In this paper, we are concerned with the global existence and blowup of smooth
solutions of the 3-D irrotational compressible Euler equation with time-dependent
damping
where
, the
frictional coefficient
with
and
,
is a
constant,
,
,
,
, and
is sufficiently small. For
, we show that there
exists a global
-smooth
solution
by introducing and establishing some uniform time-weighted energy estimates of
, while for
, in general, the smooth
solution
blows up in
finite time. Therefore,
appears to be the critical value for the global existence of small amplitude smooth
solution
.
Keywords
compressible Euler equations, damping, time-weighted energy
inequality, Klainerman–Sobolev inequality, blowup,
hypergeometric function
School of Mathematical
Sciences
Jiangsu Provincial Key Laboratory for Numerical Simulation of
Large Scale Complex Systems
Nanjing Normal University
Nanjing
China