J. Math. Fluid Mech. 20 (2018), 801–818
2017 Springer International Publishing AG
Journal of Mathematical
Global Mild Solutions for a Nonautonomous 2D Navier–Stokes Equations with Impulses
at Variable Times
E. M. Bonotto, J. G. Mesquita and R. P. Silva
Communicated by G.P. Galdi
Abstract. The present paper deals with existence and uniqueness of global mild solutions for a new model of Navier–
Stokes equations on R
subjected to impulse eﬀects at variable times. By using the framework of impulsive/nonautonomous
dynamical systems we are able to consider impulse eﬀects in the system as well relax conditions on the external forcing
term which is, in our case, non-linear and explicitly time-dependent, extending previous results on the specialized literature.
Moreover, we also introduce suﬃcient conditions on the structure of the impulse set which ensure dissipativity for the
system, i.e., uniform boundedness of global solutions starting in bounded sets, which is an indicative to the existence of
objects as attractors.
The Navier–Stokes equations, NSEs for short, are a system of evolution partial diﬀerential equations de-
rived from Newton’s laws of motion for a continuous distribution of matter in the ﬂuid state characterized
by an inability to support shear stresses . These equations allow us to determine the velocity ﬁeld as
well the inner pressure of ﬂuids conﬁned on regions of the Euclidean space. They are used as a model
to describe plenty of diﬀerent physics phenomena as water ﬂow, ocean currents, sound propagation in
viscous medium, circulation of nervous impulses throughout the nervous system, among many others.
Currently the NSEs are of fundamental importance in both: theoretical and applied point of view. They
were the cornerstone of the development of some relevant aspects of mathematical analysis and nonlinear
diﬀerential equations, and became crucial in ﬁelds as petroleum industry, plasma physics, meteorology,
thermo-hydraulics, among others, see [1,7,13,16–18,22,24,26,27] and the references therein for a rele-
vant list of applicability. Due to this fact, these equations have been attracted the attention of several
important scientists since the middle of the nineteenth century.
On the other hand, it is well known that many relevant phenomena, including some from ﬂuid dy-
namics, have their behavior drastically modiﬁed somehow after an instantaneous change on their state,
which may introduce in the model several discontinuities. Properties as velocity, density and viscosity are
discontinuous at interfaces between diﬀerent ﬂuids as presented in . Despite of the extensive literature
on NSEs and the recents progress on the impulsive dynamical systems, surprisedly models from ﬂuid
dynamics incorporating impulse eﬀects on its structure are somewhat scarce. A model of NSEs incor-
porating impulses makes sense physically and allows to describe more precisely some of the phenomena
modeled by these equations, mainly when there is an instantaneous change of conditions caused by in-
trinsic inner/outer factors of the system. Motivated by this fact, we investigate the global well posedness,
in the sense of Hadamard, as well the large time behavior of mild solutions of the nonautonomous 2D
E. M. Bonotto: Supported partially by CNPq Grant 307317/2013-7 and FAPESP Grants 2012/16709-6 and 2016/24711-1.
J. G. Mesquita: Supported by FAPESP Grant 2012/08473-2.
R.P. Silva: Supported partially by FAPESP #2014/16165-1, CNPq #440371/2014-7 and FEMAT-Funda¸c˜ao de Es-
tudos em Ciˆencias Matem´aticas #038/2016.