Global Strong Solution of a 2D coupled Parabolic-Hyperbolic Magnetohydrodynamic System

0.15 MB·

by  Ruikuan Liu

7 Global Strong Solution of a 2D coupled
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0 Parabolic-Hyperbolic Magnetohydrodynamic
2
System
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9 Ruikuan Liu∗ Jiayan Yang†
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∗,†Department of Mathematics, Sichuan University Chengdu,

] Sichuan 610064, P.R.China
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†Department of Mathematics, Southwest Medical UniversityLuzhou,
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Sichuan 646000, P.R.China
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t
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Abstract
[

1
Themainobjectiveofthispaperistostudytheglobalstrongsolu-
v
tionofthe parabolic-hyperbolicincompressiblemagnetohydrodynamic
3
(MHD)modelintwodimensionalspace. BasedonAgmon,Douglisand
5
3 Nirenberg’sestimatesforthestationaryStokesequationandtheSolon-
8 nikov’s theorem of Lp-Lq-estimates for the evolution Stokes equation,
0 it is shown that the mixed-type MHD equations exist a global strong
.
1 solution.
0
7 keywords
1
: Global strong solution, Magnetohydrodynamics, Stokes equation, Lp-
v Lq-estimates.
i
X
r 1 Introduction
a
Weconsiderthefollowing2-Dincompressiblemagnetohydrodynamic(MHD)
model,whichdescribestheinteractionbetweenmovingconductivefluidsand
electromagnetic fields in [10],
∂u 1 ρ
e
+(u·∇)u= ν∆u− ∇p+ u×rotA+f(x), in Ω×[0,T),
 ∂t ρ0 ρ0
 ∂2A 1 ρ
 = ∆A+ eu−∇Φ, in Ω×[0,T),(1.1)
 ∂t2 ǫ0µ0 ǫ0
∇·u= 0, in Ω×[0,T),


∇·A= 0, in Ω×[0,T).




∗Email:[email protected]. Supportedby NSFC(11401479)
†Corresponding author:jiayan [email protected];
1
Here Ω ⊂ R2 is a bounded smooth domain, T is any fixed time. u(x,t),
A(x,t), p(x,t) are the velocity field, the magnetic potential and the pressure
function, respectively. Φ = ∂A0 represents the magnetic pressure with the
∂t
scalar electromagnetic potential A . The constants ν, ρ , ρ , ǫ , µ denote
0 0 e 0 0
kinetic viscosity, mass density, equivalent charge density, electric permittiv-
ity and magnetic permeability of free space.
In this paper, we focus on the system (1.1) with the initial-boundary
conditions
u(0,x) = φ(x), A(0,x) = ψ(x), A (0,x) = η(x), in Ω, (1.2)
t
u(t,x) = 0, A(t,x) =0, on ∂Ω×[0,T). (1.3)
Note that the MHD model (1.1) is established based on the the New-
ton’s second law and the Maxwell equations for the electromagnetic fields
in [10]. In addition, the global weak solutions of the corresponding 3-D
MHD model(1.1) with (1.2)-(1.3) has been obtained by using the Galerkin
technique and standard energy estimates in [10]. In this paper, what we are
concerned is the global strong solution of the 2-D MHD model (1.1) with
the initial-boundary conditions (1.2)−(1.3).
It is known that there have been huge mathematical studies on the ex-
istence of solutions to the N-dimension(N ≥ 2) classical MHD model estab-
lishedbyChandrasekhar[4]. Inparticular,DuvautandLions[5]constructed
aglobalweaksolutionandthelocalstrongsolutiontothe3-DclassicalMHD
equations the initial boundary value problem, and properties of such solu-
tions have been investigated by Sermange and Temam in [15]. Furthermore,
some sufficient conditions for smoothness were presented for the weak solu-
tion to the 3-D classical MHD equations in [7] andsome sufficient conditions
oflocalregularityofsuitableweaksolutionstothe3-DclassicalMHDsystem
for the points belonging to a C3-smooth part of the boundarywere obtained
in [18]. Also, the global strong solutions for heat conducting 3-D classical
magnetohydrodynamic flows with non-negative density were proved in [21].
Moreover, let’s recall some known results for the 2-D classical and gen-
eralized MHD equations. It is noticed that the 2D classical MHD equations
admits a unique global strong solution in [5,15]. Furthermore, Ren, Wu,
et.al [14] have proved the global existence and the decay estimates of small
smooth solution for the 2-D classical MHD equations without magnetic dif-
fusion and Cao, Regmi and Wu [3] have obtained the global regularity for
the 2-D classical MHD equations with mixed partial dissipation and mag-
netic diffusion. Besides, Regmi [13] established the global weak solution
for 2-D classical MHD equations with partial dissipation and vertical dif-
fusion. There are also very interesting investigations about the existence
of strong solutions to the 2-D classical and generalized MHD equations,
see [8,9,12,15,19,20,22] and references therein.
2
However, it is worth pointing out that the incompressible MHD system
(1.1)is a mixed-typedifferential differenceequation, which is combined with
the parabolic equation (1.1) and the hyperbolic equation (1.1) . The main
1 2
challenge in obtaining global strong solution of 2-D MHD model(1.1) with
(1.2)-(1.3) is the estimate for ||u×rotA|| and ||(u·∇)u|| .
L∞(0,T;L2) L∞(0,T;L2)
The difficulty is overcome by applying the Solonnikov’s theorem [6,11,16]
of Lp − Lq-estimates for the non-stationary Stokes equations and Agmon,
Douglis and Nirenberg’s estimates [1,2,11] for the stationary Stokes equa-
tions. As is known, Solonnikov [16] first gave the proof of Maximal Lp-Lq-
estimates for the Stokes equation (2.4) using potential theoretic arguments.
Recently, Geissert, Hess, Hieber et.al [6] provided a short proof of the cor-
responding Solonnikov’s theorem in [16].
The rest of this article is organized as follows. In Section 2, we introduce
some elementary function spaces, a vital embedding theorem and some reg-
ularity results of both the non-stationary Stokes equations and stationary
Stokes equations. Section 3 is mainly devoted to the proof of global strong
solution of (1.1)−(1.3).
2 Preliminaries
2.1 Notations and definitions
First, we introduce some notations and conventions used throughout this
paper.
Let Ω ⊂ R2 be a bounded sufficiently smooth domain. Let Hτ(Ω)(τ =
1,2) be the general Sobolev space on Ω with the norm ||·||Hτ and L2(Ω) be
the Hilbert space with the usual norm ||·||. The space H1(Ω) we mean that
0
the completion of C∞(Ω) under the norm ||·|| . If ̥ is a Banach space,
0 H1
we denote by Lp(0,T;̥)(1 < p < ∞) the Banach space of the ̥-valued
functions defined in the interval (0,T) that are Lp-integrable.
We also consider the following spaces of divergence-free functions (see
Temam [17])
X = {u∈ C∞(Ω,R2) | divu= 0 in Ω},
0
Y = the closure of X in L2(Ω,R2)
= {u∈ L2(Ω,R2) | divu= 0 in Ω},
Z = the closure of X in H1(Ω,R2)
= {u∈ H1(Ω,R2) | divu= 0 in Ω}.
0
Definition 2.1. Suppose that φ,η ∈ Y, ψ ∈ Z. For any T > 0, a vector
function (u,A) is called a global weak solution of problem (1.1)−(1.3) on
(0,T)×Ω if it satisfies the following conditions:
3
1. u∈ L2(0,T;Z)∩L∞(0,T;Y),
2. A∈ L∞(0,T;Z), A ∈ L∞(0,T;Y),
t
3. For any function v ∈ X, there hold
t ρ
e
u·vdx+ (u·∇)u·v+ν∇u·∇v− (u×rotA)·vdxdt
Z Z Z ρ
Ω 0 Ω 0
t
= f ·vdxdt+ φ·vdx
Z Z Z
0 Ω Ω
and
∂A t 1 ρ
e
·vdx+ ∇u·∇v+ u·vdxdt = ηvdx.
Z ∂t Z Z ǫ µ ǫ Z
Ω 0 Ω 0 0 0 Ω
Now, we define strong solution of the problem (1.1)−(1.3).
Definition 2.2. Suppose that φ,ψ ∈ H2(Ω,R2)∩Z, η ∈ Z, Φ ∈ L2(0,T;
H1(Ω)). (u,A) is called a global strong solution to (1.1)−(1.3), if (u,A)
0
satisfy
u∈ L∞(0,∞;H2(Ω,R2)∩Z), u ∈ L∞(0,∞;Y)∩L2 (0,∞;Z)
loc t loc loc
p ∈ L∞(0,∞;H1(Ω)),
loc
A∈ L∞(0,∞;H2(Ω,R2)∩Z), A ∈L∞(0,∞;Z), A ∈ L∞(0,∞;Y).
loc t loc tt loc
Furthermore, both (1.1) and (1.3) hold almost everywhere in Ω×(0,T).
2.2 Lemmas
Some more lemmas will be frequently used later. One is the following em-
bedding result in [11].
Lemma 2.3. For any k ≥0, the following hold
Lp((0,T),Wk+1,p(Ω))∩L∞(0,T;Lr(Ω)) ⊂ Lq(0,T;Wk,q(Ω)), (2.1)
where q = (r(k + 1)p + np)/(rk + n). In the special case of k = 0, (2.1)
equals to
Lp(0,T;W1,p(Ω))∩L∞(0,T;Lr(Ω)) ⊂ Lq((Ω)×(0,T)), (2.2)
provided that q = (n+r)p/n.
Proof. From Gagliardo-Nirenberg interpolation inequality, we have
||u|| ≤ C||u||θ ||u||1−θ , 0≤ θ ≤ 1, (2.3)
Wk,q Wm,p Wj,r
4
provided that
n n n
θ m− +(1−θ) j − ≥k− ,
(cid:18) p(cid:19) (cid:18) r(cid:19) q
where C is a constant independent of u.
Inserting j =0, q ≥ p, m = k+1 and θ = p into (2.3), it is easy to see that
q
1 1 1(1−p/q)
q q r
|Dku|qdx ≤ C |Dk+1u|pdx |u|rdx ,
(cid:18)Z (cid:19) (cid:18)Z (cid:19) (cid:18)Z (cid:19)
Ω Ω Ω
(r(k+1)p+np)
where q = .
rk+n
Then we get
T T
|Dku|qdxdt ≤ C sup ||u||(q−p)r |Dk+1u|pdxdt,
Z Z Lr Z Z
0 Ω 0≤t≤T 0 Ω
which implies (2.1) and (2.2).
The other lemma is responsible for the estimates for u,p,u and follows
t
from the Lp-Lq-estimates [6,16] for non-stationary Stokes equations. For its
proof, refer to [6,16].
Let us consider the following Stokes equations
∂u
= ν∆u−∇p+f(x,t),
 ∂t
∇·u =0, (2.4)


u| = 0,
∂Ω
u(0) = u ,
 0




where ν > 0 is a constant.
Lemma 2.4. Let Ω ⊂ Rn(n = 2,3) be a domain with compact C3-boundary,
1 < r,p < ∞, 0 < T < ∞. Then for any f ∈ Lr(0,T;Lq(Ω,Rn)) and
u ∈ W2,q(Ω,Rn)) there exists a unique solution (u,p) of (2.4) satisfying
0
u ∈Lr(0,T;W2,q(Ω,Rn)), u ∈Lr(0,T;Lq(Ω,Rn)),
t
p ∈ Lr(0,T;W1,q(Ω))
such that
||u|| +||u || +||p||
Lr(0,T;W2,q) t Lr(0,T;Lq) Lr(0,T;W1,q)
≤ C(||f|| +||u || ),
Lr(0,T;Lq) 0 W2,q
where C > 0 is a constant.
5
Finally, we give some regularity results for the stationary Stokes system.
For its proof, refer to [1,2,11].
Lemma 2.5. Assume that (v,p) ∈ W2,p(Ω,Rn)×W1,p(Ω)(1 <p < ∞) is a
weak solution of the stationary Stokes equations
−ν∆v−∇p = F(x), in Ω,
∇·v = 0, in Ω,


v| = 0, on ∂Ω,
∂Ω


and F ∈ Wk,q(Ω,Rn)(k ≥ 0,1 < q < ∞). Then it holds that
(v,p) ∈ Wk+2,q(Ω,Rn)×Wk+1,q(Ω)
and
||v||Wk+2,q +||p||Wk+1,q ≤ C(||F||Wk,q +||(u,p)||Lq)
with some constant C depending on n, Ω and q.
3 Main Results
In this section, we state the global weak solution existence theorem and the
global strong solution existence one for the problem (1.1)−(1.3), and also
prove them.
Theorem 3.1. Let the initial value φ,η ∈ Y, ψ ∈ Z. If f ∈ Y,Φ ∈
L2(0,T;H1(Ω)), then there exists a global weak solution for the problem
0
(1.1)−(1.3).
Proof. By the standard Galerkin method and the similar estimates in [10],
the existence of global weak solution of (1.1)−(1.3) is also valid, we omit
it.
Theorem 3.2. Let Ω be a bounded domain with compact C3-boundary. If
φ,ψ ∈ H2(Ω,R2) ∩ Z, η ∈ Z, for any f ∈ Y,Φ ∈ L2(0,T;H1(Ω)), then
0
there exists a global strong solution for the problem (1.1)−(1.3), i.e., for any
0 < T < ∞
u ∈ L∞(0,T;H2(Ω,R2)∩Z), u ∈L∞(0,T;Y)∩L2(0,T;Z)
t
p ∈ L∞(0,T;H1(Ω)),
A ∈ L∞(0,T;H2(Ω,R2)∩Z), A ∈ L∞(0,T;Z), A ∈ L∞(0,T;Y).
t tt
Proof. The proof can be divided into 3 steps. We will use the same generic
constant C to denote various constants that depend on µ ,ρ ,ρ ,ǫ and T
0 0 e 0
only.
Step 1 The estimates and regularity for A.
6
From Theorem 3.1, for any 0 < T < ∞, we get the global weak solution
u∈ L2(0,T;Z)∩L∞(0,T;Y),
(3.1)
A∈ L∞(0,T;Z), A ∈ L∞(0,T;Y).
t
Multiplying both sides of (1.1) by −∆A and integrating over Ω, we have
2 t
1 d 1 ρ
|∇A |2+ |∆A|2dx = e ∇u∇A dx (3.2)
t t
2dt(cid:18)Z ǫ µ (cid:19) ǫ Z
Ω 0 0 0 Ω
since divA= 0 and (1.3).
Using the H¨older inequality, it is easy to see that
d 1 1 ρ2
||∇A ||2 + ||∆A||2 ≤ 2 ||∇A ||2 + ||∆A||2 + e||∇u||2 .
dt(cid:18) t L2 ǫ µ L2(cid:19) (cid:18) t L2 ǫ µ L2 ǫ2 L2(cid:19)
0 0 0 0 0
(3.3)
Then, by the Gronwall inequality, (3.3) implies
ρ2 T
||∇A ||2 +||∆A||2 ≤ e2T ||∆ψ|| +||∇η|| +2 0 ||∇u||2 ds ,
t L2 L2 (cid:18) L2 L2 ǫ2 Z L2 (cid:19)
0 0
(3.4)
for ∀ 0 < T < ∞.
Therefore, we conclude that
∇A ∈ L∞(0,T;Y), ∆A∈ L∞(0,T;Y). (3.5)
t
Next, we need to derive an estimate on ||A || .
tt L∞(0,T;Y)
Multiplying both sides of Eqs. (1.1) by A integrating over Ω lead to
2 tt
1 ρ
|A |2dx = ∆AA dx+ 0 uA dx (3.6)
tt tt tt
Z ǫ µ Z ǫ Z
Ω 0 0 Ω 0 Ω
since − ∇ΦA dx = ΦdivA dx = 0.
Ω tt Ω tt
Using thRe H¨oder inequaRlity and Young inequality, we deduce from (3.6) that
1 ρ2 1
|A |2dx ≤ |∆A|2dx+ 0 |u|2dx+ |A |2dx. (3.7)
Z tt ǫ2µ2 Z ǫ2 Z 2 Z tt
Ω 0 0 Ω 0 Ω Ω
It is easy to see that
2 2ρ2
ess sup |A |2dx ≤ sup |∆A|2dx+ sup 0 |u|2dx. (3.8)
Z tt ǫ2µ2 Z ǫ2 Z
0≤t≤T Ω 0≤t≤T 0 0 Ω 0≤t≤T 0 Ω
Putting the estimates (3.1), (3.5) and (3.8) together, we have
A ∈ L∞(0,T;Y). (3.9)
tt
Hence, (3.5) and (3.9) imply the regularity for A.
4 4
Step 2 The L3-L3-estimates for u·∇u and u×A.
7
From (3.1) and Lemma 2.3(the case that k=0), it is easy to check that
u∈ L4((0,T)×Ω). (3.10)
Note that
2 1
T |Du|43|u|43dxdt ≤ T |Du|2dxdt 3 T |u|4dxdt 3,
Z Z (cid:18)Z Z (cid:19) (cid:18)Z Z (cid:19)
0 Ω 0 Ω 0 Ω
(3.11)
which implies that
u·∇u∈ L43(0,T;L43(Ω,R2)). (3.12)
Combining (3.1) and (3.10), we get
1 2
T |u×rotA|43dxdt ≤ T |u|4dxdt 3 T |rotA|2dxdt 3
Z Z (cid:18)Z Z (cid:19) (cid:18)Z Z (cid:19)
0 Ω 0 Ω 0 Ω
1 2
T 3 T 3
≤ C |u|4dxdt |∇A|2dxdt
(cid:18)Z Z (cid:19) (cid:18)Z Z (cid:19)
0 Ω 0 Ω
< ∞,
(3.13)
which in turn implies
u×rotA ∈L43(0,T;L43(Ω,R2)). (3.14)
Recall that (u,p) satisfying the following Stokes system
∂u 1
= ν∆u− ∇p+F(x,t),
 ∂t ρ0

∇·u= 0, (3.15)



u| = 0,
∂Ω
u(0) = φ,



where F(x,t) = f −(u·∇)u+ ρe(u×rotA).
ρ0
By (3.12)and(3.14), we getF ∈ L34(0,T;L43(Ω,R2)). Applyingthisinto
Lemma 2.4, we obtain that
u∈ L43(0,T;W2,43(Ω,R2)), ut ∈ L34(0,T;L43(Ω,R2)),
(3.16)
p ∈ L34(0,T;W1,43(Ω)).
Inthenextstep,theLemma2.5willbeused,since(3.15)canberewritten
as the following stationary Stokes equations
1
−ν∆u+ ∇p = F(x,t),
 ρ0
∇·u = 0, e (3.17)



u| = 0,
∂Ω
u(0) = φ,




8
where F(x,t) = f −(u·∇)u+ ρe(u×rotA)−u .
ρ0 t
Step 3 The estimate for ||F|| .
e L∞(Ω,L2(Ω,R2))
(i) The estimate for ||∇eu||L∞(0,T;L2).
Multiplying Eq. (1.1) by u and integrating over Ω, we have
1 t
µ d ρ
|∇u|2dx+ |u |2dx = −(u·∇)u·u + e(u×rotA)u dx. (3.18)
t t t
2dt Z Z Z ρ
Ω Ω Ω 0
Note that the following continuous embeddings
W2,34(Ω,R2) ֒→ W1,4(Ω,R2) ֒→ C12(Ω,R2)֒→ C0(Ω,R2). (3.19)
Combining (3.19), H¨older inequality and ǫ−Younginequality, we derive that
|(u·∇)u·u |dx ≤ C||u || ||u|| ||∇u||
t t L2 C0 L2
Z
Ω (3.20)
1
≤ ||u ||2 +C2||u||2 ||∇u||2
4 t L2 C0 L2
and
ρ
e
|(u×rotA)u |dx ≤ C||u|| ||∇A|| ||u ||
ρ Z t C0 L2 t L2
0 Ω (3.21)
1
≤ ||u ||2 +C2||u||2 ||∇A||2 ,
4 t L2 C0 L2
which together with Gronwall’s inequality implies
ess sup ||∇u|| < ∞.
L2 (3.22)
0<t<T
(ii) The estimate for ||u || .
t L∞(0,T;L2)
Taking t-derivative of Eq. (1.1) , then one gets that
1
1 ρ ρ
e e
u −µ∆u = −(u ·∇)u−(u·∇)u − ∇p + u ×rotA+ u×rotA . (3.23)
tt t t t t t t
ρ ρ ρ
0 0 0
Multiplying (3.23) by u and integrating over Ω, we obtain
t
1 d ρ
|u |2dx+µ |∇u |2dx = −(u ·∇)u·u + e(u×rotA )u dx.
t t t t t t
2dt Z Z Z ρ
Ω Ω Ω 0
(3.24)
since
1
(u ×rotA)·u = 0, (u·∇)u ·u dx = − u2divudx = 0.
t t Z t t Z 2 t
Ω Ω
9
Next, we estimate the two terms on the right hand of (3.24). By (3.19)
and integrating by parts yield
− (u ·∇)u·u dx = uiuj∂ uj −ui∂ (ujuj)dx
t t t i t t i t
Z Z
Ω Ω
(3.25)
= uiuj∂ ujdx ≤ C||u|| ||u ||2 +||∇u ||2 .
Z t i t C0(cid:18) t L2 t L2(cid:19)
Ω
And similarly,
ρ Cρ
e e
|(u×rotA )u |dx ≤ |uu ∇A |dx
t t t t
ρ Z ρ Z
0 Ω 0 Ω
(3.26)
Cρ
≤ e||u|| ||u ||2 +||∇A ||2 .
ρ C0(cid:18) t L2 t L2(cid:19)
0
Hence, by (3.24), (3.25) and (3.26), we get that
1 d
|u |2dx+µ |∇u |2dx
t t
2dt Z Z
Ω Ω
Cρ
≤ ||u|| (1+Cρ /ρ )||u ||2 +||∇u ||2 + e||u|| ||∇A ||2 ,
C0(cid:18) e 0 t L2 t L2(cid:19) ρ C0 t L2
0
(3.27)
which together with Gronwall’s inequality completes the estimate
ess sup ||u (t)|| < ∞.
t L2 (3.28)
0<t<T
(iii) The estimates for ||(u·∇u)|| and ||u×A|| .
L∞(0,T;L2) L∞(0,T;L2)
From (3.22), it is easy to see that
∇u∈ L∞(0,T;Y). (3.29)
Hence
u ∈L∞(0,T;H1).
It is known that H1 ֒→ Lq(1 < q < ∞) when n = 2. Note that
1 1 2−r
|(u·∇)u|rdx r ≤ |∇u|2dx 2 |u|22−rrdx 2r < ∞ (3.30)
(cid:18)Z (cid:19) (cid:18)Z (cid:19) (cid:18)Z (cid:19)
Ω Ω Ω
provided that 1 < r < 2. Hence
(u·∇)u∈ L∞(0,T;Lr(Ω,R2)). (3.31)
By using the H¨older inequality and the Sobolev embedding theorem, it
follows that
|u×rotA|2dx ≤ C |u|2|∇A|2dx
Z Z
Ω Ω
≤ C |u|4dx+ |∇A|4dx (3.32)
(cid:18)Z Z (cid:19)
Ω Ω
≤ C |∇u|2dx+ |∆A|2dx .
(cid:18)Z Z (cid:19)
Ω Ω
10