Porous Dissipation Research Articles

Decay property of regularity-loss type for solutions in elastic solids with voids

In this paper, we consider the Cauchy problem for a system of elastic solids with voids. First, we show that a linear porous dissipation leads to decay rates of regularity-loss type of the solution. We show some decay estimates for initial data in Hs(R)∩L1(R). Furthermore, we prove that by restricting the initial data to be in Hs(R)∩L1,γ(R) and γ∈[0,1], we can derive faster decay estimates of the solution. Second, we show that by adding a viscoelastic damping term, then we gain the regularity of the solution and obtain the optimal decay rate.

Analyticity in porous-thermoelasticity with microtemperatures

In this note we study the analyticity of the solutions to the one-dimensional porous-elasticity problem with temperatures and microtemperatures when viscoelasticity and porous viscosity effects are also present. We show the lack of analyticity when the porous dissipation is weak, and the analyticity when it is strong.

On the decay of solutions for porous-elastic systems with history

In this paper we study the asymptotic behavior to an one-dimensional porous-elasticity problem with history. We show the lack of exponential stability when the porous dissipation or the elastic dissipation is absent. And we show the lack of analyticity and exponential stability when the porous viscosity and the elastic dissipation are present.

On the time decay of solutions in porous-thermo-elasticity of type II

In this paper we investigate the asymptotic behaviour of thesolutions of the linear theory of thermo-porous-elasticity. That is,we consider the theory of elastic materials with voids when the heatconduction is of type II. We assume that the only dissipationmechanism is the porous dissipation. First we prove that,generically, the solutions are exponentially stable on time or, inother words, the decay of solutions can be controlled by a negative exponential for a generic class of materials. The reason lies in the fact that the temperature is strongly coupled with both the microscopic and macroscopic structures of the materials and plays the roleof a 'driving belt' between the dissipation at the microscopicstructure and the macroscopic one. Later we note that the decay ofsolutions cannot be fast enough to make the solutions be zero in afinite period of time. Finally, we show that when the coupling termbetween the microscopic (or macroscopic) structure and the thermalvariable vanishes, the solutions do not decay exponentially(generically).

Impossibility of localization in thermo-porous-elasticity with microtemperatures

In this note, we prove the impossibility of the localization in-time for the solutions of the linear porous-thermo-elasticity with microtemperatures theory. This means that the only solution for this problem that vanishes after a finite time is the null solution. From a thermomechanical point of view, this result says that the combination of the porous dissipation, the temperature and the microtemperatures (in the linear theory) is not strong enough to guarantee that the thermomechanical deformations will vanish after a finite time. The main idea to prove this result is to show the uniqueness of solutions for the backward in-time problem.

Impossibility of localization in linear thermoelasticity with voids

In this note we prove the impossibility of the localization in time of the solutions of the linear thermoelasticity with voids. This means that the only solution for this problem that vanishes after a finite time is the null solution. From a thermomechanical point of view, this result says that the combination of the thermal and porous dissipation in the linear theory is not sufficiently strong to guarantee that the thermomechanical deformations will vanish after a finite time. The main idea to prove this result is to show the uniqueness of solutions for the backward in time problem.

On the time decay of solutions in porous-elasticity with quasi-static microvoids

In this paper we investigate the temporal asymptotic behavior of the solutions of the one-dimensional porous-elasticity problem with porous dissipation when the motion of microvoids is assumed to be quasi-static. This question has been recently studied in the general dynamical case. Thus, the natural question is to know if the assumption of quasi-static motion for the microvoids implies significant differences in the behavior of the solutions from the results obtained in the general dynamical case. It is worth noting that this assumption involves a qualitative change in the system of equations to be analyzed because it arises from the combination of a parabolic equation with an hyperbolic one, rather different from the well-known system of the thermo-elastic problem. First, we study the coupling of elasticity with porosity and we show that if only porous dissipation is present, the decay of solutions is slow, but if viscoelasticity is added, then the solutions decay exponentially. After that, we introduce thermal effects in the system and we show that while temperature brings exponential stability to the solutions, microtemperature does not.

On the spatial behavior of solutions for porous elastic solids with quasi-static microvoids

The spatial behavior of solutions for the dynamic problem of elasticity with voids is well known. It has been analyzed making use of the usual techniques for hyperbolic differential equations. Here we consider a related but different problem, the problem determined by the dynamic deformations of porous elasticity with porous dissipation when the motion of the microvoids is supposed to be quasi-static. We think that this kind of dynamical problem deserves some consideration. However, little attention has been paid to it in the literature. On the one hand, the problem has interest from a mechanical point of view, because this is the system that one must study when the microvoids deformations are small enough as to assume that their acceleration is nearly zero. On the other hand, the problem has also interest from its mathematical structure, because the coupling of a parabolic equation with a hyperbolic one arises in the system.

Instability, nonexistence, and uniqueness in elasticity with porous dissipation

This paper is devoted to the study of the elasticity with porous dissipation. In the context of the nonlinear problem, we prove instability and nonexistence of solutions. In the context of the linear problem, we obtain exponential growth. We also obtain uniqueness of solutions of the backward in time problem of the linear equations.

Slow decay for one-dimensional porous dissipation elasticity

This paper concerns the one-dimensional linear theory of porous elastic solids. We prove the slow decay for the solutions of two initial-boundary value problems determined by several boundary conditions.