the linearized level in "Time-periodic linearized solutions of the compressible Euler equations and a problem of small divisors," is physically realized in nearby exact solutions of the fully nonlinear compressible Euler equations.

The Pressure Gradient System

Mon, 23 May 2011 16:26 EDT

Yuxi Zheng , Zachary Robinson

Source: Methods Appl. Anal., Volume 17, Number 23, 263--278.

Abstract:
The pressure gradient system is a sub-system of the compressible Euler system. It can be obtained either through a flux splitting or an asymptotic expansion. In both derivations, the velocity field is treated as a small remnant of the original velocity of the Euler system. As such, the boundary conditions for the velocity do not necessarily follow the original ones and careful consideration is needed for the validity, integrity, and completeness of the model. We provide numerical simulations as well as basic characteristic analysis and physical considerations for the Riemann problems of the model to find out appropriate internal conditions at the origin. The study reveals subtle structures of the velocity: Both components exhibit discontinuities at the origin at traditional levels of numerical resolutions around 400×400 cells on the unit square, but they vanish at the origin with possible square root type singularity when we increase the resolutions. Comparing to the roll-up of shear waves or vortex-sheets of the Euler system, these singularities are mild and occur only along rays from the origin. The numerics is done at higher resolutions than traditionally possible via the automated clawpack that contains adaptive mesh refinement (AMR) and message passing interface (MPI), for which we provide the Riemann solvers in both the normal and transversal directions, where the Roe’s approximation has the elegant 1/2 average in the model’s original variables.

Asymptotic Stability of Viscous Shock Wave for a Onedimensional Isentropic Model of Viscous Gas with Density Dependent Viscosity

Mon, 23 May 2011 16:26 EDT

Akitaka Matsumura, Yang Wang

Source: Methods Appl. Anal., Volume 17, Number 23, 279--290.

Abstract:
In this paper we investigate the asymptotic stability of viscous shock wave for a onedimensional isentropic model of viscous gas with density dependent viscosity by a weighted energy method developed in the papers of Matsumura-Mei (1997) and Hashimoto-Matsumura (2007). Under the condition that the viscosity coefficient is given as a function of the absolute temperature which is determined by the Chapman-Enskog expansion theory in rarefied gas dynamics, any viscous shock wave is shown to be asymptotically stable for small initial perturbations with integral zero. This generalizes the previous result of Matsumua-Nishihara (1985) where the viscosity coefficient is given by a constant and a restriction on the strength of the viscous shock wave is assumed. This also analytically assures the spectral stability in the Zumbrun’s theory for any viscous shock wave in our specific case.

Vanishing Viscosity Limit for Incompressible Fluids with a Slip Boundary Condition

Mon, 23 May 2011 16:26 EDT

Xiaoqiang Xie, Changmin Li

Source: Methods Appl. Anal., Volume 17, Number 23, 291--300.

Strong Stability with Respect to Weak Limits for a Hyperbolic System arising from Gas Chromatography

Mon, 23 May 2011 16:26 EDT

C. Bourdarias, M. Gisclon, S. Junca

Source: Methods Appl. Anal., Volume 17, Number 23, 301--330.

Inverse Problems for Nonlinear Delay Systems

Tue, 24 May 2011 11:06 EDT

H. T. Banks, Keri Rehm, Karyn Sutton

Source: Methods Appl. Anal., Volume 17, Number 4, 331--356.

Abstract:
We consider inverse or parameter estimation problems for general nonlinear nonautonomous dynamical systems with delays. The parameters may be from a Euclidean set as usual, may be time dependent coefficients or may be probability distributions across a population as arise in aggregate data problems. Theoretical convergence results for finite dimensional approximations to the systems are given. Several examples are used to illustrate the ideas and computational results that demonstrate efficacy of the approximations are presented.

Simultaneous Reconstruction of Shape and Impedance in Corrosion Detection

Tue, 24 May 2011 11:06 EDT

Fioralba Cakoni, Rainer Kress, Christian Schuft

Source: Methods Appl. Anal., Volume 17, Number 4, 357--378.

The Determination of Anisotropic Surface Impedance in Electromagnetic Scattering

Tue, 24 May 2011 11:06 EDT

Fioralba Cakoni, Peter Monk

Source: Methods Appl. Anal., Volume 17, Number 4, 379--394.

Abstract:
We consider the inverse scattering problem of determining the anisotropic surface impedance of a bounded obstacle from far field measurements of the electromagnetic scattered field due to incident plane waves. Such an anisotropic boundary condition can arise from surfaces covered with patterns of conducting and insulating patches. We show that the anisotropic impedance is uniquely determined if sufficient data is available, and characterize the non-uniqueness present if a single incoming wave is used. We derive an integral equation for the surface impedance in terms of solutions of a certain interior impedance boundary value problem. These solutions can be reconstructed from far field data using the Herglotz theory underlying the Linear Sampling Method. We complete the paper with preliminary numerical results.

Subdifferential Inverse Problems for Magnetohydrodynamics

Tue, 24 May 2011 11:06 EDT

Alexander Chebotarev

Source: Methods Appl. Anal., Volume 17, Number 4, 395--406.

Abstract:
The theory of solvability of an abstract evolution inequality in a Hilbert space for the operators with the quadratic nonlinearity is presented. It is then applied for the study of an inverse problem for MHD flows. For the three-dimensional flows the global in time existence of the weak solutions to the inverse problem is proved. For the two-dimensional flows existence and uniqueness of the strong solutions are proved.

SubspaceMethods for Solving Electromagnetic Inverse Scattering Problems

Tue, 24 May 2011 11:06 EDT

Xudong Chen, Yu Zhong, Krishna Agarwal

Source: Methods Appl. Anal., Volume 17, Number 4, 407--432.

Abstract:
This paper presents a survey of the subspace methods and their applications to electromagnetic inverse scattering problems. Subspace methods can be applied to reconstruct both small scatterers and extended scatterers, with the advantages of fast speed, good stability, and higher resolution. For inverse scattering problems involving small scatterers, the multiple signal classification method is used to determine the locations of scatterers and then the least-squares method is used to calculate the scattering strengths of scatterers. For inverse scattering problems involving extended scatterers, the subspace-based optimization method is used to reconstruct the refractive index of scatterers.

On an Inverse Problem of Reconstructing an Unknown Coefficient in a Second Order Hyperbolic Equation from Partial Boundary Measurements

Tue, 24 May 2011 11:06 EDT

Christian Daveau, Abdessatar Khelifi

Source: Methods Appl. Anal., Volume 17, Number 4, 433--444.

Abstract:
We consider the inverse problem of reconstructing an unknown coefficient in a second order hyperbolic equation from partial (on part of the boundary) dynamic boundary measurements. In this paper we prove that the knowledge of the partial Cauchy data for this class of hyperbolic PDE on any open subset

On the Degree of Ill-posedness for Linear Problems with Noncompact Operators

Tue, 24 May 2011 11:06 EDT

Bernd Hofmann, Stefan Kindermann

Source: Methods Appl. Anal., Volume 17, Number 4, 445--462.

Abstract:
In inverse problems it is quite usual to encounter equations that are ill-posed and require regularization aimed at finding stable approximate solutions when the given data are noisy. In this paper, we discuss definitions and concepts for the degree of ill-posedness for linear operator equations in a Hilbert space setting. It is important to distinguish between a global version of such degree taking into account the smoothing properties of the forward operator, only, and a local version combining that with the corresponding solution smoothness. We include the rarely discussed case of non-compact forward operators and explain why the usual notion of degree of ill-posedness cannot be used in this case.