| Fluid-Structure Interaction is a branch of mechanics generated by the intersection of fluid me-chanics and solid mechanics.It is a science that studies the behaviors of solid under the action of fluid field and the interaction between solid and fluid field.An important feature of fluid-solid coupling mechanics is the interaction between two phase media.In present years,the research on fluid-structure interaction problems has attracted much attention.In this paper,we handle with the well-posedness and exponential stability of solutions about fluid-thermoelastic plate interaction sys-tem,while the heat effects are modeled by the hyperbolic Cattaneo law arising from a“second sound”effect.Moreover,we study the long time behavior of the system with source term.The contents of this thesis are as follows:The existence,uniqueness and exponential decay of the solution are studied for the fluid-structure interaction between a thermoelastic plate and a fluid without external force.Firstly,since the fluid equation takes a Neumann-type boundary condition,we use Neumann-Robin mapping to write the expression of the pressure p,and then use semigroup method to prove the well-poseness of the coupled system.Secondly,the elimination of p-?·(?)?(?)uterm in the plate equation requires the use of Stokes operator which acts as a lifting function,and then the function of Stokes opera-tor which needs to be acted on is zero average.Thus,we further construct the polished zero mean multiplier.Meanwhile,due to Cattaneo's law of heat conduction and the loss of temperature in the dissipation,we use second-order energy to control the?L2(?0)and eventually get the exponential decay of solutions.In Chapter 3,we further consider the long time behavior of the coupling problem between the fluid and the thermoelastic plate under external forces.The treatment of the coupling equation itself is the same as that of the case without external force,but the difference lies in the extra treatment of the nonlinear terms.We further give the Lipschitz continuity of the abstract nonlinear terms un-der reasonable assumptions,and finally obtain the well-poseness of the global solution.The key to the proof of global attractor is to obtain the asymptotic compactness of the dynamical system,and the key to obtain the asymptotic compactness is to obtain a stable inequality.We use the polish-ing technique in Chapter 2 to realize the zero-average multiplier and use Stokes operator to solve the matching problem of fluid and plate equation multiplier.Finally,we construct the appropriate Lyapunov functional and we derive the proof.In this paper,we use semigroup method to prove the well-poseness of solutions of fluid-thermoelastic plates with Cattaneo's law in the absence and presence of external forces respectively.For the case of no external force,the main innovation of the proof of the exponential decay is to use the second or-der method to control?L2(?0).For the proof of global attractor with external force,our innovation is to obtain asymptotic compactness by reasonable construction of the new perturbation functionals.And we obtain the final result by combining with the correlation theorem of attractor... |
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