Mathematical Analysis For Small Heat Conduction Limit In Thermoelasticity

The equations of thermoelasticity describe the behavior of an elastic heat conducting body.Different elastic materials have different thermal conductivities.This thesis discusses the asymptotic behavior of the solution for small heat conduction limit of thermoelasticity with one space variable.When the coefficient of heat conduction ? goes to zero,the hyperbolic-parabolic coupled system of thermoelasticity turns into a pure hyperbolic system.The pure hyperbolic system requires boundary conditions that are different from the coupled system.In small heat conduction limit,the solution of the classical hyperbolic-parabolic coupled system will have rapid transitions near the boundary.Such behavior is referred as boundary layer problem.In order to understand small heat conduction limit,it is crucial to discuss the behavior of the solution near the boundary and away from the boundary.This discussion also proves the well-posedness of the limit equation and the stability of the boundary layer.In the first part,an example is shown to explain the phenomenon of boundary layers and demonstrate the boundary layer problem in thermoelasticity.Then by applying multi-scale matching expansion,the thickness of the boundary layer can be derived to be?? when the heat conduction vanishes.The asymptotic expansions of the formal solutions to thermoelastic equations is obtained.There are two different behaviors can be observed from the asymptotic expansions of the formal solutions,one near the boundary and another one away from the boundary.Near the boundary,the thermal layer appears in the leading order,with the boundary layer of displacement in the order of O(??).This is described by a boundary value problem of a second order differential system.On the other hand,away from the boundary,the classical thermoelasticity can be approximately described by a system with zero heat conduction.Both linear and nonlinear cases are studied.With the help of theories in first order quasilinear symmetric hyperbolic systems with characteristic boundaries,the local existence of the solutions to the limit problem is proved.Lastly,by using energy estimate,this thesis rigorously justifies the asymptotic behavior of the small heat conduction limit in thermoelasticity,and obtains that the boundary layer is stable.