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Beyond the BKM’s criterion for the 2D resistive
magnetohydrodynamic equations
Léo Agélas
To cite this version:
Beyond the BKM’s criterion for the 2D resistive
magnetohydrodynamic equations
L´eo Ag´elas
Department of Mathematics, IFP Energies Nouvelles, 1-4, avenue de Bois-Pr´eau, F-92852 Rueil-Malmaison, France
Abstract
The question of whether the two-dimensional (2D) magnetohydrodynamic (MHD) equations with only magnetic diffusion can develop a finite time singularity from smooth initial data is a challenging open problem in fluid dynamics and mathematics. In this paper, we derive a regularity criterion less restrictive than the Beale-Kato-Madja (BKM) regularity criterion type, namely any so-lution (u, b) ∈ C([0, T [; Hr
(R2)) with r > 2 remains in Hr
(R2) up to time
T under the assumption that Z T 0 k∇u(t)k12 ∞ log(e + k∇u(t)k∞) dt < +∞. This ity criterion may stand as a great improvement over the usual BKM regular-ity criterion which states that if
Z T
0
k∇ × u(t)k∞dt < +∞ then the solution
(u, b) ∈ C([0, T [; Hr(R2)) with r > 2 remains in Hr(R2) up to time T . In-deed, in virtue of ∇u = R((∇ × u) Id) with R Riesz transform on matrix-valued functions, we expect that the blow-up rate at a time T of k∇u(t)k∞ be the
same as the one of k∇ × u(t)k∞. Furthermore, our result applies also to a
class of equations arising in hydrodynamics and studied in [14] for their L∞ ill-posedness.
Keywords: MHD; Navier-Stokes, Euler, BKM’s criterion.
Introduction
Magnetohydrodynamics equations (MHD) describes the evolution of an elec-trically conducting fluid. Examples of such fluids include plasmas, liquid metals, and salt water or electrolytes. The field of MHD was initiated by Hannes Alfv´en [2], for which he received the Nobel Prize in Physics in 1970. It addresses labo-ratory as well as astrophysical plasmas and therefore is extensively used in very different contexts. In astrophysics, its applications range from solar wind [26], to the Sun [29, 30], to the interstellar medium [27] and beyond [42]. At the same
time, MHD is also relevant to large-scale motion in nuclear fusion devices such as tokamaks [31]. A tokamak is a toroidal device in which hydrogen isotopes in the form of a plasma reaching a temperature of the order of the hundred of millions of Kelvins is confined thanks to a very strong applied magnetic field. Tokamaks are used to study controlled fusion and are considered as one of the most promising concepts to produce fusion energy in the near future. However the main problem with this approach of confinement is that hydrodynamic in-stabilities arise. Numerical simulations using the MHD models are therefore of uttermost importance. Nevertheless, without the existence of a unique solution there is no hope of guaranteeing that a numerical approximation is really an approximation in any meaningful sense, since it is not clear what is being ap-proximated.
Due to their prominent roles in modeling many phenomena in astrophysics, geo-physics and plasma geo-physics, the MHD equations have been studied extensively mathematically. Furthermore, while the differences in behaviour between the two-dimensional (2D) and three-dimensional (3D) hydrodynamical turbulence of neutral fluids are accepted to be important, those of the MHD system in both cases are conventionally believed to be non-significant [6]. Strong statements were made by some authors that 2D simulations can be safely used to model 3D situations because the properties of the 2D and the 3D MHD turbulence are essentially the same [5, 6].
Hence, the mathematical studies on the MHD equations in the two-dimensional case appear highly relevant. However up to now, the question of spontaneous apparition of singularity from a local classical solution of the partially viscous 2D MHD (2) or 2D inviscid MHD ((2) without the Laplacian term) remains a challenging open problem in the mathematical fluid mechanics. Thus, in the absence of a well-posedness theory, the development of blowup/non-blowup the-ory is of major importance for both theoretical and practical purposes. Indeed, for a mathematical or numerical test of the actual finite time blow-up of a given solution, it is important to have a good blow-up criterion. Thus, there have been many computational attempts to find finite-time singularities of the 2D MHD equations (see [3, 23, 33]). Moreover, recent works on the 2D MHD equations developed regularity criteria in terms of the velocity field and deals with the MHD equations with dissipation and magnetic diffusion given by gen-eral Fourier multiplier operators such as the fractional Laplacian operators (see [34, 35, 36, 13, 32, 19, 11, 37, 38]).
Among all the regularity criteria, one of particular interest is the Beale-Kato-Majda’s criterion well known for Euler equations, extended in [9] to the inviscid MHD equations, under the assumption on both velocity field and magnetic field: RT
0 (kω(t)kL∞+kj(t)kL∞) dt < ∞, where the vorticity ω = ∇×u and the density
j = ∇ × b. And so, the Beale-Kato-Majda’s criterion ensures that the solution (u, b) of the inviscid MHD equations is smooth up to time T .
criteria (see [16, 32, 19, 37, 20, 1, 38, 39, 15]), the global regularity issue of 2D MHD equations (2) remains a challenging open problem up to date. The main reason for the unavailability of a proof of global regularity for the sys-tem of equations (2) is due to the quadratic coupling between u and b which invalidates the vorticity conservation. Indeed, the structure of the vorticity is instantaneously altered due to the effects of the magnetic fields. This fact is the source of the main difficulty connected to the global existence of classical solutions, where no strong global a priori estimates are known till now. This difficulty is revealed through the equations of the 2D inviscid MHD governing the vorticity ω = ∂1u2− ∂2u1 and the current density j = ∂1b2− ∂2b1 ,
∂tω + u · ∇ω = b · ∇j ∂tj + u · ∇j = b · ∇ω + T (∇u, ∇b), (1) where, T (∇u, ∇b) = 2∂1b1(∂2u1+ ∂1u2) + 2∂2u2(∂2b1+ ∂1b2).
We observe that the magnetic field contributes in the last nonlinear part of the second equation with the quadratic term T (∇u, ∇b).
By virtue of this difficulty, no a priori uniform bound for kωkL∞(R2×[0,T ]) is
known for the 2D MHD equations with only magnetic diffusion (2). Further in [15, 20, 1], by considering Fourier multiplier operators magnetic diffusion slightly stronger than the Laplacian magnetic diffusion, the authors were able to obtain an uniform bound of k∇jkL1([0,T ];L∞(R2))and then from the first
equa-tion of (1) obtain an uniform bound of kωkL∞(R2×[0,T ])deriving from estimates
for transport equations (see for instance Lemma 4.1 in [21]).
However, the approach used in [15, 20, 1] and based on the properties of heat equation by using singular integral representations of equations (2) fails in the case where we have only a laplacian magnetic diffusion.
Then, in this paper, we consider the initial-value problem for the 2D incom-pressible magneto-hydrodynamics equations with Laplacian magnetic diffusion,
∂tu + (u · ∇)u = −∇p + (b · ∇)b ∂tb + (u · ∇)b − ∆b = (b · ∇)u ∇ · u = 0, ∇ · b = 0 (2)
with initial conditions
u(x, 0) = u0(x) for a.e x ∈ R2
b(x, 0) = b0(x) for a.e x ∈ R2
(3)
Let us take a new look at the main obstruction. We start by noting that we can rewrite the equation (15) satisfied by ω the vorticity of u as follows:
∂tω + (u · ∇)ω = F − b1b · ∇u2+ b2b · ∇u1, (4)
where F = b1(∆b2+ b · ∇u2) − b2(∆b1+ b · ∇u1). Furthermore, it was shown
re-cently in [40] an uniform bound of k∆b+(b·∇)ukL∞(R2×[0,T ])(in section 4 we give
a sketch of the proof). Moreover, we get an uniform bound of kbkL∞(R2×[0,T ])
deriving from some estimates for linear Stokes system (see [17]), hence we de-duce an uniform bound for kF kL∞(R2×[0,T ]). Then, we notice that our equation
(4) fits with the study made in [14] about L∞ ill-posedness for a class of equa-tions arising in hydrodynamics. Thus, by virtue of ∇u = R(ω Id), where R is the Riesz transform on 2 × 2 matrix-valued functions (see (19)), we understand that the main obstruction comes from the fact that Riesz transforms does not maps L∞ into itself.
Let us specify the way in which the obstruction is characterized. By using the following logarithmic Sobolev inequality proved in [24]:
k∇f kL∞(R2). 1 + k∇ × f kL∞(R2) 1 + log+kf kWs,p(R2) with p > 1, s > 1 +
2 p, (5) where ∇ · f = 0, ∇ × f = −∂2f1+ ∂1f2 is the vorticity of f and log+x =
max(0, log x) for any x > 0, we infer that for all t ∈ [0, T [,
k∇u(t)k∞.r1 + kω(t)k∞ 1 + log+ku(t)kHr (6)
and also
k(∇u, ∇b)(t)k∞.r1 + k(ω, j)(t)k∞ 1 + log+k(u, b)(t)kHr . (7)
Then thanks to (6) and by using estimates for transport equations (see for instance Lemma 4.1 in [21]), from (4) we infer that for all t ∈ [0, T [
kω(t)k∞ .rkω0k∞+ Z t 0 (kF (s)k∞+ kb(s)k2∞) ds + Z t 0 kb(s)k2 ∞kω(s)k∞ 1 + log+ku(s)kHr ds, (8)
where r > 2. As a consequence of Gronwall Lemma, we deduce
kω(t)k∞≤ cr kω0k∞+ Z t 0 (kF (s)k∞+ kb(s)k2∞) ds ecrR0tkb(s)k∞2 (1+log+ku(s)kHr)ds, (9) where cr> 0 is a real depending only on r. Thus, the main obstruction to get
global regularity comes from the term in logarithm which appears in (9), namely log+ku(s)kHr. Nevertheless, thanks to (6), (8) and the following estimate in
the Hilbert space Hr
k(u, b)(t)kHr ≤ k(u0, b0)kHreκr
Rt
we obtain a new estimate of k∇u(t)k∞ in Lemma 5.1 which leads to a new
regularity criterion in Theorem 5.1. Our new regularity criterion states that if Z T 0 k∇u(t)k12 ∞ log(e + k∇u(t)k∞)
dt < +∞ then the solution (u, b) of the 2D MHD
5
equations (2) remains smooth up to time T . This new regularity criterion may appear less restrictive than the BKM regularity criterion which states that if Z T
0
k∇ × u(t)k∞dt < +∞ then the solution (u, b) of the 2D MHD equations (2)
remains smooth up to time T . Indeed, in virtue of ∇u = R((∇ × u) Id) with R Riesz transform on matrix-valued functions, we get
10
k∇ukBMO(R2). k∇ × ukBMO(R2),
and for any 1 < q < ∞
k∇ukLq(R2).q k∇ × ukLq(R2).
Then, we expect that the blow-up rate at a time T of k∇u(t)k∞behaves like the
one of k∇ × u(t)k∞(log(e + k∇ × u(t)k∞))γ for a given γ ≥ 0. Thus, in this case,
due to the exponent 1
2 in our regularity criterion, we get a great improvement over the usual BKM regularity criterion.
15
Then, the paper is organized as follows:
• In section 1, we give some notations and introduce the functional spaces. • In section 2, we deal with the local well-posedness of the Cauchy problem
of the partially viscous magneto-hydrodynamic system (2).
• In section 3, we give two energy estimates and some estimates from the
20
properties of heat equation by using singular integral representations of equations.
• In section 4, we recall and give a sketch of the proof of new estimates obtained in [40] related to the term ∆b + (b · ∇)u.
• In section 5, we give a new estimate for k∇u(t)k∞in Lemma 5.1 and from
25
this estimate, we obtained a new regularity criterion in Theorem 5.1 less restrictive than the BKM regularity criterion.
1. Some notations
We use X . Y to denote the estimate X ≤ CY for an absolute constant C. If we need C to depend on a parameter, we shall indicate this by subscripts,
30
thus for instance X .sY denotes the estimate X ≤ CsY for some Csdepending
on s.
For any f ∈ Lp
(R2), with 1 ≤ p ≤ ∞, we denote by kf k
p and kf kLp, the
oscillation equipped with the norm kf kBMO def = sup x∈R2,r>0 1 |Bx,r| Z Bx,r |f (y) − 35
fBx,r|dy, where Bx,r is the ball of radius r centered at x, |Bx,r| its measure
and fBx,r =
1 |Bx,r|
R
Bx,rf (y) dy. We denote by Id the 2 × 2 identity matrix.
Given an absolutely integrable function f ∈ L1
(R2), we define the Fourier
trans-form ˆf : R27−→ C by the formula,
ˆ f (ξ) =
Z
R2
e−2πix·ξf (x) dx,
and extend it to tempered distributions. We will use also the notation F (f )
40
for the Fourier transform of f . We define also the inverse Fourier transform ˇ f : R27−→ C by the formula, ˇ f (x) = Z R2 e2πix·ξf (ξ) dξ.
For s ∈ R, we define the Sobolev norm kf kHs(R2) of a tempered distribution
f : R2 7−→ R by, kf kHs(R2)= Z R2 (1 + |ξ|2)s| ˆf (ξ)|2dξ 12 ,
and then we denote by Hs
(R2) the space of tempered distributions with finite
Hs
(R2) norm, which matches when s is a non negative integer with the classical
Sobolev space Hk
(R2
), k ∈ N. The Sobolev space Hs
(R2) can be written as
Hs(R2) = J−sL2(R2) where J = (1 − ∆)12.
For s > −1, we also define the homogeneous Sobolev norm,
kf kH˙s(R2)= Z R2 |ξ|2s| ˆf (ξ)|2dξ 12 , (11)
and then we denote by ˙Hs
(R2) the space of tempered distributions with finite
45
˙ Hs
(R2) norm. We use the Fourier transform to define the fractional Laplacian
operator (−∆)α, −1 < α ≤ 1. We define it as follows, \
(−∆)αf (ξ) = |ξ|2αf (ξ).ˆ
We denote by Hs
σ(R2) the Sobolev space Hσs(R2) def
= {ψ ∈ Hs
(R2)2: divψ = 0}.
We denote by P the projector onto divergence free vector fields given by P = Id − ∇∆−1div. The operator P, which acts on vector-valued functions, is a projection : P2= P, annihilates gradients and maps into solenoidal (divergence-free) vectors; it is a bounded operator from (vector-valued) Lq to itself for all 1 < q < ∞ and commutes with translation. We can notice that the operator P can be written under the form,
which yields to Helmholtz decomposition, indeed for all v ∈ Lq(R2)2, 1 < q < ∞, v = Pv + ∇ψ, with div Pv = 0,
ψ = ∆−1div v. (13)
2. Local regularity of solutions of the 2D MHD equation
This section is devoted to the local well-posedness of the 2D MHD equations. By using P the matrix Leray operator, the first equation of (2) can be re-written as follows,
∂u
∂t + P((u · ∇)u − (b · ∇)b) = 0. (14) For (u, b) solution of (2), let us introduce ω = ∇ × u = −∂2u1+ ∂1u2 the
vorticity and j = ∇ × b = −∂2b1+ ∂1b2 the current density. Applying ∇× to
50
the equations of (2), we obtain the governing equations for ω and j ∂tω + (u · ∇)ω = (b · ∇)j ∂tj + (u · ∇)j − ∆j = (b · ∇)ω + T (∇u, ∇b). (15) where, T (∇u, ∇b) = 2∂1b1(∂2u1+ ∂1u2) + 2∂2u2(∂2b1+ ∂1b2).
In this section we assume that the initial data (u0, b0) ∈ Hσr(R2) with r > 2.
Then, we introduce ω0= ∇ × u0the vorticity of u0and j0= ∇ × b the current
density of b0.
55
We assume that (u0, b0) ∈ Hσr(R2) with r > 2, thanks to Theorem 5.1 in [9]
valid for all integer r ≥ 3 and by using the same arguments as in Proposition 4.3 of [1] valid for all real r > 2, we get that there exists a time of existence T > 0 such that there exists an unique strong solution (u, b) ∈ C([0, T [, Hσr(R2)) to the 2D MHD equations (2)-(3).
Thanks to the Beale-Kato-Madja (BKM) criterion obtained in [9] for any integer r ≥ 3 and extended in Proposition 4.2 of [1] for any real r > 2, we get that if (u, b) 6∈ C([0, T ], Hr
σ(R2)), then we have
Z T
0
k(ω, j)(t)kL∞dt = +∞. (16)
From the first equation of (2), we can retrieve the pressure p from (u, b) with the formula,
p = −∆−1div((u · ∇)u − (b · ∇)b). (17) Since ∇ · u = 0 and ∇ · b = 0, then we get (u · ∇)u = ∇ · (u ⊗ u) and (b · ∇)b = ∇ · (b ⊗ b). Then from (17), we get,
p = −∆−1div ∇ · (u ⊗ u − b ⊗ b). (18) By introducing
the Riesz transform on 2 × 2 matrix-valued functions on R2, we get
p = −R(u ⊗ u − b ⊗ b). (20) Owing to (u, b) ∈ C([0, T [, Hr
(R2)), thanks to the L2−boundedness of the Riesz
transforms and Lemma X4 in [21], from (20) we infer that p ∈ C([0, T [, Hr
(R2)).
Similarly as Proposition 4.1 in [1], we get the following local estimates in the higher Sobolev norm Hr: there exists a real κ
r> 0 depending only on r such
that for all t ∈ [0, T [
k(u, b)(t)kHr ≤ k(u0, b0)kHreκr
Rt
0k(∇u,∇b)(τ )k∞dτ. (21)
3. Some estimates
In this section, we give some estimates related to the solutions of the 2D MHD equations 2.
3.1. Energy estimates
In this subsection, we recall some energies estimates. We give here the two following energy estimates given in [32, 25, 1]: for all t ∈ [0, T∗[
ku(t)k2 2+ kb(t)k 2 2+ 2 Z t 0 k∇b(τ )k2 2dτ = ku0k22+ kb0k22 (22)
and we get also that for all t ∈ [0, T∗[ kω(t)k2 2+ kj(t)k 2 2+ Z t 0 k∇j(τ )k2 2dτ ≤ (kω0k22+ kj0k22)e C(ku0k22+kb0k22), (23)
where C > 0 is an absolute constant.
60
3.2. Some estimates deriving from heat equation
In this subsection, in the Lemma just below, we give the details (often omit-ted) of the proof of some estimates deriving from the properties of the heat kernel.
Lemma 3.1. Let (u0, b0) ∈ Hr(R2) with r > 2 and T > 0 be such that there
65
exists (u, b) ∈ C([0, T [, Hr
σ(R2)) solution of the 2D MHD equations (2)-(3).
Then there exists a real C1 > 0 depending only on k(u0, b0)kHr, r and T such
that
kbkL∞(R2×[0,T ])≤ C1.
For any real p > 1 and q > 2, we have also three real C2 > 0, C3 > 0 and
C4> 0 depending only on k(u0, b0)kHr, p, q, r and T such that, 70
k∇bkL∞([0,T ]×Lq(R2)) ≤ C2,
kωkL∞([0,T ]×Lq(R2)) ≤ C3,
Proof. For this, we write the second equation of 2 under its integral form, then we have for all t ∈ [0, T [
b(t) = et∆b0+
Z t
0
e(t−s)∆((b · ∇)u(s) − (u · ∇)b(s)) ds. (24)
Then by using inequality (2.3) in [22], we get ke(t−s)∆((b · ∇)u(s) − (u · ∇)b(s))k
∞
. (t − s)−23k(b · ∇)u(s) − (u · ∇)b(s)k3 2
. (t − s)−23(kb(s)k6k∇u(s)k2+ ku(s)k6k∇b(s)k2).
As a consequence, from (24) we get,
kb(t)k∞. kb0k∞+
Z t
0
(t − s)−23(kb(s)k6k∇u(s)k2+ ku(s)k6k∇b(s)k2) ds. (25)
Since kb(s)k6. kb(s)kH1, ku(s)k6. ku(s)kH1 and k∇b(s)k2. kj(s)k2,
k∇u(s)k2 . kω(s)k2 (see Theorem 3.1.1 in [8]), then thanks to (22) and (23),
from (25) we deduce that there exists a real C0> 0 depending only on k(u0, b0)k2,
k(ω0, j0)k2such that for all t ∈ [0, T [,
kb(t)k∞. kb0k∞+ C0 (T )
1
3. (26)
Owing to (26) and thanks to the Sobolev embedding Hr
(R2) ,→ L∞
(R2) since
r > 2, we deduce that there exists a real C1> 0 depending only on k(u0, b0)kHr,
r and T such that for all t ∈ [0, T [
kb(t)k∞≤ C1, (27)
which concludes the first part of the proof. By virtue of (24), we get that for all t ∈ [0, T [ ∇b(t) = et∆∇b 0+ Z t 0 ∇e(t−s)∆((b · ∇)u(s) − (u · ∇)b(s)) ds. (28)
Then by using inequality (2.3’) in [22], from (28) we deduce,
k∇b(t)kq .qk∇b0kq+
Z t
0
(t − s)−(1−1q)k(b · ∇)u(s) − (u · ∇)b(s)k2ds. (29)
Thanks again to (22) and (23), from (29) we get that for all t ∈ [0, T [, k∇b(t)kq .q k∇b0kq+ k(u0, b0)k2k(ω0, j0)k2eck(u0,b0)k
2 2qT
1
q. (30)
Thanks to Gagliardo-Nirenberg inequality, for any q > 2 we have the following Sobolev embedding Hr
(R2) ,→ ˙W1,q
deduce that there exists a real C2> 0 depending only on k(u0, b0)kHr, T , r and
q such that for all t ∈ [0, T [,
k∇b(t)kq ≤ C2, (31)
which concludes the second part of the proof.
To get an estimate of kωkL∞([0,T ];Lq), we borrow some arguments used in
[20]. Thanks to the Lp− Lq maximal regularity of the Laplacian operator (see
e.g [17]), from the second equation of 2, we get that for all t ∈ [0, T [, p > 1 and q > 2 Z t 0 k∇2b(s)kp q .p,q Z t 0 k(b · ∇)u(s) − (u · ∇)b(s)kp qds .p,q Z t 0 (kb(s)kp∞kω(s)kp q+ ku(s)kp∞k∇b(s)kpq) ds, (32)
where we have used the fact that k∇u(s)kq .q kω(s)kq (see Theorem 3.1.1 in
[8]). Then, we multiply the first equation of (15) by ω|ω|q−2, integrate it over
R2 and use the fact that ∇ · u = 0 to obtain
75 1 q d dtkω(t)k q q = Z R2 b(x, t) · ∇j(x, t)ω(x, t)|ω(x, t)|q−2dx ≤ kb(t)k∞k∇j(t)kqkω(t)kq−1q ,
which yields to: for all t ∈ [0, T [, 1 2 d dtkω(t)k 2 q ≤ kb(t)k∞k∇j(t)kqkω(t)kq.
After an integration over [0, t] of the inequality just above, we obtain
kω(t)k2 q ≤ kω0k2q+ 2 Z t 0 kb(s)k∞k∇j(s)kqkω(s)kqds ≤ kω0k2q+ Z t 0 (k∇j(s)k2q+ kb(s)k2∞kω(s)k2 q) ds (33)
Then thanks to (32), from (33) we infer that for all t ∈ [0, T [
kω(t)k2 q . kω0k2q+ Z t 0 (ku(s)k2∞k∇b(s)k2 q+ kb(s)k 2 ∞kω(s)k2q) ds. (34)
By using Gagliardo-Nirenberg inequalities, Young inequalities and the fact that k∇u(s)kq .qkω(s)kq, we get
ku(s)k∞.qku(s)k2+ kω(s)kq. (35)
By virtue of (34) and (35), we get that for all t ∈ [0, T [,
Thanks to (22), (27) and (31), we deduce that there exists a real C > 0 depend-ing only on k(u0, b0)kHr, T , r and q such that for all t ∈ [0, T [
kω(t)k2 q ≤ C + C Z t 0 kω(s)k2 qds. (37)
Thanks to Gronwall inequality, we infer that for all t ∈ [0, T [, kω(t)k2q ≤ Ce
C T.
Therefore with C3 =
√
Ce12C T, we obtain a real C3 > 0 depending only on
k(u0, b0)kHr, T and q such that for all t ∈ [0, T [,
kω(t)kq ≤ C3, (38)
which concludes the third part of the proof. By using (35) into (32) and thanks to (38), (22), (27) and (31), we completes the proof.
4. Some new estimates
80
In this section, we give a sketch of the proof of the Lemma 4.1 obtained in [40] by exploiting the special structure of the 2D MHD equations (2).
Lemma 4.1. Let (u0, b0) ∈ Hr(R2) with r > 2 and T > 0 be such that there
exists (u, b) ∈ C([0, T [, Hr
σ(R2)) solution of the 2D MHD equations (2)-(3).
Then there exists a real C > 0 depending only on k(u0, b0)kHr, r and T such
that
k∆b + (b · ∇)ukL∞([0,T ];L∞(R2)) ≤ C, (39)
and we have also that for any real p ≥ 2 and q ≥ 2,
k∇(∆b + (b · ∇)u)kLp([0,T ];Lq(R2))≤ C. (40)
The proof of Lemma 4.1 given in [40] is obtained by writing the equation satisfied by F := ∆b + (b · ∇)u that is
∂tF− ∆F = −(b · ∇)P((u · ∇)u) + (b · ∇)P((b · ∇)b) − ∆((u · ∇)b)
−∇u (u · ∇)b + ∇u (b · ∇)u + ∇u ∆b. (41) This equation is obtained by applying (b · ∇) and ∆ respectively to the first equation of (14) and second equation of (2), and multiplying the second equation of (2) by ∇u and after adding the resulting equations together. Then, by writing equation (41) under its integral form and using the fact that ∇ · u = 0 and ∇ · b, we get for all t ∈ [0, T [
F(t) = et∆F(0) + Z t 0 ∇e(t−s)∆ (b(s) ⊗ P((u · ∇)u)(s) − b(s) ⊗ P((b · ∇)b)(s)) ds + Z t 0 ∇e(t−s)∆∇((u · ∇)b)(s) ds + Z t 0
Then using inequalities (2.3) and (2.3’) of [22] stated for 1 < p ≤ q < +∞ but remaining true for q = ∞, we obtain for all t ∈ [0, T [
kF(t)k∞ ≤ kF(0)k∞ + Z t 0 (t − s)−56k(b(s) ⊗ P((u · ∇)u)(s) − b(s) ⊗ P((b · ∇)b)(s))k3ds + Z t 0 (t − s)−56k∇((u · ∇)b)(s)k3ds + Z t 0
(t − s)−12k − ∇u(s) (u(s) · ∇)b(s) + ∇u(s) (b(s) · ∇)u(s) + ∇u(s) ∆b(s)k2ds.
(43) By using the fact that P is a bounded operator from (vector-valued) Lq to itself
for all 1 < q < ∞ and H¨older inequality, we get k(b(s) ⊗ P((u · ∇)u)(s) − (b(s) ⊗ P((b · ∇)b)(s)k3
. kb(s)k∞ku(s)k6k∇u(s)k6+ kb(s)k∞kb(s)k6k∇b(s)k6,
k∇((u · ∇)b)(s)k3
. k∇u(s)k6k∇b(s)k6+ ku(s)k6k∇2b(s)k6,
k − ∇u(s) (u(s) · ∇)b(s) + ∇u(s) (b(s) · ∇)u(s) + ∇u(s) ∆b(s)k2
. k∇u(s)k6ku(s)k6k∇b(s)k6+ k∇u(s)k26kb(s)k6+ k∇u(s)k6k∆b(s)k3.
(44) Furthermore, thanks to Gagliardo-Nirenberg interpolation inequality and The-orem 3.1.1 in [8], we get ku(s)k6. ku(s)k 1 3 2kω(s)k 2 3 2, kb(s)k6. kb(s)k 1 3 2kj(s)k 2 3 2, k∇u(s)k6. kω(s)k6. (45) After plugging (45) into (44) and using Lemma 3.1 with the energy inequalities (22), (23), from (43) we infer that there exists a real C0> 0 depending only on
k(u0, b0)kHr, r and T such that for all t ∈ [0, T [
kF(t)k∞. C0 1 + Z t 0 (t − s)−56(1 + k∇2b(s)k6) + (t − s)− 1 2(1 + k∆b(s)k3) ds . (46) Thanks to H¨older inequality used with the couples of exponents (76, 7) and (32, 3), from (46) we deduce that for all t ∈ [0, T [
which yields to kF(t)k∞. C0 1 + t 1 42 Z t 0 (1 + k∇2b(s)k76) ds 17 + t16 Z t 0 (1 + k∆b(s)k33) ds 13! . (47) Then, thanks again to Lemma 3.1, from (47) one obtains that there exists a real
85
C1> 0 depending only on k(u0, b0)kHr, r and T such that for all t ∈ [0, T [,
kF(t)k∞≤ C1,
which gives us (39) the first inequality of Lemma 4.1.
For the second inequality of Lemma 4.1, we use the Lp− Lq maximal regularity
of the Laplacian operator [17], one has for any 1 < p < ∞, 1 < q < ∞ and g =Rt
0e
(t−s)∆f ,
k∇2gk
Lp([0,T ];Lq(R2)).p,qkf kLp([0,T ];Lq(R2)). (48)
Then, with the expression of ∇F(t) obtained from (42) and by using (48), in-equality (2.3’) of [22], Lemma 3.1 and the energy inequalities (22), (23), we obtain in a similar way (40) the second inequality of Lemma 4.1.
5. A new blowup criterion
90
In this section, we give a new estimate for k∇u(t)k∞in Lemma 5.1 and from
this estimate, we obtain a new regularity criterion in Theorem 5.1 less restrictive than the BKM regularity criterion.
Lemma 5.1. Let (u0, b0) ∈ Hr(R2) with r > 2 and T > 0 be such that there
exists (u, b) ∈ C([0, T [, Hr
σ(R2)) solution of the 2D MHD equations (2)-(3).
Then there exists a real γ0 > 0 depending only on k(u0, b0)kHr, T and r such
that for all t ∈ [0, T [,
k∇u(t)k∞≤ exp γ0exp γ0
Z t 0 k∇u(s)k12 ∞ log(e + k∇u(s)k∞) ds !! . (49)
Proof. We begin the proof with the following logarithmic Sobolev inequality proved in [24] (see inequality (4.20)) and stands as an improved version of that in [4]:
k∇f kL∞(R2). 1 + k∇ × f kL∞(R2) 1 + log+kf kWs,p(R2) with p > 1, s > 1 +
2 p, (50) where ∇ · f = 0, ∇ × f = −∂2f1+ ∂1f2 is the vorticity of f and log+x =
max(0, log x) for any x > 0.
Thus, by virtue of (50), we get that for all t ∈ [0, T [,
where βr > 0 is a real depending only on r. Let us give an estimate of the
term 1 + log+ku(t)kHr. Thanks to (21), we get that there exists a real κr> 0
depending only on r such that for all t ∈ [0, T [ k(u, b)(t)kHr ≤ k(u0, b0)kHreκr
Rt
0k(∇u,∇b)(τ )k∞dτ. (52)
After taking the logarithm in the inequality (52), we observe that for all t ∈ [0, T [,
log+k(u, b)(t)kHr ≤ log+k(u0, b0)kHr+ κr
Z t
0
k(∇u, ∇b)(τ )k∞dτ. (53)
Thanks to Lemma 3.1 and the Sobolev embedding W2,q(R2) ,→ W1,∞(R2) with q > 2, we infer that there exists a real %0 > 0 depending only on r, T ,
k(u0, b0)kHr such that
Z T
0
k∇b(σ)k∞≤ %0. (54)
Then owing to (54), from (53) we infer that there exists a real %1≥ 1 depending
only on r, T , k(u0, b0)kHr such that for all t ∈ [0, T [,
1 + log+k(u, b)(t)kHr ≤ %1+ κr
Z t
0
k∇u(s)k∞ds. (55)
Thus, by plugging (55) into (51), we deduce that there exists a real %2 > 0
depending only on k(u0, b0)kHr, T and r such that for all t ∈ [0, T [,
k∇u(t)k∞≤ βr+ %2kω(t)k∞ 1 + Z t 0 k∇u(s)k∞ds . (56)
Now, let us estimate kω(t)k∞. We observe that the first equation of (15) can
be changed into
∂tω + u · ∇ω = F − b1b · ∇u2+ b2b · ∇u1, (57)
where F = b1(∆b2+b·∇u2)−b2(∆b1+b·∇u1). By using estimates for transport
equations (see for instance Lemma 4.1 in [21]), we obtain that for all t ∈ [0, T [
kω(t)k∞≤ kω0k∞+ c Z t 0 kF (s)k∞ds + c Z t 0 kb(s)k2 ∞k∇u(s)k∞ds, (58)
where c > 0 is a constant. Thanks to Lemmata 3.1 and 4.1, we deduce that there exist two real %3> 0 and %4> 0 depending only on k(u0, b0)kHr, T and r
such that for all t ∈ [0, T [,
Thus by virtue of (59), from (58) we infer that for all t ∈ [0, T [
kω(t)k∞≤ kω0k∞+ %3+ c%4
Z t
0
k∇u(s)k∞ds. (60)
Furthermore, thanks to the Sobolev embedding Hr
(R2) ,→ W1,∞
(R2) with
r > 2, we get
kω0k∞.rku0kHr. (61)
Hence, owing to (61), from (60) we deduce that there exists a real %5 > 0
depending only on k(u0, b0)kHr, T and r such that for all t ∈ [0, T [,
kω(t)k∞≤ %5+ c%4
Z t
0
k∇u(s)k∞ds. (62)
By plugging (62) into (56), we infer that there exists a real %6≥ 1 depending
only on r, T , k(u0, b0)kHr such that for all t ∈ [0, T [,
k∇u(t)k∞≤ %6 1 + Z t 0 k∇u(s)k∞ds 2 , (63) which yields to k∇u(t)k12 ∞≤ % 1 2 6 1 + Z t 0 k∇u(s)k∞ds . (64)
We thus introduce the real function J defined for all t ∈ [0, T [ by
J(t)def= % 1 2 6 + % 1 2 6 Z t 0 k∇u(s)k∞ds. (65)
On one hand, by virtue of (64), thanks to (65) we get that for all t ∈ [0, T [, k∇u(t)k12
∞≤ J(t). (66)
On the other hand, from (65), we infer that for any t ∈ [0, T [, J0(t) = %12 6k∇u(t)k∞ = % 1 2 6k∇u(t)k 1 2 ∞ log(e + k∇u(t)k12 ∞) k∇u(t)k12 ∞log(e + k∇u(t)k 1 2 ∞). (67)
Then, owing to (66), from (67), we infer that for all t ∈ [0, T [,
J0(t) ≤ % 1 2 6k∇u(t)k 1 2 ∞ log(e + k∇u(t)k12 ∞) J(t) log(e + J(t)). (68)
As a consequence of Gronwall Lemma, from (69) we get for all t ∈ [0, T [,
log(e + J(t)) ≤ log(e + J(0)) exp %
1 2 6 Z t 0 k∇u(s)k12 ∞ log(e + k∇u(s)k 1 2 ∞) ds ! . (70) From (65), we get J(0) = % 1 2
6 and thanks to (70), we thus obtain for all t ∈ [0, T [,
J(t) ≤ exp log(e + % 1 2 6) exp % 1 2 6 Z t 0 k∇u(s)k12 ∞ log(e + k∇u(s)k 1 2 ∞) ds !! . (71)
Owing to (66) and (71), we obtain that for all t ∈ [0, T [,
k∇u(t)k∞≤ exp 2 log(e + %
1 2 6) exp % 1 2 6 Z t 0 k∇u(s)k12 ∞ log(e + k∇u(s)k 1 2 ∞) ds !! . (72) Since e + k∇u(s)k 1 2 ∞ ≥ (e + k∇u(s)k∞) 1
2, then we get log(e + k∇u(s)k 1 2
∞) ≥ 1
2log(e + k∇u(s)k∞) and hence from (72) we infer for all t ∈ [0, T [,
95
k∇u(t)k∞≤ exp 2 log(e + %
1 2 6) exp 2% 1 2 6 Z t 0 k∇u(s)k12 ∞ log(e + k∇u(s)k∞) ds !! ,
which concludes the proof.
Remark 5.1. We observe that the expression of the estimate obtained in Lemma 5.1 for k∇u(t)k∞ makes appear a double exponential growth. This double
expo-nential growth derives from the taking into account in the estimate of the term log(e + k∇u(t)k∞). We thus point out that we have also an upper bound of
100
k∇u(t)k∞ for which we get only one single exponential growth. Indeed, from
(64), thanks to Gronwall Lemma, we obtain that for all t ∈ [0, T [
k∇u(t)k12 ∞≤ % 1 2 6 exp % 1 2 6 Z t 0 k∇u(s)k12 ∞ds , which yields to k∇u(t)k∞≤ %6exp 2% 1 2 6 Z t 0 k∇u(s)k12 ∞ds ,
where %6> 0 is a real depending only on T , r and k(u0, b0)kr.
Let us establish now, a new regularity criterion in the Theorem just below.
105
Theorem 5.1. Let (u0, b0) ∈ Hσr(R2) with r > 2 and T > 0 be such that there
exists (u, b) ∈ C([0, T [, Hr
σ(R2)) solution of the 2D MHD equations (2)-(3). If
Z T
0
k∇u(t)k12
∞
log(e + k∇u(t)k∞)
Proof. Let us assume that we have Z T 0 k∇u(t)k12 ∞ log(e + k∇u(t)k∞) dt < +∞. (73)
For a contradiction, we suppose that u 6∈ C([0, T ], Hr
σ(R2). Then we get (16).
Thanks to Lemma 3.1 and the Sobolev embedding W2,q
(R2) ,→ W1,∞
(R2) with
q > 2, we infer that Z T
0
kj(t)k∞dt < +∞. Then from (16), we get only
Z T
0
kω(t)k∞dt = +∞. (74)
Thanks to Lemma 5.1, there exists a real %1> 0 depending only on k(u0, b0)kHr,
T and r such that for all t ∈ [0, T [,
k∇u(t)k∞≤ exp %1exp %1
Z t 0 k∇u(s)k12 ∞ log(e + k∇u(s)k∞) ds !! . (75)
Then by virtue of (75) and (73), we infer that Z T 0 k∇u(t)k∞dt < +∞ which 110 implies that Z T 0
kω(t)k∞dt < +∞. Then we obtain a contradiction with (74)
and hence u ∈ C([0, T ], Hr
σ(R2) which concludes the proof.
Conclusion
In this article, we obtained a new regularity criterion for the two-dimensional resistive magneto-hydrodynamic (MHD) equations less restrictive than the BKM’s regularity criterion (see Theorem 5.1), by using the logarithmic Sobolev inequal-ity. It is important to find some criteria less restrictive than the BKM’s regular-ity criterion. Indeed, due to the quadratic nonlinearregular-ity of the MHD equations, we expect that the blow-up rate of k∇u(t)k∞at a time T be at least faster than
O 1
T − t
. Thus, if one investigates numerically on the finite time singulari-ties of the solutions of such system of equations and believe that its numerical solution computed leads to a finite-time blowup at some time T , then one may observe a blow-up rate at the time T for k∇u(t)k of the form O
1 (T − t)γ
, γ ≥ 1. Further, in all the recent numerical investigations performed to find finite time singularities of the 2D inviscid MHD equations, the results suggest blow-up rates at a time T for k∇u(t)k∞ of the form O
1 (T − t)α
the solutions of the 2D resistive MHD equations. However, with the use of our regularity criterion (see Theorem 5.1), we can confirm that in fact there is no blowup of the solution at this time T . Then, it is dangerous to interpret the blow-up of an under-resolved computation as evidence of finite-time singularities for the 2D resistive MHD equations. Indeed, computing 2D MHD singularities numerically is an extremely challenging task. First of all, it requires huge com-putational resources (see [3]). Tremendous resolutions are required to capture the nearly singular behaviour of the 2D MHD equations. Secondly, one has to perform a careful convergence study.
Furthermore, we notice also that our problem fits in the class of equations con-sidered in [14] in the study of L∞ ill-posedness problem. We thus point out that by borrowing the same arguments used in this paper, we can establish the same regularity criterion to another interesting open problem in mathematical fluid dynamics mentioned in [14], it is about the following type of equation in two-dimension:
∂tu + (u · ∇)u + ∇p = Au,
∇ · u = 0, (76)
with initial conditions u0 a free-divergence field vector and A is some constant
matrix. Namely, as Theorem 5.1, we get the following Theorem for the system
115
of equations (76):
Theorem 5.2. Let u0∈ Hσr(R
2) with r > 2 and T > 0 be such that there exists
u ∈ C([0, T [, Hσr(R2)) solution of the equations (76). If Z T 0 k∇u(t)k12 ∞ log(e + k∇u(t)k∞) dt < +∞ then there cannot be blowup of the solution u in Hr
(R2) at the time T , in
other words u ∈ C([0, T ], Hr σ(R2)).
120
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