On the existence of a generalized solution of the conjugation problem for the Navier–Stokes system

By the conjugation problem we mean the problem of finding a solution $u(t)$ of a non-stationary Navier–Stokes system in some domain on some time interval $[0,T]$ such that the solution must satisfy certain conditions. Namely, the value of the solution at the boundary is equal to zero, and a conjugation condition is given which consists in the requirement that the initial value of the solution be connected with its values on the entire time interval by some linear operator defined on solutions: $u(0)=\nobreak U[u]$. In the special case where the values of the solution at the ends of the time interval coincide, we obtain the problem about a periodic solution.
For the conjugation problem we establish the existence of a generalized solution in the case of bounded and unbounded domains of arbitrary dimension under various assumptions about the conjugation operator. A peculiarity of unbounded domains is the fact that a solution of the conjugation problem may not be quadratically integrable.
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