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Statistical Mechanics, Operator Semigroup Theory and the Problem of Differentiability in Initial Data for Nonlinear Equations
(Institute of Mathematics NAS of Ukraine)
Frequently, the development of mathematics is associated with problems emerging in physics,
which pose new questions and require a fresh viewpoint at traditional matter. In this talk, we
consider an important class of models of statistical physics, the infinite system of particles
describing an anharmonic crystal, which furnishes a typical example of an infinite-dimensional
nonlinear dynamical system. The infinite dimensionality of the system along with the
nonlinearity of the model pose a two-fold complexity to overcome. The natural question on the
regular behavior of such a dynamical system gives rise to the problem of the regularity of an
operator semigroup, which need not be strongly continuous. The question itself is related to the
differentiability with respect to initial data for operator Cauchy problems with unbounded
nonlinear operators, which appears to be yet unsolved in its full generality.