Table Of ContentTime-periodic forcing and asymptotic stability for
the Navier-Stokes-Maxwell equations
Slim Ibrahim∗, Nader Masmoudi†, and Pierre Gilles Lemari´e–Rieusset‡
6
1
0
2
n Abstract
a
J Forthe3DNavier-Stokes-Maxwellproblemonthewholespaceand
5 inthepresenceofexternaltime-periodicforces,firstwestudytheexis-
2 tence oftime-periodicsmallsolutions,andthenwe provetheirasymp-
toticstability. Weusenewtypeofspacestoaccountforaverageddecay
]
P in time.
A
. Keywords: Navier–Stokes equations, periodic solutions, Maxwell’s
h
system,dyadicdecomposition,maximalregularity,nonlinearestimates
t
a 2010MathematicsSubjectClassification: 35B10,35Q30,76D05
m
[
Introduction.
1
v
8 Physical and analytical models already exist for both electro-hydrodynamic
5 and magneto-hydrodynamic. However quite often in actual situations, both
4
combined electro and magneto-hydrodynamic effects occur. Recent works
6
0 attempted to develop fully consistent, multi-dimensional, unsteady and in-
. compressible flows of electrically conducting fluids under the simultaneous
1
0 or separate influence of externally applied or internally generated electric
6
and magnetic fields. The approach is based on the use of fundamental laws
1
: of continuum mechanics and thermodynamics. See for example [12]. How-
v
ever, because of the considerable complexity of even simpler versions of the
i
X
combined electro-magneto-hydrodynamic models, it is still hard to analyze,
r even numerically, the combined influence of electric and magnetic fields and
a
the fluid flow. In this paper, we analyze an adiabatic situation where ther-
modynamical effects are neglected, and nonlinearities are reduced to the
only action of Lorentz force.
More specifically, consider a three-dimensional incompressible, viscous and
∗Department of Mathematics and Statistics, University of Victoria; e-mail:
[email protected]
†Courant Institute,New-York University;e-mail: [email protected]
‡Laboratoire de Math´ematiques et Mod´elisation d’E´vry, UMR CNRS 9071, UEVE,
ENSIIE;e-mail: [email protected]
1
charged fluid with a velocity field ~u. Fluid charged particles motion gen-
erates an electro-magnetic field (E~,B~) satisfying Maxwell equations, and a
current J~ that acts backs on the fluid through Lorentz force. We assume
that the current is given by Ohm’s law J = σ(E~ + ~u B~). Thus, the
∧
Navier-Stokes-Maxwell system we study reads as
∂ ~u+ div (~u ~u)= ν∆~u+J~ B~ +F~ ~p
t per
⊗ ∧ −∇
∂tE~ ~ B~ = J~+G~per
−∇∧ −
∂ B~ + ~ E~ = H~ (0.1)
t ∇∧ per
div ~u= div B~ = 0
J~= σ(E~ +~u B~).
∧
Here, ~u, E~, B~ : R+ R3 R3 are vector fields defined on R3, and the
t × x −→
scalar function p stands for the pressure. The positive parameters ν and σ
represent the viscosity of the fluid and the electric resistivity, respectively.
The self-interaction force term J~ B~ in the Navier-Stokes equations comes
×
fromtheLorentzforceunderaquasi-neutrality assumptionofthenetcharge
carried by the fluid. Notice that taking into account a moving reference
frame of the fluid, yields thecorrection u B to the classical Ohm’s law and
×
keeps Faraday’s law invariant under Gallilean transformation. The second
equation in (0.1) is the Amp`ere-Maxwell equation for the electric field E~.
Thethirdequation isnothingbutFaraday’s law. Foradetailed introduction
to similar models and the theory of MHD, we refer to Davidson [9] and
Biskamp [3]. In (0.1), the external forces F~ , G~ and H~ are taken
per per per
time-periodic: for a fixed time period T > 0, we have
F~ (t+T,x)= F~ (t,x), G~ (t+T,x)= G~ (t,x), H~ (t+T,x) = H~ (t,x).
per per per per per per
Before going any further, let us emphasize that despite the possible non-
physical full relevance of (0.1), the system still captures the various mathe-
matical challenges of the full complicated original set of equations. Indeed,
(0.1)isacouplingofadissipativeequation ofparabolictype(Navier-Stokes)
with a hyperbolic system (Maxwell’s equations). Despite the presence of
damping in Ohm’s law, solutions to Maxwell’s equations do not enjoy any
smoothing effect, due to the hyperbolic nature of the equations. Moreover,
as this will be seen, a few challenges also arise when dealing with differ-
ent decay rates, for different linear parts, and also different frequency sizes
caused by the coupling.
Note that the pressure p can still be determined using Leray projection
from ~u and J~ B~ via an explicit Cald´eron-Zygmund operator (see [6] for
∧
instance):
1
~p = ~ div (J~ B~ +F~ div (~u ~u)).
per
∇ ∇∆ ∧ − ⊗
2
In all what follows, we denote the solution to (0.1) by
~Γ := (U~,E~,B~).
When no exterior forces act on the system (0.1), the initial value problem
reads
∂ ~u+ div (~u ~u)= ∆~u+J~ B~ ~p
t
⊗ ∧ −∇
∂tE~ ~ B~ = J~
−∇∧ −
∂tB~ +∇~ ∧E~ = 0 (0.2)
div ~u= div B~ = 0
J~= E~ +~u B~
∧
~u(0,.) =~u0, E~(0,.) = E~0, B~(0,.) = B~0.
Solutions to(0.2) formally enjoy the energy balance
1 d
~u 2 + B~ 2 + E~ 2 + J~ 2 + ~u 2 = 0,
2dt k kL2 k kL2 k kL2 k kL2 k∇ kL2
(cid:2) (cid:3)
which suggests that weak solutions would exist in the space
~u L∞(0, ;L2) L2(0, ;H˙1); E~, B~ L∞(0, ;L2) J~ L2(0, ;L2).
∈ ∞ ∩ ∞ ∈ ∞ ∈ ∞
Unfortunately, weak solutions are not known to exist even in two space di-
mension. In [18], Masmoudi proved the existence of global strong solutions.
Later on, Ibrahim and Keraani [15] relaxed the regularity condition on the
initial data to construct global small solutions `a la Kato. This result was
recently improved by Germain, Ibrahim and Masmoudi [11] by taking small
initial data ~u , E~ and B~ in H˙1/2, and construct a solution (~u,E~,B~) such
0 0 0
that~u L∞H˙1/2 L2H˙3/2 L2L∞,E~ L∞H˙1/2 L2H˙1/2 andB~ L∞H˙1/2.
∈ ∩ ∩ ∈ ∩ ∈
This paper is organized as follows. In section one, we introduce some
useful notation and state our results: A first Theorem about the existence
of time-periodic solutions in spaces of Sobolev type with an extra spatial
regularity. Then, we relax the hypothesis of the first Theorem and extend
it to critical Besov type spaces. A such an extension seems to us necessary
in order to prove the last result about the stability of the periodic-in time
solutions. Section two is devoted to the proof of the two existence results,
and we only give the full details in the case of Sobolev. In section three,
we start by examining a maximal regularity result adapted to the spaces
that incorporates averaged decay in time. Then, we show the decay of the
electromagnetic field where we use the full spectral properties of the weakly
damped Maxwell’s equations. Then, we list all the nonlinear estimates that
appear in the study of the nonlinear stability, and we only prove the worst
two of them when two factors have no decay in time. The manuscript is
3
then finished with an Appendix summarizing the main spectral properties
of the weakly damped Maxwell’s equations.
Acknowledgements
Slim Ibrahim is partially supported by NSERC Discovery grant # 371637-
2014. Nader Masmoudi is in part supported by NSF grant DMS-1211806#
Slim Ibrahim would like to thank the University of E´vry and the University
of Paris Diderot-Paris 7 for their hospitality to accomplish a part of this
work.
1 Notation
The well-known Littlewood-Paley decomposition and the corresponding fre-
quency cut-off operators will be of frequent use in this paper. We briefly
recall it to define the functional spaces we need.
There exists a radial positive function ϕ (Rd 0 ) such that
∈D \{ }
ϕ(2−qξ) = 1 ξ Rd 0 ,
∀ ∈ \{ }
q∈Z
X
Supp ϕ(2−q ) Supp ϕ(2−j ) = , q j 2.
· ∩ · ∅ ∀| − | ≥
For every q Z and v ′(Rd) we set
∈ ∈ S
q−1
∆ v = −1 ϕ(2−qξ)vˆ(ξ) and S = ∆ .
q q j
F
j=−∞
(cid:2) (cid:3) X
Bony’s decomposition [4] consists in splitting the product uv into three
parts1:
uv = T v+T u+R(u,v),
u v
with
1
T v = S u∆ v, R(u,v) = ∆ u∆˜ v and ∆˜ = ∆ .
u q−1 q q q q q+i
q q i=−1
X X X
For (p,r) [1,+ ]2 ands Rwedefinethehomogeneous Besov space B˙s
∈ ∞ ∈ p,r
as the set of u ′(Rd) such that u= ∆ u and
∈ S q q
P
u = (2qs ∆ u ) < .
k kB˙ps,r k q kLp q∈Z ℓr(Z) ∞
(cid:13) (cid:13)
(cid:13) (cid:13)
1Itshouldbesaidthatthisdec(cid:13)ompositionistruein(cid:13)theclassofdistributionsforwhich
P ∆ =I. For example, polynomial functions do not belong to thisclass.
q∈Z q
4
In the case p = r = 2, the space B˙s turns out to be the classical homoge-
2,2
neous Sobolev space H˙s.
In order to prove the our stability result, we need to build spaces that take
into account the different decay rates coming from the coupling of the two
types of PDEs (Navier-Stokes, and Maxwell), and also the weak decay of
the low frequencies in the Maxwell’s equations. In addition, and in order to
estimate the nonlinear terms, we need to introduce spaces that capture an
average decay in time, and not just pointwise decay. This will be crucial in
our analysis. We define the spaces that distinguish between the high and
low frequencies of a function
Definition 1.1. Let∆ denote thedyadic frequencylocalization operator de-
q
fined in section 1. We define a space that distinguishes between the high and
low frequencies of a function as follows. For s ,s R and 1 p,q ,q
1 2 1 2
define the space ˙s1,s2 by its norm ∈ ≤ ≤ ∞
p,q
B
kφkB˙(s1,s2) := 2kqs1k∆kφkqL1p q11 + 2kqs2k∆kφkqL2p q12.
p,(q1,q2) k≤0 k>0
(cid:0)X (cid:1) (cid:0)X (cid:1)
We will also use the short-hands
and B˙s := ˙(s,s) , H˙s = ˙(s,s) , and H˙s,t := ˙(s,t) .
p,(q1,q2) Bp,(q1,q2) B2,(2,2) B2,(2,2)
Finally, define the space-time functional space L˜r ˙(s1,s2) by its norm
TBp,(q1,q2)
kφkL˜rTB˙p(s,(1q,1s2,q)2) := k≤02q1s1kk∆kφkqL1rTLp q11 + k>02q2ks2k∆kφkqL2rTLp q12,
(cid:0)X (cid:1) (cid:0)X (cid:1)
with the trivial extension when r = . We also define the new spaces that
∞
take into account an averaged decay in time. Precisely, we denote
nsu∈˜pN (n+1)1−2εkukL2(n,n+1);B˙p(s,(1q,1s2,q)2) : = k≤02q1s1knsu∈pN(n+1)(1−2ε)q1k∆kukqL12(n,n+1;Lp) q11
(cid:0)X (cid:1)
+ 2q2s2knsu∈pN(n+1)(1−2ε)q2k∆kukqL22(n,n+1;Lp) q12.
k≤0
(cid:0)X (cid:1)
with the obvious generalizations in the cases q = , or L˜rH˙s etc..
j ∞ T
The space ˙(s1,s2) is nothing but the usual Besov space ˙s2 for high
Bp,(q1,q2) Bp,q2
frequencies while it behaves like ˙s1 for low frequencies. If s > s , it is
Bp,q1 1 2
not difficult to see that ˙(s1,s2) = ˙s1 + ˙s2 . The L˜ type spaces were
Bp,(q1,q2) Bp,q1 Bp,q2
first used by Chemin and Lerner [8].
In the sequel, consider a parameter 0 < ε < 1, introduce the spaces ,
1
X
, , and their “dual” counterparts and by defining their norms.
2 3 1 2
X X Y Y
1−ε 1−ε
kukX1 := stu>˜p0 (t+1) 2 ku(t)kB˙2(,32(∞−ε,1,)12) +nsu∈˜pN (n+1) 2 kukL2(n,n+1;B˙232,(∞,1))
5
1−ε 1−ε
kEkX2 := st>u˜p0 (t+1) 2 kE(t)kH21 ∼ nsu∈˜pN (n+1) 2 kEkL∞(n,n+1;H12)
1−ε 1−ε
kBkX3 := st>u˜p0 (t+1) 2 kB(t)kH˙1,21 ∼ nsu∈˜pN (n+1) 2 kBkL∞(n,n+1;H˙1,12),
and
1−ε
kFkY1 := nsu∈˜pN (n+1) 2 kFkL2(n,n+1;B˙2(−,(∞21−,1ε),−21)),
1−ε
kGkY2 := nsu∈˜pN (n+1) 2 kGkL2(n,n+1;H12).
1
Finally, let := ( L˜∞( ˙2 ) .
X X1∩ B2,(∞,1)) ×X2×X3
1.1 Results
In our first result and under a smallness assumption on the forces, we con-
struct(inSobolevspaceswithanextraδ regularity)atime-periodicsolution
~Γ to (0.1). More precisely, we have
per
Theorem 1.1.
Let 0 < δ < 1. Then there exists a positive constant ǫ such that : if the
T,δ
time-periodic forces F~ , G~ and H~ satisfy the following assumptions :
per per per
1. F~per belongs to L2perH˙−21 ∩L2perH˙−21+δ and
T T
F~ (t,.) 2 dt+ F~ (t,.) 2 dt < ǫ
sZ0 k per kH˙−21 sZ0 k per kH˙−21+δ T,δ
2. the mean value F~ = 1 T F~ (t,.)dt belongs to B˙−32 and
0 T 0 per 2,∞
R
F~ < ǫ
k 0kB˙−3/2 T,δ
2,∞
3. G~per belongs to L2perH21+δ and
T
G~ (t,.) 2 dt < ǫ
sZ0 k per kH21+δ T,δ
4. the mean value G~ = 1 T G~ (t,.)dt belongs to H˙−1 and
0 T 0 per
R G~ < ǫ ,
k 0kH˙−1 T,δ
6
5. H~per is divergence-free (div H~per = 0) and H~per belongs to L2perH21+δ
with
T
H~ (t,.) 2 dt < ǫ
sZ0 k per kH21+δ T,δ
6. the mean value H~ = 1 T H~ (t,.)dt belongs to H˙−2 and
0 T 0 per
R H~ < ǫ ,
k 0kH˙−2 T,δ
then the Navier–Stokes–Maxwell problem (0.1) has a time-periodic solution
(~u ,E~ ,B~ ) such that :
per per per
• ~uper belongs to L∞perB˙212,∞∩L2perH˙ 23+δ
• E~per and B~per belong to L∞perH21+δ.
We extend the above statement to solutions in critical spaces of Besov-
type. This will be crucial for the stability as we were not able to prove the
stability in the spaces given by Theorem 1.1. More precisely, we have
Theorem 1.2.
Let T > 0 denote the time period of three periodic forces F~ , G~ and
per per
H~ decomposed as follows into a fluctuating and zero mean parts:
per
F~ (t,x) := F~ (x) + F~ (t,x), G~ := G~ (x) + G~ (t,x) and H~ :=
per 0 f per 0 f per
H~ (x)+H~ (t,x) with
0 f
T T T
F~ dt = G~ dt = H~ dt = 0.
f f f
Z0 Z0 Z0
There exists ε > 0 such that under the following smallness assumptions
T
F~ + F ε
k perkL˜2(0,T;B˙−12 ) k 0kB˙−32 ≤ T
2,(∞,1) 2,(∞,1)
G~ + G~ ε
k perkL2(0,T;H21) k 0kH˙−1 ≤ T
and
H~ + H~ ε ,
k perkL2(0,T;H12) k 0kH˙−2 ≤ T
a unique mild solution ~Γ = (~u ,E~ ,B~ ) of (0.1) exists such that
per per per per
~uper ∈ L˜∞perB˙221,(∞,1)∩L˜2perB˙232,(∞,1) and E~per,B~per ∈ L˜∞perH21.
Another variant of the existence resultof time periodicsolutions is given
by the following theorem where we require a slightly better control of the
high frequencies of the solution ~Γ.
7
Remark 1.1. One can prove local existence if the low frequency part
• 1
of the initial data of the velocity field is in ˙2 and its high fre-
B2,(∞,1)
quency is in H˙ 21.
Compared to the results of [11] in the absence of forcing terms, the
•
statement of Theorem 1.1 requires a slightly better control of high fre-
quencies for ~u, E~ and B~, a better control of low frequencies of E~ and
B~, and a weaker control on the low frequencies of ~u.
Our proof also shows that the more regular is the forcing, the more
•
regular will be its corresponding periodic-in time solution. Indeed, if
for example the forcing is small in
F~per ∈ L2perH˙−12 ∩L2perH˙ 21, G~per ∈ L2perH23, H~per ∈ L2perH32,
then, we have
• ~uper belongs to L∞perB˙212,∞∩L2per(H˙ 23 ∩H˙ 25)
• E~per and B~per belong to L∞perH32.
Next,westudythestabilityofthetime-periodicsolutions: whathappens
when, at some time t , one takes a perturbationof thesolutions constructed
0
in above:
~Γ(t ) =~Γ (t )+~Γ ,
0 per 0 err
with ~Γ small in B˙1/2 H1/2 H1/2? Do we have a global solution
err 2,(∞,1) × ×
of (0.1) on [t ,+ ) and does the error go to 0 in suitable norms when t
0
∞
goes to + ? The main problem rises when we estimate the cross terms
∞
coming from the interactions between the periodic solution and the solution
we want to construct. The worst interaction is given by a term of the type
(U~ B~ ) B~, (1.1)
per per
∧ ∧
first because of the non-decay of U , and B , and second because we
per per
barley miss an L∞(L∞) estimate on U . To overcome such a problem,
per
we impose a strong condition on the low frequencies of the velocity field.
In doing so, we are obliged to allow an-ε loss in the time decay rate. It is
important to notice that because of this problem, we were not able to show
the stability of the the solutions given by Theorem 1.1. Hence, our second
main result is the following.
Theorem 1.3. Given three T-periodic forces F~ (t), G~ (t) and H~ (t)
per per per
satisfying the hypothesis of Theorem 1.2, and denote by ~Γ (t) the corre-
per
sponding small T-periodic solution. Consider an initial data ~Γ0 +~Γ (0)
err per
8
with ~Γ0err small in B˙221,(∞,1) ×H21 ×H12, there exists ~Γ¯ a global solution of
(0.1) with that initial data ~Γ0 +~Γ (0). Moreover, we have
err per
~Γ¯ ~Γ ,
per
− ∈X
so that ~Γ¯ converges asymptotically to ~Γ as t goes to infinity.
per
Ourproofof Theorem1.1relies onseveral linearandnonlinearestimates
(product rules in Besov and Sobolev spaces) and uses Fourier series expan-
sion of the time-periodic solution. Such an expansion was used in [13] for
the time-periodic forced Navier-Stokes. The proof of Theorem 1.3 then goes
through a fixed point argument in a suitable space. In order to have the
asymptotic convergence, the functional space has to include decay proper-
ties. Thus, we are required to exhibit the decay from the velocity and the
electro-magnetic fields. Both the dissipation coming from the viscosity of
thefluidandfromtheresistivity inOhm’slaw, enableustohavesomedecay
for the velocity ~u and the electric field E~. To qualitatively transfer such a
decay to the magnetic field is not as easy and clear as for ~u and E~. In [15],
and then [11], a weak decay of the magnetic field was proven in both space
dimension two and three. However, the decay was not used (in three space
dimension) to construct global small solution. In the contrary, here the use
of the decay is an essential fact to show asymptotic convergence.
2 Construction of time-periodic solutions
The purposeof this section is to prove Theorem 1.2 and Theorem 1.1. Since
the method is the same for both but details are much more involved in
critical spaces (the Besov case), and for the sake of simplicity, we opted to
givethefulldetailsoftheproofofTheorem1.1andonlysketchthenecessary
changes to complete the proof of Theorem 1.2. First, we introduce some
useful notation
Notation :
For 0 < δ < 1, we shall write
(~u,E~,B~) X if
• ∈
1. ~u belongs to L∞perB˙221,∞∩L2perH˙ 32+δ
2. E~ and B~ belong to L∞perH12+δ.
(F~,G~,H~) Y if
• ∈
1. F~ belongs to L2perH˙−12 ∩L2perH˙−21+δ
9
2. the mean value F~ = 1 T F~(t,.)dt belongs to B˙−23
0 T 0 2,∞
3. G~ belongs to L2perH12+δR
4. the mean value G~ = 1 T G~(t,.)dt belongs to H˙−1
0 T 0
5. H~ is divergence-free (divRH~ = 0) and H~ belongs to L2perH21+δ
6. the mean value H~ = 1 T H~(t,.)dt belongs to H˙−2.
0 T 0
R
2.1 Proof of theorem 1.1.
The problem is solved by a Picard iterative scheme: (~u ,E~ ,B~ ) will
per per per
be the limit of the time-periodic functions (U~ ,E~ ,B~ ) solving the system
n n n
∂ U~ ∆U~ = F~
t n+1 n+1 n
−
∂tE~n+1 ~ B~n+1+E~n+1 = G~n
−∇∧ (2.1)
∂ B~ + ~ E~ = H~
t n+1 n+1 n
∇∧
div ~u = div B~ = 0
n+1 n+1
with
U~ = 0, E~ = 0, B~ =0
0 0 0
F~ = P div (U~ U~ )+E~ B~ +(U~ B~ ) B~ +F~
n n n n n n n n per
− ⊗ ∧ ∧ ∧
G~n = (cid:16)U~n B~n+G~per (cid:17)
− ∧
H~ = H~
n per
(2.2)
where P is the Leray projection operator on solenoidal vector fields.
The first Lemma gives product rules in Sobolev spaces that close the
iterative scheme. More precisely, we have.
Lemma 2.1. If (~u,E~,B~) X and (F~,G~,H~) Y, define (F~ ,G~ ,H~ ) as
1 1 1
∈ ∈
F~ =P div (~u ~u)+E~ B~ +(~u B~) B~ +F~
1
− ⊗ ∧ ∧ ∧
G~ = ~u(cid:16) B~ +G~ (cid:17) (2.3)
1
− ∧
H~ =H~.
1
Then we have (F~ ,G~ ,H~ ) Y.
1 1 1
∈
Proof. Point-wise product maps B˙23,/12 × H21+δ to H21+δ. As L∞perB˙212,∞ ∩
L2perH˙ 23+δ ⊂ L2perB˙23,/12, we find that G~1 belongs to L2perH21+δ.
Moreover, pointwise product maps B˙1/2 L2 to H˙−1. Thus, we find
2,∞ ×
that ~u B~ belongs to L∞ H˙−1 and the mean value of G~ belongs to H˙−1.
∧ per 1
10
