Bounding Surface Integral Of Functions Dragged By Velocity Fields

Manuel García-Casado

Published 2017-07-28Version 1

To find bounded magnitudes is essential in dynamical systems when they evolve over time. Particularly, the problem of bounded kinetic energy for velocity fields has received increasing attention on this type of systems. Here it is reasoned how to tie down a positive function surface integral, dragged by velocity fields, when certain conditions are applied to the dynamical equation of that surface. This is possible thanks to an inequality equation that arises when surface transport theorem is applied to closed one, which is the boundary of certain volume. When the positive function that holds the inequality equation is found, the velocity field and it derivatives became bounded by constant magnitudes. As a consequence of this, the surface integral of the positive function is also bounded. The mean value theorem applied over this restrictions allows to bound the surface integral of the velocity field.