| Abstract: |
| Consider a time-independent dynamical system
\begin{equation}
\dot{u}(t) = F(u(t))\,.
\end{equation}
If solutions to (1) are unique for every initial condition, the semigroup property of the flow is naturally obtained, i.e. $$u(t + s, a) = u(t, u(s,a))\,.$$
In the cases when uniqueness is not given or is not yet proven, one can define a set-valued map $S(a)$ with all the possible solutions of (1) with the same initial condition $a$.
Together with Lev Kapitanski, we introduced in 2017 a general result in the selection of measurable semiflows motivated by the results of N.V.~Krylov [1], and D.~Stroock and S.R.S.~Varadhan\
[2].
I will present some applications of the selection of measurable semiflows to PDEs in the context of fluid dynamics, dispersive equations and the heat flow of harmonic maps.
[1] N.V. Krylov, On the selection of a Markov process from a system of processes and the construction of quasi-diffusion processes, \emph{Izv. Akad. Nauk SSSR Ser. Mat.} \textbf{37} (1973) \
No. 3, pp. 691--708.
[2] D.W. Stroock and S.R.S. Varadhan, \textit{Multidimensional Diffusion Processes}, Springer Verlag, Berlin, 1979. |
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