Volume 2008, Article ID 735846, 14 pagesdoi:10.1155/2008/735846 Research Article Global Existence and Uniqueness of Strong Solutions for the Magnetohydrodynamic Equations Jianwen Zhang S
Trang 1Volume 2008, Article ID 735846, 14 pages
doi:10.1155/2008/735846
Research Article
Global Existence and Uniqueness of Strong
Solutions for the Magnetohydrodynamic Equations
Jianwen Zhang
School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
Correspondence should be addressed to Jianwen Zhang, jwzhang@xmu.edu.cn
Received 21 June 2007; Accepted 5 October 2007
Recommended by Colin Rogers
This paper is concerned with an initial boundary value problem in one-dimensional magnetohy-drodynamics We prove the global existence, uniqueness, and stability of strong solutions for the planar magnetohydrodynamic equations for isentropic compressible fluids in the case that vacuum can be allowed initially.
Copyright q 2008 Jianwen Zhang This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
MagnetohydrodynamicsMHD concerns the motion of a conducting fluid in an electromag-netic field with a very wide range of applications The dynamic motion of the fluids and the magnetic field strongly interact each other, and thus, both the hydrodynamic and electrody-namic effects have to be considered The governing equations of the plane magnetohydrody-namic compressible flows have the following formsee, e.g., 1 5:
ρ t ρu x 0,
ρu t
ρu2 p 1
2|b|2
x
λu x
x ,
ρw t ρuw − b x μw x
x ,
bt ub − w xυb x
x ,
ρe t ρeu x−κθ x
x λu2
x μw x2 υb x2− pu x ,
1.1
where ρ denotes the density of the fluid, u ∈ R the longitudinal velocity, w w1, w2 ∈ R2
the transverse velocity, b b1, b2 ∈ R2 the transverse magnetic field, θ the temperature,
Trang 2p pρ, θ the pressure, and e eρ, θ the internal energy; λ and μ are the bulk and shear
viscosity coefficients, υ is the magnetic viscosity, κ is the heat conductivity Notice that the longitudinal magnetic field is a constant which is taken to be identically one in1.1
The equations in1.1 describe the macroscopic behavior of the magnetohydrodynamic flow This is a three-dimensional magnetohydrodynamic flow which is uniform in the trans-verse directions There is a lot of literature on the studies of MHD by many physicists and mathematicians because of its physical importance, complexity, rich phenomena, and mathe-matical challenges, see1 14 and the references cited therein We mention that, when b 0,
the system1.1 reduces to the one-dimensional compressible Navier-Stokes equations for the flows between two parallel horizontal platessee, e.g., 15
In this paper, we focus on a simpler case of 1.1, namely, we consider the magneto-hydrodynamic equations for isentropic compressible fluids Thus, instead of the equations in
1.1, we will study the following equations:
ρu t
ρu2 p 1
2|b|2
x
λu x
ρw t ρuw − b x μw x
bt ub − w xυb x
where p Rρ γ with γ ≥ 1 being the adiabatic exponent and R > 0 being the gas constant.
We will study the initial boundary value problem of1.2–1.5 in a bounded spatial domain
Ω 0, 1 without loss of generality with the initial-boundary data:
ρ, ρu, ρw, b x, 0 ρ0, m0, n0, b0
where the initial data ρ0 ≥ 0, m0, n0, b0 satisfy certain compatibility conditions as usual and
some additional assumptions below, and m0 n0 0 whenever ρ0 0 Here the boundary conditions in1.7 mean that the boundary is nonslip and impermeable
The purpose of the present paper is to study the global existence and uniqueness of strong solutions of problem 1.2–1.7 The important point here is that initial vacuum is
allowed; that is, the initial density ρ0 may vanish in an open subset of the space-domain
Ω 0, 1, which evidently makes the existence and regularity questions more difficult than the usual case that the initial density ρ0 has a positive lower bound For the latter case, one can show the global existence of unique strong solution of this initial boundary value problem
in a similar way as that in3,9,14 The strong solutions of the Navier-Stokes equations for isentropic compressible fluids in the case that initial vacuum is allowed have been studied in
16,17 In this paper, we will use some ideas developed in 16,17 and extend their results to the problems1.2–1.7 However, because of the additional nonlinear equations and the
non-linear terms induced by the magnetic field b, our problem becomes a bit more complicated.
Our main result in this paper is given by the following theoremthe notations will be defined at the end of this section
Trang 3Theorem 1.1 Assume that ρ0, m0 ρ0u0, n0 ρ0w0, and b0satisfy the regularity conditions:
ρ0∈ H1, ρ0≥ 0, u0, w0
∈ H1
0∩ H2, b0∈ H1
Assume also that the following compatibility conditions hold for the initial data:
λu0xx−
Rρ γ0 1
2b02
x
ρ 1/2
0 f for some f ∈ L2Ω, 1.9
μw0xx b0x ρ 1/2
0 g for some g∈ L2Ω. 1.10
Then there exists a unique global strong solution ρ, u, w, b to the initial boundary value problem
1.2–1.7 such that for all T ∈ 0, ∞,
ρ∈ L∞
0, T;H1
, u, w ∈ L∞
0, T;H1
0∩ H2
, b ∈ L∞0, T; H1
0,
ρ t ,√ρu
t ,√ρw
t
∈ L∞0, T; L2, u t , w t ∈ L2
0, T;H1
, bt , b xx ∈ L2
0, T;L2
. 1.11
Remark 1.2 The compatibility conditions given by1.9, 1.10 play an important role in the proof of uniqueness of strong solutions Similar conditions were proposed in16–18 when the authors studied the global existence and uniqueness of solutions of the Navier-Stokes equa-tions for isentropic compressible fluids In fact, one also can show the global existence of weak solutions without uniqueness if the compatibility conditions1.9, 1.10 are not valid
We will prove the global existence and uniqueness of strong solutions in Sections3and
4, respectively, whileSection 2is devoted to the derivation of some a priori estimates
We end this section by introducing some notations which will be used throughout the paper LetWm,pΩ denote the usual Sobolev space, and Wm,2Ω HmΩ, W0,pΩ LpΩ
For simplicity, we denote by C the various generic positive constants depending only on the data and T, and use the following abbreviation:
Lq
0, T;Wm,p
≡ Lq
0, T;Wm,pΩ, Lp ≡ Lp Ω, · p ≡ ·LpΩ. 1.12
2 A priori estimates
This section is devoted to the derivation of a priori estimates ofρ, u, w, b We begin with the
observation that the total mass is conserved Moreover, if we multiply1.3, 1.4, and 1.5 by
u, w, and b, respectively, and sum up the resulting equations, we have by using1.2 that
1
2ρ
u2w2
1
2b2
t
1
2ρu
u2w2 ub2− w·b
x
up x
λuu x μw·w x υb·b x
x−λu2x μw x2 υb x2
.
2.1
Integrating1.2 and 2.1 over 0, t × Ω, we arrive at our first lemma.
Trang 4Lemma 2.1 For any t ∈ 0, T, one has
Ωρ x, tdx
Ωρ0xdx ≤ C,
Ω
G ρ 1
2ρ
u2w2
1
2b2
x, tdx
t
0
Ω
λu2x μw x2 υb x2
x, sdx ds ≤ C,
2.2
where G ρ is the nonnegative function defined by
G ρ
⎧
⎪
⎪
Rρ γ
γ− 1 if γ > 1
R ρ ln ρ − ρ 1 if γ 1.
2.3
The next lemma gives us an upper bound of the density ρx, t, which is crucial for the
proof ofTheorem 1.1
Lemma 2.2 For any x, t ∈ Q T : Ω × 0, T, ρx, t ≤ C holds
Proof Notice that1.3 can be rewritten as
ρu t
λu x − ρu2− p − b2
2
x
Set
ψ x, t :
t
0
λu x − ρu2− p − 1
2b2
x, sds
x
0
from which and2.4, we find that ψ satisfies
ψ x ρu, ψ t λu x − ρu2− p − 1
2b2
, ψ|t0
x
0
In view ofLemma 2.1and2.6, we have by using Cauchy-Schwarz’s inequality that
ψ xL∞0,T;L1≤ C,
Ωψ x, tdx
which imply
ψL∞0,T×Ω≤ψ xL∞0,T;L1
Ωψ x, tdx
Letting D t : ∂t u∂ x denote the material derivative and choosing F exp ψ/λ, we
obtain after a straightforward calculation that
D t ρF : ∂ t ρF u∂ x ρF −1
λ
p b2
2
which, together with2.8, yieldsLemma 2.2immediately
Trang 5To be continued, we need the following lemma because of the effect of magnetic field b.
Lemma 2.3 The magnetic field b satisfies the following estimates:
sup
0≤t≤TbtL∞bx tL2
btL20,T;L2≤ C, bxxL20,T;L2≤ C. 2.10
Proof Multiplying1.5 by btand integrating over0, t × Ω, we have
1
4
t
0
Ωbt2x, sdx ds υ
2
Ωbx2x, tdx
≤ υ 2
Ωb0x2xdx
t
0
Ω
u2bx2 u2
xb2wx2
x, sdx ds
≤ C 2
t
0
Ωu2x x, sdx
Ωbx2x, sdx
ds,
2.11
where we have used Cauchy-Schwarz’s inequality,Lemma 2.1, and the following inequalities:
max
x∈Ω u2·, s ≤u x s2
L 2, max
x∈Ω b·,s2≤bx s2
Sinceu xL20,T;L2 ≤ C because ofLemma 2.1, we thus obtain the first inequality indi-cated in this lemma from2.11 by applying Gronwall’s lemma and then Sobolev’s inequality
To prove the second part, we multiply1.5 by bxxand integrate the resulting equation over0, T × Ω to deduce that
T
0
Ωbxx2x, tdx dt
≤ C
T
0
Ωbt2wx2 u2
xb2 u2bx2
x, tdx dt
≤ C C sup
t ∈0,T bt2
L ∞
T
0
u x t2
L 2dt C
T
0
u x t2
L 2bx t2
L 2dt
≤ C C sup
t ∈0,T bt2
L ∞bx t2
L 2
T
0
u x t2
L 2dt ≤ C,
2.13
where we have used Cauchy-Schwarz’s inequality, Sobolev’s inequality2.12,Lemma 2.1, and the first part of the lemma This completes the proof ofLemma 2.3
Lemma 2.4 The following estimates hold for the velocity u, w:
sup
0≤t≤T
u tL∞u x tL2
√ρu tL20,T;L2≤ C,
sup
0≤t≤TwtL∞wx tL2
√ρw tL20,T;L2≤ C. 2.14
Trang 6Proof Multiplying1.3 by u tand then integrating overΩ, by Young’s inequality we obtain
λ
2
d
dt
Ωu2x dx1
2
Ωρu2t dx≤ 1
2
Ωρu2u2x dx
Ωpu tx dx dt1
2
Ω|b|2u xt dx. 2.15
It follows from1.2 and 1.3 that
Ωpu tx dx d
dt
Ωpu x dx− R
2λ
d dt
Ωρ γ1u2dx−R γ − 2
2λ
Ωρ γ1u2u x dx
2λ
Ωp2u x dx− 1
λ
Ωpu
ρuu x b·bx
dx γ − 1
Ωpu2x dx,
1
2
Ω|b|2u xt dx 1
2
d dt
Ω|b|2u x dx−
Ωb·bt u x dx.
2.16
Thus, inserting2.16 into 2.15, and integrating over 0, t, we see that
Ωu2x x, tdx
t
0
Ωρu2t dx ds
≤ C C
Ω
pu x ρ γ1u2 |b|2u x x,tdx
C
t
0
Ω
ρu2u2x ρ γ1u2u x p2u x pub·b x pu2
xb·bt u xdx ds,
2.17
where the terms on the right-hand side can be bounded by using Lemmas2.1–2.3as follows:
Ω
pu x ρ γ1u2 |b|2u x tdx ≤ Cδ δ
Ωu2x tdx, δ > 0, 2.18 t
0
Ω
ρu2u2x ρ γ1u2u x p2u x pu2
x
dx ds
≤ C C
t
0
max
x∈Ω u2·, su x s2
L 2ds ≤ C C
t
0
u x s4
L 2ds,
2.19 t
0
Ω
pub·bx b·bt u xdx ds ≤ C t
0
Ω
ρu2bx2bt2 u2
x
dx dt ≤ C. 2.20
Therefore, taking δ appropriately small, we conclude from2.17–2.20 that
Ωu2x tdx
t
0
Ωρu2t dx ds ≤ C C
t
0
u x s4
where, combined with the fact that u xL20,T;L2 ≤ C due to Lemma 2.1, we obtain the first part of Lemma 2.4 by applying Gronwall’s lemma and then Sobolev’s inequality Similarly, multiplying1.4 by wtand integrating the resulting equation overΩ, we get that
μ
2
d
dt
Ωwx2
dx 1 2
Ωρwt2
dx≤
Ω
1
2ρu
2wx2 bt·wx
dx− d
dt
Ωb·wx dx, 2.22
Trang 7where we have also used Cauchy-Schwarz’s inequality Integration of2.22 over 0, t gives
Ωwx x, t2
dx
t
0
Ωρwt2
dx ds ≤ C 1
2
Ωwx x, t2
dx
C
t
0
Ωbt2wx2
dx ds ≤ C 1
2
Ωwx x, t2
dx,
2.23
where Lemmas2.1–2.3and the first conclusion of this lemma have been used Therefore, from the above inequality we obtain the second part, and soLemma 2.4is proved
Notice that1.3, 1.4 can be written as follows:
where G : λu x − p − |b|2/2 and K : μw x b Thus, by Lemmas2.1–2.4, we see that
G,KL∞0,T;L2 G x , K xL20,T;L2≤ C, 2.25 which immediately implies
T
0
u x t2
L ∞dt ≤ C
T
0
G2
L ∞ p2
L ∞ b4
L ∞
tdt
≤ C
T
0
G2
L 2G x2
L 2 p2
L ∞ b4
L ∞
tdt ≤ C,
T
0
wx t2
L ∞dt ≤ C
T
0
K2
L ∞ b2
L ∞
tdt
≤ C
T
0
K2
L 2Kx2
L 2 b2
L ∞
tdt ≤ C.
2.26
Hence, we have the following lemma
Lemma 2.5 There exists a positive constant C, such that
T
0
u x t2
L ∞G x t2
L 2
dt ≤ C,
T
0
wx t2
L ∞Kx t2
L 2
where G : λu x − p − |b|2/2 and K : μw x b.
To prove the uniqueness of strong solutions, we still need the following estimates
Lemma 2.6 The pressure pρ Rρ γ satisfies sup0≤t≤Tp x ·, tL2 ≤ C Furthermore, if the
compati-bility conditions1.9, 1.10 hold, then
sup
0≤t≤T
Ωρ
u2t wt2
x, tdx
T
0
Ω
u2txwtx2
dx dt ≤ C. 2.28
Trang 8Proof It follows from the continuity equation1.2 that p satisfies
which, differentiated with respect to x, leads to
Multiplying the above equation by p xand integrating overΩ, we deduce that
d
dtp x t2
L 2 ≤ C
Ω
p x2u x pp xu xx x,tdx
≤ Cu x tL∞p x t2
L 2p x t2
L 2bx t2
L 2G x t2
L 2
,
2.31
where we have used the inequality
u xx2
L 2 ≤ CG x2
L 2p x2
L 2bx2
L 2
which follows from the definition of G Therefore, applying the previous Lemmas2.1–2.5and Gronwall’s lemma, one has
sup
0≤t≤T
which proves the first part of the lemma
We are now in a position to prove the second part We first derive the estimate for the
longitudinal velocity u To this end, we firstly rewrite1.3 as
ρu t ρuu x − λu xx
pb2
2
x
Differentiation of 2.34 with respect to t gives
ρu tt ρuu xt − λu xxt
p1
2b2
xt
−ρ tu t uu x
which, multiplied by u tand integrated by parts overΩ, yields
1
2
d
dt
Ωρu2t dx λ
Ωu2xt dx−
Ω
p1
2b2
t
u xt dx −
Ω
ρu
u2t uu x u t
x ρu2
t u x
dx.
2.36
On the other hand, by virtue of1.2 we have
−
Ωp t u tx dx
Ωp x uu tx dxγ
2
d dt
Ωpu2x dx− γ
2
Ωp t u2x dx
γ 2
d dt
Ωpu2x dx
Ωp x uu tx dx γ
2
Ω
− puu2x
x γ − 1pu3
x
dx,
2.37
Trang 9from which and2.36 we see that
d
dt
Ω
1
2ρu
2
t γ
2pu
2
x
dx λ
Ωu2tx dx
≤
Ω
2ρ|u|u tu tx ρ|u|u tu x2 ρ|u|2u tu xx ρ|u|2u xu tx
ρu t2u x p x |u|u tx γp|u|u xu xx γγ − 1
2 pu x3 |b|btu txdx
≡9
j1
I j
2.38
Using the previous lemmas and Young’s inequality, we can estimate each term on the right-hand side of2.38 as follows with a small positive constant :
I1 ≤ 2ρ 1/2
L ∞uL∞ρu tL2u txL2 ≤ u tx2
L 2 C −1ρu t2
L 2, I2 ≤ ρ 1/2
L ∞uL∞ρu tL2u xL2u xL∞ ≤ Cu x2
L ∞ Cρu t2
L 2, I3 ≤ ρ 1/2
L ∞u2
L ∞ρu tL2u xxL2 ≤ Cu xx2
L 2 Cρu t2
L 2, I4 ≤ ρL ∞u2
L ∞u xL2u txL2 ≤ u tx2
L 2 C −1, I5 ≤ u xL∞ρu t2
L 2,
I6≤ uL∞p xL2u txL2 ≤ u tx2
L 2 C −1, I7 ≤ γpL∞uL∞u xL2u xxL2 ≤ C Cu xx2
L 2, I8 ≤ CpL∞u xL∞u x2
L 2 ≤ C Cu x2
L ∞, I9≤ bL∞btL2u txL2 ≤ u tx2
L 2 C −1bt2
L 2.
2.39
Putting the above estimates into2.38 and taking sufficiently small, we arrive at
d
dt
Ω
ρu2t pu2
x
dx
Ωu2tx dx
≤ C1
ρu t2
L 2u xx2
L 2 u x2
L ∞bt2
L 2u xL∞
ρu t2
L 2
,
2.40
so that, using the relation between G x and u xxagain, one infers from2.40 that
d
dt
Ω
ρu2t pu2
x
dx
Ωu2tx dx
≤ C1ρu t2
L 2G x2
L 2u x2
L ∞bt2
L 2
Cu xL∞
ρu t2
L 2,
2.41
Trang 10where the first term on the right-hand side of2.41 is integrable on 0, T due to the previous
lemmas Thus, integrating2.41 over τ, t ⊂ 0, T, we deduce from 1.3 that
Ωρu2t x, tdx
t
τ
Ωu2tx dx ds
≤
Ωρu2t x, τdx C
1
t
0
u x sL∞
ρu t s2
L 2ds
Ωρ−1
λu xx−
p 1
2b2
x
− ρuu x
2
x, τdx
C
1
t
0
u x sL∞
ρu t s2
L 2ds
.
2.42
Letting τ → 0 and using the compatibility condition 1.9, we easily obtain from 2.42 that
Ωρu2t x, tdx
t
0
Ωu2tx dx ds ≤ C
1
t
0
u x sL∞
ρu t s2
L 2ds
which, together withu xL1
0,T;L∞ ≤ C and Gronwall’s lemma, immediately yields
sup
0≤t≤T
ρu t t2
L 2u tx2
L 20,T;L2 ≤ C. 2.44
In a same manner as that in the derivation of2.44, we can show the analogous
esti-mate for the transverse velocity w by using the previous lemmas,2.44, and the compatibility condition1.10 as well Thus, we complete the proof ofLemma 2.6
Remark 2.7 From the a priori estimates established above, one sees that the compatibility
con-ditions are used to obtain the second part ofLemma 2.6 only However, this is crucial in the proof of the uniqueness of strong solutions
3 Global existence of strong solutions
In this section, we prove the global existence of strong solutions to the problem1.2–1.7 by applying the a priori estimates given in the previous section As usual, we first mollify the initial data to get the existence of smooth approximate solutions For this purpose, we choose
the smooth approximate functions ρ
0and b
0such that
ρ 0∈ C2Ω, 0 < ≤ ρ
0≤ρ
0L∞ 1, ρ
0−→ ρ0 in H1,
b
0∈ C2Ω, b
0≤b0L∞ 1, b
0−→ b0 in H1
0.
3.1
Letu
0, w 0 ∈ C1
0Ω ∩ C3Ω, satisfying u
0, w 0 → u0, w0 in H1
0∩ H2, be the unique solution to the boundary value problems
λu 0xx
R
ρ 0γ
1
2b
02
x
ρ 01/2
f inΩ, u
0|x 0,1 0,
μw 0xx −b
0xρ 01/2
g inΩ, w
0|x 0,1 0,
3.2
... crucial in the proof of the uniqueness of strong solutions3 Global existence of strong solutions< /b>
In this section, we prove the global existence of strong solutions to the problem1.2–1.7...
Trang 10where the first term on the right-hand side of 2.41 is integrable on 0, T due to the previous
lemmas... applying the a priori estimates given in the previous section As usual, we first mollify the initial data to get the existence of smooth approximate solutions For this purpose, we choose
the