Long-time behavior of a coupled heat-wave system arising in fluid-structure interaction. (English) Zbl 1178.74075
The authors study the long-time behavior of a coupled system consisting of wave and heat equations coupled through transmission conditions along a steady interface. This model is a linearized and simplified version of more sophisticated models which arise in the theory of fluid-structure interaction.
One of the main difficulties in the analysis of these models is that they are in fact free boundary problems, with the free boundary being the interface between the fluid and elastic body, and this interface evolves as time increases. In fact, very few results exist on the uniqueness and global existence of solutions because of the possible collision of the interface with exterior boundaries. The authors refer to J.-L. Vázquez and E. Zuazua [Commun. Partial Differ. Equations 28, No. 9–10, 1705–1738 (2003; Zbl 1071.74017)] for some results in this direction, concerning a one-dimensional model which couples the Burgers equation with the motion of a finite number of solid point masses.
The authors deal with a simplified model, and their main goal is to perform the asymptotic analysis of solutions as \( t \rightarrow \infty\). After linearization around the trivial solution, the interface does not move and is independent of time. The model under consideration consists of two coupled equations in two adjacent domains separated by this interface. This model couples the wave and heat equations as a prototype of more realistic models in which the system of elasticity is coupled with the Navier-Stokes or Stokes equations.
The novelty of the model consists in transmission conditions at the interface which are much more realistic, and for which the time derivative of the wave solution is coupled with the heat solution. This type of interface condition is indeed more physical in fluid-structure interaction, since both the fluid solution and the time derivative of elastic solution constitute velocity fields of motion of the fluid and deformations of the elastic body, respectively.
The main results are the following: (i) regardless of the geometric configuration, the rate of decay is never uniform, and (ii) under suitable geometric conditions smooth solutions decay polynomially.
A comparison to the results by J. Rauch, X. Zhang and E. Zuazua [J. Math. Pures Appl., IX. Sér. 84, No. 4, 407–470 (2005; Zbl 1077.35030)] is given. In particular, it is shown that for the system considered in the cited work smooth solutions decay logarithmically without any geometric restrictions.
One of the main difficulties in the analysis of these models is that they are in fact free boundary problems, with the free boundary being the interface between the fluid and elastic body, and this interface evolves as time increases. In fact, very few results exist on the uniqueness and global existence of solutions because of the possible collision of the interface with exterior boundaries. The authors refer to J.-L. Vázquez and E. Zuazua [Commun. Partial Differ. Equations 28, No. 9–10, 1705–1738 (2003; Zbl 1071.74017)] for some results in this direction, concerning a one-dimensional model which couples the Burgers equation with the motion of a finite number of solid point masses.
The authors deal with a simplified model, and their main goal is to perform the asymptotic analysis of solutions as \( t \rightarrow \infty\). After linearization around the trivial solution, the interface does not move and is independent of time. The model under consideration consists of two coupled equations in two adjacent domains separated by this interface. This model couples the wave and heat equations as a prototype of more realistic models in which the system of elasticity is coupled with the Navier-Stokes or Stokes equations.
The novelty of the model consists in transmission conditions at the interface which are much more realistic, and for which the time derivative of the wave solution is coupled with the heat solution. This type of interface condition is indeed more physical in fluid-structure interaction, since both the fluid solution and the time derivative of elastic solution constitute velocity fields of motion of the fluid and deformations of the elastic body, respectively.
The main results are the following: (i) regardless of the geometric configuration, the rate of decay is never uniform, and (ii) under suitable geometric conditions smooth solutions decay polynomially.
A comparison to the results by J. Rauch, X. Zhang and E. Zuazua [J. Math. Pures Appl., IX. Sér. 84, No. 4, 407–470 (2005; Zbl 1077.35030)] is given. In particular, it is shown that for the system considered in the cited work smooth solutions decay logarithmically without any geometric restrictions.
Reviewer: Peter A. Velmisov (Ul’yanovsk)
MSC:
| 74H40 | Long-time behavior of solutions for dynamical problems in solid mechanics |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
| 74F05 | Thermal effects in solid mechanics |
| 35Q74 | PDEs in connection with mechanics of deformable solids |
| 35Q79 | PDEs in connection with classical thermodynamics and heat transfer |
| 80A20 | Heat and mass transfer, heat flow (MSC2010) |
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