Аннотация:

We address the system of Navier--Stokes equations that models a homogeneous viscous incompressible fluid in the presence of a short-term intense pulse starting at an initial moment in time. The existence of a weak Leray--Hopf solution to the initial-boundary value problem for this system in the case when the pulse duration is fixed is guaranteed by the well-known theory. On a rigorous mathematical level, we carry out the limiting transition in the problem as the pulse duration tends to zero, while the cumulative impact of the pulse remains constant. As a result, we prove that the family of weak Leray--Hopf solutions to the problem under consideration has a subsequence converging to a weak solution of the initial-boundary value problem for the system of classical Navier--Stokes equations supplemented by the ``corrected'' initial velocity field inheriting complete information about the profile and cumulative impact of the original pulse. The ``corrected'' initial velocity field is found out as the solution of an additional limit system of equations of inviscid fluid derived at the microscopic (``fast'') timescale, which is the characteristic timescale of the pulse duration. At the end of the article, we identify two particular cases in which this system can be solved explicitly, which leads to an explicit algebraic expression of the ``corrected'' initial velocity.