ISSN:

1531-3492

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1553-524X

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### Volume 10, 2008

## Discrete & Continuous Dynamical Systems - B

September 2014 , Volume 19 , Issue 7

Special issue dedicated to Mauro Fabrizio's 70th birthday

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2014, 19(7): i-i
doi: 10.3934/dcdsb.2014.19.7i

*+*[Abstract](497)*+*[PDF](74.7KB)**Abstract:**

This Special Issue is dedicated to celebrate Mauro Fabrizio's 70th Birthday.

It is a pleasure and an honour for us to devote it to Mauro with deep appreciation and friendship for the scientist as well as for the man.

Mauro's wide and intense research activity touched many branches of Mathematical Physics. This fact is testified by the variety of subjects studied in the contributions collected here.

Mauro was born on December 17, 1940. He graduated in Bologna in 1965. Dario Graffi, a renowned italian mathematical physicist, was his advisor. He has been full professor in the Universities of Salerno and Ferrara before returning to his Alma Mater. Since 1967 to present he published over 160 papers and 5 books.

For over 45 years, he has been greatly influential through his research contributions in a several areas of Mechanics and Thermodynamics. In particular, the development of the mathematical modeling of Complex Systems.

In these areas, starting from Dario Graffi's ideas, Mauro obtained a number of important results in mathematical modeling in continuous thermomechanics, materials with fading memory and hereditary system, electromagnetism of continuous media, first and second order phase transition models.

Mauro has always been a stimulating and open minded Colleague as well as a reliable mentor to many young scientists within the mathematical community. His deep questions and sharp remarks are well known among all the people who had the chance to have him in the audience.

Among the several Mauro's recognitions, we only recall that, on June 22, 2012, the prestigious ``Premio Linceo per la Meccanica e applicazioni e Matematica" was bestowed upon him by Giorgio Napolitano, the President of the Italian Republic.

The study of complex systems is a multi-faceted area where many different mathematical tools come into play. From functional analysis to calculus of variations, from geometric analysis to semigroup theory and, of course, numerical methods.

The present volume collects 31 peer reviewed contributions of a number of leading scholars in the analysis of mathematical models. It aims to present an overview of some challenging research lines and to stimulate further investigations.

We are grateful to all the authors. They did a great job.

May 23, 2014

2014, 19(7): 1815-1835
doi: 10.3934/dcdsb.2014.19.1815

*+*[Abstract](707)*+*[PDF](494.6KB)**Abstract:**

Some expressions for the free energy in the case of incompressible viscoelastic fluids are given. These are derived from free energies already introduced for other viscoelastic materials, adapted to incompressible fluids. A new free energy is given in terms of the minimal state descriptor. The internal dissipations related to these different functionals are also derived. Two equivalent expressions for the minimum free energy are given, one in terms of the history of strain and the other in terms of the minimal state variable. This latter quantity is also used to prove a theorem of existence and uniqueness of solutions to initial boundary value problems for incompressible fluids. Finally, the evolution of the system is described in terms of a strongly continuous semigroup of linear contraction operators on a suitable Hilbert space. Thus, a theorem of existence and uniqueness of solutions admitted by such an evolution problem is proved.

2014, 19(7): 1837-1853
doi: 10.3934/dcdsb.2014.19.1837

*+*[Abstract](667)*+*[PDF](408.0KB)**Abstract:**

We study the effect of intracellular ion diffusion on ionic currents permeating through the cell membrane. Ion flux across the cell membrane is mediated by specific channels, which have been widely studied in recent years with remarkable results: very precise measurements of the true current across a single channel are now available. Nevertheless, a complete understanding of this phenomenon is still lacking, though molecular dynamics and kinetic models have provided partial insights. In this paper we demonstrate, by analyzing the KcsA current-voltage currents via a suitable lattice model, that intracellular diffusion plays a crucial role in the permeation phenomenon. We believe that the interplay between the channel behavior and the ion diffusion in the cell is a key ingredient for a full explanation of the current-voltage curves.

2014, 19(7): 1855-1867
doi: 10.3934/dcdsb.2014.19.1855

*+*[Abstract](622)*+*[PDF](374.8KB)**Abstract:**

In this paper we generalize the notion of hypercontractivity for nonlinear semigroups allowing the functions to belong to mixed spaces. As an application of this notion, we consider a class of Hamilton-Jacobi equations and we establish functional inequalities. More precisely, we get hypercontractivity for viscosity solutions given in terms of Hopf-Lax type formulas. In this framework, we consider different measures associated with the variables; consequently, using mixed norms, we find new inequalities. The novelty of this approach is the study of functional inequalities with mixed norms for semigroups.

2014, 19(7): 1869-1888
doi: 10.3934/dcdsb.2014.19.1869

*+*[Abstract](724)*+*[PDF](401.9KB)**Abstract:**

This paper deals with the multiscale modeling of vehicular traffic according to a kinetic theory approach, where the microscopic state of vehicles is described by position, velocity and activity, namely a variable suitable to model the quality of the driver-vehicle micro-system. Interactions at the microscopic scale are modeled by methods of game theory, thus leading to the derivation of mathematical models within the framework of the kinetic theory. Macroscopic equations are derived by asymptotic limits from the underlying description at the lower scale. This approach shows the hypothesis under which macroscopic models known in the literature can be derived and how new models can be developed.

2014, 19(7): 1889-1909
doi: 10.3934/dcdsb.2014.19.1889

*+*[Abstract](636)*+*[PDF](904.2KB)**Abstract:**

A unified phase-field continuum theory is developed for transition and separation phenomena. A nonlocal formulation of the second law which involves an extra-entropy flux gives the basis of the thermodynamic approach. The phase-field is regarded as an additional variable related to some phase concentration, and its evolution is ruled by a balance equation, where flux and source terms are (unknown) constitutive functions. This evolution equation reduces to an equation of the rate-type when the flux is negligible, and it takes the form of a diffusion equation when the source term is disregarded. On this background, a general model for first-order transition and separation processes in a compressible fluid or fluid mixture is developed. Upon some simplifications, we apply it to the liquid-vapor phase change induced either by temperature or by pressure and we derive the expression of the vapor pressure curve. Taking into account the flux term, the sign of the diffusivity is discusssed.

2014, 19(7): 1911-1934
doi: 10.3934/dcdsb.2014.19.1911

*+*[Abstract](865)*+*[PDF](685.0KB)**Abstract:**

The `tipping point' phenomenon is discussed as a mathematical object, and related to the behaviour of non-linear discontinuity waves in the dynamics of topical sociological and biological problems. The theory of such waves is applied to two illustrative systems in particular: a crowd-continuum model of pedestrian (or traffic) flow; and an hyperbolic reaction-diffusion model for the spread of the hantavirus infection (a disease carried by rodents). In the former, we analyse propagating acceleration waves, demonstrating how blow-up of the wave amplitude might indicate formation of a `human-shock', that is, a `tipping point' transition between safe pedestrian flow, and a state of overcrowding. While in the latter, we examine how travelling waves (of both acceleration and shock type) can be used to describe the advance of a hantavirus infection-front. Results from our investigation of crowd models also apply to equivalent descriptions of traffic flow, a context in which acceleration wave blow-up can be interpreted as emergence of the `phantom congestion' phenomenon, and `stop-start' traffic motion obeys a form of wave propagation.

2014, 19(7): 1935-1953
doi: 10.3934/dcdsb.2014.19.1935

*+*[Abstract](540)*+*[PDF](446.6KB)**Abstract:**

A thermodynamic model describing phase transitions with thermal memory, in terms of an entropy equation and a momentum balance for the microforces, is adressed. Convergence results and error estimates are proved for the related integrodifferential system of PDE as the sequence of memory kernels converges to a multiple of a Dirac delta, in a suitable sense.

2014, 19(7): 1955-1967
doi: 10.3934/dcdsb.2014.19.1955

*+*[Abstract](526)*+*[PDF](443.2KB)**Abstract:**

We consider on $L^2(\mathbb{R})$ the Schrödinger operator family $H(g)$ with domain and action defined as follows $$ D(H(g))=H^2(\mathbb{R})\cap L^2_{2M}(\mathbb{R}); \quad H(g) u=\bigg(-\frac{d^2}{dx^2}+\frac{x^{2M}}{2M}-g\,\frac{x^{M-1}}{M-1}\bigg)u $$ where $g\in\mathbb{C}$, $M=2,4,\ldots\;$. $H(g)$ is self-adjoint if $g\in\mathbb{R}$, while $H(ig)$ is $\mathcal{PT}$-symmetric. We prove that $H(ig)$ exhibits the so-called $\mathcal{PT}$-symmetric phase transition. Namely, for each eigenvalue $E_n(ig)$ of $H(ig)$, $g\in\mathbb{R}$, there exist $R_1(n)>R(n)>0$ such that $E_n(ig)\in\mathbb{R}$ for $|g| < R(n)$ and turns into a pair of complex conjugate eigenvalues for $|g| > R_1(n)$.

2014, 19(7): 1969-1985
doi: 10.3934/dcdsb.2014.19.1969

*+*[Abstract](634)*+*[PDF](2379.2KB)**Abstract:**

We establish uniform estimates in weighted Sobolev spaces for the solutions of the Dirichlet problems on snowflake pre-fractal domains.

2014, 19(7): 1987-2011
doi: 10.3934/dcdsb.2014.19.1987

*+*[Abstract](670)*+*[PDF](468.7KB)**Abstract:**

Wave equation defined on a compact Riemannian manifold $(M, \mathfrak{g})$ subject to a combination of locally distributed viscoelastic and frictional dissipations is discussed. The viscoelastic dissipation is active on the support of $a(x)$ while the frictional damping affects the portion of the manifold quantified by the support of $b(x)$ where both $a(x)$ and $b(x)$ are smooth functions. Assuming that $a(x) + b(x) \geq \delta >0 $ for all $x\in M$ and that the relaxation function satisfies certain nonlinear differential inequality, it is shown that the solutions decay according to the law dictated by the decay rates corresponding to the slowest damping. In the special case when the viscoelastic effect is active on the entire domain and the frictional dissipation is differentiable at the origin, then the overall decay rates are dictated by the viscoelasticity. The obtained decay estimates are intrinsic without any prior quantification of decay rates of both viscoelastic and frictional dissipative effects. This particular topic has been motivated by influential paper of Fabrizio-Polidoro [15] where it was shown that viscoelasticity with poorly behaving relaxation kernel destroys exponential decay rates generated by linear frictional dissipation. In this paper we extend these considerations to: (i) nonlinear dissipation with unquantified growth at the origin (frictional) and infinity (viscoelastic) , (ii) more general geometric settings that accommodate competing nature of frictional and viscoelastic damping.

2014, 19(7): 2013-2026
doi: 10.3934/dcdsb.2014.19.2013

*+*[Abstract](669)*+*[PDF](420.5KB)**Abstract:**

In this paper, we are interested in the study of the asymptotic behavior of a generalization of the Cahn-Hilliard equation with a proliferation term and endowed with Neumann boundary conditions. Such a model has, in particular, applications in biology. We show that either the average of the local density of cells is bounded, in which case we have a global in time solution, or the solution blows up in finite time. We further prove that the relevant, from a biological point of view, solutions converge to $1$ as time goes to infinity. We finally give some numerical simulations which confirm the theoretical results.

2014, 19(7): 2027-2038
doi: 10.3934/dcdsb.2014.19.2027

*+*[Abstract](522)*+*[PDF](322.2KB)**Abstract:**

In this paper we study the spatial behavior of the amplitude of the steady-state vibrations in a thermoviscoelastic porous beam. Here we take into account the effects of the viscoelastic and thermal dissipation energies upon the corresponding harmonic vibrations in a right cylinder made of a thermoviscoelastic porous isotropic material. In fact, we prove that the positiveness of the viscoelastic and thermal dissipation energies are sufficient for characterizing the spatial decay and growth properties of the harmonic vibrations in a cylinder.

2014, 19(7): 2039-2045
doi: 10.3934/dcdsb.2014.19.2039

*+*[Abstract](497)*+*[PDF](341.8KB)**Abstract:**

A Neumann problem in the strip for the Fitzhugh Nagumo system is considered. The transformation in a non linear integral equation permits to deduce a priori estimates for the solution. A complete asymptotic analysis shows that for large $ t $ the effects of the initial data vanish while the effects of boundary disturbances $ \varphi_1 (t), $ $ \varphi_2(t) $ depend on the properties of the data. When $ \varphi_1,\,\, \varphi_2 $ are convergent for large $ t $, the solution is everywhere bounded and depends on the asymptotic values of $ \varphi_1 , $ $ \varphi_2 $. More, when $ \varphi_i \in L^1 (0,\infty) (i=1,2)$ too, the effects are vanishing.

2014, 19(7): 2047-2064
doi: 10.3934/dcdsb.2014.19.2047

*+*[Abstract](661)*+*[PDF](427.7KB)**Abstract:**

We show the existence of bounded solutions $u\in L^2(0,T;H^1_0(\Omega))$ for a class of parabolic equations having a lower order term $b(x,t,u,\nabla u)$ growing quadratically in the $\nabla u$-variable and singular in the $u$-variable on the set $\{u=0\}$.

We refer to the model problem $$\left\{ \begin{array}{ll} u_t - \Delta u = b(x,t) \frac{|\nabla u|^2}{|u|^k} + f(x,t) & in \Omega \times (0,T)\\ u(x,t) = 0 & on \partial\Omega\times(0,T)\\ u(x,0) = u_0 (x) &

in \Omega \end{array}\right. $$ where $\Omega$ is a bounded open subset of $\mathbb{R}^N, N \geq 2, 0 < T < + \infty$ and $0 < k < 1$. The data $f(x,t), u_0(x)$ can change their sign, so that the possible solution $u$ can vanish inside $Q_T=\Omega\times(0,T)$ even in a set of positive measure. Therefore, we have to carefully define the meaning of solution. Also $b(x,t)$ can have a quite general sign.

2014, 19(7): 2065-2089
doi: 10.3934/dcdsb.2014.19.2065

*+*[Abstract](664)*+*[PDF](1974.2KB)**Abstract:**

The widespread interest on the hereditary behavior of biological and bioinspired materials motivates deeper studies on their macroscopic ``minimal" state. The resulting integral equations for the detected relaxation and creep power-laws, of exponent $\beta$, are characterized by fractional operators. Here strains in $SBV_{loc}$ are considered to account for time-like jumps. Consistently, starting from stresses in $L_{loc}^{r}$, $r\in [1,\beta^{-1}], \, \, \beta\in(0,1)$ we reconstruct the corresponding strain by extending a result in [42]. The ``minimal" state is explored by showing that different histories delivering the same response are such that the fractional derivative of their difference is zero for all times. This equation is solved through a one-parameter family of strains whose related stresses converge to the response characterizing the original problem. This provides an approximation formula for the state variable, namely the residual stress associated to the difference of the histories above. Very little is known about the microstructural origins of the detected power-laws. Recent rheological models, based on a top-plate adhering and moving on functionally graded microstructures, allow for showing that the resultant of the underlying ``microstresses" matches the action recorded at the top-plate of such models, yielding a relationship between the macroscopic state and the ``microstresses".

2014, 19(7): 2091-2109
doi: 10.3934/dcdsb.2014.19.2091

*+*[Abstract](757)*+*[PDF](517.2KB)**Abstract:**

We consider a thermodynamic model for fatigue accumulation in an oscillating elastoplastic Kirchhoff plate based on the hypothesis that the fatigue accumulation rate is proportional to the plastic part of the dissipation rate. For the full model with periodic boundary conditions we prove existence of a solution in the whole time interval.

2014, 19(7): 2111-2132
doi: 10.3934/dcdsb.2014.19.2111

*+*[Abstract](677)*+*[PDF](427.5KB)**Abstract:**

In this paper we have proved exponential asymptotic stability for the corotational incompressible diffusive Johnson-Segalman viscolelastic model and a simple decay result for the corotational incompressible hyperbolic Maxwell model. Moreover we have established continuous dependence and uniqueness results for the non-zero equilibrium solution.

In the compressible case, we have proved a Hölder continuous dependence theorem upon the initial data and body force for both models, whence follows a result of continuous dependence on the initial data and, therefore, uniqueness.

For the Johnson-Segalman model we have also dealt with the case of negative elastic viscosities, corresponding to retardation effects. A comparison with other type of viscoelasticity, showing short memory elastic effects, is given.

2014, 19(7): 2133-2143
doi: 10.3934/dcdsb.2014.19.2133

*+*[Abstract](575)*+*[PDF](747.6KB)**Abstract:**

In this work, we compare different constitutive models of heat flux in a rigid heat conductor. In particular, we investigate the relation between the solutions of the Green-Naghdi type III equation and those of the classical Fourier heat equation. The latter is often referred to as a limit case of the former one, as (formally) obtained by letting certain small positive parameter $\epsilon$ vanish. In presence of steady heat sources, we prove that the type III equation may be considered as a perturbation of the Fourier one only if the solutions are compared on a finite time interval of order $1/\epsilon$, whereas significant differences occur in the longterm. Moreover, for a bar with finite length and prescribed heat flux at its ends, the solutions to the type III equation do not converge asymptotically in time to the steady solutions to the corresponding Fourier model. This suggests that the Green-Naghdi type III theory is not to be viewed as comprehensive of the Fourier theory, at least when either asymptotic or stationary phenomena are involved.

2014, 19(7): 2145-2157
doi: 10.3934/dcdsb.2014.19.2145

*+*[Abstract](636)*+*[PDF](390.1KB)**Abstract:**

We consider the free fall of slender rigid bodies in a viscous incompressible fluid. We show that the dimensional reduction (DR), performed by substituting the slender bodies with one-dimensional rigid objects, together with a hyperviscous regularization (HR) of the Navier--Stokes equation for the three-dimensional fluid lead to a well-posed fluid-structure interaction problem. In contrast to what can be achieved within a classical framework, the hyperviscous term permits a sound definition of the viscous force acting on the one-dimensional immersed body. Those results show that the DR/HR procedure can be effectively employed for the mathematical modeling of the free fall problem in the slender-body limit.

2014, 19(7): 2159-2168
doi: 10.3934/dcdsb.2014.19.2159

*+*[Abstract](611)*+*[PDF](365.9KB)**Abstract:**

In this paper we study an inverse problem for strongly degenerate differential equations in Banach spaces. Projection method on suitable subspaces will be used to solve the given problem. A number of concrete applications to ordinary and partial differential equations is described.

2014, 19(7): 2169-2187
doi: 10.3934/dcdsb.2014.19.2169

*+*[Abstract](673)*+*[PDF](366.6KB)**Abstract:**

In this paper we present a strain gradient theory of thermoelastic porous solids with initial stresses and initial heat flux. First, we establish the equations governing the infinitesimal deformations superposed on large deformations. Then, we derive a linear theory of prestressed porous bodies with initial heat flux. The theory is capable to describe the deformation of chiral materials. A reciprocity relation and a uniqueness result with no definiteness assumption on the elastic constitutive coefficients are presented.

2014, 19(7): 2189-2205
doi: 10.3934/dcdsb.2014.19.2189

*+*[Abstract](581)*+*[PDF](550.1KB)**Abstract:**

We consider the propagation of acoustic and thermal waves in a class of inviscid, thermally relaxing gases wherein the flow of heat is described by the Maxwell--Cattaneo law, i.e., in Cattaneo--Christov gases. After first considering the start-up piston problem under the linear theory, we then investigate traveling wave phenomena under the weakly-nonlinear approximation. In particular, a shock analysis is carried out, comparisons with predictions from classical gases dynamics theory are performed, and critical values of the parameters are derived. Special case results are also presented and connections to other fields are noted.

2014, 19(7): 2207-2225
doi: 10.3934/dcdsb.2014.19.2207

*+*[Abstract](779)*+*[PDF](766.6KB)**Abstract:**

We study the effect of the length scales $\alpha$ and $\beta$ on the performance of the Navier--Stokes-$\alpha\beta$ equations for numerical simulations of turbulence over coarse discretizations. To this end, we rely on the strained spiral vortex model and take advantage of the dimensional reduction allowed by that model. In particular, the three-dimensional energy spectrum is reformulated so that it can be calculated from solutions of the two-dimensional unstrained Navier--Stokes-$\alpha\beta$ equations. A similarity theory for the spiral vortex model shows that the Navier--Stokes-$\alpha\beta$ model is better equipped than the Navier--Stokes-$\alpha$ model to capture smaller-scale behavior. Numerical experiments performed using a pseudo-spectral discretization along with the second-order Adams--Bashforth time-stepping algorithm yield results indicating that the fidelity of the energy spectrum in both the inertial and dissipation ranges is significantly improved for $\beta<\alpha$.

2014, 19(7): 2227-2246
doi: 10.3934/dcdsb.2014.19.2227

*+*[Abstract](569)*+*[PDF](839.7KB)**Abstract:**

A phase-field system of coupled Allen--Cahn type PDEs describing grain growth is analyzed and simulated. In the periodic setting, we prove the existence and uniqueness of global weak solutions to the problem. Then we investigate the long-time behavior of the solutions within the theory of infinite-dimensional dissipative dynamical systems. Namely, the problem possesses a global attractor as well as an exponential attractor, which entails that the global attractor has finite fractal dimension. Moreover, we show that each trajectory converges to a single equilibrium. A time-adaptive numerical scheme based on trigonometric interpolation is presented. It allows to track the approximated long-time behavior accurately and leads to a convergence rate. The scheme exhibits a physically consistent discrete free energy dissipation.

2014, 19(7): 2247-2265
doi: 10.3934/dcdsb.2014.19.2247

*+*[Abstract](524)*+*[PDF](404.6KB)**Abstract:**

We consider the problem of recovering a polarization kernel in an axially inhomogeneous cylindrical dielectric, the polarization depending on time and the axial variable, but being constant on each cross section of the cylinder.

For this problem, under some additional measurement, we prove an existence and uniqueness result.

2014, 19(7): 2267-2278
doi: 10.3934/dcdsb.2014.19.2267

*+*[Abstract](909)*+*[PDF](1144.1KB)**Abstract:**

We analyse some peculiar properties of the function of the Mittag-Leffler (M-L) type, $e_\alpha(t) := E_\alpha(-t^\alpha)$ for $0<\alpha<1$ and $t>0$, which is known to be completely monotone (CM) with a non-negative spectrum of frequencies and times, suitable to model fractional relaxation processes. We first note that (surprisingly) these two spectra coincide so providing a universal scaling property of this function, not well pointed out in the literature. Furthermore, we consider the problem of approximating our M-L function with simpler CM functions for small and large times. We provide two different sets of elementary CM functions that are asymptotically equivalent to $e_\alpha(t)$ as $t\to 0$ and $t\to +\infty$. The first set is given by the stretched exponential for small times and the power law for large times, following a standard approach. For the second set we chose two rational CM functions in $t^\alpha$, obtained as the Pad\`e Approximants (PA) $[0/1]$ to the convergent series in positive powers (as $t\to 0$) and to the asymptotic series in negative powers (as $t\to \infty$), respectively. From numerical computations we are allowed to the conjecture that the second set provides upper and lower bounds to the Mittag-Leffler function.

2014, 19(7): 2279-2296
doi: 10.3934/dcdsb.2014.19.2279

*+*[Abstract](464)*+*[PDF](520.4KB)**Abstract:**

Via a new approach, ternary fluid mixtures saturating rotating horizontal porous layers, heated from below and salted from above and below, are investigated. With or without the presence of Brinkman viscosity, the absence of subcritical instabilities is shown together with the coincidence of linear and non-linear global stability of the thermal conduction solution. The stability-instability conditions are found to be given by simple algebraic conditions in closed forms.

2014, 19(7): 2297-2312
doi: 10.3934/dcdsb.2014.19.2297

*+*[Abstract](652)*+*[PDF](415.5KB)**Abstract:**

Integrable reductions in non-isothermal spatial gasdynamics are isolated corresponding to q-Gaussian density distributions. The availability of a Tsallis parameter q in the reductions permits the construction via a Madelung transformation of wave packet solutions of a class of associated q-logarithmic nonlinear Schrödinger equations involving a de Broglie-Bohm quantum potential term.

2014, 19(7): 2313-2333
doi: 10.3934/dcdsb.2014.19.2313

*+*[Abstract](606)*+*[PDF](561.9KB)**Abstract:**

A thermodynamically consistent mathematical model for hydrogen adsorption in metal hydrides is proposed. Beside hydrogen diffusion, the model accounts for phase transformation accompanied by hysteresis, swelling, temperature and heat transfer, strain, and stress. We prove existence of solutions of the ensuing system of partial differential equations by a carefully-designed, semi-implicit approximation scheme. A generalization for a drift-diffusion of multi-component ionized ``gas'' is outlined, too.

2014, 19(7): 2335-2352
doi: 10.3934/dcdsb.2014.19.2335

*+*[Abstract](652)*+*[PDF](365.6KB)**Abstract:**

In this paper the linear theory of viscoelasticity for Kelvin-Voigt materials with double porosity is presented and the basic partial differential equations are derived. The system of these equations is based on the equations of motion, conservation of fluid mass, the effective stress concept and Darcy's law for materials with double porosity. This theory is a straightforward generalization of the earlier proposed dynamical theory of elasticity for materials with double porosity. The fundamental solution of the system of equations of steady vibrations is constructed by elementary functions and its basic properties are established. Finally, the properties of plane harmonic waves are studied. The results obtained from this study can be summarized as follows: through a Kelvin-Voigt material with double porosity three longitudinal and two transverse plane harmonic attenuated waves propagate.

2014, 19(7): 2353-2364
doi: 10.3934/dcdsb.2014.19.2353

*+*[Abstract](476)*+*[PDF](405.8KB)**Abstract:**

The authors study the small oscillations of a floating body in a bounded tank containing an incompressible viscous fluid.

Using the variational formulation of the problem, they obtain an operator equation from which they can study the spectrum of the problem.

The small motions are strongly and weakly damped aperiodic motions and, if the viscosity is sufficiently small, there is also at most finite number of damped oscillatory motions.

The authors give also an existence and uniqueness theorem for the solution of the associated evolution problem.

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