Font Size:
Well-posedness and long time behavior of finite energy solutions corresponding to 3-d flows interacting with dynamic elasticity
Irena Lasiecka
Last modified: 2010-04-17
Abstract
We shall discuss finte energy solutions corresponding to 3-D flows (gas or fluid) interacting on an interface with dynamic nonlinear elasticity. Specific examples include : fluid structure interactions (Navier Stokes equation coupled with wave equation), structural acoustic interaction (wave equation coupled with nonlinear dynamic plate or shell equation), flow-gas interaction (linearised Euler equation interacting with a nonlinear dynamic plate). The interaction in these models takes place on a manifold of lower dimension.
Of particular interest to this talk are evolutions with "supercritical" (with respect to Sobolev's embeddings) nonlinearities. These include supercritical semilinear wave equations, von Karman evolutions, Kirchhoff Bousinesq equations and other non-linear plate equations. This class of models, while motivated by a multitude of physical applications, does not seem to have well established mathematical methodology for dealing with the issues such as existence, uniqueness and stability (including long time behavior) of solutions posed in physically significant finite energy space. The issue at hand is a well recognized dychotomy between global existence and uniqueness. Existence for weak (finite energy) solutions is known, while uniqueness holds for strong solutions that are defined only locally.
In this talk we shall single out several classes of models for which rather general techniques have been recently developed. (see Ref. [1-4] and references therein). These techniques lead -ultimately- to the resolution of the dychotomy and also provide a treatment for the study of long time behavior (global attractors) for the resulting dynamics. The results presented will supply positive answers to several open questions raised in the context of "supercritical" dynamic elesticity.
The methods discussed are based on tools such as : harmonic analysis, compensated compactness and logarithmic control of Sobolev's embeddings.
References:
[1] J. Ball, Global attractors for semilinear wave equations, Discr. Cont. Dyn. Sys. 10 (2004), 31–52.
[2] G. Sell and Y. You. Dyanamics of Evolutionary Equations. Springer Verlag, 2002.
[3] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Memoirs of AMS, vol.195, no. 912, AMS, Providence, RI, 2008.
[4] I. Chueshov and I. Lasiecka. Von Karman Evolutions-Wellposedness and Long Time Dynamics, Springer Verlag, to appear 2009.
Of particular interest to this talk are evolutions with "supercritical" (with respect to Sobolev's embeddings) nonlinearities. These include supercritical semilinear wave equations, von Karman evolutions, Kirchhoff Bousinesq equations and other non-linear plate equations. This class of models, while motivated by a multitude of physical applications, does not seem to have well established mathematical methodology for dealing with the issues such as existence, uniqueness and stability (including long time behavior) of solutions posed in physically significant finite energy space. The issue at hand is a well recognized dychotomy between global existence and uniqueness. Existence for weak (finite energy) solutions is known, while uniqueness holds for strong solutions that are defined only locally.
In this talk we shall single out several classes of models for which rather general techniques have been recently developed. (see Ref. [1-4] and references therein). These techniques lead -ultimately- to the resolution of the dychotomy and also provide a treatment for the study of long time behavior (global attractors) for the resulting dynamics. The results presented will supply positive answers to several open questions raised in the context of "supercritical" dynamic elesticity.
The methods discussed are based on tools such as : harmonic analysis, compensated compactness and logarithmic control of Sobolev's embeddings.
References:
[1] J. Ball, Global attractors for semilinear wave equations, Discr. Cont. Dyn. Sys. 10 (2004), 31–52.
[2] G. Sell and Y. You. Dyanamics of Evolutionary Equations. Springer Verlag, 2002.
[3] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Memoirs of AMS, vol.195, no. 912, AMS, Providence, RI, 2008.
[4] I. Chueshov and I. Lasiecka. Von Karman Evolutions-Wellposedness and Long Time Dynamics, Springer Verlag, to appear 2009.