Methods and Applications of Analysis Articles (Project Euclid)
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The latest articles from Methods and Applications of Analysis on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTTue, 24 May 2011 11:06 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Lyapunov's Functions and Existence of Integral Manifolds for Impulsive Differental Systems with Time-varying Delay
http://projecteuclid.org/euclid.maa/1273002793
<strong>Gani Stamov</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 16, Number 3, 291--298.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.maa/1273002793_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTCompleteness of Eigenfunctions of Sturm-Liouville Problems with Transmission Conditions
http://projecteuclid.org/euclid.maa/1273002794
<strong>Aiping Wang</strong>, <strong>Jiong Sun</strong>, <strong>Xiaoling Hao</strong>, <strong>Siqin Yao</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 16, Number 3, 299--312.</p><p><strong>Abstract:</strong><br/>
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 $H$.
</p>projecteuclid.org/euclid.maa/1273002794_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTLocal Time Decay for a Quasilinear Schrodinger Equation
http://projecteuclid.org/euclid.maa/1273002795
<strong>J. E. Lin</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 16, Number 3, 313--320.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.maa/1273002795_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTConcentration in Lotka-Volterra Parabolic or Integral Equations: A General Convergence Result
http://projecteuclid.org/euclid.maa/1273002796
<strong>Guy Barles</strong>, <strong>Sepideh Mirrahimi</strong>, <strong>Benoît Perthame</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 16, Number 3, 321--340.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.maa/1273002796_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTA Paradigm for Time-periodic Sound Wave Propagation in the Compressible Euler Equations
http://projecteuclid.org/euclid.maa/1273002797
<strong>Blake Temple</strong>, <strong>Robin Young</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 16, Number 3, 341--364.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.maa/1273002797_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTParabolic and Elliptic Systems with VMO Coefficients
http://projecteuclid.org/euclid.maa/1273002798
<strong>Hongjie Dong</strong>, <strong>Doyoon Kim</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 16, Number 3, 365--388.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.maa/1273002798_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTConvergence Rate to the Nonlinear Waves for Viscous Conservation Laws on the Half Line
http://projecteuclid.org/euclid.maa/1273002799
<strong>Itsuko Hashimoto</strong>, <strong>Yoshihiro Ueda</strong>, <strong>Shuichi Kawashima</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 16, Number 3, 389--402.</p>projecteuclid.org/euclid.maa/1273002799_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTOn the Convergence of Fully-discrete High-Resolution Schemes with van Leer's Flux Limiter for Conservation Laws
http://projecteuclid.org/euclid.maa/1273002800
<strong>Nan Jiang</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 16, Number 3, 403--422.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.maa/1273002800_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTCascade of Phase Shifts and Creation of Nonlinear Focal Points for Supercritical Semiclassical Hartree Equationhttp://projecteuclid.org/euclid.maa/1286890988<strong>Satoshi Masaki </strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 16, Number 4, 403--458.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.maa/1286890988_Tue, 12 Oct 2010 09:43 EDTTue, 12 Oct 2010 09:43 EDTAsymptotic Behavior of Oscillating Radial Solutions to Certain Nonlinear Equations, Part IIhttp://projecteuclid.org/euclid.maa/1286890989<strong>Changfeng Gui</strong>, <strong>Xue Luo</strong>, <strong>Feng Zhou</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 16, Number 4, 459--468.</p>projecteuclid.org/euclid.maa/1286890989_Tue, 12 Oct 2010 09:43 EDTTue, 12 Oct 2010 09:43 EDTSecant-like Method for Solving Generalized Equationshttp://projecteuclid.org/euclid.maa/1286890990<strong>Ioannis K. Argyros</strong>, <strong>Saïd Hilout </strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 16, Number 4, 469--478.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.maa/1286890990_Tue, 12 Oct 2010 09:43 EDTTue, 12 Oct 2010 09:43 EDTOn Breakdown of Solutions to the Full Compressible Navier-Stokes Equationshttp://projecteuclid.org/euclid.maa/1286890991<strong>Xiangdi Huang</strong>, <strong>Jing Li</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 16, Number 4, 479--490.</p>projecteuclid.org/euclid.maa/1286890991_Tue, 12 Oct 2010 09:43 EDTTue, 12 Oct 2010 09:43 EDTA Blowup Criterion for the Full Compressible Navier-Stokes Equationshttp://projecteuclid.org/euclid.maa/1286890992<strong>Xiangdi Huang</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 16, Number 4, 491--506.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.maa/1286890992_Tue, 12 Oct 2010 09:43 EDTTue, 12 Oct 2010 09:43 EDTBounds on Front Speeds for Inviscid and Viscous G-equationshttp://projecteuclid.org/euclid.maa/1286890993<strong>James Nolen</strong>, <strong>Jack Xin</strong>, <strong>Yifeng Yu</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 16, Number 4, 507--520.</p>projecteuclid.org/euclid.maa/1286890993_Tue, 12 Oct 2010 09:43 EDTTue, 12 Oct 2010 09:43 EDTSpectrum and Functions of Operators on Direct Families of Banach Spaceshttp://projecteuclid.org/euclid.maa/1286890994<strong>M. I. Gil</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 16, Number 4, 521--534.</p>projecteuclid.org/euclid.maa/1286890994_Tue, 12 Oct 2010 09:43 EDTTue, 12 Oct 2010 09:43 EDTSingular Support of the Scattering Kernel for the Rayleigh Wave in Perturbed Half-Spaceshttp://projecteuclid.org/euclid.maa/1291644607<strong>Mishio Kawashita</strong>, <strong>Hideo Soga</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 17, Number 1, 1--48.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.maa/1291644607_Mon, 06 Dec 2010 09:10 ESTMon, 06 Dec 2010 09:10 ESTDynamical Laws of the Coupled Gross-Pitaevskii Equations for Spin-1 Bose-Einstein Condensateshttp://projecteuclid.org/euclid.maa/1291644608<strong>Weizhu Bao</strong>, <strong>Yanzhi Zhang</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 17, Number 1, 49--80.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.maa/1291644608_Mon, 06 Dec 2010 09:10 ESTMon, 06 Dec 2010 09:10 ESTSmoothness Criterion for Navier-Stokes Equations in Terms of Regularity along the Streamlineshttp://projecteuclid.org/euclid.maa/1291644609<strong>Chi Hin Chan</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 17, Number 1, 81--104.</p>projecteuclid.org/euclid.maa/1291644609_Mon, 06 Dec 2010 09:10 ESTMon, 06 Dec 2010 09:10 ESTOn Bellman's Equations with VMO Coefficientshttp://projecteuclid.org/euclid.maa/1291644610<strong>Nicolai V. Krylov</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 17, Number 1, 105--122.</p>projecteuclid.org/euclid.maa/1291644610_Mon, 06 Dec 2010 09:10 ESTMon, 06 Dec 2010 09:10 ESTOn the Differential Operators of the Generalized Fifth-order Korteweg-de Vries Equationhttp://projecteuclid.org/euclid.maa/1291644611<strong>Chun-Te Lee</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 17, Number 1, 123--136.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.maa/1291644611_Mon, 06 Dec 2010 09:10 ESTMon, 06 Dec 2010 09:10 ESTSemi-classical Analysis of a Conjoint Crossing of Three Symmetric Modeshttp://projecteuclid.org/euclid.maa/1298298164<strong>Clotilde Fermanian Kammerer</strong>, <strong>Vidian Rousse</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 17, Number 2, 137--164.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.maa/1298298164_Mon, 21 Feb 2011 09:22 ESTMon, 21 Feb 2011 09:22 ESTWave Interactions for the Pressure Gradient Equationshttp://projecteuclid.org/euclid.maa/1298298165<strong>T. Raja Sekhar</strong>, <strong>V. D. Sharma</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 17, Number 2, 165--178.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.maa/1298298165_Mon, 21 Feb 2011 09:22 ESTMon, 21 Feb 2011 09:22 ESTSpectral Asymptotics and Quasiclassical Analysis of Schrödinger Type Operatorshttp://projecteuclid.org/euclid.maa/1298298166<strong>Andrea Ziggioto</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 17, Number 2, 179--190.</p>projecteuclid.org/euclid.maa/1298298166_Mon, 21 Feb 2011 09:22 ESTMon, 21 Feb 2011 09:22 ESTGlobal Smooth Solutions of the Three-dimensional Modified Phase Field Crystal Equationhttp://projecteuclid.org/euclid.maa/1298298167<strong>Cheng Wang</strong>, <strong>Steven M. Wise</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 17, Number 2, 191--212.</p>projecteuclid.org/euclid.maa/1298298167_Mon, 21 Feb 2011 09:22 ESTMon, 21 Feb 2011 09:22 ESTNonexistence Results for Hessian Inequalityhttp://projecteuclid.org/euclid.maa/1298298168<strong>Qianzhong Ou</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 17, Number 2, 213--224.</p><p><strong>Abstract:</strong><br/>
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 .
</p>projecteuclid.org/euclid.maa/1298298168_Mon, 21 Feb 2011 09:22 ESTMon, 21 Feb 2011 09:22 ESTA Liapunov-Schmidt Reduction for Time-Periodic Solutions of the Compressible Euler Equationshttp://projecteuclid.org/euclid.maa/1306182405<strong>Blake Temple</strong>, <strong>Robin Young</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 17, Number 23, 225--262.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.maa/1306182405_Mon, 23 May 2011 16:26 EDTMon, 23 May 2011 16:26 EDTThe Pressure Gradient Systemhttp://projecteuclid.org/euclid.maa/1306182406<strong>Yuxi Zheng </strong>, <strong>Zachary Robinson</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 17, Number 23, 263--278.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.maa/1306182406_Mon, 23 May 2011 16:26 EDTMon, 23 May 2011 16:26 EDTAsymptotic Stability of Viscous Shock Wave for a Onedimensional Isentropic Model of Viscous Gas with Density Dependent Viscosityhttp://projecteuclid.org/euclid.maa/1306182407<strong>Akitaka Matsumura</strong>, <strong>Yang Wang</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 17, Number 23, 279--290.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.maa/1306182407_Mon, 23 May 2011 16:26 EDTMon, 23 May 2011 16:26 EDTVanishing Viscosity Limit for Incompressible Fluids with a Slip Boundary Conditionhttp://projecteuclid.org/euclid.maa/1306182408<strong>Xiaoqiang Xie</strong>, <strong>Changmin Li</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 17, Number 23, 291--300.</p>projecteuclid.org/euclid.maa/1306182408_Mon, 23 May 2011 16:26 EDTMon, 23 May 2011 16:26 EDTStrong Stability with Respect to Weak Limits for a Hyperbolic System arising from Gas Chromatographyhttp://projecteuclid.org/euclid.maa/1306182409<strong>C. Bourdarias</strong>, <strong>M. Gisclon</strong>, <strong>S. Junca</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 17, Number 23, 301--330.</p>projecteuclid.org/euclid.maa/1306182409_Mon, 23 May 2011 16:26 EDTMon, 23 May 2011 16:26 EDTInverse Problems for Nonlinear Delay Systemshttp://projecteuclid.org/euclid.maa/1306249556<strong>H. T. Banks</strong>, <strong>Keri Rehm</strong>, <strong>Karyn Sutton</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 17, Number 4, 331--356.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.maa/1306249556_Tue, 24 May 2011 11:06 EDTTue, 24 May 2011 11:06 EDTSimultaneous Reconstruction of Shape and Impedance in Corrosion Detectionhttp://projecteuclid.org/euclid.maa/1306249557<strong>Fioralba Cakoni</strong>, <strong>Rainer Kress</strong>, <strong>Christian Schuft</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 17, Number 4, 357--378.</p>projecteuclid.org/euclid.maa/1306249557_Tue, 24 May 2011 11:06 EDTTue, 24 May 2011 11:06 EDTThe Determination of Anisotropic Surface Impedance in Electromagnetic Scatteringhttp://projecteuclid.org/euclid.maa/1306249558<strong>Fioralba Cakoni</strong>, <strong>Peter Monk</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 17, Number 4, 379--394.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.maa/1306249558_Tue, 24 May 2011 11:06 EDTTue, 24 May 2011 11:06 EDTSubdifferential Inverse Problems for Magnetohydrodynamicshttp://projecteuclid.org/euclid.maa/1306249559<strong>Alexander Chebotarev</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 17, Number 4, 395--406.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.maa/1306249559_Tue, 24 May 2011 11:06 EDTTue, 24 May 2011 11:06 EDTSubspaceMethods for Solving Electromagnetic Inverse Scattering Problemshttp://projecteuclid.org/euclid.maa/1306249560<strong>Xudong Chen</strong>, <strong>Yu Zhong</strong>, <strong>Krishna Agarwal</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 17, Number 4, 407--432.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.maa/1306249560_Tue, 24 May 2011 11:06 EDTTue, 24 May 2011 11:06 EDTOn an Inverse Problem of Reconstructing an Unknown Coefficient in a Second Order Hyperbolic Equation from Partial Boundary Measurementshttp://projecteuclid.org/euclid.maa/1306249561<strong>Christian Daveau</strong>, <strong>Abdessatar Khelifi</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 17, Number 4, 433--444.</p><p><strong>Abstract:</strong><br/>
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
</p>projecteuclid.org/euclid.maa/1306249561_Tue, 24 May 2011 11:06 EDTTue, 24 May 2011 11:06 EDTOn the Degree of Ill-posedness for Linear Problems with Noncompact Operatorshttp://projecteuclid.org/euclid.maa/1306249562<strong>Bernd Hofmann</strong>, <strong>Stefan Kindermann</strong><p><strong>Source: </strong>Methods Appl. Anal., Volume 17, Number 4, 445--462.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.maa/1306249562_Tue, 24 May 2011 11:06 EDTTue, 24 May 2011 11:06 EDT