16. Computer assisted proofs in dynamics of dissipative PDEs
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Polish supervisor:
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Piotr Zgliczyński
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Cooperating partners:
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Konstantin Mischaikow (Rutgers University)
Warwick Tucker (Uppsala University)
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In the study of nonlinear PDEs there is a huge gap between what we
can observe in the numerical simulations and what we can prove
rigorously. One possibility to overcome this problem are the
computer assisted proofs. In the project we will focus on
dissipative PDEs, which includes Navier-Stokes eq, Ginzburg-Landau
eq., reaction-diffusion eq. and many others. For this class of
equations the nonrigorous numerical simulations clearly show that
the dynamics is 'essentially' finite dimensional and it is quite
natural to expect that a lot of topological and geometrical tools
developed in the theory of dynamical system should be also
applicable, after suitable modification, in the context of
dissipative PDEs. The student is expected to work on
- the theory - overcoming problem of the infinite dimension, required for application of tools from the finite-dimensional dynamics
- the algorithms for rigorous numerics for dissipative PDEs,
which should result in computer assisted proofs of interesting
dynamical phenomena for such systems as the Navier-Stokes eq. or
the Ginzburg-Landau eq.
It should be stressed that the field is wide open, the number of
rigorous results which go beyond the existence of the global
attractor and bifurcations from some trivial solutions rather
small.
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