Nonlinear instability of the solutions of the Navier-Stokes equations: Formulas for constructing exact solutions

Several classes of exact solutions of the two-dimensional and three-dimensional nonstationary Navier-Stokes equations are considered. Useful formulas are given that make it possible to construct exact solutions of one of the determining equations using the solutions of other equations. The problems of the nonlinear stability (instability) of the derived solutions are studied. It is found that the characteristic feature of many solutions of the Navier-Stokes equations is instability. To prove the instability of the solutions, a new exact method (that does not use any assumptions and approximations) is applied, which can be useful for analyzing other nonlinear physical models and phenomena. It is shown that instability can occur not only at sufficiently large Reynolds numbers, but also at arbitrarily small Reynolds numbers (and can be independent of the profile of the velocity of a fluid).