On the problem of B₀-reduction for Navier-Stokes-Maxwell equations
In this paper a range of applicability of B₀-reduction for a rather wide class of MHD-flows in bounded plane domains is investigated. Besides effective formulas of estimates of absolute error are removed.
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irk-123456789-1692872020-06-10T01:26:13Z On the problem of B₀-reduction for Navier-Stokes-Maxwell equations Britov, N.A In this paper a range of applicability of B₀-reduction for a rather wide class of MHD-flows in bounded plane domains is investigated. Besides effective formulas of estimates of absolute error are removed. 1999 Article On the problem of B₀-reduction for Navier-Stokes-Maxwell equations / N.A. Britov // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 156-161. — Бібліогр.: 2 назв. — англ. 0236-0497 http://dspace.nbuv.gov.ua/handle/123456789/169287 en Нелинейные граничные задачи Інститут прикладної математики і механіки НАН України |
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In this paper a range of applicability of B₀-reduction for a rather wide class of MHD-flows in bounded plane domains is investigated. Besides effective formulas of estimates of absolute error are removed. |
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Article |
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Britov, N.A |
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Britov, N.A On the problem of B₀-reduction for Navier-Stokes-Maxwell equations Нелинейные граничные задачи |
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Britov, N.A |
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Britov, N.A |
| title |
On the problem of B₀-reduction for Navier-Stokes-Maxwell equations |
| title_short |
On the problem of B₀-reduction for Navier-Stokes-Maxwell equations |
| title_full |
On the problem of B₀-reduction for Navier-Stokes-Maxwell equations |
| title_fullStr |
On the problem of B₀-reduction for Navier-Stokes-Maxwell equations |
| title_full_unstemmed |
On the problem of B₀-reduction for Navier-Stokes-Maxwell equations |
| title_sort |
on the problem of b₀-reduction for navier-stokes-maxwell equations |
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Інститут прикладної математики і механіки НАН України |
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1999 |
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http://dspace.nbuv.gov.ua/handle/123456789/169287 |
| citation_txt |
On the problem of B₀-reduction for Navier-Stokes-Maxwell equations / N.A. Britov // Нелинейные граничные задачи: сб. науч. тр. — 1999. — Т. 9. — С. 156-161. — Бібліогр.: 2 назв. — англ. |
| series |
Нелинейные граничные задачи |
| work_keys_str_mv |
AT britovna ontheproblemofb0reductionfornavierstokesmaxwellequations |
| first_indexed |
2025-07-15T04:02:40Z |
| last_indexed |
2025-07-15T04:02:40Z |
| _version_ |
1837684134862913536 |
| fulltext |
On the problem of B0-reduction for
Navier-Stokes-Maxwell equations
c© N.A.Britov
A question of investigation of the range of significances of parameters, at which to
description of the flow of conducting fluid is possible to apply B0-reduction of Navier-
Stokes-Maxwell equations (inductionless approximation) constantly involved attention of
specialists in magnetic hydrodynamics. It was considered that application of B0-reduction
is possible, when parameter Rm is reasonably small. A.B.Tsinober has paid attention
that the intensity of magnetic field decrease with growth of Hartmann parameter. It has
allowed to formulate hypothesis about of simultaneous admissibility of B0-reduction and
Stokes approximation. On the necessity of the mathematical analysis of admissibility of
B0-reduction author’s attention has paid H.E.Kalis.
In this paper a range of applicability of B0-reduction for a rather wide class of MHD-
flows in bounded plane domains is investigated. Besides effective formulas of estimates of
absolute error are removed. Everywhere in this paper it should Hartmann and Reynolds
numbers H, R are greater that 1.
Let D = D+ ∪ D− be a bounded plane domain. Boundary S =
n⋃
i=0
Si be a join of
contours Si ∈ Cα
1 , α > 0. It separates domains D+ and D−. Domain D− is bounded
by Se =
m⋃
j=0
Se
j and contours Se
j ∈ Cα
1 . In D ∪ S Cartesian coordinates x = (x1, x2) are
entered. Let D = D+ ∪D− be a bounded plane domain. Boundary S =
n⋃
i=0
Si be a join
of contours Si ∈ Cα
1 , α > 0. It separates domains D+ and D−. Domain D− is bounded
by Se =
m⋃
j=0
Se
j and contours Se
j ∈ Cα
1 .
Let consider the Navier-Stokes-Maxwell equations in the dimensionless and coordi-
nateless form:
in the domain D+:
(∇×)2 U +∇
(
P + R
2
|U|2
)
= RU× (∇×U)−
−H2R−1
m B× (∇×B);
∇×B = Rm (E + U×B) ;
∇ ·U = 0, ∇ ·B = 0; ∇× E = 0;
(1)
in the domain D−:
∇×B = I0(x), ∇ ·B = 0, ∇× E = 0. (2)
Vector field I0(x) is assumed known. Vector fields U,B,E satisfy to boundary value
conditions:
U|S = U0, U0 · ν = 0; B× ν |Se
0
= 0; E× ν |S = 0; (3)
where ν be a unit normal to S vector field. Let τ be a unit tangential to S vector field
and κ = ν × τ . Vector field B is assumed continuous on the S. In this conditions E ≡ 0.
Let B0 be a solution of the problem (1)-(3) at Rm = 0. Later on will be assumed that
vector field B0 satisfies to conditions:
0 < m ≤ |B0| ≤ m < ∞;
∫
S0
|B0 · ν| dS0 > 0. (B)
Let β1 = |B0|−1 B0; β2 ⊥ β1, |β2| = 1, µ = mm−1 < 1. Vector fields (β1, β2) form in
D∪S a local basis connected with vector field B0. Later on next designation will be used
µ = m−1m ≤ 1.
By nonlinear inductionless approximation of the solution of the problem (1)-(3) is
named a triple (Ul, Pl,El), which satisfy in the domain D+ to reduced Navier-Stokes-
Maxwell system:
(∇×)2 Ul +∇
(
Pl + R
2
|Ul|2
)
= RUl × (∇×Ul)−
−H2B0 × (El + Ul×B0);
∇ ·Ul = 0; ∇× El = 0;
(4)
and boundary value conditions (3).
By the error of B0-reduction is named a triple (δU = U−Ul, δP = P−Pl, δB = B−B0)
. This triple satisfies to next system
(∇×)2 δU +∇
[
δP + R
2
(2Ul + |δU|2)
]
= R {δU× (∇×Ul) +
+Ul × (∇×δU) + δU× (∇×δU)}−
−H2 [B0 × (Ul × δB + δU×B0 + δU×δB) +
+δB× (Ul ×B0 + Ul × δB + δU×B0 + δU×δB)] ;
∇×δB = Rm (Ul ×B0 + Ul × δB + δU×B0 + δU× δB) ;
∇·δU = 0, ∇·δB = 0.
(5)
and homogeneous boundary value conditions.
1. Some auxiliary results. Let Lp(D) - be a Banach space of vector fields a, which
have p-degree integrable module in D. If p = 2 this is a Hilbert’s space and index p in
that case will be missed. The scalar product in L2 is designated as <,>. Let Hν(D) - be
a Hilbert space of solenoidal in D vector fields, which normal components are equal to a
zero on S− and have a finite norm:
‖a‖Hν
= ‖∇ × a‖ .
Theorem 1 If f ∈ Hν(D) and q < 2n/(n−2), n be a dimension of D then next inequality
is valid ‖f‖q ≤ Mq ‖∇ × f‖ .
A subspace of Hν(D), consisting of vector fields which are equal to a zero on S is
denoted as H(D+).
It is well known an outstanding role of the embedding theorems and it’s expressions
as multiplicative inequalities [1] in the theory of Navier- Stokes equations. Here some
vector analog of this inequalities will be formulated without proof. Let (ξ1, ξ2) be some
nondegenerate system of curvilinear coordinates in D+ ∪ S. Let Lame coefficients of this
coordinate system are Λ1 = Λ2 = Λ
(
0 ≤ λ ≤ Λ ≤ λ
)
and d be a diameter of D+.
Theorem 2 Let q > 2 and f(f1, f2)∈ H(D+). Next inequalities are valid
‖f1‖ ≤ d1/2 ‖f2‖1/2 ‖∇ × f‖1/2 ,
‖f1‖r ≤ 4λλ−1 ‖f1‖1−3α ‖f2‖α ‖∇ × f‖2α , 2 < r < 6;
‖f1‖r ≤ βλλ−1 ‖f2‖α ‖∇ × f‖1−α , 6 < r < ∞;
α = 1/2− 1/r, β = max (4, r/2) .
(6)
Theorem 1 is given without proof.
Later on one special extension of vector field U0 into domain D+ will be need. Let aδ
be a twice differentiable in D+ vector field, which is equal to U0 on S and to a zero in the
points of D+, distended from S greater then on δ. This vector field can be constructed
by means of well known Hopf’s cutoff function or it generalizations. Later on will be
used one form of aδ, constructed in [2]. Estimates for ‖aδ‖p, ‖∇ × aδ‖, p > 2 contain
parameter δ. For the purposes of this paragraph it is possible to choose δ = δ0H
−1, δ0 =
min(1/3, K−1), K - be a maximum of curvatures of contours Sk. In view of this, estimates
for ‖aδ‖p, ‖∇ × aδ‖ will accept next form:
‖aδ‖p ≤ 21−2/pM∇
[
1 + 9(1 + p)1/p
]
Ls
1/pH−1/p = CapH
−1/p;
‖aδ‖ ≤ 18M∇L1/2
s H−1/2 = CaH
−1/2;
‖∇ × aδ(x)‖ ≤ (26M∇ + Mδ)(Ls/δ0)
1/2H1/2 = Ca∇H1/2,
(7)
where M∇,Mδ - some positive constants, Ls be a length of S.
For solutions of the boundary value problems (1)-(3) and it B0 -reduction next a priory
estimates are valid
‖∇ ×U‖ ≤ C∇H1/2, ‖U×B‖ ≤ C⊥H−1/2,
‖∇ ×B‖ = ‖∇ × δB‖ ≤ RmCbH
−1/2;
‖∇ ×Ul‖ ≤ Cl∇H1/2, ‖Ul × β2‖ ≤ Cl⊥H−1/2,
‖Ul‖p ≤ ClpH
1/4−1/2p.
(8)
Lemma 1 Let S ∈ Cα
1 , 0 < α < 1, vector field B0 satisfies in D+ ∪ S to condition (B)
and H ≥ H01, where
H01 >
McCcRmL−1
S0
max
(
mLS0
,
∫
S0
|B0·ν|2dS0
)
11/5
,
Cc = m (Cap + Cb) + 4MrC2CqRm,
p > 2, 1/r + 1/q = 1
and LS0− be a length of contour S0, Mc be a constant of embedding of Sobolev’s space
W 1
p (D+) into C(D+). Then vector field Bs satisfies in D+∪S to condition (B) with some
constants M, M : M ≤ |B| ≤ M.
In conditions of Lemma 1 next a priory estimate for solutions of the problem (1)-(3)
is valid
‖U‖p ≤ CpH
1/4−1/2p.
A problem of finding of estimates of the error and conditions of admissibility of B0-
reduction is reduced to obtaining of a priori estimates of solutions of the system (5) with
homogenious boundary value conditions. As far as for ‖∇ × δB‖ estimate (8) is valid,
reasonably will establish estimates of norms: ‖∇ × δU‖, ‖δU×B0‖, ‖δU‖.
For obtaining of a priori estimates of solutions of the problem (5) by the standard way
an equation of the balance of energy is made out:
‖∇ × δU‖2 + H2‖δU×B0‖2 = R 〈∇ × δU, δU×Ul〉−
−H2 [〈δU×B0,U× δB〉+ 〈δU×δB,U×B〉] .
In this equation vector fields Ul, δU, δB are decomposed by vectors of the basis (β1, β2):
Ul = Ul1β1 + Ul2β2, δU = w1β1 + w2β2, δB = δB1β1 + δB2β2.
Summands, worth in both parts of equation of the balance of energy, are estimated by
means of Hölder’s inequality:
‖∇ × δU‖2 + H2m2‖w2κ‖2 ≤ R‖∇ × δU‖ (‖w1Ul2κ‖+ ‖w2Ul1κ‖) +
+H2
(
m ‖w2κ‖ ‖U× δB‖+ C⊥H−1/2 ‖δU× δB‖
)
.
(9)
Norms of products of components of vector fields worth in round brackets are estimated
by means of Hölder’s and multiplicative inequalities:
‖w1Ul2κ‖ ≤ ‖w1‖q‖Ul2‖p ≤ 4βµ
(
p
2
)1−2/p ◦
◦ ‖w2β2‖1/2−1/q ‖∇×δU‖1/2+1/q ‖Ul2β2‖2/p ‖∇ ×Ul‖1−2/p =
= 4βµ
(
p
2
)1−2/p ‖w2β2‖1/p ‖∇×δU‖1−1/p ‖Ul2β2‖2/p ‖∇ ×Ul‖1−2/p ;
‖w2Ul1‖ ≤ ‖w2‖p‖Ul1‖q ≤ 4βµ
(
p
2
)1−2/p ◦
◦ ‖w2β2‖2/p ‖∇×δU‖1−2/p ‖Ul2β2‖1/2−1/q ‖∇ ×Ul‖1/2+1/q =
= 4βµ
(
p
2
)1−2/p ‖w2β2‖2/p ‖∇×δU‖1−2/p ‖Ul2β2‖1/2p ‖∇ ×Ul‖1−1/2p ;
‖U× δB‖ ≤ ‖U‖p‖δB‖q ≤ Mq ‖U‖p‖∇ × δB‖;
‖δU× δB‖ ≤ ‖δU‖p‖δB‖q ≤ Mq‖δU‖p‖∇ × δB‖;
2 < p ≤ 3, 1/p + 1/q = 1/2, 6 ≤ q < ∞, β = max
(
4, p
p−2
)
.
Here
‖δU‖p ≤ 4µd3/2p−1/4 ‖w2β2‖1/4+1/2p ‖∇×δU‖3/4−1/2p +
+
(
p
2
)1−2/p ‖w2β2‖2/p ‖∇×δU‖1−2/p ≤
≤ µd3/2p−1/4
(
p
p+2
ε
−1−2/p
1 ‖w2β2‖+ p
3p−2
ε
3−2/p
1 ‖∇×δU‖
)
+
+
(
p
2
)1−2/p (
2
p
ε
−p/2
2 ‖w2β2‖+ p−2
p
ε
p/(p−2)
2 ‖∇×δU‖
)
=
=
[
µd3/2p−1/4 p
p+2
ε
−1−2/p
1 +
(
2
p
)2/p
ε
−p/2
2
]
‖w2β2‖+
+
[
µd3/2p−1/4 p
3p−2
ε
3−2/p
1 + p−2
p
(
p
2
)1−2/p
ε
p/(p−2)
2
]
‖∇×δU‖ .
Established estimates and estimates (8) are substituted into (9):
‖∇ × δU‖2 + H2m2‖w2κ‖2 ≤ 4βµ
(
p
2
)1−2/p
R‖∇ × δU‖◦
◦
(
C
2/p
l⊥ C
1−2/p
l∇ H1−2/p ‖w2β2‖1/p ‖∇×δU‖1−1/p +
+C
1/2p
l⊥ C
1−1/2p
l∇ H1/2−1/2p ‖w2β2‖2/p ‖∇×δU‖1−2/p
)
+
+H3/2Rm
{
mMq Cp ‖w2β2‖H1/4−1/2p+
+C⊥MqµH−1/2
{[
d3/2p−1/4 p
p+2
ε
−1−2/p
1 +
(
2
p
)2/p
ε
−p/2
2
]
‖w2β2‖+
+
[
d3/2p−1/4 p
3p−2
ε
3−2/p
1 + p−2
p
(
p
2
)1−2/p
ε
p/(p−2)
2
]
‖∇×δU‖
}}
.
(10)
From the inequality (10) the required estimates are removed. Availability of estimate
for ‖∇ × δB‖ permits to bypass only by “direct” estimates. For their establishing in
expressions in the right part of the inequality (10) by means of Jung’s inequality estimated
norms are allocated. This summands are estimated so:
‖∇×δU‖2−1/p ‖w2β2‖1/p ≤
≤ 2p−1
2p
ε
2p/(2p−1)
3 ‖∇ × δU‖2 + 1
2p
ε−2p
3 ‖w2β2‖2 ;
‖∇×δU‖2−2/p ‖w2β2‖2/p ≤
≤ p−1
p
ε
p/(p−1)
4 ‖∇ × δU‖2 + 1
p
ε−p
4 ‖w2β2‖2 .
This estimates are substituted into (10). After grouping of summands of the same
type the main energetic inequality is established:
{
1− 4βµ
(
p
2
)1−2/p
R
[(
1− 1
2p
)
C
2/p
l⊥ C
1−2/p
l∇ ε
2p/(2p−1)
3 H1−2/p+
(
1− 1
p
)
C
1/2p
l⊥ C
1−1/2p
l∇ ε
p/(p−1)
4 H1/2−1/2p
]}
‖∇ × δU‖2+
+H2
{
m2 − 4µ
(
p
2
)1−2/p
R
[
1
2p
C
2/p
l⊥ C
1−2/p
l∇ ε−2p
3 H−1−2/p+
+1
p
C
1/2p
l⊥ C
1−1/2p
l∇ ε−p
4 H−3/2−1/2p
]}
‖w2κ‖2 ≤
≤ H3/2Rm
{
mMq Cp ‖w2β2‖H1/4−1/2p+
+C⊥MqµH−1/2
{[
d3/2p−1/4 p
p+2
ε
−1−2/p
1 +
(
2
p
)2/p
ε
−p/2
2
]
‖w2β2‖+
+
[
d3/2p−1/4 p
3p−2
ε
3−2/p
1 + p−2
p
(
p
2
)1−2/p
ε
p/(p−2)
2
]
‖∇×δU‖
}}
.
(11)
Numbers ε1, ε2 in this inequality are chosen from the condition of the positiveness of
the multiplier at ‖∇ × δU‖2. Let
ε3 =
[
p
2p−1
· H−1+2/p
8βµR( p
2)
1−2/p
C
2/p
l⊥ C
1−2/p
l∇
] 2p−1
2p
;
ε4 =
[
p
p−1
· H−1/2+1/2p
8βµR( p
2)
1−2/p
C
1/2p
l⊥ C
1−1/2p
l∇
] p−1
p
.
Sufficient conditions of the admissibility of B0-reduction are obtained from the condi-
tions of positiveness of multiplier at ‖w2‖ in (11). Let H ≥ max(H01, H02), where H02 is
determined from next condition:
24p+1−2/pp−4+2/p(2p− 1)2p−1 (βµ)2p−1 C4
l⊥C2p−4
l∇ R2p−1H2p−6
02 +
+22p−2/pp−2+ 2
p (p− 1)p−1 (βµ)p−1 C
1/2
l⊥ C
p−1/2
l∇ Rp−1H
−(5−p)/2
02 ≤ 0.75m2.
(12)
Then from (11) follows inequality
0.25‖∇ × δU‖2 + 0.25m2H2‖w2κ‖2 ≤
≤ H3/2Rm
{
mMq Cp ‖w2β2‖H1/4−1/2p+
+C⊥MqµH−1/2
{[
d3/2p−1/4 p
p+2
ε
−1−2/p
1 +
(
2
p
)2/p
ε
−p/2
2
]
‖w2β2‖+
+
[
d3/2p−1/4 p
3p−2
ε
3−2/p
1 + p−2
p
(
p
2
)1−2/p
ε
p/(p−2)
2
]
‖∇×δU‖
}}
.
(13)
In this inequality it should to determine values of numbers ε2, ε2. Let
ε1 = H− p
2(3p−2) , ε
p/(p−2)
2 = H− p−2
2p .
After allocation of the complete squares from the inequality (13) next estimates are es-
tablished:
‖∇ × δU‖ ≤ 2C⊥MqµRmH1/2
[
d3/2p−1/4 p
3p−2
+ (p− 2)
(
2
p
)2/p
]
+
+RmH1/2m−1
{
mMq CpH
3/4−1/2p+
+C⊥Mqµ
[
d3/2p−1/4 p
p+2
H
p+2
2(3p−2) +
(
2
p
)2/p
H
p−2
4
]}
,
‖w2κ‖ ≤ C⊥MqµRmH−1
[
d3/2p−1/4 p
3p−2
+ p−2
p
(
p
2
)1−2/p
]
+
+2Rmm−1
{
mMq CpH
−1/4−1/2p+
+C⊥MqµH−1
[
d3/2p−1/4 p
p+2
H
p+2
2(3p−2) +
(
2
p
)2/p
H
p−2
4
]}
.
(14)
In this estimates number p, obviously, should choose reasonably close two. From the
estimates (14) and first inequality (6) follows next estimate for residual of the vector field
δU:
‖δU‖ ≤ (2.5d)1/2 Rm
{
2C⊥Mqµ
[
d3/2p−1/4 p
3p−2
+ p−2
p
(
p
2
)1−2/p
]
+
+m−1
{
mMq CpH
3/4−1/2p+
+C⊥MqµH−1/2
[
d3/2p−1/4 p
p+2
H
p+2
2(3p−2) +
(
2
p
)2/p
H
p−2
4
]}
.
(15)
For the sufficiently large significances of H obtained estimates can be copied so:
‖∇ × δU‖ ≤ RmCδ∇(R,D, S)H1/2+γ;
‖w2κ‖ ≤ RmCδ⊥(R, D, S)H−1/2+γ;
‖δU‖ ≤ RmCδ(R,D, S)Hγ,
(16)
where γ there is no matter how small positive number.
From (12) follows that conditions of allowability of B0−reduction depend not only from
numbers Rm and H, but from from the Reynolds number R too. Conditions (12) give
following estimate of the significances of R and H0, at which ininductional approximation
is allowable: H0 = ChR
2+δ δ > 0. From estimates (15) follows that there are MHD-flows,
for which A.B. Tsinober’s hypothesis, as appear, is incorrect.
REFERENCES
[1] Ladygenskaya O.A. Mathematical questions of the dynamics of the viscous incom-
pressible fluid, Moscow, Nauka (1971) (Russian).
[2] Britov N.A. The effective a priori estimates of the solution of the first boundary
value problem of magnetic hydrodynamics, Kiev, Naukova Dumka, Nonlinear boundary
value problems, 3, (1991), 13-16 (Russian).
Donetsk, Institute of Applied Mathematics
and Mechanics NAS of Ukraine
E-mail: britoviamm.ac.donetsk.ua
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