Spatial Decay Of Solutions Research Articles

On stability in the thermoelastostatics of dipolar bodies

Our study is concerned with the initial boundary value problem in the context of the thermoelastostatics of dipolar bodies. We will derive a result which describes the exponential spatial decay of solutions of this problem. We will also find a superior limit for the amplitude, which is dependent on the initial and boundary conditions.

Decay Estimates for Linearized Unsteady Incompressible Viscous Flows Around Rotating and Translating Bodies

We consider the time-dependent Oseen sytem with rotational terms. This system is a linearized model for the ow of a viscous incompressible uid around a rigid body moving at a constant velocity and rotating with constant angular velocity. We present results on temporal and spatial decay of solutions to this system in the whole space. The spatial asymptotics we establish exhibit a wake.

Infinite Energy Solutions for Damped Navier–Stokes Equations in $${\mathbb{R}^2}$$

We study the so-called damped Navier-Stokes equations in the whole 2D space. The global well-posedness, dissipativity and further regularity of weak solutions of this problem in the uniformly-local spaces are verified based on the further development of the weighted energy theory for the Navier-Stokes type problems. Note that any divergent free vector field $u_0\in L^\infty(\mathbb R^2)$ is allowed and no assumptions on the spatial decay of solutions as $|x|\to\infty$ are posed. In addition, applying the developed theory to the case of the classical Navier-Stokes problem in R2, we show that the properly defined weak solution can grow at most polynomially (as a quintic polynomial) as time goes to infinity.

Asymptotic and other estimates for semilinear parabolic problem in semi-infinite cylinder

The spatial decay of solutions to initial-boundary value problems for a semilinear parabolic equation in a semi-infinite cylinder of variable cross-section subject to zero condition on the lateral boundaries is investigated. A second-order differential inequality that shows the spatial decay O(exp(−z2/(4(t+t0)))) for an L2p cross-sectional measure of the solution is obtained. A first-order differential inequality leading to growth or decay is also derived. In the case of growth, an upper bound for blow-up in space is obtained, while in the case of decay an upper bound for the total energy in terms of data is obtained.

Spatial Stability for the Quasi-Static Problem in Thermoelastic Diffusion Theory

In this paper we derive some spatial stability results for the quasi-static problem in thermoelastic diffusion theory for anisotropic media. The coupled system of equations of thermoelasticity with diffusion is a coupling of an elliptic equation with two parabolic equations. It poses some new mathematical difficulties. Here we study the exponential spatial decay of solutions. An upper bound for the amplitude in terms of the boundary and initial conditions is obtained. The extension of the spatial stability results is also treated.

Saint-Venant decay rates for an anisotropic and non-homogeneous mixture of elastic solids in anti-plane shear

The purpose of this research is to investigate the influence of material inhomogeneity and anisotropy on the decay of Saint-Venant end effects in anti-plane shear deformations of linear mixtures of elastic solids. The spatial decay of solutions of a boundary value problem with variable coefficients on a semi-infinite strip is investigated. The results may be interpreted in terms of a Saint-Venant principle for anti-plane shear deformations of linear anisotropic mixtures of elastic solids. As our first results have a very general point of view, we study some examples in detail.

Qualitative aspects in dual-phase-lag heat conduction

We investigate the equation of the dual-phase-lag heat conduction proposed by Tzou. To describe this equation, we use the phase lag of the heat flux and the phase lag of the gradient of the temperature. We analyse the basic properties of the solutions of this problem. First, we prove that when both parameters are positive, the problem is well posed and the spatial decay of solutions is controlled by an exponential of the distance. When the phase lag of the gradient of the temperature is bigger than the phase lag of the heat flux, the problem is exponentially stable (which is a natural property to expect for a heat equation) and the spatial behaviour is controlled by an exponential of the square of the distance. Also, a uniqueness result for unbounded solutions is proved in this case.

Spatial decay in a double diffusive convection problem in Darcy flow

This paper deals with a model of time-dependent double diffusive convection in Darcy flow. In particular it is concerned with the spatial decay of solutions when the flow is confined to a semi-infinite cylinder. Decay bounds for an energy expression are derived.

Saint-Venant decay rates for a non-homogeneous isotropic mixture of elastic solids in anti-plane shear

The purpose of this research is to investigate the influence of material inhomogeneity on the decay of Saint-Venant end effects in anti-plane shear deformations of linear isotropic mixtures of elastic solids. The work is motivated by the recent research activity on functionally graded materials (FGMs), i.e. materials with spatially varying properties tailored to satisfy particular engineering applications. The spatial decay of solutions of a boundary value problem with variable coefficients on a semi-infinite strip is investigated. The results may be interpreted in terms of a Saint-Venant principle for anti-plane shear deformations of linear isotropic mixtures of elastic solids.

On the spatial decay of solutions in the theory of swelling porous thermoelastic soils

In this paper we consider the linear theory of swelling porous thermoelastic soils. The formulation belongs to the general theory of mixtures for porous elastic solids filled with fluid and gas with thermal conduction and by considering the time derivative of temperature as a variable in the set of constitutive equations. The spatial decay is studied for the solutions of the initial-boundary value problems associated with the swelling theory of porous thermoelastic soils. Appropriate estimates are established for describing the spatial behaviour by means of a first-order differential inequality. The method of proof is based on the time-weighted method applied in appropriate classes of solutions.

Anti-plane shear deformations of swelling porous elastic soils

This paper is concerned with the theory of anti-plane shear deformations of swelling porous elastic soils in the case of fluid saturation or gas saturation. The formulation belongs to the theory of mixtures for porous elastic solids filled with fluid or gas. These problems propose some new mathematical difficulties. We prove the exponential decay of solutions, a uniqueness result and the spatial decay of solutions (in the dynamical and the quasi-static cases). Some remarks are presented in the last section.

ON THE ASYMPTOTIC SPATIAL BEHAVIOR IN LINEAR THERMOELASTICITY OF MATERIALS WITH VOIDS

The comparison principle involving solutions of the one-dimensional heat equation is used to investigate the spatial decay of solutions to initial-boundary value problems for the linear theory of thermoelasticity with voids. A spatial decay estimate is established for an appropriate volumetric measure of the solution in an unbounded body. This proves that at large distances from the support of the external given data the decay of the mechanical and thermal effects is controlled only by the thermal characteristic coefficients of the material.

On the Spatial Decay of Solutions to a Quasi-Linear Parabolic Initial-Boundary Value Problem and Their Derivatives

In this paper, we study the spatial decay of the solution of a quasi-linear heat equation in a long cylindrical region if the far end and the lateral surface are held at a zero temperature and a nonzero temperature is applied at the near end. Our result follows from the maximum principle applied to an auxiliary function $\Phi$ defined on the solution u and its derivatives.

ON THE SPATIAL DECAY OF ILL-POSED PARABOLIC PROBLEMS

This paper investigates the spatial decay of solutions to two ill-posed parabolic initialboundary value problems: 1) the backward heat equatoin defined on a semi-infinite cylinder in ℝN and 2) the backward Stokes flow equations defined on a semi-infinite strip in ℝ2.

On Saint-Venant's principle in finite anti-plane shear: An energy approach

Some time ago a version of Saint-Venant’s principle was formulated and established for finite elastostatics [1]. As was discussed in [1], the issues of concern in connection with Saint-Venant’s principle in the nonlinear theory of elasticity are considerably more involved then those arising in the linear theory. (For a survey of results on Saint-Venant’s principle, primarily for linear theories, see e.g. [2–5].) One difficulty is that the appropriate Saint-Venant solutions need to be carefully characterized (see e.g. [6–13] and the references cited therein). Secondly, in the absence of superposition, consideration of self-equilibrated end loads is no longer sufficient. Furthermore, instabilities may have to be taken into account. Also the decay rate for end effects, even if exponential, might depend on the overall loading as well as on geometry and material characteristics. Several of these issues have been considered in recent studies in the nonlinear elasticity context [1, 14–23] as well as in investigations of spatial decay of solutions of nonlinear elliptic partial differential equations [24–32].

On the spatial decay of solutions in the nonlinear theory of heat conduction

On the spatial decay of solutions in the nonlinear theory of heat conduction

On the spatial decay of solutions of parabolic equations

Es wird ein von Edelstein erzieltes Ergebnis uber den exponentiellen Temperaturabfall eines Korpers an von Warmezufuhrquellen entfernten Stellen auf eine erweiterte Klasse von Randwertproblemen verallgemeinert.