# American Institute of Mathematical Sciences

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2163-2480

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## Evolution Equations & Control Theory

2014 , Volume 3 , Issue 3

Special issue on mathematical models and analytical problems in modern continuum thermomechanics

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2014, 3(3): i-ii doi: 10.3934/eect.2014.3.3i +[Abstract](540) +[PDF](7910.0KB)
Abstract:
1.Mauro Fabrizio. This volume entitled Mathematical Models and Analytical Problems in Modern Continuum Thermomechanics" is dedicated to Mauro Fabrizio on the occasion of his retirement.

2014, 3(3): 355-361 doi: 10.3934/eect.2014.3.355 +[Abstract](553) +[PDF](335.9KB)
Abstract:
We are concerned with an inverse problem for first order linear evolution equations. We indicate sufficient conditions for existence and uniqueness of a solution to these problems. All the results apply well to inverse problems for equations from mathematical physics. Indeed, as a possible application of the abstract theorems, some examples of partial differential equations are given.
2014, 3(3): 363-372 doi: 10.3934/eect.2014.3.363 +[Abstract](492) +[PDF](412.7KB)
Abstract:
We prove a lower semicontinuity theorem for a polyconvex functional of integral form, related to maps $u:\Omega \subset \mathbb{R}^{n}\rightarrow \mathbb{R}^{m}$ in $W^{1,n}(\Omega ;\mathbb{R}^{m})$ with $n\geq m\geq 2$, with respect to the weak $W^{1,p}$-convergence for $p>m-1$, without assuming any coercivity condition.
2014, 3(3): 373-397 doi: 10.3934/eect.2014.3.373 +[Abstract](520) +[PDF](523.2KB)
Abstract:
In this paper we discuss the asymptotic behavior of a doubly nonlinear problem describing the vibrations of a coupled suspension bridge. The single-span road-bed is modeled as an extensible viscoelastic beam which is simply supported at the ends. The main cable is modeled by a viscoelastic string and is connected to the road-bed by a distributed system of one-sided elastic springs. A constant axial force $p$ is applied at one end of the deck, and time-independent vertical loads are allowed to act both on the road-bed and on the suspension cable. For this general model we obtain original results, including the existence of a regular global attractor for all $p\in\mathbb{R}$.
2014, 3(3): 399-410 doi: 10.3934/eect.2014.3.399 +[Abstract](535) +[PDF](350.4KB)
Abstract:
The existence and uniqueness of solution to an integro-differential problem arising in heat conduction with memory is here considered. Specifically, a singular kernel problem is analyzed in the case of a multi-dimensional rigid heat conductor. The choice to investigate a singular kernel material is suggested by applications to model a wider variety of materials and, in particular, new materials whose heat flux relaxation function may be superiorly unbounded at the initial time $t=0$. The present study represents a generalization to higher dimensions of a previous one concerning a $1$-dimensional problem in the framework of linear viscoelasticity with memory. Specifically, an existence theorem is here proved when initial homogeneous data are assumed. Indeed, the choice of homogeneous data is needed to obtain the a priori estimate in Section 2 on which the subsequent results, are based.
2014, 3(3): 411-427 doi: 10.3934/eect.2014.3.411 +[Abstract](550) +[PDF](482.8KB)
Abstract:
We address the thermal control of the quasi-static evolution of a polycrystalline shape memory alloy specimen. The thermomechanical evolution of the body is described by means of an extension of the phenomenological Souza-Auricchio model [6,7,8,57] accounting also for permanent inelastic effects [9,11,27]. By assuming to be able to control the temperature of the body in time we determine the corresponding quasi-static evolution in the energetic sense. In a similar way as in [28], using results by Rindler [49,50] we prove the existence of optimal controls for a suitably large class of cost functionals.
2014, 3(3): 429-445 doi: 10.3934/eect.2014.3.429 +[Abstract](490) +[PDF](422.6KB)
Abstract:
The paper is devoted to the study of the motion of one-dimensional rigid bodies during a free fall in a quasi-Newtonian hyperviscous fluid at low Reynolds number. We show the existence of a steady solution and furnish sufficient conditions on the geometry of the body in order to get purely translational motions. Such conditions are based on a generalized version of the so-called Reciprocal Theorem for fluids.
2014, 3(3): 447-483 doi: 10.3934/eect.2014.3.447 +[Abstract](513) +[PDF](584.5KB)
Abstract:
The free energy for most materials with memory is not unique. There is a convex set of free energy functionals with a minimum and a maximum element. Various functionals have been shown to have the properties of a free energy for materials with particular types of relaxation behaviour. Also, over the last decade or more, forms have been given for the minimum and related free energies. These are all quadratic functionals which yield linear memory terms in the constitutive equations for the stress.
A difficulty in constructing free energy functionals arises in making choices that ensure a non-negative quadratic form both for the free energy and for the rate of dissipation. We propose a technique which renders this task more straightforward. Instead of constructing the free energy and determining from this the rate of dissipation, which may not have the required non-negativity, the procedure is reversed, which guarantees a satisfactory free energy functional.
Certain results for quadratic functionals in the time and frequency domains are derived, providing a platform for this alternative approach, which produces new free energies, including a family of functionals that are generalizations of the minimum and related free energies.
2014, 3(3): 485-497 doi: 10.3934/eect.2014.3.485 +[Abstract](418) +[PDF](347.2KB)
Abstract:
Heat equations with memory of Gurtin-Pipkin type (i.e. Eq. (1) with $\alpha=0$) have controllability properties which strongly resemble those of the wave equation. Instead, recent counterexamples show that when $\alpha>0$ the control properties do not parallel those of the (memoryless) heat equation, in the sense that there are square integrable initial conditions which cannot be controlled to zero. The proof of this fact, in previous papers, consists in the construction of two quite special examples of systems with memory which cannot be controlled to zero. Here we prove that lack of controllability holds in general, for every smooth memory kernel $M(t)$.
2014, 3(3): 499-524 doi: 10.3934/eect.2014.3.499 +[Abstract](465) +[PDF](456.0KB)
Abstract:
Via Carleman's estimates we prove uniqueness and continuous dependence results for a severely ill-posed linear integro-differential singular parabolic problems without initial conditions.
2014, 3(3): 525-539 doi: 10.3934/eect.2014.3.525 +[Abstract](551) +[PDF](455.2KB)
Abstract:
Research on convective-diffusive fluid motions in porous media has a notable relevance (increasing with the number of salts dissolved in the fluid) either for geophysical applications (engineering geology, subsurface and structural geology, subsurface contaminant transport, underground water flow, ...) or because porous materials occur very frequently in real life (fiber materials for insulating purposes, metallic foams in heat transfer devices ([1,13]). In the present paper porous horizontal layers, heated from below and salted from above and below, in the Darcy-Boussinesq scheme, are investigated. By virtue of the absence of subcritical instabilities ([20]), the bifurcating competition of Rayleigh and Prandtl numbers for promoting or inhibiting the onset of convection, investigated through the linearized equations, allows to show ([20]) that this competition can be restricted to the inequalities (2) which are necessary and sufficient for inhibiting the onset of convection. But while the onset of convection requires that only one of (3) holds, the stability requires that, at least, all the reverse of (3) hold. Therefore, either for theoretical reasons or for practical use of stability conditions, the problem of overcoming this gap arises. We call one stability condition (OSC) problem the looking for inequalities of type (4) able to inhibit the onset of convection for large sets of values of the bifurcating parameters with special attention to the case {$\alpha=3,g=0$} since in this case (4) is necessary and sufficient for inhibiting the onset of convection. To this goal new fields for the salts densities are introduced. These transformations allow to: i) discover skew-symmetries hidden in the Darcy-Boussinesq equations; ii) obtain meaningful contributions to the OSC problem.
2014, 3(3): 541-555 doi: 10.3934/eect.2014.3.541 +[Abstract](685) +[PDF](371.6KB)
Abstract:
The use of discontinuity surface propagation (e.g. shock waves and acceleration waves) is well known in modern continuum mechanics and yields a very useful means to obtain important information about a fully nonlinear theory with no approximation whatsoever. A brief review of some of the recent uses of such discontinuity surfaces is given and then we mention modelling of some social problems where the same mathematical techniques may be used to great effect. We specifically show how to develop and analyse models for evolution of one language overtaking use of another leading to possible extinction of the former language. Then we analyse shock transmission in a model for the evolutionary transition from the human period when hunter-gatherers transformed into farming. Finally we address modelling discontinuity waves in the context of diffusion of an innovation.

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