Showing papers in "Quarterly of Applied Mathematics in 2011"

Qualitative properties of solutions to a time-space fractional evolution equation

Abstract: In this article, we analyze a spatio-temporally nonlocal nonlinear parabolic equation. First, we valid the equation by an existence-uniqueness result. Then, we show that blowing-up solutions exist and study their time blow-up profile. Also, a result on the existence of global solutions is presented. Furthermore, we establish necessary conditions for local or global existence.

Shock wave formation process for a multidimensional scalar conservation law

Abstract: We construct a global smooth approximate solution to a multidimensional scalar conservation law describing the shock wave formation process for initial data with small variation. In order to solve the problem, we modify the method of character- istics by introducing "new characteristics" | nonintersecting curves along which the (approximate) solution to the problem under study is constant. The procedure is based on the weak asymptotic method, a technique which appeared to be rather powerful for investigating nonlinear waves interactions

Delta shock waves for the chromatography equations as self-similar viscosity limits

Abstract: The Riemann problem for the changed form of the chromatography system is considered here. It can be shown that the delta shock wave appears in the Riemann solution for exactly specified initial states. The generalized Rankine-Hugoniot relation of the delta shock wave is derived in detail. The existence and uniqueness of solutions involving the delta shock wave for the Riemann problem is proven by employing the self-similar viscosity vanishing approach.

Global smooth solutions for the compressible viscous and heat-conductive gas

Abstract: This paper is concerned with the global existence of smooth solutions to a system of equations describing one-dimensional motion of a self-gravitating, radiative and chemically reactive gas. We have proved that for any arbitrary large smooth initial data, the problem under consideration admits a unique globally smooth (classical) solution. Our results have improved those results by Umehara and Tani ([J. Differential Equations, 234(2007), 439-463; Proc. Japan Acad., 84, Ser. A(2008), 123-128]) and also by Qin, Hu, Huang, and Ma.

Representation of divergence-free vector fields

Abstract: This paper focuses on a representation result for divergence-free vector fields. Known results are recalled, namely the representation of divergence-free vector fields as curls in two and three dimensions. The representation proposed in the present paper expresses the vector field as an exterior product of gradients and remains valid in arbitrary dimensions. Links to computer graphics and to partial differential equations are discussed.

Multiple solutions for hydromagnetic flow of a second grade fluid over a stretching or shrinking sheet

Abstract: We study a class of fourth-order nonlinear differential equations arising in the hydromagnetic flow of a second grade fluid over a stretching or shrinking sheet. Explicit exact solutions are obtained. Furthermore we show that the differential equation may admit zero or one or two physically meaningful solutions depending on the values of the physical parameters of the model. As a special case, we recover the single or the dual solutions and compare them with the available results in the literature. Also, the obtained multiple solutions for several sets of values of the parameters are presented through tables and graphs, and the qualitative behaviors are discussed.

Instantaneous shock location and one-dimensional nonlinear stability of viscous shock waves

Abstract: We illustrate in a simple setting the instantaneous shock tracking approach to stability of viscous conservation laws introduced by Howard, Mascia, and Zumbrun. This involves a choice of the definition of instanteous location of a viscous shock‐ we show that this choice is time-asymptotically equivalent both to the natural choice of least

Global weak solutions to the Euler-Boltzmann equations in radiation hydrodynamics

Abstract: The Cauchy problem for the one-dimensional Euler-Boltzmann equations in radiation hydrodynamics is studied. The global weak entropy solutions are constructed using the Godunov finite difference scheme. The global existence of weak entropy solutions in L∞ with arbitrarily large initial data is established with the aid of the compensated compactness method.

Global attractors for the suspension bridge equations with nonlinear damping

Abstract: In this paper, we prove the existence of a global attractor for the suspension bridge equations with nonlinear damping.

Mathematical modelling of avascular ellipsoidal tumour growth

Abstract: Breast cancer is the most frequently diagnosed cancer in women. From mammography, Magnetic Resonance Imaging (MRI), and ultrasonography, it is well documented that breast tumours are often ellipsoidal in shape. The World Health Organisation (WHO) has established a criteria based on tumour volume change for classifying response to therapy. Typically the volume of the tumour is measured on the hypothesis that growth is ellipsoidal. This is the Calliper method, and it is widely used throughout the world. This paper initiates an analytical study of ellipsoidal tumour growth based on the pioneering mathematical model of Greenspan. Comparisons are made with the more commonly studied spherical mathematical models. 1. Tumour biology. Cell proliferation is normally a highly regulated process, such that only the required numbers of cells populate a given tissue. If control of proliferation is altered or lost, cells may continue to divide leading to an abnormal mass of tissue – a tumour. The most common cause of tumours is genetic mutation resulting in uncontrolled cell division. Although a single mutation can account for this loss of control, it is far more common for a series of mutations in a number of genes to accumulate, eventually resulting in loss of proliferative control. This increased cell mass can be due to increased cell division or a decrease in programmed cell death, which normally occurs as part of limiting cell numbers, or a combination of both. Two classes of genes that are commonly mutated in tumours are oncogenes and tumour suppressors. Oncogenes are mutated forms of proto-oncogenes,which normally encode Received March 18, 2010. 2010 Mathematics Subject Classification. Primary 92C05, 92C50. c ©2011 Brown University 1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/license/jour-dist-license.pdf 2 G. DASSIOS, F. KARIOTOU, M. N. TSAMPAS, AND B. D. SLEEMAN proteins involved in growth promoting signal transduction and mitogenesis. Oncogenes are more active, hence increasing the rate of proliferation. Tumour suppressor genes, as their name suggests, are normally involved in slowing cell growth and division; mutations release this control, again increasing proliferation. Mutations in either class of gene, or in both, result in reduced control of cell growth and division, giving these cells a growth advantage over neighbouring healthy cells and leading to the development of a tumour. Tumours can be benign or malignant, dependent on their aggressiveness. A benign tumour stays as a noninvasive cluster, without spreading into its surroundings. The margin of the tumour is very distinct and the whole tumour can usually be removed by surgery. By aquiring more mutations, a benign tumour can become malignant. Malignant tumours grow aggressively and invade into surrounding tissue. Tumour cells that break away from the parent tumour, and move via lymphatic or blood vessels to a distant site to form secondary tumours or metastases, are characteristic of a malignant tumour. Most tumours start as a small mass of rapidly proliferating cells, where nutrients and oxygen are acquired by passive diffusion from the surrounding tissues, the size of the tumour is limited to about 2mm in diameter and tumours can stay in this diffusion limited state, where cell proliferation is balanced by cell death, for months or even years. It was over thirty years ago that Judah Folkman [10] first proposed that in order to develop beyond this dormant state, the tumour must induce the growth of new blood vessels in order to supply the increasing metabolic demands. At that time, the mechanism of new vessel growth, called angiogenesis, was not known, but Folkman suggested that the switch in a tumour towards a pro-angiogenic state is a specific stage in tumour development. Once the tumour becomes vascularised, diffusion no longer limits size, and the tumour can grow and develop. It is now known that control of angiogenesis is orchestrated by a large number of pro-angiogenic and anti-angiogenic factors, and it is the shift in balance from antito pro-angiogenic that elicits the so-called angiogenic switch and induces growth of blood vessels towards the tumour. Many factors contribute to the switch towards angiogenesis, one of which is oxygen deficiency within the tumour. Tissues deprived of oxygen become hypoxic, and express a range of factors to help them survive, some of which are proangiogenic factors, driving the growth of new blood vessels towards that tissue. This whole orchestration of complex events leads to a micro-vascular structure that eventually reaches and penetrates the tumor, vastly improving its blood supply and allowing for rapid and unconstrained growth. For an up-to-date account of the biochemistry of tumour angiogenesis, we refer to Plank and Sleeman [14] as well as the references [3], [13] and [16]. In this paper we consider the growth of avascular tumours, the first step being in understanding the growth of complex processes involved in angiogenesis and vascular structures. There exist in the literature several mathematical models of avascular tumour growth. These include (i) models that describe continuum cell populations and their growth by considering the interactions between cell density and the chemical species that provide nutrients as well as inhibitors, (ii) models that describe mechanical interactions between tumour cells and their surroundings and (iii) individual cell based models that allows one to track cells in both space and time. From in vitro experiments and License or copyright restrictions may apply to redistribution; see https://www.ams.org/license/jour-dist-license.pdf MATHEMATICAL MODELLING OF AVASCULAR ELLIPSOIDAL TUMOUR GROWTH 3 some observed in vivo studies, it is known that avascular tumours may grow as symmetric spheroids wherein growth is essentially radial. In this situation many of the above mathematical models admit to analytical treatment and enable one to determine cell movement, track the spheroidal boundary and to assess the roles of growth inhibitors and growth promoters. The stability of tumour growth can also be analysed; see [12] for a review and cited references. A solid mass growing in healthy tissue produces stress. Models [11] have been developed to study this form of mechanical stress in which the tumour deforms the surrounding tissue due to the stress it imposes on the environment, and the environment in turn alters tumour growth dynamics. In these models, tumour growth inhibition depends on the stiffness of the surrounding environment. In an in vivo setting this corresponds to the stiffness of the extracellular matrix environment. In an in vitro setting, this corresponds to the stiffness of the agarose gel. The effects of physical confinement on tumour growth have been studied experimentally. In [8], human colon adenocarcinoma cells were grown in cylindrical glass tubes with a radius that was much smaller than the length of the tube. It was found that cell aggregates in 0.7% gel placed in a capillary tube grew to take on an ellipsoidal shape driven by the geometry of the capillary tube. The same cells grown outside a capillary tube developed into a spherical shape. This experiment highlights that geometric confinement alters the shape and growth dynamics of a developing tumour. Mathematical models which treat avascular tumours as visco-elastic materials and discuss the effects of mechanical stress are considered in [4, 15]. Breast cancer is the most frequently diagnosed cancer in women. From mammography, magnetic resonance imaging (MRI) and ultrasonography it is well documented that breast tumours are often ellipsoidal in shape. Indeed the World Health Organisation (WHO) established in 1979 criteria based on tumour volume change for classifying response to therapy as progressive disease, partial recovery or no change. Typically the volume of the tumour is measured on the hypothesis that growth is ellipsoidal. This is the socalled calliper method and is widely used throughout the world in assessing and grading gliomas. See [2, 5, 17]. In this paper we initiate an analytical study of ellipsoidal tumour growth based on the pioneering mathematical model of avascular tumour growth due to Greenspan [6, 7, 15]. In section 2 we formulate the mathematical model in terms of ellipsoidal geometry and explicitly solve for the pressure field and nutrient concentration within a growing avascular ellipsoidal tumour made up of a live cell layer, a quiescent layer and a necrotic core. Because the analysis depends extensively on the use of Lamé functions, their relevant properties together with a description of ellipsoidal coordinates are outlined in Appendix A. Section 3 contains a resumé of the well-known spherical tumour problem and emphasises the modelling differences with the Greenspan model. In section 4 we carry out a number of numerical simulations. The paper concludes in section 5 with a discussion of the results. 2. The mathematical model. The tumour is assumed to have a three-layer structure consisting of a thin outer layer of live proliferating cells that envelops an inner layer of quiescent live but not proliferating cells which in turn envelops a large necrotic core of License or copyright restrictions may apply to redistribution; see https://www.ams.org/license/jour-dist-license.pdf 4 G. DASSIOS, F. KARIOTOU, M. N. TSAMPAS, AND B. D. SLEEMAN dead cells and debris. Cells proliferate as long as the available concentration of nutrient supply, denoted by σ(r, t), remains above a critical level σ1. A cell dies due to apoptosis or otherwise when σ falls below a critical level σ2. In the quiescent region, nutrient supply varies over the interval σ2 ≤ σ ≤ σ1. The characteristic thickness s of the layer of live proliferating cells depends on σ1 and the value of σ at the outer surface of the tumour. It is assumed that the tumour boundaries evolve as members

Global solvability for the heat equation with boundary flux governed by nonlinear memory

Abstract: We introduce the study of global existence and blowup in finite time for the heat equation with flux at the boundary governed by a nonlinear memory term. Via a simple transformation, the model may be written in a form which has been introduced in previous studies of tumor-induced angiogenesis. The present study is also in the spirit of extending work on models of the heat equation with local, nonlocal, and delay nonlinearities present in the boundary flux. Additionally, we provide a brief summary of related studies regarding heat equation models where memory terms are incorporated within reaction or diffusion.

Stability of ZND detonations for Majda’s model

Abstract: We evaluate by direct calculation the Lopatinski determinant for ZND detonations in Majda's model for reacting flow and show that on the nonstable (nonnegative real part) complex half-plane it has a single zero at the origin of multiplicity one, implying stability. Together with results of Zumbrun on the inviscid limit, this recovers the result of Roquejoffre―Vila that viscous detonations of Majda's model also are stable for sufficiently small viscosity, for any fixed detonation strength, heat release, and rate of reaction.

Existence, uniqueness and exponential decay: An evolution problem in heat conduction with memory

Abstract: A rigid linear heat conductor with memory conductor is considered. An evolution problem which arises in studying the thermodynamical state of the material with memory is considered. Specically, the time evolution of the temperature distribution within a rigid heat conductor with memory is investigated. The constitutive equations which characterize heat conduction with memory, involve an integral term since the temperature’s time derivative is connected to the heat ux gradient. The integro-dierenti al problem, when initial and boundary conditions are assigned, is studied to obtain existence and uniqueness results. Key tools, turn out to be represented by suitable expressions of the minimum free energy which allow to construct functional spaces which are both meaningful under the physical as well as the analytic viewpoint since therein the existence and uniqueness results can be established. Finally, conditions which guarantee exponential decay at innity are obtained.

Unilateral dynamic contact of two viscoelastic beams

Abstract: This work is focused on a dynamic unilateral contact problem between two viscoelastic beams. Global-in-time existence of weak solutions describing the dynamics of the system is established. In addition, asymptotic longtime behavior of weak solutions is discussed: it is shown that the energy solutions decay exponentially to zero under suitable decay properties of the memory kernels.

Three–phase eccentric annulus subjected to a potential field induced by arbitrary singularities

Abstract: An infinite planar, three-component heterogeneous medium with a pair of circles as interfaces between homogeneous zones forming an eccentric annulus is considered for refraction of a potential field on the two interfaces. The velocity field is generated by an arbitrary system of singularities of arbitrary order, in congruity with the MilneThomson case of a two-component medium and a single circular interface. An exact analytical solution of the corresponding R-linear conjugation problem of two Laplacian fields in the eccentrical annulus structure is derived in the class of piecewise meromorphic functions with fixed principal part. Three general cases of loci of the singularities with respect to the interfaces are investigated. Flow nets (isobars and streamlines) are presented.

“Driving forces” and radiated fields for expanding/shrinking half-space and strip inclusions with general eigenstrain

Abstract: A half-space constrained Eshelby inclusion (in an infinite elastic matrix) with general uniform eigenstrain (or transformation strain) is analyzed when the plane boundary is moving in general subsonic motion starting from rest. The radiated fields are calculated based on the Willis expression for constrained time-dependent inclusions, which involves the three-dimensional dynamic Green’s function in an infinite tractionfree body, and they constitute the unique elastodynamic solution, with initial condition the Eshelby static fields obtained as the unique minimum energy solutions by a limiting process from the spherical inclusion. The mechanical energy-release rate and associated “driving force” to create dynamically an incremental region of eigenstrain (due to any physical process) is calculated for general uniform eigenstrain. For dilatational eigenstrain the solution coincides with the one obtained by a limiting process from a spherically expanding inclusion, while for shear eigenstrain the fields are due to the propagation of the rotation. The “driving force” has the same expression both for expanding and shrinking motions, resulting in expenditure of the energy rate for motion of the boundary in both cases. By superposition from the half-space inclusions, the fields and “driving force” for a strip inclusion with both boundaries moving are obtained. The “driving force” consists also of a contribution from the other boundary when it has time to arrive. The presence of applied loading contributes the counterpart of the Peach-Koehler force of dislocations, in addition to the self-force. Introduction. In a recent publication, Markenscoff and Ni (2010) obtained the energy-release rate required to create dynamically an incremental region of dilatational Received February 1, 2010. 2000 Mathematics Subject Classification. Primary 74N20, 74H05, 74B99. Partial support by NSF (grant # CMS 0555280) is gratefully acknowledged. The first author wishes to thank Professors Rodney Clifton and Lev Truskinovsky for suggesting the problem of the moving plane boundary and initial comments. c ©2011 Brown University 529 License or copyright restrictions may apply to redistribution; see https://www.ams.org/license/jour-dist-license.pdf 530 X. MARKENSCOFF AND L. NI uniform eigenstrain by an expanding spherical inclusion, as well as by an expanding plane boundary, through a limiting process from the sphere. Here, the energy release rate to create an incremental region of general uniform eigenstrain εij by a moving plane boundary of a constrained inclusion is obtained. Markenscoff and Ni (2010) obtained the radiated fields from a spherical inclusion with dilatational eigenstrain (constrained in an infinite linearly elastic matrix) expanding in a general subsonic motion based on the analysis of Willis (1965) for inclusions with time-dependent eigenstrain (transformation strain) constrained in an elastic matrix that is traction-free at the boundary at infinity. It results in an expression for the displacement in terms of the dynamic Green’s function, analogous to the Eshelby one (1957) for static inclusions. The static Eshelby solution for a spherical inclusion (1957) was obtained from this elastodynamic expression when evaluated from t = −∞ to t = 0, and the Hadamard jump conditions were shown to be satisfied. Using these fields, the energy-release rate required to create an incremental volume of eigenstrain as the spherical inclusion expands was computed. It may be noted here that the energy-release rate expression of Atkinson and Eshelby (1968), Rice (1968), and Freund (1972), derived initially for moving cracks when evaluated for a singularity that is a jump discontinuity (Stolz, 2003), gives an expression which coincides with that of the associated “driving force” in the thermodynamic literature (Truskinovky, 1982) for a system that is purely mechanical. The energy-release rate is equivalent to the pathindependent dynamic J integral derived on the basis of Noether’s theorem (Freund (1990), Maugin (1990), Gupta and Markenscoff (in preparation)) for an “elastic singularity” for which the integrals involved exist (as Cauchy Principal Values). The radiated fields and energy-release rate to move a plane boundary with dilatational eigenstrain were obtained by Markenscoff and Ni (2010) by a limiting process from the spherically expanding inclusion, as the radius of the sphere tends to infinity, and that solution, radiated fields and self-force is recovered here as a special case of eigenstrain. The energy-release rate, and associated “driving force”, or “self-force” of the moving plane boundary, has a static part coinciding with the one based on the expression given by Eshelby (1970, 1977) and independently calculated by Gavazza (1977) for a spherical inclusion. The dynamic part of the self-force for a plane boundary depends only on the current value of the velocity, and not the acceleration, and thus the plane phase boundary has no effective mass, in contrast to the dislocation (Ni and Markenscoff, 2008). However, for a spherical inclusion the furthermost point of the back of the inclusion, where a discontinuity occurs, also contributes to the “driving force” on the front boundary. In the present treatment, the radiated fields from a constrained (in an elastic matrix) three-dimensional linearly elastic inclusion occupying x1 ≤ R0 for t ≤ 0, and expanding/shrinking in a general subsonic motion of the plane inclusion boundary according to x1 = R0 + (t), are calculated based on Willis (1965, equation (26)) for inclusions with time-dependent boundaries constrained in an elastic matrix that is traction-free on the boundary. The Willis expression involves the three-dimensional dynamic Green’s function for a point force in an infinite elastic body, and is the exact dynamic analog to the static Eshelby expression (1957). The eigenstrain is general, but due to antisymmetries in some terms of the dynamic Green’s function, the evaluation of the integrals is simplified. The solution for the displacement is obtained (modulo rigid body motion), License or copyright restrictions may apply to redistribution; see https://www.ams.org/license/jour-dist-license.pdf “DRIVING FORCES” AND RADIATED FIELDS 531 from which the strains, rotations, jumps thereof, and “driving force” are obtained for general uniform eigenstrain. In the dynamic case, here as well as in Markenscoff and Ni (2010), for the same reason as in the static half-plane inclusion (Dundurs and Markenscoff, 2009), the obtained solution is unique, since it is derived by the elasticity solution for a constrained inclusion in an infinite medium with zero tractions on the boundary at infinity, having as initial condition the Eshelby static fields. No superposed compatible externally applied fields at infinity are allowed (which would increase the energy, e.g. Mura (1982), and, which were called by Dundurs and Markenscoff, 2009, “rogue states”). The “driving force” has the same expression both for expanding and shrinking motion, resulting in expenditure of energy for motion of the boundary both cases. The case of shear eigenstrain ε12, which is frequently of interest in phase transformations (e.g. Mura, 1982), is part of the solution. By superposition of the half-space fields, the radiated fields for a strip inclusion with shear eigenstrain, expanding and shrinking in either direction, are obtained, and the “driving force” computed. The “driving force” has a contribution also from the jump discontinuity at the other boundary, when it has the time to arrive, similar to the contribution to the front boundary from the back of the spherically expanding inclusion (Markenscoff and Ni, 2010). In the present treatment, the radiated fields from a constrained (in an elastic matrix) three-dimensional linearly elastic inclusion occupying x1 ≤ R0 for t ≤ 0, and expanding/shrinking in a general subsonic motion of the plane inclusion boundary according to x1 = R0 + (t), are calculated based on Willis (1965, equation (26)) for inclusions with time-dependent boundaries constrained in an elastic matrix that is traction-free on the boundary. The Willis expression involves the three-dimensional dynamic Green’s function for a point force in an infinite elastic body, and is the exact dynamic analog to the static Eshelby expression (1957). The eigenstrain is general, but due to antisymmetries in some terms of the dynamic Green’s function, the evaluation of the integrals is simplified. The solution for the displacement is obtained (modulo rigid body motion), from which the strains, rotations, jumps thereof, and “driving force” are obtained for general uniform eigenstrain. In the dynamic case, here as well as in Markenscoff and Ni (2010), for the same reason as in the static half-plane inclusion (Dundurs and Markenscoff, 2009), the obtained solution is unique, since it is derived by the elasticity solution for a constrained inclusion in an infinite medium with zero tractions on the boundary at infinity, having as initial condition the Eshelby static fields. The static Eshelby fields for the half-space inclusion are unique minimum energy ones, as derived from the minimum energy solution of the spherical inclusion by a limiting process. No superposed compatible externally applied fields at infinity are allowed (which would increase the energy, e.g. Mura (1982), and which were called by Dundurs and Markenscoff, 2009, “rogue states”). The “driving force” has the same expression both for expanding and shrinking motion, resulting in expenditure of energy for motion of the boundary in both cases. The case of shear eigenstrain ε12, which is frequently of interest in phase transformations (e.g. Mura, 1982), is part of the solution. By superposition of the half-space fields, the radiated fields for a strip inclusion with shear eigenstrain, expanding and shrinking in either direction, are obtained, and the “driving force” computed. The

Laplace transform of products of Bessel functions: A visitation of earlier formulas

Abstract: This note deals with the Laplace transforms of integrands of the form x J (ax) J (bx), which are found in numerous fields of application. Specifically, we provide herein both a correction and a supplement to the list of integrals given in 1997 by Hanson and Puja, who in turn extended the formulas of Eason, Noble and Sneddon of 1956. The paper concludes with an extensive tabulation for particular cases and range of parameters. 1. Introduction. In a classic 1956 paper, Eason, Noble and Sneddon - henceforth ENS for short - presented a general methodology for finding integrals involving products of Bessel functions, and provided a set of closed-form formulas for cases commonly en- countered in engineering science and in applied mathematics. Although these integrals extended considerably the repertoire of exact formulas available in standard tables such as Oberhettinger's or Gradshteyn & Ryzhik's, even to this day programs such as Math- ematica, Maple and Matlab's symbolic tool seem to have remained unaware of the ENS paper, for they are unable to provide answers to such integrals. Some four decades later, Hanson and Puja (1997) - henceforth denoted as HP - reconsidered the ENS paper and not only extended considerably the formulas therein, but by changing the arguments to the functions, they arrived at alternative forms which allegedly avoided discontinuities at certain values of the parameters. Unfortunately, once the arguments in the HP formulas exceed some threshold value, the computations for some of the integrals suffer a com- plete breakdown and become useless. This led us to investigate both the reasons for the erroneous results and also seek a corrected set of formulas, which constitutes the subject of this paper. To avoid repetitions, the presentation herein will be rather terse, avoiding needless explanations of well-known facts and/or of details which can be found in the originals of the papers referred to. Also, to distinguish clearly between the equations and parameters

Asymptotic solution of a nonlinear advection-diffusion equation

Abstract: We carry out an asymptotic analysis as t → ∞ for the nonlinear advection-diffusion equation, ∂ t u = 2αu∂ x u + ∂ x (u∂ x u), where α is a constant. This equation describes the movement of a buoyancy-driven plume in an inclined porous medium, with α having a specific physical significance related to the bed inclination. For compactly supported initial data, the solution is characterized by two moving boundaries propagating with finite speed and spanning a distance of O(√t). We construct an exact outer solution to the PDE that satisfies the right boundary condition. The vanishing condition at the left boundary is enforced by introducing a moving boundary layer, for which we obtain a closed-form expression. The leading-order composite solution is uniformly correct to O(1/√t). A higher-order correction to the inner and the composite solutions is also derived analytically. As a result, we obtain late-time asymptotic expansions for the two moving boundaries, correct to O(1), as well as a composite solution correct to O(1/t). The findings of this paper are illustrated and verified by numerical computations.

Stability criteria of 3D inviscid shears

Abstract: Recent numerical studies in the area of transition to turbulence discovered that the classical plane Couette flow, plane Poiseuille flow, and pipe Poiseuille flow share some universal 3D steady coherent structure in the form of a "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.

Lack of contact in a lubricated system

Abstract: We consider the problem of a rigid surface moving over a flat plane. The surfaces are separated by a small gap filled by a lubricant fluid. The relative position of the surfaces is unknown except for the initial time t = 0. The total load applied over the upper surface is a known constant for t > 0. The mathematical model consists of a coupled system formed by the Reynolds variational inequality for incompressible fluids and Newton's Second Law. We study the steady states of the problem and the global existence and uniqueness of the time-dependent problem. We assume one degree of freedom for the position of the surface. We consider different cases depending on the geometry of the upper surface.

Thermodynamics of a non-simple heat conductor with memory

Abstract: A generalization is given of the linearized constitutive equation, proposed by Gurtin and Pipkin for the heat flux in a rigid heat conductor, which includes the effects of both the histories of the temperature gradient g and of ∇g. This new contribution yields a non-simple material, for which the Second Law of Thermodynamics assumes a modified form, characterized by an extra flux, depending on the material. Some standard free energy functionals are adapted to these new materials, including an explicit formula for the minimum free energy.

Uniform stabilization of a class of coupled systems of KdV equations with localized damping

Abstract: We study the stabilization of solutions of a coupled system of Kortewegde Vries (KdV) equations in a bounded interval under the effect of a localized damping term. We use multiplier techniques combined with the so-called “compactness-uniqueness argument”. The problem is then reduced to proving a unique continuation property (UCP) for weak solutions. The exponential decay of solutions was previously obtained in Bisognin, Bisognin, and Menzala (2003) when the damping was effective simultaneously in neighborhoods of both extremes of the bounded interval. In this work we address the general case using a different approach to obtain the UCP and stabilize the system.

Global regularity for a coupled Cahn-Hilliard-Boussinesq system on bounded domains

Abstract: We study an initial-boundary value problem (IBVP) for a coupling of the Cahn-Hilliard equation with the 2D inviscid heat-conductive Boussinesq equations. For large initial data with finite energy, we prove global existence and uniqueness of classical solutions to the IBVP, together with some uniform-in-time and decay estimates of the solution.

Remark on a regularity criterion in terms of pressure for the Navier-Stokes equations

Abstract: In this note we establish a Serrin-type regularity criterion in terms of pressure for Leray weak solutions to the Navier-Stokes equation in R d . It is known that if a Leray weak solution u belongs to L2 1―r ((0, T); Ld r) for some 0 ≤ r ≤ 1, (0.1) then u is regular. It is proved that if the pressure p associated to a Leray weak solution u belongs to L2 2―r ((0,T); M 2,d r (R d ) d ), (0.2) where M 2, d r (R d ) is the critical Morrey-Campanato space (a definition is given in the text) for 0 < r < 1, then the weak solution is actually regular. Since this space M 2,d r is wider than Ld r and Ẋ r , the above regularity criterion (0.2) is an improvement of Zhou's result.

Field behavior near the edge of a microstrip antenna by the method of matched asymptotic expansions

Abstract: The cavity model is a wide-spread powerful empirical approach for the numerical simulation of microstrip antennas. It is based on several hypotheses assumed a priori: a dimension reduction in the cavity, that is, the zone limited by a metallic patch and the ground plane in which is fed the antenna, supplied by the additional condition that the open sides of the cavity act as magnetic walls. An additional important assumption of this model consists in an adequate description of the singular field behavior in the proximity of the edge of the patch. A simplified two-dimensional problem incorporating the main features of the field behavior near the edge of the patch and inside the cavity is addressed. The method of matched asymptotic expansions is used to carry out a two-scale asymptotic analysis of the field relatively to the thickness of the cavity. All the empirical hypotheses at the basis of the derivation of the cavity model can thus be recovered. Proved error estimates are given in a simplified framework where the dielectric constants of the substrate are assumed to be 1 in order to avoid some unimportant technical difficulties.

Multiple scales analysis of water hammer attenuation

Abstract: A multiple scales analysis is used to construct a uniformly accurate approximation to water hammer pressure wave attenuation that is initiated by a sudden valve opening. The method of analysis is well suited to the study of a water hammer that possesses several time scales and is applied to a mild generalization of the classical equations. It should prove useful for finding attenuation curves when effects such as unsteady friction and fluid-structure interaction are added. The analytical results are numerically verified using the method of characteristics.

Well-posedness and regularity for non-uniform Schrödinger and Euler-Bernoulli equations with boundary control and observation

Abstract: The open-loop systems of a Schrödinger equation and an Euler-Bernoulli equation with variable coefficients and boundary controls and collocated observations are considered. It is shown, with the help of a multiplier method on a Riemannian manifold, that both systems are well-posed in the sense of D. Salamon and regular in the sense of G. Weiss. The feed-through operators are found to be zero. The results imply particularly that the exact controlability of each open-loop system is equivalent to the exponential stability of the associated closed-loop system under the output proportional feedback.

Modeling the flash-heat experiment on porous domains

Abstract: : We discuss several models for a ash-heat experiment in homogeneous isotropic media. We use these to investigate the use of homogenization techniques in approximating models for interrogation via flash-heating in porous materials. We represent porous materials as both randomly perforated domains and periodically perforated domains.

A multi-dimensional blow-up problem due to a concentrated nonlinear source in ℝ^{ℕ}

Abstract: Let B be an N-dimensional ball {x ∈ ℝ N : |x| 0, f (u) and f' (u) are positive for u > 0, f" (u) > 0 for u > 0, and ψ is nontrivial on ∂B, nonnegative, and continuous such that ψ → 0 as |x| → oo, ∫ ℝN ψ (x) dx 0 in ℝ N It is shown that the problem has a unique nonnegative continuous solution before blowup occurs We assume that ψ (x) = M (0) > ψ (y) for x ∈ ∂B and y ∉ ∂B, where M (t) = sup x∈ℝN u (x, t) It is proved that if u blows up in a finite time, then it blows up everywhere on ∂B If, in addition, ψ is radially symmetric about the origin, then we show that if u blows up, then it blows up on ∂B only Furthermore, if f (u) ≥ κu p , where κk and p are positive constants such that p > 1, then it is proved that for any α, u always blows up in a finite time for N 3, it is shown that there exists a unique number α* such that u exists globally for α α* A formula for computing α* is given