Abstract
Invariant measures are the basic tool in the theory of dynamical systems. In the case when a dynamical system is finite-dimensional (more precisely, when the phase space of a dynamical system is a compact metric space) the classical result by N.N. Bogoliubov and N.M. Krylov states the existence of a nonnegative normalized invariant measure (see [13,52] and, also, [72]). By analogy, as it is well known, a finite-dimensional Hamiltonian system possesses a Liouville invariant measure. Another situation occurs when the phase space of a dynamical system is infinite-dimensional. Of course, there is a natural interest whether invariant measures exist in this case, too. Numerous recent papers are devoted to proving the existence or constructing such measures in the infinite-dimensional case when a dynamical system is generated by a nonlinear partial differential equation. In this chapter, we illustrate these investigations for the NLSE and KdVE with our two results. Some other literature on this subject is indicated in Additional remarks.
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© 2001 Springer-Verlag Berlin Heidelberg
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(2001). Invariant measures. In: Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory. Lecture Notes in Mathematics, vol 1756. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45276-1_6
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DOI: https://doi.org/10.1007/3-540-45276-1_6
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