1,219 research outputs found
Asymptotic behaviour of the non-autonomous 3D Navier-Stokes problem with coercive force
We construct pullback attractors to the weak solutions of the
three-dimensional Dirichlet problem for the incompressible Navier-Stokes
equations in the case when the external force may become unbounded as time goes
to plus or minus infinity.Comment: 22 page
Inertial manifolds and finite-dimensional reduction for dissipative PDEs
These notes are devoted to the problem of finite-dimensional reduction for
parabolic PDEs. We give a detailed exposition of the classical theory of
inertial manifolds as well as various attempts to generalize it based on the
so-called Man\'e projection theorems. The recent counterexamples which show
that the underlying dynamics may be in a sense infinite-dimensional if the
spectral gap condition is violated as well as the discussion on the most
important open problems are also included.Comment: This manuscript is an extended version of the lecture notes taught by
the author as a part of the crash course in the Analysis of Nonlinear PDEs at
Maxwell Center for Analysis and Nonlinear PDEs (Edinburgh, November, 8-9,
2012
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