University of Toronto
                  PDE/Applied Math/Analysis Seminar
                  Friday, 21 September, 1:10-2:00pm
                          6183 Bahen Center      



SPEAKER:  Tadahiro Oh (University of Toronto)

TITLE:    Diophantine conditions in the well-posedness 
          theory of a coupled KdV-type system and its 
          invariant Gibbs measure


ABSTRACT: We consider the well-posedness issues of a (parametric) 
          system of two coupled nonlinear equations of KdV-type 
          arising in atmospheric sciences, introduced by A. Majda 
          and J. Biello.  In this talk, we will concentrate on the 
          periodic setting.

          Sharp local and global well-posedness results are known 
          for a single KdV equation. These, however, do not apply 
          at all to the systems under consideration as the 
          presence of the coupling parameter makes the system 
          exhibit nontrivial resonant interactions which are not 
          present in the single KdV.

          In the LWP theory, we need to establish several crucial 
          bilinear estimates. If a certain constant $C$ related 
          to the coupling parameter $0<\alpha<1$ is rational, we 
          show that particular resonances occur and the estimates 
          fail completely.  If the constant $C$ is irrational, 
          then these estimates hold but they vary in a sensitive 
          manner according to "how irrational" $C$ is.  We employ 
          the Diophantine conditions used in Dynamical Systems 
          to exactly quantify this notion.  While these kinds of 
          phenomena occur in mechanical systems, especially for 
          the persistence of periodic solutions (KAM theory), it 
          is somewhat  peculiar for the well-posedness theory of 
          a partial differential equation to be so sensitive to 
          number theoretical character of a coupling parameter.  

          Next, we will briefly discuss global well-posedness. 
          For the GWP theory, we use the method of almost 
          conservation laws (I-method) developed by Colliander 
          et al. to obtain global solutions below the energy 
          space.  Unlike the KdV situation, however, and depending 
          on "how irrational" $C$ is, a serious small denominator 
          problem arises, which stops the I-method before reaching 
          a global theory that matches the local theory.

          Finally, we show the existence of Gibbs measures 
          invariant under the flow of the system.  Using the 
          invariance of the measure and the finite dimensional 
          approximation, we establish the almost surely GWP in a 
          regularity where the I-method fails.  This part involves 
          defining a new Sobolev class and establishing the new 
          LWP there.  We will sketch the idea, comparing with the 
          original LWP result.

          If time permits, we will briefly discuss the results in 
          the non-periodic setting and the ill-posedness results.
	  

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University of Toronto PDE/Applied Math/Analysis Seminar 
http://www.math.toronto.edu/appmath/

2007-2008 organizers:
          Walid Abou-Salem      walid@ math.toronto.edu
          Almut Burchard        almut@ math.toronto.edu
          Robert McCann		mccann@ math.toronto.edu
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