Some nonstandard problems in generalized heat conduction

The Maxwell-Cattaneo system of equations for generalized heat conduction is considered where the temperature and heat flux, respectively, are subject to auxiliary conditions which prescribe a combination of their values initially and at a later time. By means of differential inequalities, L₂ exponential decay bounds for the temperature and heat flux are determined in terms of data for a range of values of the parameter in the nonstandard auxiliary condition. Decay bounds are also obtained in two related problems.     

Improved bounds for some nonstandard problems in generalized heat conduction

We consider the Maxwell-Cattaneo system of equations for generalized heat conduction where the temperature and heat flux satisfy a nonstandard auxiliary condition which prescribes a combination of their values initially and at a later time. We obtain L₂ bounds for the temperature and heat flux by means of Lagrange identities. These bounds extend the range of validity for the parameter in the nonstandard condition under a constraint on the coefficients in the differential equations.     

On a nonstandard problem for heat conduction in a cylinder

Decay bounds are derived for the solution of a heat conduction problem in a semi-infinite cylinder when the lateral surface is held at zero temperature, a nonzero temperature is prescribed on the finite base, and the temperature at time T is prescribed to be a constant multiple of the temperature at initial time. Both energy and pointwise decay bounds are computed for a range of values of the constant multiple. Such problems were originally introduced as a means of stabilizing the backward-in-time problem for the heat equation.     

Spatial decay estimates for the Maxwell-Cattaneo equations with mixed boundary conditions

In this paper the authors derive spatial decay bounds for the temperature and heat flux as defined by the Generalized Maxwell-Cattaneo equations for heat conduction in a semi-infinite cylinder when the temperature and the tangential components of the heat flux vector vanish on the lateral surface of the cylinder. The results here supplement those previously found by the authors [5] when the heat flux vector was assumed to be zero on the lateral surface but no condition was imposed on the temperature.Received: February 7, 2002; revised: June 3, 2002     

Similarity solution of boundary layer stagnation-point flow towards a heated porous stretching sheet saturated with a nanofluid with heat absorption/generation and suction/blowing: A Lie group analysis

In this paper, heat and mass transfer analysis for boundary layer stagnation-point flow over a stretching sheet in a porous medium saturated by a nanofluid with internal heat generation/absorption and suction/blowing is investigated. The governing partial differential equation and auxiliary conditions are converted to ordinary differential equations with the corresponding auxiliary conditions via Lie group analysis. The boundary layer temperature, concentration and nanoparticle volume fraction profiles are then determined numerically. The influences of various relevant parameters, namely, thermophoresis parameter Nt, Brownian motion parameter Nb, Lewis number Le, suction/injection parameter S, permeability parameter k₁, source/sink parameter λ and Prandtl parameter Pr on temperature and concentration as well as wall heat flux and wall mass flux are discussed. Comparison with published results is presented.     

Blow-up phenomena for a system of semilinear heat equations with nonlinear boundary flux

This paper deals with the blow-up for a system of semilinear r-Laplace heat equations with nonlinear boundary flux. It is shown that, under certain conditions on the nonlinearities and data, blow-up will occur at some finite time, and when blow-up does occur upper and lower bounds for the blow-up time are obtained.     

Improved spatial decay bounds in generalized heat conduction

This paper deals with heat conduction in a semi-infinite cylinder using the generalized Maxwell-Cattaneo equations. Spatial decay bounds for the temperature and heat flux under two different types of boundary conditions are derived. For fixed time it is shown that in each case the solutions decay in appropriate measure like the exponential of a quadratic function of the distance from the base of the cylinder, whereas in previous work they had been shown to decay only at least as fast as the exponential of a linear function.Received: January 13, 2004     

The integrable nonlinear degenerate diffusion equationu t=[f(u)u x −1 x and its relatives

The nonlinear diffusion equationu _t=[f(u)g(u _x )] arises in recent models of turbulent transport and of stress dissipation in rock blasting. A Lie point symmetry analysis produces many similarity reductions of exponential and power-law forms, and reveals that for all choices off the equation is always integrable wheng(u _x )=1/u _x . We identify the functionsf(u) which guarantee equivalence to the linear heat equation. For all other choices off, the linear canonical form leads to a self-adjoint differential equation by separation of variablesx andt. We construct a number of explicit solutions with simple boundary conditions, which illustrate behavior in the vicinity of the degenerate region withu _x =. If zero flux and constant concentration are maintained on free boundaries, then steep concentration gradients may evolve from smooth initial conditions. For other boundary conditions, unlike the examples of strong degeneracy, smoothing will occur at initial step discontinuities.     

Complementary variational principles for radiation heat transfer in an absorbing medium

The radiation heat flux in a plane slab of an absorbing medium is evaluated by determining upper and lower bounds via complementary variational principles. The results obtained are in excellent agreement with those corresponding to other more cumbersome solution procedures (for small values of the spacing parameterd).

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Riassunto Si valuta il trasporto di energia raggiante in una lastra piana di un mezzo assorbente, determinandone limiti superiori ed inferiori, mediante principi variazionali complementari. I risultati ottenuti sono in eccellente accordo con quelli ottenuti mediante tecniche risolutive più complesse (per piccoli valori del parametrod).

     

On the transversal Helly numbers of disjoint and overlapping disks

A family of disks is said to have the property T(k) if any k members of the family have a common line transversal. We call a family of unit diameter disks t-disjoint if the distances between the centers are greater than t. We consider for each natural number k≧ 3 the infimum t_k of the distances t for which any finite family of t-disjoint unit diameter disks with the property T(k) has a line transversal. We determine exact values of t₃ and t₄, and give general lower and upper bounds on the sequence t_k, showing that t_k = O(1/k) as k → ∞. In honour of Helge Tverberg’s seventieth birthday Received: 9 June 2005     

The asymmetric random cluster model and comparison of Ising and Potts models

We introduce the asymmetric random cluster (or ARC) model, which is a graphical representation of the Potts lattice gas, and establish its basic properties. The ARC model allows a rich variety of comparisons (in the FKG sense) between models with different parameter values; we give, for example, values (β, h) for which the 0‘s configuration in the Potts lattice gas is dominated by the “+” configuration of the (β, h) Ising model. The Potts model, with possibly an external field applied to one of the spins, is a special case of the Potts lattice gas, which allows our comparisons to yield rigorous bounds on the critical temperatures of Potts models. For example, we obtain 0.571 ≤ 1 − exp(−β _c ) ≤ 0.600 for the 9-state Potts model on the hexagonal lattice. Another comparison bounds the movement of the critical line when a small Potts interaction is added to a lattice gas which otherwise has only interparticle attraction. ARC models can also be compared to related models such as the partial FK model, obtained by deleting a fraction of the nonsingleton clusters from a realization of the Fortuin-Kasteleyn random cluster model. This comparison leads to bounds on the effects of small annealed site dilution on the critical temperature of the Potts model. Received: 27 August 2000 / Revised version: 31 August 2000 / Published online: 8 May 2001     

Analysis of fluid flow and heat transfer over an unsteady stretching surface

This article considers two situations involving unsteady laminar boundary layer flow due to a stretching surface in a quiescent viscous incompressible fluid. In one configuration, the surface is impermeable with prescribed heat flux, in the other, the surface is permeable with prescribed temperature. The boundary value problems governing a similarity reduction for each of these situations are investigated and the existence of a solution is proved for all relevant values of physical parameters. The uniqueness of the solution is also proved for some (but not all) values of the parameters. Finally, a priori bounds are obtained for the skin friction coefficient and local Nusselt number.     

Heat kernel expansions on the integers and the Toda lattice hierarchy

We consider the heat equation u _t = Lu where L is a second-order difference operator in a discrete variable n. The fundamental solution has an expansion in terms of the Bessel functions of imaginary argument. The coefficients α _k (n, m) in this expansion are analogs of Hadamard’s coefficients for the (continuous) Schr?dinger operator. We derive an explicit formula for α _k in terms of the wave and the adjoint wave functions of the Toda lattice hierarchy. As a first application of this result, we prove that the values of these coefficients on the diagonals n = m and n = m + 1 define a hierarchy of differential-difference equations which is equivalent to the Toda lattice hierarchy. Using this fact and the correspondence between commutative rings of difference operators and algebraic curves we show that the fundamental solution can be summed up, giving a finite formula involving only two Bessel functions with polynomial coefficients in the time variable t, if and only if the operator L belongs to the family of bispectral operators constructed in [18].      

Square function and heat flow estimates on domains

The first purpose of this note is to provide a proof of the usual square function estimate on L^p(Ω). It turns out to follow directly from a generic Mikhlin multiplier theorem obtained by Alexopoulos, and we provide a sketch of its proof in the Appendix for the reader’s convenience. We also relate such bounds to a weaker version of the square function estimate which is enough in most instances involving dispersive PDEs and relies on Gaussian bounds on the heat kernel (such bounds are the key to Alexopoulos’result as well). Moreover, we obtain several useful L^p(Ω;H) bounds for (the derivatives of) the heat flow with values in a given Hilbert space H.     

Ideal weights: Asymptotically optimal versions of doubling, absolute continuity, and bounded mean oscillation

Sharp inequalities between weight bounds (from the doubling, A_p, and reverse Hölder conditions) and the BMO norm are obtained when the former are near their optimal values. In particular, the BMO norm of the logarithm of a weight is controlled by the square root of the logarithm of its A_∞ bound. These estimates lead to a systematic development of asymptotically sharp higher integrability results for reverse Hölder weights and extend Coifman and Fefferman's formulation of the A_∞ condition as an equivalence relation on doubling measures to the setting in which all bounds become optimal over small scales.     

Convergence to equilibrium for a fully hyperbolic phase‐field model with Cattaneo heat flux law

We consider a fully hyperbolic phase‐field model in this paper. Our model consists of a damped hyperbolic equation of second order with respect to the phase function χ(t) , which is coupled with a hyperbolic system of first order with respect to the relative temperature θ(t) and the heat flux vector q (t). We prove the well‐posedness of this system subject to homogeneous Neumann boundary condition and no‐heat flux boundary condition. Then, we show that this dynamical system is a dissipative one. Finally, using the celebrated ?ojasiewicz–Simon inequality and by constructing an auxiliary functional, we prove that the solution of this problem converges to an equilibrium as time goes to infinity. We also obtain an estimate of the decay rate to equilibrium. Copyright © 2008 John Wiley & Sons, Ltd.     

Homogenization of a non‐linear degenerate parabolic problem in a highly heterogeneous periodic medium

We consider the homogenization of a time‐dependent heat transfer problem in a highly heterogeneous periodic medium made of two connected components having comparable heat capacities and conductivities, separated by a third material with thickness of the same order ε as the basic periodicity cell but having a much lower conductivity such that the resulting interstitial heat flow is scaled by a factor λ tending to zero with a rate λ=λ(ε). The heat flux vectors a_j, j=1,2,3 are non‐linear, monotone functions of the temperature gradient. The heat capacities c_j(x) are positive, but may vanish at some subsets such that the problem can be degenerate (parabolic–elliptic). We show that the critical value of the problem is δ=lim_ε→0ε^p/λ and identify the homogenized problem depending on whether δ is zero, strictly positive finite or infinite. Copyright © 2005 John Wiley & Sons, Ltd.     

Shared values of meromorphic functions on compact Riemann surfaces

Let S be a compact Riemann surface of genus g and gonality d. We derive upper bounds (in terms of g and/or d) for the number of values that two non-constant meromorphic functions on S can share. The case d = 2 (i.e., the surface is hyperelliptic or elliptic) is studied in more detail.Received: 14 April 2004     

Temporal decay bounds in generalized heat conduction

In this paper the authors derive exponential decay bounds for the temperature and heat flux as defined by the generalized Maxwell-Cattaneo equations for heat conduction in a bounded region when the temperature and the tangential components of the heat flux vanish on the boundary. They also derive bounds in for the heat flux and temperature when the heat flux is assumed to vanish on the boundary but no boundary condition is imposed on the temperature.     

An inverse source problem for Maxwell's equations in anisotropic media

In this article, we consider nonstationary Maxwell's equations in an anisotropic medium in the (x ₁,?x ₂,?x ₃)-space, where equations of the divergences of electric and magnetic flux densities are also unknown. Then we discuss an inverse problem of determining the x ₃-independent components of the electric current density from observations on the plane x ₃?=?0 over a time interval. Our main aim is, study conditional stability in the inverse problem provided the permittivity and the permeability are independent of x ₃. The main tool is a new Carleman estimate.