# Quarterly of Applied Mathematics

Published by American Mathematical Society

Print ISSN: 0033-569X

Published by American Mathematical Society

Print ISSN: 0033-569X

Publications

We consider equilibrium problems for elastic bodies in domains with cracks. Inequality type boundary conditions are imposed at the crack describing a mutual nonpenetration between the crack faces. A new formulation for such problems is proposed in smooth geometrical domains for two-dimensional elasticity and for Kirchhoff plates.

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The classical plane Couette flow, plane Poiseuille flow, and pipe Poiseuille flow share some universal 3D steady coherent structure in the form of "streak-roll-critical layer". As the Reynolds number approaches infinity, the steady coherent structure approaches a 3D limiting shear of the form ($U(y,z), 0, 0$) in velocity variables. All such 3D shears are steady states of the 3D Euler equations. This raises the importance of investigating the stability of such inviscid 3D shears in contrast to the classical Rayleigh theory of inviscid 2D shears. Several general criteria of stability for such inviscid 3D shears are derived. In the Appendix, an argument is given to show that a 2D limiting shear can only be the classical laminar shear. Comment: This is a short note. The relevant manuscript is arXiv:0811.0383 (Y. Li, D. Viswanath, Exact and asymptotic conditions on traveling wave solutions of the Navier-Stokes equations, Physics of Fluids 21 (2009), 101703

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In this paper a model problem is considered that simulates an atmospheric acoustic wave propagation situation that is nonlinear. The model is derived from the basic Euler equations for the atmospheric flow and from the regular perturbations for the acoustic part. The nonlinear effects are studied by obtaining two successive linear problems in which the second one involves the solution of the first problem. Well-posedness of these problems is discussed and approximations of the radiation boundary conditions that can be used in numerical simulations are presented.

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The refraction of acoustic duct waveguide modes emitted from the open end of a semiinfinite rectangular duct by a jet-like exhaust flow is studied theoretically. The problem is formulated as a Wiener-Hopf problem and is ultimately solved by an approximate method due to Carrier and Koiter. Continuity of transverse acoustic particle displacement and of acoustic pressure is assumed at the jet/still-air interface. The solution exhibits several features of the acoustics of moving media such as a source convection effect, zones of relative silence, and simple refraction. Plots of far-field directivity patterns are presented for several cases and show refraction effects to be important even at modest exhaust Mach numbers of order 0.3. Only subsonic exhaust Mach numbers are considered.

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Flexible finite cylindrical shell undergoing arbitrary temporal and spatial motion, determining fluid /aerodynamic/ forces acting on shell

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A class of singular integral equations is considered which arise in various two-dimensional mixed boundary-value problems with simple harmonic time variation. A problem typical of this class is that of determining the lifting pressure distribution on an oscillating airfoil in an unbounded incompressible potential flow. It is shown that Theodorsen's (1935) solution to this problem, with some modification, is valid for a general class of unsteady kernel functions. The technique employed is to consider an equivalent steady problem and then show that the unsteady resolvent and unsteady homogeneous solution can be written directly in terms of the steady solutions and a single frequency-dependent function which reduces to the Theodorsen function for the steady kernel.

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Almost periodic solutions of two nonlinear Volterra integral equations

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We study the emergent flocking behavior in a group of Cucker-Smale flocking
agents under rooted leadership with alternating leaders. It is well known that
the network topology regulates the emergent behaviors of flocks. All existing
results on the Cucker-Smale model with leader-follower topologies assume a
fixed leader during temporal evolution process. The rooted leadership is the
most general topology taking a leadership. Motivated by collective behaviors
observed in the flocks of birds, swarming fishes and potential engineering
applications, we consider the rooted leadership with alternating leaders; that
is, at each time slice there is a leader but it can be switched among the
agents from time to time. We will provide several sufficient conditions leading
to the asymptotic flocking among the Cucker-Smale agents under rooted
leadership with alternating leaders.

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The extreme characteristics of long wave run-up are studied in this paper.
First we give a brief overview of the existing theory which is mainly based on
the hodograph transformation (Carrier & Greenspan, 1958). Then, using numerical
simulations, we build on the work of Stefanakis et al. (2011) for an infinite
sloping beach and we find that resonant run-up amplification of monochromatic
waves is robust to spectral perturbations of the incoming wave and resonant
regimes do exist for certain values of the frequency. In the setting of a
finite beach attached to a constant depth region, resonance can only be
observed when the incoming wavelength is larger than the distance from the
undisturbed shoreline to the seaward boundary. Wavefront steepness is also
found to play a role in wave run-up, with steeper waves reaching higher run-up
values.

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Spatially periodic large amplitude solutions of the von Karman model are obtained in the neighborhood of singularities. These singularities correspond to vortex clusters in the physical plane. The quasi-periodic and unbounded solutions found analytically confirm earlier numerical work and show qualitative agreement with experimental observations of large-scale phenomena of vortex trails. Separatrices or heteroclinic orbits were explicitly found for an integrable approximate equation, which indicate that the von Karman model itself supports chaotic solutions.

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The well-known analytical solution of Burgers' equation is extended to curvilinear coordinate systems in three dimensions by a method that is much simpler and more suitable to practical applications than that previously used. The results obtained are applied to incompressible flow with cylindrical symmetry, and also to the decay of an initially linearly increasing wind.

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We present a family of numerical implementations of Kato’s ODE propagating global bases of analytically varying invariant subspaces of which the first-order version is a surprisingly simple “greedy algorithm” that is both stable and easy to program and the second-order version a relaxation of a first-order scheme of L. Q. Brin and K. Zumbrun [Mat. Contemp. 22, 19–32 (2002; Zbl 1044.35057)]. The method has application to numerical Evans function computations used to assess stability of traveling-wave solutions of time-evolutionary PDE.

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We analyze cloaking due to anomalous localized resonance in the quasistatic
regime in the case when a general charge density distribution is brought near a
slab superlens. If the charge density distribution is within a critical
distance of the slab, then the power dissipation within the slab blows up as
certain electrical dissipation parameters go to zero. The potential remains
bounded far away from the slab in this limit, which leads to cloaking due to
anomalous localized resonance. On the other hand, if the charge density
distribution is further than this critical distance from the slab, then the
power dissipation within the slab remains bounded and cloaking due to anomalous
localized resonance does not occur. The critical distance is shown to strongly
depend on the the rate at which the dissipation outside of the slab goes to
zero.

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The asymptotic behavior of the micromagnetic free energy governing a
ferromagnetic film is studied as its thickness gets smaller and smaller
compared to its cross section. Here the static Maxwell equations are treated as
a Murat's constant-rank PDE constraint on the energy functional. In contrast to
previous work this approach allows to keep track of the induced magnetic field
without solving the magnetostatic equations. In particular, the mathematical
results of Gioia and James [Proc. R. Soc. Lond. A 453 (1997), pp. 213-223]
regarding convergence of minimizers are recovered by giving a characterization
of the corresponding Gamma-limit.

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In this paper, we introduce a method of imposing asymmetric conditions on the velocity vector with respect to independent variables and a method of moving frame for solving the three dimensional Navier-Stokes equations. Seven families of non-steady rotating asymmetric solutions with various parameters are obtained. In particular, one family of solutions blow up at any point on a moving plane with a line deleted, which may be used to study turbulence. Using Fourier expansion and two families of our solutions, one can obtain discontinuous solutions that may be useful in study of shock waves. Another family of solutions are partially cylindrical invariant, contain two parameter functions of $t$ and structurally depend on two arbitrary polynomials, which may be used to describe incompressible fluid in a nozzle. Most of our solutions are globally analytic with respect to spacial variables.

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The problem of injectivity of the parameter-to-state map is discussed for Galerkin approximations of a linear parabolic equation. A necessary and sufficient condition is derived and illustrated by means of simple examples. Finally, output least squares identifiability of the Galerkin approximations is discussed.

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A kernel method is proposed to estimate the condensed density of the
generalized eigenvalues of pencils of Hankel matrices whose elements have a
joint noncentral Gaussian distribution with nonidentical covariance. These
pencils arise when the complex exponentials approximation problem is considered
in Gaussian noise. Several moments problems can be formulated in this framework
and the estimation of the condensed density above is the main critical step for
their solution. It is shown that the condensed density satisfies approximately
a diffusion equation, which allows to estimate an optimal bandwidth. It is
proved by simulation that good results can be obtained even when the
signal-to-noise ratio is so small that other methods fail.

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We provide a precise description of the set of residual boundary conditions
generated by the self-similar viscous approximation introduced by Dafermos et
al. We then apply our results, valid for both conservative and non conservative
systems, to the analysis of the boundary Riemann problem and we show that,
under appropriate assumptions, the limits of the self-similar and the classical
vanishing viscosity approximation coincide. We require neither genuinely
nonlinearity nor linear degeneracy of the characteristic fields.

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The two-dimensional steady flow past an ellipse of arbitrary aspect ratio is investigated analytically, applying a linearized version of the Navier-Stokes equation based on the approximation of Burgers (1928). The resulting infinite system of linear equations is truncated to give reliable results for Reynolds numbers between zero and five, and separation, drag, and boundary-layer phenomena are characterized and illustrated with graphs. The Burgers approximation is found to provide good qualitative results near the ellipse, with reasonable quantitative accuracy for the special case of a circular cylinder.

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We use the compactness result of A. Burchard and Y. Guo [J. Funct. Anal. 214, No. 1, 40–73 (2004; Zbl 1065.49006)] to analyze the reduced ‘energy’ functional arising naturally in the stability analysis of steady states of the Vlasov-Poisson system (cf. Sánchez and Soler, to appear, and Hadzic, 2005). We consider the associated variational problem and present a new proof that puts it in the general framework for tackling the variational problems of this type, given by Y. Guo and G. Rein [cf. G. Rein, Nonlinear stability of Newtonian galaxies and stars from a mathematical perspective, nonlinear dynamics in astronomy and physics. Ann. New York Acad. Sci. 1045, 103–119 (2005); SIAM J. Math. Anal. 33, No.4, 896-912 (2001; Zbl 1019.35003)].

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This paper deals with the approximation of the weak solutions of the incompressible Navier Stokes Fourier system. In particular it extends the artificial compressibility method for the Leray weak solutions of the Navier Stokes equation, used by Temam, in the case of a bounded domain and later in the case of the whole space. By exploiting the wave equation structure of the pressure of the approximating system the convergence of the approximating sequences is achieved by means of dispersive estimate of Strichartz type. It will be proved that the projection of the approximating velocity fields on the divergence free vectors is relatively compact and converges to a weak solution of the incompressible Navier Stokes Fourier system.

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This paper is an examination of special nonlinearities of the Jeffcott equations in rotordynamics. The immediate application of this analysis is directed toward understanding the excessive vibrations recorded in the LOX pump of the SSME during hot-firing ground testing. Deadband, side force, and rubbing are three possible sources of inducing nonlinearity in the Jeffcott equations. The present analysis initially reduces these problems to the same mathematical description. A special frequency, named the nonlinear natural frequency, is defined and used to develop the solutions of the nonlinear Jeffcott equations as singular asymptotic expansions. This nonlinear natural frequency, which is the ratio of the cross-stiffness and the damping, plays a major role in determining response frequencies.

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Asymptotic behavior of viscoelastic Green's functions near the wavefront is
expressed in terms of a causal function $g(t)$ defined in \cite{SerHanJMP} in
connection with the Kramers-Kronig dispersion relations. Viscoelastic Green's
functions exhibit a discontinuity at the wavefront if $g(0) < \infty$.
Estimates of continuous and discontinuous viscoelastic Green's functions near
the wavefront are derived.

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Buckling of long cylinders with homogeneous random axisymmetric geometric imperfections under axial compression, using truncated hierarchy technique

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An analytical method based on Kummer's series transformation is presented, which allows for the evaluation of Fourier-Bessel series with poor convergence properties and, in addition, yields the singularities of the series in closed form. This method is applied to the Fourier-Bessel series which arise as solutions of the linearized gas dynamic potential equation for the cylindrical flow field of a supersonic free jet. The presented method is applied to D. C. Pack's classical solution for the axisymmetric free jet with initial homogeneous pressure perturbation, which in its original form cannot be evaluated directly. It is shown that in contrast to the plane jet, the flow field of the axisymmetric jet does not exhibit a strictly periodic behaviour, whereas its singularities are distributed periodically. (from Authors)

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We review results on the spherically symmetric, asymptotically flat
Einstein-Vlasov system. We focus on a recent result where we found explicit
conditions on the initial data which guarantee the formation of a black hole in
the evolution. Among these data there are data such that the corresponding
solutions exist globally in Schwarzschild coordinates. We put these results
into a more general context, and we include arguments which show that the
spacetimes we obtain satisfy the weak cosmic censorship conjecture and contain
a black hole in the sense of suitable mathematical definitions of these
concepts which are available in the literature.

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This paper is concerned with the lifespan and the blowup mechanism for smooth
solutions to the 2-D nonlinear wave equation
$\p_t^2u-\ds\sum_{i=1}^2\p_i(c_i^2(u)\p_iu)$ $=0$, where $c_i(u)\in
C^{\infty}(\Bbb R^n)$, $c_i(0)\neq 0$, and $(c_1'(0))^2+(c_2'(0))^2\neq 0$.
This equation has an interesting physics background as it arises from the
pressure-gradient model in compressible fluid dynamics and also in nonlinear
variational wave equations. Under the initial condition $(u(0,x),
\p_tu(0,x))=(\ve u_0(x), \ve u_1(x))$ with $u_0(x), u_1(x)\in C_0^{\infty}(\Bbb
R^2)$, and $\ve>0$ is small, we will show that the classical solution $u(t,x)$
stops to be smooth at some finite time $T_{\ve}$. Moreover, blowup occurs due
to the formation of a singularity of the first-order derivatives
$\na_{t,x}u(t,x)$, while $u(t,x)$ itself is continuous up to the blowup time
$T_{\ve}$.

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We are concerned with a class of two-dimensional nonlinear wave equations
$\p_t^2u-\div(c^2(u)\na u)=0$ or $\p_t^2u-c(u)\div(c(u)\na u)=0$ with small
initial data $(u(0,x),\p_tu(0,x))=(\ve u_0(x), \ve u_1(x))$, where $c(u)$ is a
smooth function, $c(0)\not =0$, $x\in\Bbb R^2$, $u_0(x), u_1(x)\in
C_0^{\infty}(\Bbb R^2)$ depend only on $r=\sqrt{x_1^2+x_2^2}$, and $\ve>0$ is
sufficiently small. Such equations arise in a pressure-gradient model of fluid
dynamics, also in a liquid crystal model or other variational wave equations.
When $c'(0)\not= 0$ or $c'(0)=0$, $c"(0)\not= 0$, we establish blowup and
determine the lifespan of smooth solutions.

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The problem of estimating discontinuous coefficients, including locations of discontinuities, that occur in second-order hyperbolic systems typical of those arising in one-dimensional surface seismic problems, is considered. The problem of identifying unknown parameters that appear in boundary conditions for the system is addressed. A spline-based approximation theory is presented as well as related convergence findings and representative numerical examples.

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Under consideration is the hyperbolic relaxation of a semilinear
reaction-diffusion equation on a bounded domain, subject to a dynamic boundary
condition. We also consider the limit parabolic problem with the same dynamic
boundary condition. Each problem is well-posed in a suitable phase space where
the global weak solutions generate a Lipschitz continuous semiflow which admits
a bounded absorbing set. We prove the existence of a family of global
attractors of optimal regularity. After fitting both problems into a common
framework, a proof of the upper-semicontinuity of the family of global
attractors is given as the relaxation parameter goes to zero. Finally, we also
establish the existence of exponential attractors.

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The identification of the geometrical structure of the system boundary for a two-dimensional diffusion system is reported. The domain identification problem treated here is converted into an optimization problem based on a fit-to-data criterion and theoretical convergence results for approximate identification techniques are discussed. Ressults of numerical experiments to demonstrate the efficacy of the theoretical ideas are reported.

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The size of the shock-layer governed by a conservation law is studied. The conservation law is a parabolic reaction-convection-diffusion equation with a small parameter multiplying the diffusion term and convex flux. Rigorous upper and lower bounding functions for the solution of the conservation law are established based on maximum-principle arguments. The bounding functions demonstrate that the size of the shock-layer is proportional to the parameter multiplying the diffusion term.

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We consider Markov processes in continuous time with state space $\posint^N$
and provide two sufficient conditions and one necessary condition for the
existence of moments $E(\|X(t)\|^r)$ of all orders $r \in \nat$ for all $t \geq
0$. The sufficient conditions also guarantee an exponential in time growth
bound for the moments. The class of processes studied have finitely many state
independent jumpsize vectors $\nu_1,\dots,\nu_M$. This class of processes arise
naturally in many applications such as stochastic models of chemical kinetics,
population dynamics and queueing theory for example. We also provide a
necessary and sufficient condition for stochiometric boundedness of species in
terms of $\nu_j$.

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In the last few years a one-dimensional, time-dependent and nonlinear approximation to the Navier-Stokes equations, (1) below, has found applications in fields as diverse as number theory, gas dynamics, heat conduction, elasticity, etc. Probably the most important reason for this is that the complete and explicit solution of this equation became known in 1950. That solution, however, applies only to the homogeneous part of Eq. (1). In an attempt to tackle the nonhomogeneous case, we relate Eq. (1) to a Riccati equation, through a similarity transformation. Via this route, it is shown that solutions to the nonhomogeneous equation can be obtained.

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The vector Burgers equation is extended to include pressure gradients and gravity. It is shown that within the framework of the Cole-Hopf transformation there are no physical solutions to this problem. This result is important because it clearly demonstrates that any extension of Burgers equation to more interesting physical situations is strongly limited.

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Phase transitions are in the focus of the modeling of multiphase flows. A
large number of models is available to describe such processes. We consider
several different two phase models that are based on the Euler equations of
compressible fluid flows and which take into account phase transitions between
a liquid phase and its vapor. Especially we consider the flow of liquid water
and water vapor. We give a mathematical proof, that all these models are not
able to describe the process of condensation by compression. This behavior is
in agreement with observations in experiments, that simulate adiabatic flows,
and shows that the Euler equations give a fairly good description of the
process. The mathematical proof is valid for the official standard {\em
IAPWS-IF97} for water and for any other good equation of state. Also the
opposite case of expanding the liquid phase will be discussed.

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A numerical technique is developed analytically to solve a class of singular integral equations occurring in mixed boundary-value problems for nonhomogeneous elastic media with discontinuities. The approach of Kaya and Erdogan (1987) is extended to treat equations with generalized Cauchy kernels, reformulating the boundary-value problems in terms of potentials as the unknown functions. The numerical implementation of the solution is discussed, and results for an epoxy-Al plate with a crack terminating at the interface and loading normal to the crack are presented in tables.

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An Ito-Skorokhod bi-linear equation driven by infinitely many independent colored noises is considered in a normal triple of Hilbert spaces. The special feature of the equation is the appearance of the Wick product in the definition of the Ito-Skorokhod integral, requiring innovative approaches to computing the solution. A chaos expansion of the solution is derived and several truncations of this expansion are studied. A recursive approximation of the solution is suggested and the corresponding approximation error bound is computed.

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In this paper we analyze the Hilbert boundary-value problem of the theory of analytic functions for an $(N+1)$-connected circular domain. An exact series-form solution has already been derived for the case of continuous coefficients. Motivated by the study of the Hall effect in a multiply connected plate we extend these results by examining the case of discontinuous coefficients. The Hilbert problem maps into the Riemann-Hilbert problem for symmetric piece-wise meromorphic functions invariant with respect to a symmetric Schottky group. The solution to this problem is derived in terms of two analogues of the Cauchy kernel, quasiautomorphic and quasimultiplicative kernels. The former kernel is known for any symmetry Schottky group. We prove the existence theorem for the second, quasimultiplicative, kernel for any Schottky group (its series representation is known for the first class groups only). We also show that the use of an automorphic kernel requires the solution to the associated real analogue of the Jacobi inversion problem which can be bypassed if we employ the quasiautomorphic and quasimultiplicative kernels. We apply this theory to a model steady-state problem on the motion of charged electrons in a plate with $N+1$ circular holes with electrodes and dielectrics on the walls when the conductor is placed at right angle to the applied magnetic field. Comment: 28 pages, 2 figures

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The objectives of the paper are to solve the problem of a circumferentially-cracked cylindrical shell by taking into account the effect of transverse shear, and to obtain the stress intensity factors for the bending moment as well as the membrane force as the external load. The formulation of the problem is given for a specially orthotropic material within the framework of a linearized shallow shell theory. The particular theory used permits the consideration of all five boundary conditions as to moment and stress resultants on the crack surface. The effect of Poisson's ratio on the stress intensity factors and the nature of the out-of-plane displacement along the edges of the crack, i.e., bulging, are also studied.

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For propagation of surface shallow-water waves on irrotational flows, we
derive a new two-component system. The system is obtained by a variational
approach in the Lagrangian formalism. The system has a non-canonical
Hamiltonian formulation. We also find its exact solitary-wave solutions.

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We represent three generations of students: Bob Glassey, Walter's student
finishing at Brown in 1972, Jack Schaeffer, Bob's student finishing at Indiana
University in 1983, and Steve Pankavich, Jack's student finishing at Carnegie
Mellon in 2005. We have all thrived professionally from our association with
Walter and are delighted to dedicate this note to him on the occasion of his
70th birthday. The problem we study concerns the asymptotic behavior of
solutions to Vlasov equations, an area to which Walter has contributed greatly.

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This paper is concerned with global existence of weak solution for a periodic
two-component $\mu$-Hunter-Saxton system. We first derive global existence for
strong solutions to the system with smooth approximate initial data. Then, we
show that the limit of approximate solutions is a global weak solution of the
two-component $\mu$-Hunter-Saxton system.

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Differential equations for composite radiation and convection heat exchange along constant area fins

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Derivation of governing equations for multiphase flow on the base of
thermodynamically compatible systems theory is presented. The mixture is
considered as a continuum in which the multiphase character of the flow is
taken into account. The resulting governing equations of the formulated model
belong to the class of hyperbolic systems of conservation laws. In order to
examine the reliability of the model, the one-dimensional Riemann problem for
the four phase flow is studied numerically with the use of the MUSCL-Hancock
method in conjunction with the GFORCE flux.

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Noniterative algorithm for generalized inverse of arbitrary rectangular matrix computation

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The classical (inviscid) stability analysis of shock waves is based on the
Lopatinski determinant, \Delta---a function of frequencies whose zeros
determine the stability of the underlying shock. A careful analysis of \Delta\
shows that in some cases the stable and unstable regions of parameter space are
separated by an open set of parameters. Zumbrun and Serre [Indiana Univ. Math.
J., 48 (1999) 937--992] have shown that, by taking account of viscous effects
not present in the definition of \Delta, it is possible to determine the
precise location in the open, neutral set of parameter space at which stability
is lost. In particular, they show that the transition to instability under
suitably localized perturbations is determined by an "effective viscosity"
coefficient. Here, in the simplest possible setting, we propose and implement
two new approaches toward the practical computation of this coefficient.
Moreover, in a special case, we derive an exact solution of the relevant
differential equations.

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Mechanical and thermodynamical theories of material behavior based on rate independence concept, discussing equivalence of various theories

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Linear comparison systems stability condition proved with Kotelianskii and Bailey theorems

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