Ibrahim, Slim

EPrint Series of Department of Mathematics, Hokkaido University

ABOUT THE LOCAL AND GLOBAL WELLPOSEDENSS OF THE
NAVIER-STOKES-MAXWELL COUPLED SYSTEM
SLIM IBRAHIM
Department of Mathematics and Statistics, University of Victoria
PO Box 3060 STN CSC, Victoria, BC, Canada, V8W 3R4
1. Introduction
This is the summary of joint works with S. Keraani (University of Lille 1, France) [8], and
more recently with N. Masmoudi (Courant Institute, NYU, USA) [9].
We investigate the wellposedness of solutions of a full Magneto-Hydro-Dynamic system
(MHD) in the space dimension two and three. The full MHD system is a coupling of a forced
Navier-Stokes equations with Maxwell equations. It reads as follows
(1.1)

∂tv + v · ∇v − ν∆v +∇p = j ×B
∂tE −∇×B = −j
∂tB +∇× E = 0
divv = divB = 0
σ(E + v ×B) = j
with the initial data
v|t=0 = v0, B|t=0 = B0, E|t=0 = E0.
Here, v,E,B : R+t × Rdx −→ R3 are vector fields defined on Rd (d = 2 or 3). The vector
field v = (v1, ..., vd) represents the velocity of the fluid, and the positive constants ν and σ are
its viscosity and resistivity, respectively. The scalar function p stands for the pressure. The
vector fields E and B are the electric and magnetic fields of the fluid, respectively. The last
equation in the system expresses Ohm’s law for the electric current j. The force term j ×B
in the Navier-Stokes equations comes from Lorentz force under a quasi-neutrality assumption
of the net charge carried by the fluid. Note that the pressure p can be recovered from v and
j × B via an explicit Calde´ron-Zygmund type operator (for example, see [4]). The second
equation in (1.1) is the Ampe`re-Maxwell equation for an electric field E. The third equation
is nothing but Farady’s law. For a detailed introduction to the MHD, we refer to Davidson
[6] and Biskamp [1].
Our main goal is to solve the system of equation (1.1). Before going any further, we first
recall a few fundamental known results for the standard Navier-Stokes equations.
The incompressible Navier-Stokes equations are
∂v
∂t
+ v · ∇v − ν∆v +∇p = 0, ∇ · v = 0.(1.2)
From the one hand, multiplying (1.2) by v and integrating in space formally gives the energy
identity
1
2
d
dt
‖v‖2L2 + ν‖∇v‖2L2 = 0,(1.3)
which shows that the viscosity dissipates the energy. Given an L2 initial data, J. Leray [11]
constructed a global weak solution v ∈ L∞((0,∞), L2(Rd)) ∩ L2((0,∞), H˙1(Rd)) satisfying
the energy inequality.
1
2 SLIM IBRAHIM
On the other hand, applying Leray’s projection P to (1.2) the pressure disappears and the
solution of (1.2) can be written in the integral form (mild solution)
v(t) = eνt∆v0 −
∫ t
0
eν(t−t
′)∆P∇(v ⊗ v)(t′) dt′.
With H˙
d
2
−1 initial data, Fujita and Kato [7] constructed a unique (in Ct(H˙ d2−1) ∩ L2t (H˙
d
2 ))
local mild solution which is global if the data is small. Moreover, u ∈ L4t (H˙
d−1
2 ). When
the space dimension is two, the two above solutions coincide and therefore we have both
uniqueness and regularity of the solution. However, when the space dimension is three, the
questions of the uniqueness of Leray’s solutions and the global regularity of mild solutions
remain outstanding open problems in contemporary Mathematics.
Similarly, for the full MHD system (1.1), one can formally get the following energy identity
1
2
d
dt
[‖v‖2L2 + ‖B‖2L2 + ‖E‖2L2]+ ‖j‖2L2 + ν‖∇v‖2L2 = 0.
showing that both the viscosity and the resistivity effects dissipate the energy. Therefore,
one may wonder to extend Leray’s result of the existence of global weak solutions to (1.1).
Unfortunately, a such result remain an interesting open problem in both space dimension two
and three. Indeed, one cannot have compactness for the term E × B in the Lorentz force.
Recently, Masmoudi [12] constructed, in dimension two, a unique global strong solutions to
(1.1) starting from initial data (v0, E0, B0) ∈ L2(R2) × (Hs(R2))2 with s > 0. This extra
regularity on the electromagnetic field was needed to get an L1t (L
∞
x ) bound of the velocity
field through a standard bi-dimensional logarithmic estimate. One of our main goals in this
work is to reduce as much as possible the regularity required on the electromagnetic field.
Notice that in dimension two, if the electromagnetic field is just an L2 function, then the
term E ×B in the Lorentz force is just integrable, and therefore one cannot gain regularity
using the parabolic regularization of the Stokes operator. In addition, to define (in the
distributional sense) the trilinear term (v × B) × B, the vector field should (heuristically)
be a bounded function. Our first result then requires less regularity on the electromagnetic
field than Masmoudi’s one. The regularity we impose is the minimal that enables us to have
parabolic regularization. However, our condition on the velocity vector field is a little bit
more restrictive in order for us to define the trilinear term. To be more precise, let us first
set the functional spaces we work in.
Definition 1.1. • For s ∈ R, define the space H˙slog as the set of tempered distributions
that satisfy
‖ψ‖2
H˙slog
:=
∑
q≤0
22qs‖∆qψ‖2L2 +
∑
q>0
q22qs‖∆qψ‖2L2 <∞.
Here, ∆q stands for the dyadic localization operator in the frequency space.
• Recall the Besov space B˙sp,q defined by
‖u‖q
B˙sp,q
= Σj∈Z2qjs‖∆ju‖qLp
• For every r ∈ [1,+∞] the space L˜rTL2log is endowed with the norm
‖φ‖2
L˜rTL
2
log
:=
∑
q≤0
‖∆qφ‖2LrTL2 +
∑
q>0
q‖∆qφ‖2LrTL2 .
Our main Theorem is
Theorem 1.2. Let d = 2, and set
X2 := B˙02,1(R2)× L2log(R2)× L2log(R2).
ON THE LOCAL AND GLOBAL WELLPOSEDENSS OF THE NAVIER-STOKES-MAXWELL SYSTEM 3
There exists a small constant δ > 0 such that to any initial data (v0, E0, B0) ∈ X2 satisfying
‖(v0, E0, B0)‖X2 ≤ δ,
corresponds a unique global solution (v,E,B) of (1.1)
v ∈ C(R+; B˙02,1) ∩ L˜2(R+; B˙12,1), E, B ∈ L˜∞(R+;L2log).
The proof of the above result goes by compactness. It basically relies on two major steps.
The first one is a microlocal refinement of the parabolic regularization given by the following
Lemma
Lemma 1.3. Let u be a smooth divergence free vectors fields solving
∂tu−∆u+∇p = F ×B, u|t=0 = 0,
on some time interval [0, T ]. Then,
‖u‖L˜2T B˙12,1 . ‖F‖L2TL2log
(
‖B‖L˜∞T L2log + ‖B‖L2TL2H
)
.(1.4)
The norm in L2H is given by
‖ψ‖2L2H :=
∑
q≤0
22q‖∆qψ‖2L2 +
∑
q≥0
‖∆qψ‖2L2 <∞.
To use this Lemma, one has to prove a decay (in time) of the magnetic field. This is given
by the following result
Lemma 1.4. Let (E,B) be a smooth solution of
∂tE − curl B + E = f × g,
∂tB + curl E = 0
on some interval [0, T ]. Then, we have
‖E‖L˜∞T L2log + ‖E‖L2TL2log + ‖B‖L˜∞T L2log . ‖(E0, B0)‖L2log + ‖f × g‖L2TL2log ,
and
(1.5)
‖B‖L2TL2H . ‖(E0, B0)‖L2 + ‖g‖L∞T L2‖f‖L˜2T B˙12,1 +
(
‖∇f‖L2TL2 + ‖f‖L∞T L2
)
‖g‖L2TL2H .
Observe that the factor in front of ‖g‖L2TL2H in (1.5) can be made arbitrary small when the
data is small. Thus, the term in (1.5) containing ‖g‖L2TL2H can be absorbed in the left hand
side giving an a` priori bound on ‖g‖L2TL2H .
The proof of this lemma uses the representation of solutions of the damped wave equation
satisfied by B. For high frequencies, such an equation behaves like the damped wave and
therefore solutions decay exponentially in time. However, for low frequencies, it behaves like
the heat equation, and the decay rate is rather weak.
In the three dimensional case, we construct mild solution using a fixed point argument
based on the following nonlinear estimates
Lemma 1.5. There exists a constant C > 0 such that
‖FB‖
L˜2T B˙
− 12
2,1
≤ C‖F‖
L2H˙
1
2
‖B‖
L˜∞T H˙
1
2
,
‖vB‖
L2T H˙
1
2
≤ C‖v‖
L˜2T B˙
3
2
2,1
‖B‖
L˜∞T H˙
1
2
,
‖uv‖
L1T B˙
3
2
2,1
≤ C‖u‖
L˜2T B˙
3
2
2,1
‖v‖
L˜2T B˙
3
2
2,1
,
for all smooth functions F,B, v and u defined on some interval [0, T ].
4 SLIM IBRAHIM
With a more careful study of the full MHD system, we recently obtained with N. Masmoudi
[9] the following result about the local and global wellposedness of (1.1).
Theorem 1.6. Let d = 2, 3, and set
Xd := H˙
d
2
−1 × H˙
d
2
−1
log (R
d)× H˙
d
2
−1
log (R
d).
For any Γ0 := (v0, E0, B0) ∈ Xd, there exists T > 0 and a unique mild solution Γ of (1.1)
with initial data Γ0 and
v ∈ C((0, T ); H˙ d2−1) ∩ L˜2((0, T ); H˙ d2 ) + L˜2((0, T ); B˙
d
2
2,1), E, B ∈ L˜∞(R+; H˙
d
2
−1
log ).
Moreover, the solution is global (i.e. T =∞) if the initial data is sufficiently small.
References
[1] D. Biskamp. Nonlinear magnetohydrodynamics, volume 1 of Cambridge Monographs on Plasma Physics.
Cambridge University Press, Cambridge, 1993.
[2] J.-M. Bony. Calcul symbolique et propagation des singularite´s pour les e´quations aux de´rive´es partielles
non line´aires. Ann. Sci. E´cole Norm. Sup. (4), 14(2):209–246, 1981.
[3] M. Cannone, Q. Chen, and C. Miao. A losing estimate for the ideal MHD equations with application to
blow-up criterion. SIAM J. Math. Anal., 38(6):1847–1859 (electronic), 2007.
[4] J. -Y. Chemin, Perfect incompressible Fluids, Clarendon press, Oxford, 1998.
[5] J.-Y. Chemin and N. Lerner. Flot de champs de vecteurs non lipschitziens et e´quations de Navier-Stokes.
J. Differential Equations, 121(2):314–328, 1995.
[6] P. A. Davidson. An introduction to magnetohydrodynamics. Cambridge Texts in Applied Mathematics.
Cambridge University Press, Cambridge, 2001.
[7] H. Fujita and T. Kato. On the Navier-Stokes initial value problem I, Archiv for Rationnal Mechanic
Analysis, 16, (1964), pages 269315.
[8] S. Ibrahim and S. Keraani Global small solutions of the Navier-Stokes Maxwell system, to appear in
SIAM Journal on Math. Analysis.
[9] S. Ibrahim and N. Masmoudi On the wellposedness of the Navier-Stokes Maxwell system, preprint SIAM
Journal on Math. Analysis.
[10] P. G. Lemarie´-Rieusset. Recent developments in the Navier-stokes problem. CRC Press, (2002).
[11] J. Leray. Essai sur le mouvement dun liquide visqueux emplissant lespace, Acta Mathematica, 63, (1933),
pages 193248.
[12] N. Masmoudi. Global well posedness for the Maxwell-Navier-Stokes system in 2D. Journal de Math.
Pures et Appl. 93 (2010) 559571.
E-mail address: S. Ibrahim: ibrahim@math.uvic.ca
URL: http://www.math.uvic.ca/~ibrahim/


About the local and global wellposedenss of the Navier-Stokes-Maxwell coupled system

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EPrint Series of Department of Mathematics, Hokkaido University