Nonlinear Analysis 103 (2014) 87–97
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Nonlinear Analysis
journal homepage: www.elsevier.com/locate/na
On the exponential type explosion of
Navier–Stokes equations
Jamel Benameur
∗
Department of Mathematics, College of Sciences, King Saud University, Riyadh 11451, Saudi Arabia
article info
Article history:
Received 2 May 2013
Accepted 18 March 2014
Communicated by Enzo Mitidieri
MSC:
35-XX
35Qxx
35Q30
35D35
Keywords:
Incompressible fluids
Navier–Stokes equations
Regularity of generalized solutions
Sobolev spaces
Blow-up criterion
abstract
The classical results on the explosion of the maximal solution of incompressible
Navier–Stokes equations are of type c (T
∗
− t )
−σ
0
for some σ
0
> 0. Inspired by the works
Benameur and Selmi (2012) [15], Chemin (2004) [16], we use the Sobolev–Gevrey spaces to
get better explosion results, precisely if e
a|D|
1/σ
u
0
∈ H
s
(R
3
), then |e
a|D|
1/σ
u(t )|
H
s is at least
of the order (T
∗
− t )
−σ
1
exp(c (T
∗
− t )
−σ
2
) for some σ
1
> 0 and σ
2
> 0. Fourier analysis
and standard techniques are used.
© 2014 Elsevier Ltd. All rights reserved.
1. Introduction
The incompressible Navier–Stokes system in Cartesian coordinates is given by:
∂
t
u − ν u + (u ·∇)u = −∇p, in R
+
× R
3
,
div u = 0 in R
+
× R
3
,
u(0) = u
0
in R
3
,
(NS)
where ν> 0 is the viscosity of the fluid, u = u(t , x) = (u
1
, u
2
, u
3
) and p = p(t , x) denote respectively, the unknown
velocity and the unknown pressure of the fluid at the point (t , x) ∈ R
+
× R
3
,(u ·∇u) := u
1
∂
1
u + u
2
∂
2
u + u
3
∂
3
u, and
u
0
= (u
0
1
(x), u
0
2
(x), u
0
3
(x)) is a given initial velocity. If u
0
is quite regular, the pressure p is determined. The system (NS)
has the scaling property: If u(t , x) is a solution of the initial data u
0
(x), then for any λ> 0,λu(λ
2
t ,λx) is a solution of
(NS) with the initial data λu
0
(λx). Our problem is the type of the blow-up criterion of the solution if the maximal time
T
∗
is finite. Precisely, the question posed by K. Ammari [1]: Is the type of explosion due to the chosen space or to the
non-linear part of the Navier–Stokes equations? In the literature, there are several authors who have studied the problem
of explosion of a non-global solution of (NS) system, and all the obtained results do not exceed C (T
∗
− t )
−σ
for some σ ≥ 0,
precisely: if u is a maximal solution of (NS) in C([0, T
∗
), B) with T
∗
< ∞, then C (T
∗
− t )
−σ
≤∥u(t )∥
B
. The classical
result due to J. Leray [2,3]: if u is a non regular solution of the system (NS) and T
∗
is the first time where u is not regular,
∗
Tel.: +966 533087448; fax: +966 14676512.
E-mail addresses: jamelbenamer@yahoo.fr, jbenameur@ksu.edu.sa.
http://dx.doi.org/10.1016/j.na.2014.03.011
0362-546X/© 2014 Elsevier Ltd. All rights reserved.