Viscoelastic boundary stabilization for a transmission problem in elasticity

We consider an anisotropic body which is constituted of twodifferent types of materials supporting a memory boundary conditionand we show that its energy decays uniformly as time goes toinfinity with the same rate as the relaxation function g, thatis, the energy decays exponentially when g decays exponentially,and polynomially when g decays polynomially.     

A remark on the stabilization of the 1-d wave equation

We consider the wave equation on an interval of length 1 with an interior damping at ξ and with Dirichlet boundary condition at the two ends. It is well known that, if ξ is rational, the energy does not decay to 0. In this case, we prove that the energy decays exponentially to a constant which we identify.     

Exponential decay in thermoelastic materials with voids and dissipative boundary without thermal dissipation

We study the energy decay of the solutions of a linear homogeneous anisotropic porous thermoelastic system in the context of Green and Naghdi model of type II with the following boundary condition with memory for the displacement ${{\bf T}(x,t)n(x) = -\gamma_0v(x,t) - \int_0^\infty \lambda(s)v^t(x,s) {{d}}s}$ . By introducing a boundary free energy, we prove that if the kernel λ exponentially decays in time, then also the energy exponentially decays when porosity viscosity is present.     

A counterexample to uniqueness and regularity for harmonic maps between hyperbolic spaces

We construct a harmonic diffeomorphism from the Poincaré ballH ⁿ⁼¹ to itself, whose boundary value is the identity on the sphereS ⁿ, and which is singular at a boundary point, as follows: The harmonic map equations between the corresponding upper-half-space models reduce to a nonlinear o.d.e. in the transverse direction, for which we prove the existence of a solution on the whole R₊ that grows exponentially near infinity and has an expansion near zero. A conjugation by the inversion brings the singularity at the origin, and a conjugation by the Cayley transform and an isometry of the ball moves the singularity at any point on the sphere.     

Exponentially stable equilibria to an indefinite nonlinear Neumann problem in smooth domains

We prove existence of two nonconstant exponentially stable equilibria to the heat equation supplied with a nonlinear Neumann boundary condition in any smooth n-dimensional domain (n ≥ 2), independently of its geometry. The Neumann boundary condition reflects the fact that the flux on the boundary is proportional to the product of a prescribed bistable function of the density or concentration with an indefinite weight. Such solutions are obtained via variational methods, by minimizing the corresponding energy functional on suitable invariant sets to the semiflow generated by the parabolic problem. But this is possible only if the parameter in the boundary condition is sufficiently large, otherwise we prove using the Implicit Function Theorem the uniqueness of constant equilibrium solutions. The same theorem allows us to derive isolation and smooth dependence on the parameter for nonconstant exponentially stable equilibria found.     

Effects of thermal radiation on the boundary layer flow of a Jeffrey fluid over an exponentially stretching surface

In the present investigation we have analyzed the boundary layer flow of a Jeffrey fluid over an exponentially stretching surface. The effects of thermal radiation are carried out for two cases of heat transfer analysis known as (1) Prescribed exponential order surface temperature (PEST) and (2) Prescribed exponential order heat flux (PEHF). The highly nonlinear coupled partial differential equations of Jeffrey fluid flow along with the energy equation are simplified by using similarity transformation techniques based on boundary layer assumptions. The reduced similarity equations are then solved analytically by the homotopy analysis method (HAM). The convergence of the HAM series solution is obtained by plotting (^h/_2p)\hbar-curves for velocity and temperature. The effects of physical parameters on the velocity and temperature profiles are examined by plotting graphs.     

Spectral analysis and stability of thermoelastic Bresse system with second sound and boundary viscoelastic damping

In this paper, we consider the energy decay rate of a thermoelastic Bresse system with variable coefficients. Assume that the thermo-propagation in the system satisfies the Cattaneo's law, which can eliminate the paradox of infinite speed of thermal propagation in the assumption of the Fourier's law in the classical theory of thermoelasticity. Meanwhile, we also discuss the effect of a boundary viscoelastic damping on the stability of this system. By a detailed spectral analysis, we obtain the expressions of the spectrum and deduce some spectral properties of the system. Then based on the distribution of the spectrum, we prove that the energy of the system with a boundary viscoelastic damping decays exponentially. However, it no longer decays exponentially if there is no boundary damping. Copyright © 2013 John Wiley & Sons, Ltd.     

Nonlinear stability of traveling wave fronts for delayed reaction diffusion systems

This paper is concerned with nonlinear stability of traveling wave fronts for a delayed reaction diffusion system. We prove that the traveling wave front is exponentially stable to perturbation in some exponentially weighted L^∞ spaces, when the difference between initial data and traveling wave front decays exponentially as x→−∞, but the initial data can be suitable large in other locations. Moreover, the time decay rates are obtained by weighted energy estimates.     

The longtime behavior of a nonlinear Reissner-Mindlin plate with exponentially decreasing memory kernels

In this paper we investigate the longtime behavior of the mathematical model of a homogeneous viscoelastic plate based on Reissner-Mindlin deformation shear assumptions. According to the approximation procedure due to Lagnese for the Kirchhoff viscoelastic plate, the resulting motion equations for the vertical displacement and the angle deflection of vertical fibers are derived in the framework of the theory of linear viscoelasticity. Assuming that in general both Lame's functions, λ and μ, depend on time, the coupling terms between the equations of displacement and deflection depend on hereditary contributions. We associate to the model a nonlinear semigroup and show the behavior of the energy when time goes on. In particular, assuming that the kernels λ and μ decay exponentially, and not too weakly with respect to the physical properties considered in the model, then the energy decays uniformly with respect to the initial conditions; i.e., we prove the existence of an absorbing set for the semigroup associated to the model.     

Uniform stabilization of a thermo piezoelectric/piezomagnetic model

We consider an evolution system describing an electro/magneto/thermoelastic phenomenon in a bounded domain of R³${\mathbb{R}}^3$. The resulting anisotropic model is a coupling between two hyperbolic equations, two elliptic equations and one parabolic equation. Assuming that a dissipative mechanism is present on the boundary and suitable boundary conditions are given we prove that the total energy decays exponentially as t?+∞$t?+\infty$ provided the region satisfies a geometric condition.     

A finite element method with a large mesh-width for a stiff two-point boundary value problem

We describe a finite element method for computation of numerical approximations of the solution of the second order singularly perturbed two-point boundary value problem on [?1, 1]

$\text{? u″ + pu′ = f, u(?1) = u(1) = 0, 0 < ? ∠ 1, (′ =}\text{d}\text{dx}\text{)}$

On a quasi-uniform mesh we construct exponentially fitted trial spaces which consist of piece-wise polynomials and of exponentials which fit locally to the singular solution of the equation or its adjoint. We discretise the Galerkin form for the boundary problem using such exponentially fitted trial spaces. We derive rigorous bounds for the error of discretisation with respect to the energy norm and we obtain superconvergence at the mesh-points, the error depending on ?, the mesh-width and the degree of the piece-wise polynomials.     

具有动态边界的非均匀Euler-Bernoulli梁的边界反馈镇定

本讨论一端带有重物的Euler-Bernoulli梁的边界反馈镇定问题。在生物的质量忽略不计而只考虑重物的转动惯量的情况下，证明了同时在梁的自由端施加力和力矩反馈，闭环系统的能量可被指数镇定。进而对于系统只有力反馈或只有力矩反馈的情况，得到了闭环系统（指数）稳定的充分必要条件。     

Low-Temperature Localisation for Continuous Spin-Systems

We study the free energy of continuous spin-systems on Z ^d , in the framework of Laplace integrals and transfer operators. Under a weak coupling condition, we show that the free energy in the low-temperature limit is determined, up to an exponentially small error, by the restriction to a neighbourhood of global minima of the energy. We have results for some single- and double-well problems.     

Solvability of the Stokes Immersed Boundary Problem in Two Dimensions

We study coupled motion of a 1-D closed elastic string immersed in a 2-D Stokes flow, known as the Stokes immersed boundary problem in two dimensions. Using the fundamental solution of the Stokes equation and the Lagrangian coordinate of the string, we write the problem into a contour dynamic formulation, which is a nonlinear nonlocal equation solely keeping track of evolution of the string configuration. We prove existence and uniqueness of local-in-time solution starting from an arbitrary initial configuration that is an H^5/2-function in the Lagrangian coordinate satisfying the so-called well-stretched assumption. We also prove that when the initial string configuration is sufficiently close to an equilibrium, which is an evenly parametrized circular configuration, then a global-in-time solution uniquely exists and it will converge to an equilibrium configuration exponentially as t → + ∞. The technique in this paper may also apply to the Stokes immersed boundary problem in three dimensions. © 2018 Wiley Periodicals, Inc.     

Slow dynamics for the hyperbolic Cahn‐Hilliard equation in one‐space dimension

The aim of this paper is to study the metastable properties of the solutions to a hyperbolic relaxation of the classic Cahn‐Hilliard equation in one‐space dimension, subject to either Neumann or Dirichlet boundary conditions. To perform this goal, we make use of an “energy approach," already proposed for various evolution PDEs, including the Allen‐Cahn and the Cahn‐Hilliard equations. In particular, we shall prove that certain solutions maintain a N‐transition layer structure for a very long time, thus proving their metastable dynamics. More precisely, we will show that, for an exponentially long time, such solutions are very close to piecewise constant functions assuming only the minimal points of the potential, with a finitely number of transition layers, which move with an exponentially small velocity.     

Stability analysis of Rayleigh�CB��nard convection in a porous medium

In this paper we study the problem of Rayleigh?CBénard convection in a porous medium. Assuming that the viscosity depends on both the temperature and pressure and that it is analytic in these variables we show that the Rayleigh?CBénard equations for flow in a porous media satisfy the idea of exchange of stabilities. We also show that the static conduction solution is linearly stable if and only if the Rayleigh number is less than or equal to a critical Rayleigh number. Finally, we show that a measure of the thermal energy of the fluid decays exponentially which in turn implies that the L² norm of the perturbed temperature and velocity also decay exponentially.     

Boundary output feedback stabilization for a class of coupled parabolic systems

In this paper, the boundary output feedback stabilization problem is addressed for a class of coupled nonlinear parabolic systems. An output feedback controller is presented by introducing a Luenberger‐type observer based on the measured outputs. To determine observer gains, a backstepping transform is introduced by choosing a suitable target system with nonlinearity. Furthermore, based on the state observer, a backstepping boundary control scheme is presented. With rigorous analysis, it is proved that the states of nonlinear closed‐loop system including state estimation and estimation error of plant system are locally exponentially stable in the L²norm. Finally, a numerical example is proposed to illustrate the effectiveness of the presented scheme.     

Chern–Simons solitons in quantum potential

The self-dual Chern–Simons solitons under the influence of the quantum potential are considered. The single-valuedness condition for an arbitrary integer number N⩾0 of solitons leads to quantization of Chern–Simons coupling constant κ=m(e²/g), and the integer strength of quantum potential s=1−m². As we show, the Jackiw–Pi model corresponds to the first member (m=1) of our hierarchy of the Chern–Simons gauged nonlinear Schrödinger models, admitting self-dual solitons. New types of exponentially localized Chern–Simons solitons for the Bloch electrons near the hyperbolic energy band boundary are found.     

A highly accurate numerical solution of a biharmonic equation

The coefficients for a nine–point high–order accuracy discretization scheme for a biharmonic equation ∇ ⁴u = f(x, y) (∇² is the two–dimensional Laplacian operator) are derived. The biharmonic problem is defined on a rectangular domain with two types of boundary conditions: (1) u and ∂²u/∂n² or (2) u and ∂u/part;n (where ∂/part;n is the normal to the boundary derivative) are specified at the boundary. For both considered cases, the truncation error for the suggested scheme is of the sixth-order O(h⁶) on a square mesh (h_x = h_y = h) and of the fourth-order O(h⁴_xh²_xh²_y h⁴_y) on an unequally spaced mesh. The biharmonic equation describes the deflection of loaded plates. The advantage of the suggested scheme is demonstrated for solving problems of the deflection of rectangular plates for cases of different boundary conditions: (1) a simply supported plate and (2) a plate with built-in edges. In order to demonstrate the high–order accuracy of the method, the numerical results are compared with exact solutions. © John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 375–391, 1997     

Long Time Behavior of Solutions to the 3D Compressible Euler Equations with Damping

Abstract

The effect of damping on the large-time behavior of solutions to the Cauchy problem for the three-dimensional compressible Euler equations is studied. It is proved that damping prevents the development of singularities in small amplitude classical solutions, using an equivalent reformulation of the Cauchy problem to obtain effective energy estimates. The full solution relaxes in the maximum norm to the constant background state at a rate of t ^?(3/2). While the fluid vorticity decays to zero exponentially fast in time, the full solution does not decay exponentially. Formation of singularities is also exhibited for large data.