bluuebunny/crossref_metadata_2025 huggingface dataset:
On a method to evaluate Fourier-Bessel series with poor convergence properties and its application to linearized supersonic free jet flow
Authors: Dillmann A. , Grabitz G.
Published: December 2016

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. As an illustrative example, 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.
The modeling of piezoceramic patch interactions with shells, plates, and beams
Authors: Banks H. T. , Smith R. C. , Wang Yun
Published: December 2016

General models describing the interactions between one or a pair of piezoceramic patches and elastic substructures consisting of a cylindrical shell, plate, or beam are presented. In each case, the contributions to the internal moments and forces due to the presence of the patches are carefully discussed. In addition to these material contributions, the input of voltage to the patches produces mechanical strains that lead to external moments and forces. These external loads depend on the material properties of the patch, the geometry of patch placement, and the voltage. The internal and external moments and forces due to the patches are then incorporated into the equations of motion, which yields models describing the dynamics of the combined structure. These models are sufficiently general to allow for potentially different patch voltages, which implies that they can be suitably employed when using piezoceramic patches for controlling system dynamics when both extensional and bending vibrations are present.
Shock-layer bounds for a singularly perturbed equation
Authors: Scroggs Jeffrey S.
Published: December 2016

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.
The effective electrical conductivity of nonlinear laminate composites
Authors: Li Guoan , Douglas Andrew S.
Published: December 2016

This paper studies the effective nonlinear electrical conductivity of sequentially laminated materials in both two and three dimensions. The exact nonlinear conductive behavior is obtained through a variational procedure [1] that expresses the nonlinear properties of the laminate composite in terms of an optimization with respect to the properties of a series of linear comparison materials. Multiphase laminate composites are discussed and these results are compared with nonlinear Hashin-Shtrikman (H-S) bounds. It is found that some of the laminates possess extremal microstructures that attain the nonlinear H-S lower bound while others are very close to extremal microstructures. The results apply also to the nonlinear thermal conductivity and dielectric and magnetic behavior of laminated heterogeneous materials.
On finite displacements of curved annular elastic membranes without wrinkling
Authors: Beck Armin , Grabmüller Hans
Published: December 2016

Axisymmetric deformations of curved annular elastic membranes subjected to vertical surface loads and radial edge loads or displacements are considered within Reissner’s finite-rotation theory of thin shells of revolution, assuming linear stress-strain relations. The principal stresses in the membrane are determined by the solutions of a nonlinear second-order ODE, dependent on a geodesic variable. Analytical methods are used in order to determine the range of those boundary data for which the solutions of the differential equation are wrinkle-free in the sense that both the radial and the circumferential stress components are nonnegative everywhere.
On the second solution of flow of viscoelastic fluid over a stretching sheet
Authors: Ariel P. D.
Published: December 2016

For the boundary layer flow of a viscoelastic fluid caused by the stretching of a sheet, it is demonstrated that, besides the well-known solution, a second solution exists for all nonzero values of

k
k

, the viscoelastic fluid parameter. This solution is obtained in closed form. It exhibits an oscillatory behavior with the oscillations increasing as

k
k

approaches the value zero.
Some criteria for the disappearance of the mushy region in the Stefan problem
Authors: Götz I. G. , Zaltzman B.
Published: December 2016

The disappearance of the mushy region in a multidimensional one-phase Stefan problem is discussed. In the case of a piecewise-smooth boundary of the domain and bounded initial-boundary data, sufficient conditions for the disappearance of the mushy zone in a finite time are presented. For a

C
2

{C^2}

-smooth boundary and appropriately smooth boundary data both necessary and sufficient conditions for the mush to vanish are obtained. Possible behaviors of the transient phase for a twodimensional solution near a corner point of the domain are also investigated.
An exact solution to a problem of axisymmetric torsion of an elastic space with a spherical crack
Authors: Godin Yuri A.
Published: December 2016

It is a well-known fact that the three-dimensional axisymmetric dilatation problem for an elastic space containing a spherical crack has an explicit solution, found and analyzed in [1]—[3]. The corresponding torsion problem, however, also of interest, has not apparently been investigated. To remedy this shortage we present here an exact solution to this problem, which incidentally reveals a remarkable effect concerning the stress intensity.
Electromagnetic field in the source region of continuously varying current density
Authors: Fikioris John G.
Published: December 2016

Continuity, analyticity, and the singular points of the vector potential A and the field vectors H, E in a spherical source region

ν
\nu

are investigated thoroughly for, practically, any continuous current density distribution J in

ν
\nu

. In other words, this is a study of the inhomogeneous Helmholtz equation in

ν
\nu

. Explicit results for A, H, E are obtained by direct integration, extending previous results for constant density in

ν
\nu

to continuously varying ones. The importance of imposing the Hölder condition on J to insure existence of E and of certain second derivatives of A is explicitly demonstrated by a specific continuous J, violating this condition at a point; it is then seen that E and some second derivatives of A do not exist, tending to infinity, at that point.
Limits to extensions of Burgers’ equation
Authors: Nerney Steven , Schmahl Edward J. , Musielak Z. E.
Published: December 2016

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.
Structure for nonnegative square roots of unbounded nonnegative selfadjoint operators
Authors: Yao Peng-Fei , Feng De-Xing
Published: December 2016

It is well known that, for an unbounded nonnegative selfadjoint operator

A
A

on a Hilbert space, there is a unique nonnegative square root

A

1

/

2

{A^{1/2}}

, which is frequently associated with the structural damping in many practical vibration systems. In this paper we develop a general theory for the structure of

A

1

/

2

{A^{1/2}}

, which includes the expression of

A

1

/

2

{A^{1/2}}

and a program to find the domain of

A

1

/

2

{A^{1/2}}

explicitly from the domain of

A
A

. The relationship between

A

1

/

2

{A^{1/2}}

and related differential operators is determined for the selfadjoint differential operator

A
A

. Finally, the theoretical results given in this paper are applied to fourth-order “beam” operators and

n
n

-dimensional “wave” operators with sufficient complexity for applications to elastic vibration systems.
Time-dependent drop deformation in a rotating high viscosity fluid
Authors: Stone H. A. , Bush J. W. M.
Published: December 2016

An exact solution is presented which describes the time-dependent deformation of a nearly spherical drop suspended on the rotation axis of a more dense rotating viscous fluid. The solution is demonstrated to be similar, though not identical, to that derived from the commonly invoked assumption that the external flow field is purely extensional.
General contact-angle conditions with and without kinetics
Authors: Angenent Sigurd , Gurtin Morton E.
Published: December 2016

The classical condition for the contact angle of a phase interface at a container wall is generalized to include both anisotropy and kinetics. The derivation, which does not involve an assumption of local equilibrium, is based on a capillary force balance, a dissipation inequality representing the second law, and suitable constitutive assumptions.
Dissipation of energy for magnetoelastic waves in a conductive medium
Authors: Andreou Elias , Dassios George
Published: December 2016

We consider the propagation of magnetoelastic waves within a homogeneous and isotropic elastic medium exhibiting finite electric conductivity. An appropriate physical analysis leads to a decoupling of the governing system of equations which in turn effects an irreducible factorization of the ninth-degree characteristic polynomial into a product of first, third, and fifth-degree polynomials. Regular and singular perturbation methods are then used to deduce asymptotic expansions of the characteristic roots which reflect the low and the high frequency dependence of the frequency on the wave number. Dyadic analysis of the spacial spectral equations brings the general solution into its canonical dyadic form. Extensive asymptotic analysis of the quadratic forms that define the kinetic, the strain, the magnetic and the dissipation energy provides the rate of dissipation of these energies as the time variable approaches infinity. The rate of dissipation obtained coincides with the corresponding rate for thermoelastic waves. Therefore, a similarity between the dissipative effects of thermal coupling and that of finite conductivity upon the propagation of elastic waves is established.
The one-phase supercooled Stefan problem with a convective boundary condition
Authors: Tarzia Domingo A. , Turner Cristina V.
Published: December 2016

We consider the supercooled one-phase Stefan problem with convective boundary condition at the fixed face. We analyse the relation between the heat transfer coefficient and the possibility of continuing the solution for arbitrarily large time intervals.
Discontinuities in velocity gradients and temperature in the Stokes’ first problem with nonclassical heat conduction
Authors: Puri P. , Kythe P. K.
Published: December 2016

The nonclassical heat conduction equation based on the MCF model is used to study the discontinuities in velocity gradients and temperature in fluid flows induced by impulsive or sudden heating of a plate. The influence of the thermal relaxation time in the temperature and velocity fields is investigated.
Exact analysis of a nonlinear partial differential equation of gas dynamics
Authors: Sachdev P. L. , Dowerah S. , Vaganan B. Mayil , Philip Varughese
Published: December 2016

A new second-order nonlinear partial differential equation is derived from one-dimensional unsteady non-isentropic gas-dynamic equations through the introduction of three “potential” functions. Appropriate boundary conditions at the shock and at the piston in terms of the new functions are obtained. The nonlinear partial differential equation is analysed in great detail. Intermediate integrals and generalized Riemann invariants are discovered. Using the classical Lie group method, the direct similarity method due to Clarkson and Kruskal (1989), and equation-splitting etc., large families of new solutions are found. The direct similarity method is found to yield the most general results. Solutions with shocks (both finite and strong) are constructed to illustrate the applicability of the solutions.
High-order essentially non-oscillatory scheme for viscoelasticity with fading memory
Authors: Shu Chi-Wang , Zeng Yanni
Published: December 2016

In this paper we describe the application of high-order essentially nonoscillatory (ENO) finite difference schemes to the viscoelastic model with fading memory. ENO schemes can capture shocks as well as various smooth structures in the solution to a high-order accuracy without spurious numerical oscillations. We first verify the stability and resolution of the scheme. We apply the scheme to a nonlinear problem with a known smooth solution and check the order of accuracy. Then we apply the scheme to a linear problem with initial discontinuities. Discontinuity locations and strengths in the solutions of such problems can be found explicitly by making use of a pointwise estimate obtained in this paper for the Green’s function of the equations, which contains two Dirac

δ
\delta

-functions decaying exponentially. We check the resolution of the discontinuities by the scheme. After verifying that the scheme is indeed high-order accurate, produces sharp, non-oscillatory shocks with the correct location and strength, we then proceed in applying it to the nonlinear case with discontinuous or smooth initial conditions, and study the local properties (in time) as well as the long time behavior of the solutions. We conclude that the ENO scheme is a robust, accurate numerical tool to supplement theoretical analysis to study such equations with memory terms. It should also provide an efficient and reliable practical tool when such equations must be solved numerically in applications.
A dynamical model for multilayered plates with independent shear deformations
Authors: Hansen Scott W.
Published: December 2016

In this paper a dynamic model for an

n
n

-layered plate is developed based upon the assumptions of Reissner-Mindlin plate theory. Each plate layer is assumed to be transversely isotropic, transversely homogeneous and of a uniform thickness; however, no symmetry in the material properties or thicknesses of each plate is assumed. The layers are assumed to be perfectly bonded so that no slip occurs along the interface. No additional a priori kinematic restrictions are imposed upon the motion of the plates. The equations of motion are derived by the principle of virtual work. Existence and uniqueness results are obtained. In the case where the layers are symmetric we show that all solutions decouple into a bending solution (with antisymmetric displacements about the mid-plane) and an in-plane solution (with symmetric displacements).
Penny-shaped crack bridged by fibres
Authors: Movchan N. V. , Willis J. R.
Published: December 2016

An axi-symmetric problem for a penny-shaped crack bridged by fibres is considered. The study is reduced to the analysis of a hypersingular integral equation with respect to the relative crack-face separation over a circular domain occupied by the crack. The special case of a large crack subjected to Mode-I loading is examined. A matched asymptotic expansions technique is used in order to estimate the stress intensity factor at the crack edge and to evaluate the magnitude of the applied load which provides the critical opening for failure of fibres at the centre of the crack.
Morphological instability of similarity solution to the Stefan problem with undercooling and surface tension
Authors: Rubinstein I. , Zaltzman B.
Published: December 2016

This paper concerns morphological stability of a similarity solution to the Stefan problem with surface tension and initial supercooling. The linear stability analysis shows that for a nonzero surface tension each perturbation mode with a nonzero wave number is stable. However, the solution is unstable with respect to perturbations with a zero wave number limit point in their Fourier spectrum.
Dynamics of a viscoelastic spherical shell with a nonconvex strain energy function
Authors: Fosdick Roger , Ketema Yohannes , Yu Jang-Horng
Published: December 2016

We study the radial motion of an incompressible viscoelastic spherical shell with a nonconvex strain energy function that models a material that can undergo a phase transition. In addition to the classical Newtonian viscosity for viscoelastic materials, we consider a material with two microstructural coefficients that are supposed to sense local configurational changes that take place during a deformation. Conditions necessary to show the effect of the nonconvexity of the strain energy function during a phase transition of the material, are determined, and the resulting dynamics is analyzed. It is shown that, though small periodic vibrations are possible, the system can easily revert into a mode of large amplitude motion as a result of small external excitation. Such motion may be transient to periodic motion or to chaotic motion. Boundaries in parameter space for the occurrence of this type of motion are determined and examples are given.
The complicated dynamics of heavy rigid bodies attached to deformable rods
Authors: Antman Stuart S. , Marlow Randall S. , Vlahacos Constantine P.
Published: December 2016

We study the motion in space of light nonlinearly elastic and viscoelastic rods with heavy rigid attachments. The rods, which can suffer flexure, extension, torsion, and shear, are described by a general geometrically exact theory. We pay particular attention to the leading term of the asymptotic expansion of the governing equations as the inertia of the rod goes to zero. When the rods are elastic and weightless, and when they have appropriate initial conditions, they move irregularly through a family of equilibrium states parametrized by time; the motion of the rigid body is governed by an interesting family of multivalued ordinary differential equations. These ordinary differential equations for a heavy mass point attached to an elastica undergoing planar motion are explicitly treated. These problems illuminate such phenomena as snap-buckling. On the other hand, when the rods are viscoelastic and weightless, the rigid body is typically not governed by ordinary differential equations, but, as we show, the motion of the system is well-defined for arbitrary initial conditions. This analysis relies critically on the careful use of our properly invariant constitutive hypotheses.
Eigenoscillations of mechanical systems with boundary conditions containing the frequency
Authors: Belinskiy B. P. , Dauer J. P.
Published: December 2016

The problem of eigenoscillations of beam-mass systems is investigated and four examples are developed. For such systems the corresponding Sturm-Liouville problems contain the eigenvalue parameter in the boundary conditions. It is shown that the eigenfunctions for the systems considered form a basis of the appropriate Hilbert space. Rayleigh-Ritz formulas are also developed. Some lower bound estimations for eigenfrequencies are also found.
Boundary value problems for degenerate von Kármán equations
Authors: Root Robert G.
Published: December 2016

This article presents regularity results that admit a weak formulation for degenerate von Kármán boundary value problems modeling the deformation of clamped plates that lose stiffness in one direction. These boundary value problems are derived in the companion article, A Derivation of Degenerate von Kármán Equations for Strongly Anisotropic Plates, by the author. The equations are a fourth-order elliptic-parabolic system of weakly coupled nonlinear equations. The article includes the weak formulation and a brief description of the appropriate existence results for the formulation.
Global existence and asymptotic behavior of weak solutions to nonlinear thermoviscoelastic systems with clamped boundary conditions
Authors: Shen Weixi , Zheng Songmu , Zhu Peicheng
Published: December 2016

This paper is concerned with global existence, uniqueness, and asymptotic behavior, as time tends to infinity, of weak solutions to nonlinear thermoviscoelastic systems with clamped boundary conditions. The constitutive assumptions for the Helmholtz free energy include the model for the study of phase transitions in shape memory alloys. To describe phase transitions between different configurations of crystal lattices, we work in a framework in which the strain

u
u

belongs to

L
∞

{L^\infty }

. It is shown that for any initial data of (strain, velocity, absolute temperature)

(

u
0

,

v
0

,

θ
0

)

∈

L
∞

×

W
0

1
,
∞

×

H
1

\left ( {u_0}, {v_0}, {\theta _0} \right ) \in \\ {L^\infty } \times W_0^{1, \infty } \times {H^1}

, there is a unique global solution
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis u comma v comma theta right-parenthesis element-of upper C left-parenthesis left-bracket 0 comma plus normal infinity right-bracket semicolon upper L Superscript normal infinity Baseline right-parenthesis times upper C left-parenthesis 0 comma plus normal infinity right-parenthesis semicolon upper W 0 Superscript 1 comma normal infinity Baseline right-parenthesis intersection upper L Superscript normal infinity Baseline left-parenthesis left-bracket 0 comma plus normal infinity right-parenthesis semicolon upper W Superscript 1 comma normal infinity Baseline right-parenthesis times upper C left-parenthesis left-bracket 0 comma plus normal infinity right-parenthesis semicolon upper H Superscript 1 Baseline right-parenthesis">

(
u
,
v
,
θ
)

∈
C

(

[
0
,
+
∞
]

;

L
∞

)

×
C

(
0
,
+
∞
)

;

W
0

1
,
∞

)

∩

L
∞

(

[
0
,
+
∞
)
;

W

1
,
∞

)

×
C

(

[
0
,
+
∞
)

;

H
1

)

\left ( u, v, \theta \right ) \in C\left ( \left [ 0, + \infty \right ]; {L^\infty } \right ) \times C\left ( 0, + \infty \right ); \\ \left . W_0^{1, \infty } \right ) \cap {L^\infty }\left ( \left [ 0, + \infty ); {W^{1, \infty }} \right ) \times C\left ( \left [ 0, + \infty \right ); {H^1} \right ) \right .

. Results concerning the asymptotic behavior as time goes to infinity are obtained.
On a nonlocal dispersive equation modeling particle suspensions
Authors: Zumbrun Kevin
Published: December 2016

We study a nonlocal, scalar conservation law

u
t

+

(

(

K
a

∗
u
)

u
)

x

=
0

{u_t} + {\left ( \left ( {K_a} * u \right )u \right )_x} = 0

, modeling sedimentation of particles in a dilute fluid suspension, where

K
a

(
x
)

=

a

−
1

K

(
x

/

a
)

{K_a}\left ( x \right ) = {a^{ - 1}}K\left ( x/a \right )

is a symmetric smoothing kernel, and

∗
\ast

represents convolution. We show this to be a dispersive regularization of the Hopf equation,

u
t

+

(

u
2

)

x

=
0

{u_t} + {\left ( {u^2} \right )_x} = 0

, analogous to KdV and certain dispersive difference schemes. Using the smoothing property of convolution and the physical principle of conservation of mass, we establish the global existence of smooth solutions.
Matched asymptotic expansion calculation of the equilibrium shape of a hole in a thin liquid film
Authors: O’Brien S. B. G.
Published: December 2016

We consider the occurrence of small axisymmetric pinholes in an otherwise uniform infinite thin liquid film. Corresponding to any particular undisturbed film thickness there exists precisely one (unstable) equilibrium solution reflecting a balance between surface tension and gravity effects. If a pinhole is smaller than this critical size the pinhole tends to close over and “heal". If a pinhole is larger it tends to open out. So determination of this critical hole size is crucial. We examine this problem in the case of a “small” pinhole where the fundamental length-scale in the film is much smaller than the capillary length. Solutions are obtained using matched asymptotic expansions for which several different scalings are necessary.
A study of the effective properties of mass and stiffness microstructures—a multiresolution approach
Authors: Steinberg Ben Zion , McCoy John J.
Published: December 2016

The theory of multiresolution decomposition and wavelets is used to study the effective properties of a thin elastic plate with surface mass density or stiffness heterogeneity, subjected to time-harmonic forcing. The heterogeneity possesses micro- and macro-scale variations, and has a macroscale outer dimension. It is shown that the microscale mass variation has practically no effect on the macroscale plate response, whereas microscale stiffness variation can have a significant effect. We derive an effective constitutive relation pertaining to a microscale stiffness variation. It is shown that it is possible to synthesize classes of different stiffness microstructures that have the same footprint on the macroscale component of the plate response. An effective, smooth, stiffness heterogeneity associated with the classes is developed. The results are first derived analytically and then supported by numerical simulations.
Existence of solution of a coupled problem arising in the thermoelectrical simulation of electrodes
Authors: Bermúdez Alfredo , Muñoz-Sola Rafael
Published: December 2016

In this paper we prove the existence of a solution to a system of partial differential equations arising from the thermoelectrical modeling of electrodes for electric furnaces. It consists of Maxwell equations coupled with the heat transfer equation through the Joule effect and the fact that thermal conductivity and electrical resistivity depend on temperature. The problem is formulated in cylindrical coordinates to take advantage of its axisymmetry.
Multiple existence and linear stability of equilibrium balls in a nonlinear free boundary problem
Authors: Taniguchi M.
Published: December 2016

This paper studies construction and linear stability of spherical interfaces in an equilibrium state in a two-phase boundary problem arising in activator-inhibitor models in chemistry. By studying the linearized eigenvalue problem near a given equilibrium ball, we show that the eigenvalues with nonnegative real parts are all real, and that they are characterized as values of a strictly convex function for specific discrete values of its argument. The stability is determined by the location of the zero points of this convex function. Using this fact, we present a criterion of stability in a useful form. We show examples and illustrate that stable equilibrium balls and unstable ones coexist near saddle-node bifurcation points in the bifurcation diagram, and a given equilibrium ball located far from bifurcation points is unstable and the eigenfunction associated with the largest eigenvalue consists of spherically harmonic functions of high degrees.
Existence and regularity of solutions of non-Newtonian flow
Authors: Bae Hyeong-Ohk
Published: December 2016

The existence and regularity of Young measure-valued solutions and weak solutions to non-Newtonian flows are considered. Galerkin approximation and an

L

2

{L^{2}}

compactness theorem are main ingredients for the proof of the existence of Young measure-valued solutions. Under a certain convexity condition for the energy, we prove that Young measure-valued solutions are weak solutions. Also, for the limited cases, we prove a regularity theorem.
The onset of convection in horizontal cylinders
Authors: McHugh John P.
Published: December 2016

The convective instability of a fluid that fills a horizontal cylindrical cavity is considered. The boundaries of the cavity are conducting, and the driving force is a linear temperature gradient far from the cylinder. Critical Rayleigh numbers governing the onset of convection are determined for neutral stability. The results show that the critical Rayleigh number depends on the ratio of thermal conductivities of the solid to the fluid

(
λ
)

\left ( \lambda \right )

, and a wavenumber. Both two- and three-dimensional disturbances are included. The disturbances are separated into even modes and odd modes. The most unstable odd modes have been found to be two-dimensional, while the most unstable even modes are three-dimensional. The two-dimensional odd modes are most unstable in the vicinity of

λ
=
1

\lambda = 1

. The three-dimensional even modes are more unstable for other values of

λ
\lambda

. The results are compared with the previous results of Gershuni and Zhukhovitskii.
Uniform attractors for a non-autonomous semilinear heat equation with memory
Authors: Giorgi Claudio , Pata Vittorino , Marzocchi Alfredo
Published: December 2016

In this paper we investigate the asymptotic behavior, as time tends to infinity, of the solutions of a non-autonomous integro-partial differential equation describing the heat flow in a rigid heat conductor with memory. Existence and uniqueness of solutions is provided. Moreover, under proper assumptions on the heat flux memory kernel and on the magnitude of nonlinearity, the existence of uniform absorbing sets and of a global uniform attractor is achieved. In the case of quasiperiodic dependence of time of the external heat supply, the above attractor is shown to have finite Hausdorff dimension.
Qualitative behavior of conservation laws with reaction term and nonconvex flux
Authors: Mascia Corrado
Published: December 2016

The aim of the paper is to study qualitative behavior of solutions to the equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction partial-differential u Over partial-differential t EndFraction plus StartFraction partial-differential f left-parenthesis u right-parenthesis Over partial-differential x EndFraction equals g left-parenthesis u right-parenthesis comma">

∂
u

∂
t

+

∂
f

(
u
)

∂
x

=
g

(
u
)

,

\frac {{\partial u}}{{\partial t}} + \frac {{\partial f\left ( u \right )}}{{\partial x}} = g\left ( u \right ) ,

\]
where

(
x
,
t
)

∈

R

×

R

+

,
u
=
u

(
x
,
t
)

∈

R

\left ( x, t \right ) \in \mathbb {R} \times {\mathbb {R}_ + }, u = u\left ( x, t \right ) \in \mathbb {R}

. The main new feature with respect to previous works is that the flux function

f
f

may have finitely many inflection points, intervals in which it is affine, and corner points. The function

g
g

is supposed to be zero at 0 and 1, and positive in between.
Load maximum behavior in the inflation of hollow spheres of incompressible material with strain-dependent damage
Authors: Huntley H. E. , Wineman A. S. , Rajagopal K. R.
Published: December 2016

Carroll has shown three qualitatively different cases of behavior in the load-expansion relation for the inflation of hollow incompressible isotropic elastic spheres. Each of these cases was related to material response in uniaxial compression (or equal biaxial extension). For “type A” materials, load increases monotonically with expansion; for “type B” materials, load increases monotonically and then decreases; for “type C” materials, load increases monotonically, decreases, and again increases. The present work discusses the monotonicity properties of the load-expansion relation when rubbery materials undergo microstructural change or damage. The analysis is carried out using a constitutive equation for materials undergoing continuous scission and reformation of macromolecular junctions. Results are presented for the case when this leads to softening of response. For “type A", sufficient softening can cause loss of monotonicity; for “type B", the softening leads to loss of monotonicity at smaller levels of inflation and lower loads.
Well-posedness and longtime behavior of the phase-field model with memory in a history space setting
Authors: Giorgi Claudio , Grasselli Maurizio , Pata Vittorino
Published: December 2016

A thermodynamically consistent phase-field model with memory, based on the linearized version of the Gurtin-Pipkin heat conduction law, is considered. The formulation of an initial and boundary value problem for the phase-field evolution system is framed in a history space setting. Namely, the summed past history of the temperature is regarded itself as a variable along with the temperature and the phase-field. Well-posedness results are discussed, as well as longtime behavior of solutions. Under suitable conditions, the existence of an absorbing set can be achieved.
A remark on the existence of global BV solutions for a nonlinear hyperbolic wave equation
Authors: Dias João-Paulo , Figueira Mário
Published: December 2016

By means of a suitable change of variables we obtain, by application of a general result by Dafermos and Hsiao, cf. [2], an existence theorem in

L
∞

∩

B

V

l
o
c

{L^\infty } \cap {BV_{loc}}

of a weak solution of the system corresponding to the quasilinear hyperbolic equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi Subscript t t Baseline minus p prime left-parenthesis phi Subscript x Baseline right-parenthesis phi Subscript x x Baseline plus phi Subscript t Baseline plus upper F left-parenthesis phi right-parenthesis equals 0 i n double-struck upper R times left-bracket 0 comma plus normal infinity left-bracket">

ϕ

t
t

−

p
′

(

ϕ
x

)

ϕ

x
x

+

ϕ
t

+
F

(
ϕ
)

=
0

i
n

R

×

[

0
,
+
∞

[

{\phi _{tt}} - p’\left ( {\phi _x} \right ){\phi _{xx}} + {\phi _t} + F\left ( \phi \right ) = 0 \qquad in \qquad \mathbb {R} \times \left [ {0, + \infty } \left [ \right . \right .

\]
, for small initial data in BV. This theorem is a partial extension of Dafermos’s result for the case with

F

(
ϕ
)

≡
0

F\left ( \phi \right ) \equiv 0

, proved in [1].
Initial-boundary value problem to systems of conservation laws with relaxation
Authors: Xin Zhouping , Xu Wen-Qing
Published: December 2016

In this paper we consider the initial-boundary value problem (IBVP) for the one-dimensional Jin-Xin relaxation model. The main interest is to study the boundary layer behaviors in the solutions to the IBVP of the relaxation system and their asymptotic convergence to solutions of the corresponding hyperbolic conservation laws in the limit of small relaxation rate. First we develop a general expansion theory for the relaxation IBVP using a matched asymptotic analysis. This formal procedure determines a unique equilibrium limit, and also reveals rich initial and boundary layer structures in the solutions of the relaxation system. Arbitrarily accurate solutions to the IBVP of the relaxation system are then constructed by combining the various orders of the equilibrium solutions, the initial and boundary layer solutions. The validity of the initial and boundary layers and the asymptotic convergence results are rigorously justified through a stability analysis for a broad class of boundary conditions in the case when the relaxation system is

2
×
2

2 \times 2

.
Maximum and minimum free energies for a linear viscoelastic material
Authors: Fabrizio M. , Golden J. M.
Published: December 2016

Certain results about free energies of materials with memory are proved, using the abstract formulation of thermodynamics, both in the general case and as applied within the theory of linear viscoelasticity. In particular, an integral equation for the strain continuation associated with the maximum recoverable work from a given linear viscoelastic state is shown to have a unique solution and is solved directly, using the Wiener-Hopf technique. This leads to an expression for the minimum free energy, previously derived by means of a variational technique in the frequency domain. A new variational method is developed in both the time and frequency domains. In the former case, this approach yields integral equations for both the minimum and maximum free energies associated with a given viscoelastic state. In the latter case, explicit forms of a family of free energies, associated with a given state of a discrete spectrum viscoelastic material, are derived. This includes both maximum and minimum free energies.
Large-time stability of travelling waves for a class of fully nonlinear parabolic equations
Authors: Camilli Fabio , Molinari Manuela
Published: December 2016

In this paper we prove existence and

L

1

{L^{1}}

stability of travelling waves for a class of second-order nonlinear parabolic equations in divergence form. As a consequence of the previous result, we get stability in the

L

∞

{L^{\infty }}

norm of travelling waves for a class of fully nonlinear second-order equations.
The Riemann problem for an elastic string with a linear Hooke’s law
Authors: Hanche-Olsen Harald , Holden Helge , Risebro Nils Henrik
Published: December 2016

We solve the Riemann problem for vibrations of an infinite, planar, perfectly elastic string with a linear relation between the string tension and the local stretching, i.e., with a linear Hooke’s law. The motion is governed by a

4
×
4

4 \times 4

system of conservation laws that is linearly degenerate in all wave families. Conservation of energy leads to

L

2

{L^{2}}

estimates that are used in the analysis.
Quasiperiodicity and chaos in the nonlinear evolution of the Kelvin-Helmholtz instability of supersonic anisotropic tangential velocity discontinuities
Authors: Choudhury S. Roy , Brown Kevin G.
Published: December 2016

A nonlinear stability analysis using a multiple-scales perturbation procedure is performed for the instability of two layers of immiscible, strongly anisotropic, magnetized, inviscid, arbitrarily compressible fluids in relative motion. Such configurations are of relevance in a variety of astrophysical and space configurations. For modes near the critical point of the linear neutral curve, the nonlinear evolution of the amplitude of the linear fields on the slow first-order scales is shown to be governed by a complicated nonlinear Klein-Gordon equation. The nonlinear coefficient turns out to be complex, which is, to the best of our knowledge, unlike previously considered cases and leads to completely different dynamics from that reported earlier. Both the spatially dependent and space-independent versions of this equation are considered to obtain the regimes of physical parameter space where the linearly unstable solutions either evolve to final permanent envelope wave patterns resembling the ensembles of interacting vortices observed empirically, or are disrupted via nonlinear modulation instability. In particular, the complex nonlinearity allows the existence of quasiperiodic and chaotic wave envelopes, unlike in earlier physical models governed by nonlinear Klein-Gordon equations. In addition, numerical diagnostics reveal the onset of chaos as a consequence of modulation of the external magnetic field.
Finite speed propagation in the relaxation of vortex patches
Authors: Rosier Carole , Rosier Lionel
Published: December 2016

A degenerate parabolic equation has been proposed by Robert and Sommeria to describe the relaxation towards a statistical equilibrium state for a two-dimensional incompressible perfect fluid with a vortex patch as initial vorticity. In this paper, flows obtained by numerical integration of the Robert-Sommeria equation over a long-time interval are compared with those obtained for the Navier-Stokes equation at high Reynolds number. A finite speed propagation for the extremal values of the vorticity is numerically shown to hold for the Robert-Sommeria equation. A rigorous proof of this (fine) property is also provided.
Asymptotic behavior of solutions in linear 2- or 3-D thermoelasticity with second sound
Authors: Racke Reinhard
Published: December 2016

We consider thermoelastic systems in two or three space dimensions where thermal disturbances are modeled propagating as wavelike pulses traveling at finite speed. This is done using Cattaneo’s law for heat conduction instead of Fourier’s law. For Dirichlet type boundary conditions, the exponential stability of the now purely, but slightly damped, hyperbolic system is proved in the radially symmetric case.
Global weak solutions to the Navier-Stokes equations for a 1D viscous polytropic ideal gas
Authors: Jiang Song , Zhang Ping
Published: December 2016

We prove the existence of global weak solutions to the Navier-Stokes equations for a one-dimensional viscous polytropic ideal gas. We require only that the initial density is in

L
∞

∩

L

l
o
c

2

{L^\infty } \cap L_{loc}^2

with positive infimum, the initial velocity is in

L

l
o
c

2

L_{loc}^2

, and the initial temperature is in

L

l
o
c

1

L_{loc}^1

with positive infimum. The initial density and the initial velocity may have differing constant states at

x
=
±
∞

x = \pm \infty

. In particular, piecewise constant data with arbitrary large jump discontinuities are included. Our results show that neither vacuum states nor concentration states can form and the temperature remains positive in finite time.
On an elastic dissipation model for a cantilevered beam
Authors: van Horssen W. T. , Zarubinskaya M. A.
Published: December 2016

In this paper we will study an elastic dissipation model for a cantilevered beam. This problem for a cantilevered beam has been formulated by D. L. Russell as an open problem in [1, 2]. To determine the relationship between the damping rates and the frequencies we will use a recently developed, adapted form of the method of separation of variables. It will be shown that the dissipation model for the cantilevered beam will not always generate damping. Moreover, it will be shown that some solutions can become unbounded.
Surface waves of non-Rayleigh type
Authors: Kuznetsov Sergey V.
Published: December 2016

Existence of surface waves of non-Rayleigh type propagating on some anisotropic elastic half-spaces is proved. Conditions for originating the non-Rayleigh type waves are analyzed. An example of a transversely isotropic material admitting a surface wave of the non-Rayleigh type is constructed.
On unique solvability and regularity in the linearized two-dimensional wave resistance problem
Authors: Pierotti Dario
Published: December 2016

We discuss existence, uniqueness, and regularity of the solutions of a boundary value problem in a strip, which is obtained by linearization of the equations of the wave-resistance problem for a cylinder semisubmerged in a heavy fluid of constant depth

H
H

and moving at uniform velocity

c
c

in the direction orthogonal to its generators. We show that the problem has a unique solution, rapidly decreasing at infinity, for every

c
>

g
H

c > \sqrt {gH}

, where

g
g

is the acceleration of gravity. For

c
>

g
H

c > \sqrt {gH}

, we prove unique solvability provided

c
≠

c
k

c \ne {c_k}

, where

c
k

{c_k}

is a known sequence monotonically decreasing to zero. In this case, the related flow has in general nontrivial oscillations at infinity downstream.
Far field boundary condition for convection diffusion equation at zero viscosity limit
Authors: Liu Jian-Guo , Xu Wen-Qing
Published: December 2016

In this paper, we give a systematic study of the boundary layer behavior for linear convection-diffusion equation in the zero viscosity limit. We analyze the boundary layer structures in the viscous solution and derive the boundary condition satisfied by the viscosity limit as a solution of the inviscid equation. The results confirm that the Neumann type of far-field boundary condition is preferred in the outlet and characteristic boundary condition. Under some appropriate regularity and compatibility conditions on the initial and boundary data, we obtain optimal error estimates between the full viscous solution and the inviscid solution with suitable boundary layer corrections. These results hold in arbitrary space dimensions and similar statements also hold for the strip problem.
The role of material non-homogeneities on the formation and evolution of strain non-uniformities in thermoviscoplastic shearing
Authors: Baxevanis Theocharis , Charalambakis Nicolas
Published: December 2016

In this paper we present the effect of macroscopic non-homogeneities on the distribution and evolution of strain non-uniformities during the shearing of thermoviscoplastic materials. The thermomechanical parameters (strain hardening, strain rate sensitivity, and thermal softening), as well as all of the material parameters are supposed to depend explicitly on the space variable. We show that, even under stability conditions, the strain exhibits intense, time-increasing non-uniformities, following the non-homogeneities, in a specific rate, which depends on the degree of non-homogeneity exhibited by the thermomechanical parameters. By considering both the isothermal and anisothermal cases, we obtain results indicating that non-uniformity measures, based on the control of the strain gradient, are more suitable to give stability conditions of non-homogeneous materials. Moreover, we present numerical results concerning the interplay between material non-homogeneities, initial defects, and boundary conditions for two specific cases: the shearing of a reinforced slab and the shearing of a plate composed of several thin layers of periodic structure. The results are in complete agreement with the analytical behavior of the model.
Dynamic behavior of the inextensible string
Authors: Dickey R. W.
Published: December 2016

The two dimensional dynamic behavior of a geometrically exact inextensible string is discussed. A variety of exact solutions are described and various asymptotic theories are derived. The similarity between the motion of the inextensible string and galactic motion is described.
Shrinking similar solutions of a convection diffusion equation
Authors: Yin Jingxue , Wang Chunpeng
Published: December 2016

In this paper we study the convection diffusion equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction partial-differential u Over partial-differential t EndFraction equals normal upper Delta u Superscript m Baseline minus x dot nabla u Superscript q Baseline comma left-parenthesis x comma t right-parenthesis element-of double-struck upper R Superscript n Baseline times left-parenthesis 0 comma plus normal infinity right-parenthesis">

∂
u

∂
t

=
Δ

u
m

−
x
⋅
∇

u
q

,

(
x
,
t
)
∈

R

n

×
(
0
,
+
∞
)

\frac {{\partial u}}{{\partial t}} = \Delta {u^m} - x \cdot \nabla {u^q}, \qquad (x, t) \in {\mathbb {R}{^n}} \times (0, + \infty )

\]
, where

m
>
1
,
1
>
q
≤
m

m > 1, 1 > q \le m

.
A non-standard boundary value problem related to geomagnetism
Authors: Kaiser Ralf , Neudert Michael
Published: December 2016

Consider the following boundary value problem in the exterior space

S
^

d
−
1

=

{

x
∈

d

:

|
x
|

>
1

}

{\hat S^{d - 1}} = \left \{ {x \in {^d}:\left | x \right | > 1} \right \}

of a sphere

S

d
−
1

{S^{d - 1}}

in two and three dimensions

(
d
=
2
,
3
)

\left ( d = 2, 3 \right )

: Given a vector field

D
:

S

d
−
1

→

d

D:{S^{d - 1}} \to {^d}

we ask for all harmonic vector fields

B
:

S
^

d
−
1

→

d

B:{\hat S^{d - 1}} \to {^d}

which decay at least as fast as a dipole field at infinity and are parallel to D on

S

d
−
1

{S^{d - 1}}

i.e. there is

f
:

S

d
−
1

→

f:{S^{d - 1}} \to

such that

B
=
f
D

B = fD

. For

d
=
3

d = 3

, this problem is related to the problem of reconstructing the geomagnetic field outside the earth from directional data measured on the earth’s surface. The question for uniqueness or non-uniqueness is of particular interest here.