========================================================= INFINITE ANALYSIS NEWS LETTER No. 263 July 14, 1997 ========================================================= Secretariat, Juten Ryoiki Kenkyu 231 Tel. & Fax: 075-753-3707 Email: juten::kusm.kyoto-u.ac.jp We welcome related information. ============================================== Seminar Information ============================================== Infinite Analysis Seminar Date : 14:00--15:00, July 22 (Tue), 1997 Place : Rm. 402(4th floor), RIMS, Kyoto University Speaker: Prof Tudor Ratiu Title : The doubel bracket equation, the dispersionless Toda PDE, and the Kostant convexity theorem Abstract: The double bracket equation is a gradient vector field on adjoint orbits of compact Lie algebras whose dynamics is completely described in Lie theoretic terms. In addition, the finite non-periodic Toda lattice can be written as a double bracket equation. In the limit, one gets the dispersionless Toda PDE, aslo a completely integrable system. In trying to analyze the scattering behavior of its solutions, one is led to a generalization of Kostant's convexity theorem for the group of area preservig diffeomorphisms of the annulus. ---------------------------------------------------------------------- Date : 15:30--16:00, July 22 (Tue), 1997 Place : Rm. 402(4th floor), RIMS, Kyoto University Speaker: Prof Tudor Ratiu Title : Dissipation induced instability Abstract: In the Lyapunov stability analysis by energy methods, only sufficient conditions can be obtained by determining the definiteness of the Hessian at a critical point. If the Hessian is indefinite, the criterion is inconclusive. It will be shown that for quite general types of dissipations, these equlibria become spectrally unstable (exponentially growing modes) for the perturbed dissipative system. Applications include the rigid body with rotors and a certain beam equation. The mathematical techniques are based on normal forms for the Hessian and the symplecitc form at the relative equilibrium, combined with a judicious choice of Lyapunov function. ---------------------------------------------------------------------- Date : 13:30--15:00, July 25 (Fri), 1997 Place : Rm. 102(1st floor), RIMS, Kyoto University Speaker: Prof Tudor Ratiu Title : Shadowing curves for two dimensional Navier-Stokes equation Abstract: The Euler euqations for two dimensional homogeneous incompressible fluids admit a large family of stationary solutions. Among these are the vorticities that are eigenfucntions of the Laplacian. These flows are the two dimensional analogs of the Arnold-Beltrami-Childress (ABC) flows. For the first eigenvalue, the corresponding vorticity is a global minimum of the enstrophy restricted to the level set of the energy and is hence stable in the sense of Lyapunov. It turns out that for any initial condition of the Navier-Stokes equation in a conical domain determined by the second eigenvalue of the Laplacian, the solution if the Navier-Stokes equation is attracted by the manifold of these stable ABC flows along a shadowing curve in such a way that one can give a precise exponential type estimate of the error. ==============================================