Uniform existence of the 1-d complete equations for an electromagnetic fluid

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J. Math. Anal. Appl. ••• (••••) •••–•••

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Uniform existence of the 1-d complete equations for an electromagnetic fluid Jishan Fan a , Yuxi Hu b,∗ a b

Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, PR China Institute of Applied Physics and Computational Mathematics, Beijing 100088, PR China

a r t i c l e

i n f o

Article history: Received 13 November 2013 Available online xxxx Submitted by D. Wang Keywords: MHD Complete equations Dielectric constant

a b s t r a c t In this paper, we prove the global-in-time and uniform-in- existence of the 1-d complete equations for an electromagnetic fluid in a bounded interval without vacuum, where  is the dielectric constant. Consequently, the limit as  → 0 can be established. This approximation is usually referred to as the magnetohydrodynamic approximation, and is equivalent to the neglect of the displacement current. © 2014 Elsevier Inc. All rights reserved.

1. Introduction In this paper, we consider the complete equations for an electromagnetic fluid, where the unknowns  := are the mass density ρ, the velocity u := (u1 , u2 , u3 ), the absolute temperature θ, the electric field E 1 2 3 1 2 3  := (B , B , B ) and the electric charge density ρe . We refer (E , E , E ), the magnetic flux density B the reader to [2] for the explicit form of this system. However, there is no local existence result for this system. The reason is as follows: When the hydrodynamic quantities (ρ, u, θ) are regarded as known smooth  B,  ρe ) form a first order hyperbolic system functions, the equations for the electromagnetic quantities (E, (nonlinear Maxwell system), which is neither symmetric hyperbolic nor strictly hyperbolic in the three  B,  ρe ) are independent of the dimensional case. For this reason, we assume that all the unknowns (ρ, u, θ, E, third component of the space variable (x1 , x2 , x3 ) and that   u := u1 , u2 , 0 ,

   := 0, 0, E 3 , E

   := B 1 , B 2 , 0 . B

In this case, we have ρe = 0. The reduced system is symmetric hyperbolic–parabolic. Kawashima and Shizuta [5–7] proved the local existence of smooth solutions for large data and global existence of smooth solutions for small data and studied the limit as  → 0 when Ω := R2 . Similar results have been obtained * Corresponding author. E-mail addresses: [email protected] (J. Fan), [email protected] (Y. Hu). http://dx.doi.org/10.1016/j.jmaa.2014.04.052 0022-247X/© 2014 Elsevier Inc. All rights reserved.

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in [3,4,8–10]. Very recently, Fan and Hu [1] proved the local-in-time and uniform-in- existence of strong solutions with vacuum and Ω is a bounded domain in R2 .  B,  ρe ) The aim of this paper is to study the 1-d problem; we assume that all the unknowns (ρ, u, θ, E, are independent of the second and the third components of space variable (x1 , x2 , x3 ) and that u := (u, 0, 0),

 := (0, 0, E), E

 := (0, b, 0). B

In this case, we have ρe = 0. We consider therefore the following symmetric system of the unknowns (ρ, u, θ, E, b): ρt + (ρu)x = 0,  2  (ρu)t + ρu + p x = (λ + 2μ)uxx − (E + ub)b,

(1.1)

Et − bx + E + ub = 0,

(1.3)

bt − Ex = 0,  (ρθ)t + (ρuθ)x − k(θ)θx x + pux = (λ + 2μ)u2x + (E + ub)2 ,

(1.4)



(1.2)

(1.5)

in QT := Ω × (0, T ), with the initial and boundary conditions (u, E, b, θx ) = (0, 0, 0, 0) on ∂Ω × (0, T ), (ρ, u, E, b, θ)(·, 0) = (ρ0 , u0 , E0 , b0 , θ0 )

in Ω := (0, 1).

(1.6) (1.7)

Here the pressure p := ρθ and λ and μ are viscosity constants satisfying λ + 2μ > 0 and for simplicity we will take λ + 2μ = 1.  is the dielectric constant and we will assume 0 <  < 1. The heat conductivity k(θ) is a smooth function of θ and we will assume that    1 1 + θq ≤ k(θ) ≤ C 1 + θq C

(1.8)

for some positive constants C ≥ 1 and q > 1. For the initial data, we assume that 0 < inf ρ0 ≤ ρ0 (x) ≤ sup ρ0 < ∞, u0 , E0 , b0 ∈ H01 (Ω)

ρ0 , θ0 ∈ H 1 (Ω),

and θ0 ≥ 0 in Ω.

(1.9)

Now we are ready to state the main result of this paper. Theorem 1.1. Assume that the initial data satisfy (1.9) and the heat conductivity satisfies (1.8). Then for any T > 0 independent of , the problem (1.1)–(1.7) has a unique strong solution (ρ, u, θ, E, b) such that     ρt ∈ L∞ 0, T ; L2 , ρ ≥ C > 0 in QT , ρ ∈ L∞ 0, T ; H 1 ,       ∞ 1 2 2 ut ∈ L2 0, T ; L2 , u ∈ L 0, T ; H0 ∩ L 0, T ; H ,       E, b ∈ L∞ 0, T ; H01 , Et ∈ L2 0, T ; L2 , bt ∈ L∞ 0, T ; L2 ,       θ ∈ L∞ 0, T ; H 1 ∩ L2 0, T ; H 2 , θt ∈ L2 0, T ; L2 , θ ≥ 0 in QT ,

(1.10)

with the corresponding norms that are uniformly bounded with respect to  > 0. Since the local well-posedness has been proved in [11], we only need to establish the uniform-in- estimates.

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2. Uniform a priori estimates This section is devoted to the derivation of a priori estimates of (ρ, u, E, b, θ) which are independent of . To begin with, we notice that the total mass and energy are conserved. In fact, let 1 1 1 E := ρθ + ρu2 + E 2 + b2 2 2 2

and s := ln θ − ln ρ

denote the total energy and the entropy, respectively, then we have    1 2 Et + ρθ + ρu + p u = (uux )x + (kθx )x + (bE)x , 2 x   kθ2 u2 + (E + ub)2 k θx + 2x . (ρs)t + (ρus)x − = x θ θ θ x

(2.1) (2.2)

Integrating (1.1), (2.1) and (2.2) over Ω × (0, t), we get Lemma 2.1. 





Edx =

ρ0 dx =: m > 0,

ρdx = 

T 

 E0 dx,

k(θ)θx2 u2x + (E + ub)2 + dxdt ≤ C. θ θ2

ρ ln ρ + ρ|ln θ|dx + 0

Here and later on C will denote a positive constant independent of . The following lemma gives point wise uniform upper and lower bounds of the density. Lemma 2.2. 1 ≤ ρ(x, t) ≤ C, C

(2.3)

T  u2x + (E + ub)2 dxdt ≤ C.

(2.4)

0

Proof. Firstly, we note that there exists an x ˜ ∈ (0, 1) such that √

√ x √  ρ θdx 1 |θx | √ dy θ− = ( θ )x dy ≤ 2 ρdx θ x ˜

≤

1 2

 k 

≤C

θx2 dy θ2

1/2 

k(θ)θx2 dy θ2

θ dy k

1/2

1/2 ,

which gives T θ L∞ dt ≤ C 0

by Lemma 2.1.

(2.5)

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Now we set A :=

x 0

(E + ub)bdy, p˜ := ux − ρu2 − p − A, and t φ :=

x p˜(x, τ )dτ +

0

ρ0 (y)u0 (y)dy, 0

we find that x φx = ρu,

φx |∂Ω = 0,

φt = p˜,

φ|t=0 =

ρ0 u0 dy.

(2.6)

0

By virtue of Lemma 2.1 and the Cauchy–Schwartz inequality,







φx L∞ (0,T ;L1 ) ≤ C,

φdx ≤ C, which gives φ L∞ (QT ) ≤ C.

(2.7)

Now, denoting F := eφ and using (2.6), we have after a straightforward calculation that Dt (ρF ) := ∂t (ρF ) + u∂x (ρF ) = ρF (−p − A) ≤ −ρF A, which implies  T ρF ≤ ρ0 F0 exp − Adt 0

T  ≤ ρ0 F0 exp



(E + ub)b dxdt



0

T 

1 (E + ub)2 + b2 θ L∞ (Ω) dxdt θ 4

≤ ρ0 F0 exp



0

≤C

(2.8)

by (2.5) and Lemma 2.1. Similarly, a calculation gives  Dt

1 ρF

 =

1 (p + A), ρF

which gives 1 1 ≤ exp ρF ρ0 F0

T

T ≤ C exp

0

|θ|L∞ + |A|L∞ dt 0

by (2.5), (2.8) and Lemma 2.1.

|p + A|L∞ dt ≤C

(2.9)

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(2.8) and (2.9) give (2.3). As a consequence of Lemma 2.1, (2.3) and (2.5), we have θ L2 (QT ) ≤ C,

(2.10)

which, by integrating (1.5) over QT and making use of the boundary conditions (1.6), yields (2.4).

2

Lemma 2.3. b L∞ (0,T ;H 1 ) ≤ C,

E L2 (0,T ;H 1 ) ≤ C.

(2.11)

Proof. Taking ∂x to (1.3) and (1.4), testing by Ex and bx and summing up the result and using the boundary condition (1.6) and (2.4), we obtain 1 d 2 dt





 Ex2 dx = −

Ex2 + b2x dx +

≤

1 2

≤

1 2

(ub)x Ex dx 

 Ex2 dx +

(ub)2x dx

 Ex2 dx + C ux 2L2 bx 2L2 ,

which leads to (2.11). 2 Lemma 2.4. T 

(1 + θq )θx2 dxdt ≤ C, θ1+α

(2.12)

0

T θ q+1−α dt ≤ C L∞

(2.13)

0

for any 0 < α < 1. Proof. Testing (1.5) by θ−α for any 0 < α < 1, using Lemma 2.1, (2.3) and (2.10), we see that T 

u2x + (E + ub)2 dxdt + α θα

0

T 

k(θ)θx2 dxdt θ1+α

0

T  =



(ρθ)t + (ρuθ)x + pux θ−α dxdt

0

=

=

1 1−α 1 1−α

T 

   1−α  + ρuθ1−α x dxdt + ρθ t

0

T 

pux θ−α dxdt

0



 ρθ1−α dx −

 T  ρ0 θ01−α dx + pux θ−α dxdt 0

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T  ≤C+

ρθ1−α ux dxdt 0

1 ≤C+ 2

T 

u2x dxdt + C θα

0

T  θ2−α dxdt, 0

which leads to (2.12).  q−α+1      θ (·, t)L∞ ≤ C θq−α+1 (·, t)L1 + C ∂x θq−α+1 (·, t)L1 q−α+1    ≤ C θ(·, t)Lq−α+1 + C θq−α θx (·, t)L1  q 2 q−α+1 1  q−α+1  θ θx     θ(·, t) L∞ ≤ C θ(·, t) L1 + +C dx, 2 θ1+α which proves (2.13). 2 Lemma 2.5.  sup t

ρ2x dx ≤ C.

Proof. Using Eq. (1.1), Eq. (1.2) can be rewritten as 



ρx ρ 2 +u ρ Testing the above equation by

ρx ρ2



 ρx + ρu 2 + u = −px − (E + ub)b. ρ t x

+ u, using (2.3), Lemma 2.1, (2.5), (2.11) and (2.12), we have

2 ρx + u dx ρ2       ρx ρx = − px 2 + u dx − (E + ub)b 2 + u dx ρ ρ           ρx ρx ρx ρx = − ρ2 θ 2 + u − u + u dx − ρθ + u dx − (E + ub)b + u dx x ρ ρ2 ρ2 ρ2          ρx ρx ρx ≤ ρ2 θu 2 + u dx − ρθx 2 + u dx − (E + ub)b 2 + u dx ρ ρ ρ       √ ρx  √ ρx  √    ≤ C  ρ ρ2 + u  2 θ L∞ ρu L2 + C  ρ ρ2 + u  2 θx L2 L L    √ ρx     + C E L2 + u L2 b L∞ b L∞   ρ ρ2 + u  2 L      √ ρx  ≤ C θ L∞ + θx L2 + E L2 + 1   ρ ρ2 + u  2 , L

1 d 2 dt





ρ

which proves (2.14). 2

(2.14)

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Lemma 2.6. 

T  u2x dx

sup t

u2t + u2xx dxdt ≤ C,

+

(2.15)

0

 ρ2t dx ≤ C.

sup t

(2.16)

Proof. Testing (1.2) by ut , using (2.3), (2.11), Lemma 2.1, (2.12), (2.13) and (2.14), we derive 1 d 2 dt



 u2x dx +

 ρu2t dx = −

 ρuux ut dx −

 (E + ub)but dx −

(ρx θ + ρθx )ut dx

  √ √ ≤ ρ L∞ ρut L2 u L∞ ux L2 + E L2 + u L2 b L∞ b L∞ ut L2   + ρx L2 θ L∞ + ρ L∞ θx L2 ut L2    √  √ √ ≤ C ux 2L2 ρut L2 + C 1 + E L2 ρut L2 + C θ L∞ + θx L2 ρut L2 ≤

  1 √ ρut 2L2 + C ux 4L2 + C 1 + E 2L2 + θ 2L∞ + θx 2L2 , 2

which yields the first part of (2.15). Using Eq. (1.2), it is easy to show that   ρut + ρuux + px + (E + ub)b ∈ L2 0, T ; L2 and thus proves (2.15). (2.16) follows from (2.14), (2.15) and Eq. (1.1): ρt = −(ρu)x = −ρx u − ρux .

2

Lemma 2.7.  sup t

T  Ex2

+

b2t dx

Et2 dxdt ≤ C.

+

(2.17)

0

Proof. Applying ∂t to (1.3) and (1.4), testing by Et and bt and summing up the result and using (2.15) and (2.11), we have 1 d 2 dt



 Et2 + b2t dx +

 Et2 dx = − 1 ≤ 2

This proves (2.17).

2

≤

1 2

≤

1 2

∂t (ub) · Et dx 

 Et2 dx

+



∂t (ub) 2 dx

 Et2 dx + C ut 2L2 b 2L∞ + C u 2L∞ bt 2L2  Et2 dx + C ut 2L2 + C bt 2L2 .

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Lemma 2.8. θ L∞ (0,T ;H 1 ) + θ L2 (0,T ;H 2 ) + θt L2 (0,T ;L2 ) ≤ C.

(2.18)

Proof. Testing (1.5) by θq+1 , using (1.1), (1.8), (2.3), (2.17) and (2.15), we have d dt





 k 2 θx2 dx ≤ C

ρθq+2 dx + C

2

ux + (E + ub)2 − pux θq+1 dx

    ≤ C ux 2L∞ + E + ub 2L∞ θq+1 dx + C ux L∞ θq+2 dx     2 q+2 ≤ C 1 + uxx L2 ρθ dx + 1 ,

which implies  θq+2 dx ≤ C.

sup t

(2.19)

Testing (1.5) by k(θ)θt , using (1.1), (2.3), (2.15), (2.19), (2.17), (2.14) and (2.11), we have 1 d 2 dt



 k 2 (θ)θx2 dx + 

=− 1 ≤ 4 ≤

ρk(θ)θt2 dx

1 4





ρuθx k(θ)θt dx − 

ρθux k(θ)θt dx +

 ρk(θ)θt2 dx

 ρk(θ)u2 θx2 dx

+C

 ρk(θ)θ2 u2x dx

+C







+C



2 u2x + (E + ub)2 k(θ)dx

 k(θ)θx2 dx + C ux 2L∞

ρk(θ)θt2 dx + C u 2L∞

  + C ux 4L∞ + E + ub 4L∞ 1 ≤ 4

 2  ux + (E + ub)2 k(θ)θt dx

k(θ)θ2 dx

 k(θ)dx

 ρk(θ)θt2 dx

+C

k(θ)θx2 dx + C uxx 2L2 + C,

which implies 

T 

sup

k

t

2

(θ)θx2 dx

k(θ)θt2 dxdt ≤ C.

+

(2.20)

0

Here we have used the Gagliardo–Nirenberg inequality ux 2L∞ ≤ C ux L2 uxx L2 . It follows from (1.5), (2.20), (2.3), (2.21), (2.11), (2.15) and (2.17) that  θxx 2L2 ≤ C

ρθt2 + u2 θx2 + θx4 + u4x + (E + ub)4 dx + C   ≤ C ρθt2 dx + C θx4 dx + C

(2.21)

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J. Fan, Y. Hu / J. Math. Anal. Appl. ••• (••••) •••–•••

 ≤C

9

 ρθt2 dx

+C

θx2 dx θx 2L∞ + C

 ≤C

ρθt2 dx + C θxx L2 + C,

which gives T θxx 2L2 ≤ C. 0

This completes the proof. 2 Acknowledgment J. Fan is partially supported by NSFC (No. 11171154). References [1] J. Fan, Y. Hu, Uniform existence of the complete equations for an electromagnetic fluid, preprint, 2013. [2] I. Imai, General principles of magneto-fluid dynamics, Chapter I, in: H. Yukawa (Ed.), Magneto-Fluid Dynamics, in: Progr. Theoret. Phys. Suppl., vol. 24, RIFP, Kyoto University, 1962, pp. 1–34. [3] S. Jiang, F.C. Li, Rigorous derivation of the compressible magnetohydrodynamic equations from the electromagnetic fluid system, Nonlinearity 25 (2012) 1735–1752. [4] S. Jiang, F.C. Li, Convergence of the complete electromagnetic fluid system to the full compressible magnetohydrodynamic equations, arXiv:1309.3668v1 [math. AP], 14 September 2013. [5] S. Kawashima, Smooth global solutions for two-dimensional equations of electro–magneto-fluid dynamics, Jpn. J. Appl. Math. 1 (1984) 207–222. [6] S. Kawashima, Y. Shizuta, Magnetohydrodynamic approximation of the complete equations for an electromagnetic fluid, Tsukuba J. Math. 10 (1) (1986) 131–149. [7] S. Kawashima, Y. Shizuta, Magnetohydrodynamic approximation of the complete equations for an electromagnetic fluid II, Proc. Japan Acad. Ser. A Math. Sci. 62 (1986) 181–184. [8] A. Milani, On a singular perturbation problem for the linear Maxwell equations, Rend. Semin. Mat. Univ. Politec. Torino 38 (3) (1980) 99–110. [9] A. Milani, Local in time existence for the complete Maxwell equations with monotone characteristic in a bounded domain, Ann. Mat. Pura Appl. (4) 131 (1982) 233–254. [10] A. Milani, The quasi-stationary Maxwell equations as singular limit of the complete equations: the quasi-linear case, J. Math. Anal. Appl. 102 (1984) 251–274. [11] A.I. Vol’pert, S.I. Hudjaev, On the Cauchy problem for composite systems of nonlinear differential equations, Math. USSR Sb. 16 (1972) 517–544.