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Smooth solutions to diffusion approximation radiation hydrodynamics equations
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Doctopic: Partial Differential Equations
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J. Math. Anal. Appl. ••• (••••) •••–•••
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Smooth solutions to diffusion approximation radiation hydrodynamics equations Peng Jiang ∗ , Yulei Zhou Department of Mathematics, College of Science, Hohai University, Nanjing 210098, PR China
a r t i c l e
i n f o
Article history: Received 18 July 2017 Available online xxxx Submitted by W. Layton MSC: 35K05 47J40 26A39
a b s t r a c t In this paper, we consider the smooth solutions to the Cauchy problem for the diffusion approximation radiation hydrodynamics equations in R3 . The model describes the interaction of an inviscid gas with photons. The local existence and uniqueness of smooth solutions is obtained by using the energy method with more subtle energy estimates. © 2018 Elsevier Inc. All rights reserved.
Keywords: Radiation hydrodynamics Diffusion approximation Smooth solutions
1. Introduction The key aim of the radiation hydrodynamics is to include the radiation effects into hydrodynamics. The importance of thermal radiation in physical problems increases as the temperature is raised. At moderate temperatures (say, thousands of degrees Kelvin), the role of the radiation is transporting energy by radiative processes. At higher temperatures (say, millions of degrees Kelvin), the energy and momentum densities of the radiation field may become dominate the corresponding fluid quantities. In this case, the radiation field significantly affects the dynamics of the fluid. In this paper, we are interested diffusion approximation (also called the Eddington approximation) model in radiation hydrodynamics, which is the energy flow due to radiative process in a semi-quantitative sense, and is particularly accurate if the specific intensity of radiation is almost isotropic (cf. [15]). Such systems can be used to simulate, for instance, nonlinear stellar pulsation, supernova explosions and stellar winds in astrophysics and so on (cf. [10,14,17]). The diffusion approximation is valid for optically thick regions where the photons emitted by the gas have a high probability of reabsorption within the region (cf. [1]). Based on the standard hydrodynamics, the governing * Corresponding author. E-mail addresses: [email protected] (P. Jiang), [email protected] (Y. Zhou). https://doi.org/10.1016/j.jmaa.2018.05.071 0022-247X/© 2018 Elsevier Inc. All rights reserved.
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P. Jiang, Y. Zhou / J. Math. Anal. Appl. ••• (••••) •••–•••
equations of the diffusion approximation in radiation hydrodynamics for 3-D flow of a inviscid gas, can be written in terms of Euler coordinates as follows: ρt + div(ρu) = 0,
(1.1)
ρ(ut + u · ∇u) + ∇P = −∇n,
(1.2)
cν ρ(θt + u · ∇θ) + P divu = n − θ4 + u · ∇n,
(1.3)
nt − Δn = θ4 − n.
(1.4)
Here, the unknowns are (ρ, u, θ, n), where ρ = ρ(x, t) > 0, θ = θ(x, t), u = u(x, t) = (u1 (x, t), u2 (x, t), u3 (x, t)), for t > 0, x ∈ R3 denote the mass density, temperature and velocity field of the fluid respectively, and n = n(x, t) > 0 for t > 0, x ∈ R3 denote the radiation field, P = Rρθ is the material pressure, cν > 0 being the heat conductivity, R = cν (γ − 1), and γ > 1 being the specific heat ratio. More discuss of this system can be seen [1,5,15]. Due to its physical importance, complexity, rich phenomena, and mathematical challenges, there is large of literature on the studies of radiation hydrodynamics from the mathematical/physical point of view, see, for example [2,3,6,18]. Specially, for diffusion approximation model, in [7], Jiang et al. showed the global existence of smooth solutions for one-dimensional case with viscosity. However, the effect of radiation on the momentum (i.e. ∇n in (1.2)) is omitted for technical reason, and for full model, the global existence results can be seen [4]. The global well-posedness and large time behavior of classical solutions for multi-dimensional case see [5]. For inviscid case, Lin, Coulombel and Goudon consider a simplified model as follows (see [12]): ρt + (ρu)x = 0, (ρu)t + (ρu2 + P )x = 0, 1 2 1 2 ρθ + ρu + ρθ + ρu u + P u = n − θ4 , 2 2 t x
(1.5)
− nxx = θ4 − n, which describes the interaction between an inviscid gas and photons. They showed the existence of smooth traveling waves, called “shock profiles”, when the strength of shock is small. While, system (1.5) can be seen as a stationary radiation field case and excluded the effect of radiative pressure from the momentum equation of (1.1)–(1.4). As a deep simplified model of (1.5), the “radiating gases” model reads as follows: ut + (
u2 )x = −qx , 2
(1.6)
−qxx + q = −ux . The thorough study on (1.6) motivated a lot of works, see, for example, [8,9,11,16] and the references cited therein. In this paper, we are interested in the local existence of smooth solutions to system (1.1)–(1.4) with the initial conditions: (ρ, u, θ, n)|t=0 = (ρ0 , u0 , θ0 , n0 )(x),
x ∈ R3 .
(1.7)
To obtain this result, we use an iteration and the Banach contraction mapping principle, a standard procedure see, e.g., [13]. However, we should point out here that the main difficulties in the proof lie in dealing with the nonlinear and non-local terms in (1.1)–(1.4). Especially for the nonlinear term θ4 and radiation
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effect term ∇n, which requires appropriate iteration scheme and delicate energy estimates to control the terms and some new ideas and new ingredients in the proof. The rest of this paper is organized as follows. In Section 2, we reformulate the problem and state the main result. In Section 3, we consider the corresponding linearized problem of (1.1)–(1.4) and give some useful estimates. Finally, in section 4 we prove the main result of this paper. Through the paper we use · H s , · W 1,∞ to denote norm of H s (R3 ), W 1,∞ ([0, T ] × R3 ) respectively, B(·) denotes the smooth functions with respect to its argument and C denotes a positive constant which may take different values in different place. 2. Main results and reformulation Firstly, we introduce some notations. For positive number T and s, we define space X s (T ) = C 0 ([0, T ], H s (R3 )) ∩ C 1 ([0, T ], H s−1 (R3 )), and Y s (T ) = L2 ([0, T ], H s+1 (R3 )) ∩ H 1 ([0, T ], H s (R3 )). We also use the notation |u(t)|s = u(t)H s + ∂t u(t)H s−1 . Denote by V = (ρ, u1 , u2 , u3 , θ)t , then, we state the main result of this paper: Theorem 2.1. Assume that there is a positive constant C0 such that 0 < C0−1 ≤ ρ0 , θ0 ≤ C0 , x ∈ R3 and V0 (x) ∈ H s (R3 ), n0 (x) ∈ H s+1 (R3 ), where s > 32 + 1. Then, there exists a T > 0, such that the problem (1.1)–(1.4), (1.7) have unique smooth solutions (V, n) on [0, T ] satisfying V ∈ C 1 ([0, T ] × R3 ), n ∈ C 1 ([0, T ] × R3 ) ∩
H j ([0, T ]; H s+1−j (R3 ))
with
Δn ∈ C 0 ([0, T ] × R3 ),
j=0,1
and
0
−1
≤ ρ, θ ≤ C
for any
(x, t) ∈ R3 × [0, T ].
Remark 2.1. Moreover, under the same conditions in Theorem 2.1, we can prove the solutions which establish above have V ∈
s−1
C j ([0, T ]; H s−j (R3 )),
j=0
n∈
s−1
C j ([0, T ]; H s−j (R3 )) ∩
j=0
s−1
H j ([0, T ]; H s+1−j (R3 )).
j=0
For the solutions (V, n) defined in Theorem 2.1, there is a positive definite symmetric matrix A0 (V ) defined by ⎛
ρ−1 ⎜ 0 ⎜ ⎜ A0 (V ) = ⎜ 0 ⎜ ⎝ 0 0 with A0 (V ) is a smooth function of V satisfying
0 ρ Rθ
0 0 0
0 0 ρ Rθ
0 0
0 0 0 ρ Rθ
0
0 0 0 0 cν ρ Rθ 2
⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎠
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CId5 ≤ A0 (V ) ≤ C −1 Id5
(2.1)
for some positive constant C. Now, we rewrite system (1.1)–(1.3) to a symmetric hyperbolic system of first order. Firstly, (1.1)–(1.3) can be rewritten as ∂V j (V ) ∂V = H(V, n), + A ∂t ∂xj j=1 3
(2.2)
j (V ) = { where A amn }5×5 , aii = uj , a1(j+1) = ρ, a(j+1)1 = Rθ a(j+1)5 = R, a5(j+1) = ρ , 1, 2, 3, 4, 5 and the remaining elements of aij equal to 0, and
Rθ cν
for j = 1, 2, 3, i =
H(V, n) = (0, g1 , g2 , g3 , g4 )t , 1 1 (n − θ4 + u · ∇n). Then, we symmetrize (2.2) by multiplying with gj = − nxj for j = 1, 2, 3, and g4 = ρ cν ρ A0 (V ). Therefore, (2.2) can be represented by the quasilinear symmetric system ∂V ∂V + Aj (V ) = F (V, n), ∂t ∂xj j=1 3
A0 (V ) j (V ), with where Aj (V ) = A0 (V )A ⎛
u1 ρ
⎜ 1 ⎜ ⎜ A1 (V ) = ⎜ 0 ⎜ ⎝ 0 0
1 ρu1 Rθ
0 0 ρ θ
0 0 ρu1 Rθ
0 0
0 0 0
0 ρ θ
0 0
ρu1 Rθ
0
⎛
⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎠
cν ρu1 Rθ 2 u3 ρ
⎜ 0 ⎜ ⎜ A3 (V ) = ⎜ 0 ⎜ ⎝ 1 0
0 ρu3 Rθ
0 0 0
⎛
u2 ρ
⎜ 0 ⎜ ⎜ A2 (V ) = ⎜ 1 ⎜ ⎝ 0 0 0 0 ρu3 Rθ
0 0
1 0 0
0 0 0
ρu3 Rθ ρ θ
ρ θ cν ρu3 Rθ 2
0 ρu2 Rθ
0 0 0 ⎞
1 0 ρu2 Rθ
0 ρ θ
0 0 0 ρu2 Rθ
0
0 0 ρ θ
0
⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎠
cν ρu2 Rθ 2
⎟ ⎟ ⎟ ⎟, ⎟ ⎠
F (V, n) = A0 (V )H(V, n) = (0, f1 , f2 , f3 , f4 )t with fj = −
1 ∂n , Rθ ∂xj
for j = 1, 2, 3,
and f4 =
1 (n − θ4 + u · ∇n). Rθ2
So the problem we consider in this paper can be written as follows: ⎧ 3 ⎪ ∂V ∂V ⎪ ⎪ + (V ) Aj (V ) = F (V, n), A 0 ⎪ ⎨ ∂t ∂xj j=1
⎪ nt − Δn = θ4 − n, ⎪ ⎪ ⎪ ⎩ (ρ, u, θ, n)|t=0 = (ρ0 , u0 , θ0 , n0 )(x),
x ∈ R3 .
(2.3)
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3. Energy estimates of linearized problems In this section, we consider the corresponding linearized problem of (2.3) as follows: ⎧ 3 ⎪ ∂U ∂U ⎪ ⎪ + (h) Aj (h) = F (h, l, ∇η) + F (h, l, ∇n00 ) + G, A ⎪ ⎨ 0 ∂t ∂xj j=1
⎪ ηt − Δη = h45 − l + Δn00 , ⎪ ⎪ ⎪ ⎩ (U, η)|t=0 = (0, 0),
(3.1)
and then give some useful energy estimates. Here U = V − V00 , η = n − n00 , V00 = j0 ∗ V0 , n00 = j0 ∗ n0 x with j(x) ∈ C0∞ (R3 ), suppj ⊆ {x : |x| ≤ 1}, j ≥ 0 and j(x)dx = 1, j0 = −3 0 j( 0 ), where 0 > 0 will be R3
˜ h5 )t , l = l(x, t) are the smooth functions of (x, t), and A0 (h) chosen later, h = (h1 , h2 , h3 , h4 , h5 )t = (h1 , h, ˜ f˜ )t with f˜ = − 1 ∇η, f˜ = 1 (l − h4 + h ˜ · ∇η), satisfy (2.1), F (h, l, ∇η) = (0, f˜1 , f˜2 , f˜3 , f˜4 )t = (0, f, 4 4 5 Rh5 Rh25 3 ∂V 0 G=− Aj (h) 0 . First, we introduce a useful lemma ∂xj j=1 Lemma 3.1. If s > 32 , then for f, g ∈ H s and |α| ≤ s, we have Dα (f g)L2 ≤ Cs f H s gH s . 3.1. L2 estimate Theorem 3.1. Given h ∈ W 1,∞ ([0, T ]×R3 )∩L∞ ([0, T ]; L2 (R3 )), l ∈ L∞ ([0, T ]; L2 (R3 ))∩L2 ([0, T ]; H 1 (R3 )). If U ∈ L∞ ([0, T ]; L2 (R3 )), η ∈ L∞ ([0, T ]; L2 (R3 )) ∩ L2 ([0, T ]; H 1 (R3 )) are solutions of (3.1), then, the following inequality holds: T max (U 2L2 0≤t≤T
+
η2L2 )
+
∇η2L2 dt
0 C B(hW 1,∞ )+1 T
≤e
(3.2)
0 2 V0 H 1 + η00 2H 2 + B (h, l)L∞ ([0,T ];L2 ) , hW 1,∞ T .
Proof. Firstly, taking the dot product between the first equation of (3.1) and U , integrating over R3 , we deduce that d (U, A0 (h)U )dx dt R3
3 ≤ |(U, A0 (h)t U )|dx + U, Aj (h)xj U dx j=1 R3 R3 0 + 2 |U · F (h, l, ∇η)|dx + 2 |U · F (h, l, ∇n0 )|dx + 2 |U · G|dx
R3
≤ A0 (h)t L∞ U 2L2
R3
3 + A (h) j xj j=1
R3
U 2L2 + B((h, l)L∞ )n00 H 1 U L2 L∞
(3.3)
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3 ˜ + |(θ − θ0 )f˜ | dx + 0 − u +2 ) · f A (h) (u j xj 0 0 4 j=1 3 R
V00 H 1 U L2 . L∞
Next, we estimate the fourth term of (3.3). By Hölder’s and Young’s inequality, we have u − u00 2 2 2 2 (u − u00 ) · f˜ dx ≤ C h5 2 + ∇ηL2 ≤ CB(hL∞ )U L2 + ∇ηL2 , L
R3
|(θ − θ00 )f˜4 |dx ≤ CB(hL∞ )(h2L2 + l2L2 + U 2L2 ) + ∇η2L2 , R3
with > 0 a sufficiently small constant. By (3.3), it follows immediately that d dt
(U, A0 (h)U )dx ≤ CB(hW 1,∞ )(V00 2H 1 + n00 2H 1 + h2L2 + l2L2 + U 2L2 ) + ∇η2L2 .
(3.4)
R3
Then, multiplying the second equation of (3.1) by η, and integrating over R3 we get d η2L2 + dt
|∇η|2 dx ≤ CηL2 h45 L2 + lL2 + n00 H 2
R3
≤C
η2L2
+
n00 2H 2
+
h45 2L2
+
l2L2
(3.5) .
For h45 2L2 , we have h45 2L2 =
h85 dx ≤ h5 6L∞
R3
h25 dx ≤ CB(hL∞ )h2L2 .
(3.6)
R3
Substituting (3.6) into (3.5), we get d η2L2 + dt
|∇η|2 dx ≤ C η2L2 + n00 2H 2 + B(hL∞ )h2L2 + l2L2 .
(3.7)
R3
According to (3.4), (3.7), we have d dt
(U, A0 (h)U )dx + R3
d η2L2 + dt
≤ C(B(hW 1,∞ ) +
|∇η|2 dx (3.8)
R3
1)(U 2L2
+
η2L2
+
V00 2H 1
+
n00 2H 2
+
h2L2
+
l2L2 ),
applying Gronwall’s inequality to (3.8), and use (2.1), we get (3.2). 2 3.2. H s estimate Theorem 3.2. Given h ∈ X s (T ), l ∈ X s (T ) ∩ Y s (T ), s > solutions of (3.1), then, the following inequality holds:
3 2
+ 1. If U ∈ X s (T ), η ∈ X s (T ) ∩ Y s (T ) are
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P. Jiang, Y. Zhou / J. Math. Anal. Appl. ••• (••••) •••–•••
T max (|U |2s + |η|2s ) +
C B( max |(h,l)|s )+1 T
|∇η|2s dt ≤ e
0≤t≤T
7
0≤t≤T
0
0 2 0 2 0 2 0 2 × V0 H s + n0 H s + V0 H s+1 + n0 H s+2 + B( max |(h, l)|s ) T .
(3.9)
0≤t≤T
Proof. First we estimate (U, η)H s . Differentiating the second equation of (3.1) α (|α| ≤ s) times with respect to x, multiplying the resulting equation by Dα η and integrating over R3 , we deduce that d Dα η2L2 + dt
|Dα (∇η)|2 dx ≤ CDα ηL2 Dα h45 L2 + Dα lL2 + Dα Δn00 L2
R3
≤ C Dα η2L2 + Dα Δn00 2L2 + Dα h45 2L2 + Dα l2L2 .
Then, we give the estimate of Dα h45 2L2 , take |α| = 3 for example, 2 Dα h45 2L2 ≤ C h5 (Dh5 )3 + h25 D2 h5 Dh5 + h35 D3 h5 L2 2 ≤ C h5 L∞ Dh5 3L∞ + h5 2L∞ h5 H s Dh5 L∞ + h5 3L∞ h5 H s ≤ CB(h5 H s ). For other case of |α| > 1, we can obtain similar conclusion. Thus d η2H s + ∇η2H s ≤ C(η2H s + n00 2H s+2 + B((h, l)H s )). dt
(3.10)
Multiplying the first equation of (3.1) by A−1 0 (h), differentiating α times with respect to x, and multiplying the resulting equation by A0 (h), we have ∂(Dα U ) ∂(Dα U ) + A0 (h) Aj (h) ∂t ∂xj j=1 3
−1 −1 0 = A0 (h)Dα (A−1 0 (h)F (h, l, ∇η) + A0 (h)F (h, l, ∇n0 ) + A0 (h)G) + Fα ,
with Fα is defined by the commutator terms as
Fα =
∂(Dα U ) ∂U −1 α A . A0 (h) A−1 (h)A (h) − D (h)A (h) j j 0 0 ∂xj ∂xj j=1
3
Now, multiplying (3.11) by Dα U , and then integrating over R3 , by Lemma 3.1, we deduce that d dt
(Dα U, A0 (h)Dα U )dx R3
3 α α α α ≤ |(D U, A0 (h)t D U )| dx + (D U, Aj (h)xj D U ) dx j=1 R3 R3 α + 2 (A0 (h)Dα (A−1 0 (h)F (h, l, ∇η)), D U ) dx
R3
(3.11)
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+2
(A0 (h)Dα (A−1 (h)F (h, l, ∇n00 )), Dα U ) dx 0
R3
+2
(A0 (h)Dα (A−1 (h)G), Dα U ) dx + 2
0
R3
(3.12)
|(Fα , Dα U )| dx
R3
≤ A0 (h)t L∞ + Aj (h)xj L∞ Dα U 2L2 + CDα U 2L2 + CFα 2L2 + B((h, l)H s )n00 H s+1 Dα U L2 + B(hH s )V00 H s+1 Dα U L2 h1 α cν h1 α 0 α 0 α + Rh5 D (u − u0 ) · D g˜ + Rh2 D (θ − θ0 )D g˜4 dx, 5 R3
where g˜ = − 1 ∇η, h1
g˜4 =
1 ˜ · ∇η). (l − h45 + h cν h1
For the last term of (3.12), h1 α 0 α dx ≤ C D (u − u ) · D g ˜ 0 Rh5
R3
R3
0≤|β|≤|α|
h1 α−β 1 β α 0 − D ∇η · D (u − u0 ) dx. Rh5 D h1
When |β| = |α|, α α 0 − 1 D ∇η · D (u − u0 ) dx ≤ CB(hL∞ )U 2H s + ∇η2H s ; Rh5
R3
when |β| = |α| − 1, h1 1 α−1 α 0 2 2 D D − ∇η · D (u − u ) 0 dx ≤ CB(hW 1,∞ (R3 ) )U H s + CηH s ; Rh5 h1
R3
when 0 < |β| < |α| − 1, using Sobolev’s inequality, we have h1 α−β 1 β α 0 − D D ∇η · D (u − u ) 0 dx Rh5 h1
R3
h1 α−β 1 D Dβ ∇ηL4 Dα U L2 ≤ Rh5 ∞ 4 h 1 L L ≤ CU 2H s + CB(hH s )η2H s ;
when |β| = 0, h1 h1 α 1 α 0 dx ≤ − ∇η · D D (u − u ) 0 Rh5 Rh5 h1
R3
α 1 ∇ηL∞ Dα U L2 D 2 h ∞ 1 L L
≤ CU 2H s + CB(hH s )η2H s . According to (3.13), we obtain
(3.13)
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h1 α 0 α 2 2 2 Rh5 D (u − u0 ) · D g˜ dx ≤ C (B(hH s ) + 1) (U H s + ηH s ) + ∇ηH s .
(3.14)
R3
h1 α 0 α Similar to the discuss of D (u − u0 ) · D g˜ dx and by Lemma 3.1, we have Rh5 R3
cν h1 α 0 α Rh2 D (θ − θ0 )D g˜4 dx 5
R3
≤C
cν h1 α 0 α 1 4 D (θ − θ )D (l − h ) 0 5 dx Rh2 cν h1 5
R3
+C
R3 0≤|β|≤|α|
˜ cν h1 α h 0 α−β β · D ∇η dx Rh2 D (θ − θ0 )D c h ν 1 5
(3.15)
≤ CB((h, l)H s )U 2H s + C (B(hH s ) + 1) (U 2H s + η2H s ) + ∇η2H s ≤ C (B((h, l)H s ) + 1) (U 2H s + η2H s ) + ∇η2H s . For Fα 2L2 , we have
Fα 2L2
2 3 ∂U −1 α−β β = −A0 (h) Cα,β D (A0 Aj (h))D ∂x j 2 j=1 0<|β|≤|α| ≤C
3
j=1 0<|β|≤|α|
2 A0 (h)Dα−β (A−1 Aj (h))Dβ ∂U . 0 ∂xj L2
L
We also divide three cases to discuss. When |β| = 1, 2 2 A0 (h)D(A−1 Aj (h))Dα−1 ∂U ≤ CA0 (h)2 2 D(A−1 Aj (h))2L∞ Dα−1 ∂U L 0 0 ∂xj 2 ∂xj 2 L
L
≤ CB(hH s )U 2H s ; when 1 < |β| < |α|, 2 2 A0 (h)Dβ (A−1 Aj (h))Dα−β ∂U ≤ CA0 (h)2L∞ Dβ (A−1 Aj (h))2 4 Dα−β ∂U L 0 0 ∂xj L2 ∂xj L4 2 −1 2 β 2 α−β ∂U ≤ CA0 (h)L∞ D (A0 Aj (h))H 1 D ∂xj H 1 ≤ CB(hH s )U 2H s ; when |β| = |α|, 2 2 A0 (h)Dα (A−1 Aj (h)) ∂U ≤ CA0 (h)2 2 Dα (A−1 Aj (h))2 2 ∂U L L 0 0 ∂xj 2 ∂xj ∞ L
L
≤ Therefore, we can conclude that
CB(hH s )U 2H s .
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Fα 2L2 ≤ CB(hH s )U 2H s .
(3.16)
Utilizing (3.12), (3.14)–(3.16), we deduce that d dt
(Dα U, A0 (h)Dα U )dx ≤ C (B(|(h, l)|s ) + 1) (U 2H s + η2H s ) + ∇η2H s .
(3.17)
R3
Applying Gronwall’s inequality to (3.10) and (2.1), (3.17) we have T max (U 2H s 0≤t≤T
+
η2H s )
+
∇η2H s dt
0 C B( max |(h,l)|s )+1 T
≤e
0≤t≤T
(3.18)
V00 2H s+1 + n00 2H s+2 + B( max (h, l)H s ) T . 0≤t≤T
Now we estimate (Ut , ηt )H s−1 . Acting the operator Dα−1 ∂t on the second equation of (3.1), multiplying the resulting equation by Dα−1 ∂t η, and then integrating over R3 , we obtain d Dα−1 ηt 2L2 + dt
|Dα−1 (∇ηt )|2 dx ≤ CDα−1 ηt L2 Dα−1 (h45 )t L2 + Dα lt L2
R3
≤ C Dα−1 ηt 2L2 + Dα−1 (h45 )t 2L2 + Dα−1 lt 2L2 ≤ C Dα−1 ηt 2L2 + CB(|h|s ) + Dα−1 lt 2L2 .
Therefore, d ηt 2H s−1 + ∇ηt 2H s−1 ≤ C(ηt 2H s−1 + B(|(h, l)|s )). dt
(3.19)
α−1 Next, multiplying the second equation of (3.1) by A−1 ∂t , and multiplying the 0 (h), acting the operator D resulting equation by A0 (h), we deduce that
∂(Dα−1 Ut ) ∂(Dα−1 Ut ) + Aj (h) = A0 (h)Dα−1 ∂t (A−1 0 (h)F (h, l, ∇η)) ∂t ∂x j j=1 3
A0 (h)
(3.20)
0 α−1 ∂t (A−1 + A0 (h)Dα−1 ∂t (A−1 0 (h)F (h, l, ∇n0 )) + A0 (h)D 0 (h)G) + Fα ,
with Fα =
∂(Dα−1 Ut ) ∂U −1 α−1 A . A0 (h) A−1 (h)A (h) − D ∂ (h)A (h) j t j 0 0 ∂xj ∂xj j=1
3
Multiply (3.20) by Dα−1 ∂t , similarly to the proof of U H s , we get d dt
(Dα−1 Ut , A0 (h)Dα−1 Ut )dx (3.21)
R3
≤ C (B(|(h, l)|s ) + 1) (Ut 2H s−1 + ηt 2H s−1 + V00 2H s + n00 2H s ) + ∇ηt 2H s−1 . By (2.1), (3.19), (3.21) and Gronwall’s inequality we have
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T max (Ut 2H s−1 + ηt 2H s−1 ) +
C B( max |(h,l)|s )+1 T
∇ηt 2H s−1 dt ≤ e
0≤t≤T
11
0≤t≤T
0
(3.22)
× V00 2H s + n00 2H s + V00 2H s + n00 2H s + B( max |(h, l)|s ) T . 0≤t≤T
2
Together (3.18) with (3.22) we obtain (3.9). At last, we give the estimate of Δη.
Corollary 3.1. Under the assumptions of Theorem 3.2, and assume (V, η) are solutions of (3.1), we have following estimate:
max Δη2H s−1 ≤ C B( max |(h, l)|s ) + n00 2H s+1
0≤t≤T
0≤t≤T
×
+e
C B( max |(h,l)|s )+1 T 0≤t≤T
V00 2H s
+
n00 2H s
+
V00 2H s+1
+
n00 2H s+2
+ B( max |(h, l)|s ) T 0≤t≤T
(3.23)
.
Proof. By ηt − Δη = h45 − l + Δn00 , we can get that Dα−1 Δη = Dα−1 ηt − Dα−1 h45 + Dα−1 l − Dα−1 Δn00 . Therefore max Δη2H s−1 ≤ max ηt 2H s−1 + B(|h|s ) + l2H s−1 + n00 2H s+1 ,
0≤t≤T
0≤t≤T
by (3.9), we can deduce (3.23). 2 4. Proof of the main result In this section, we will give the proof of Theorem 2.1. Firstly, we construct an approximate solution sequence to (2.3) through the following iteration scheme. First we set V 0 = V00 , n0 = n00 and define V k+1 (x, t) and nk+1 (x, t) inductively as the solution of the following linearized problem: ⎧ 3 k+1 k+1 ⎪ k ∂V k ∂V ⎪ ⎪ + (V ) A (V ) = F (V k , nk , ∇nk+1 ), A 0 j ⎪ ⎨ ∂t ∂xj j=1
⎪ nk+1 − Δnk+1 = (θk )4 − nk , ⎪ t ⎪ ⎪ ⎩ k+1 |t=0 = V0k+1 (x), nk+1 |t=0 = nk+1 (x), V 0
x ∈ R3 ,
where V0k = jk ∗ V0 , nk0 = jk ∗ n0 with k = 2−k 0 . For a positive constant C, we can choose appropriate 0 such that (V0 − V0k , n0 − nk0 )H s ≤ C4 , k = 0, 1, 2, · · · and (V0k+1 − V0k , nk+1 − nk0 )L2 ≤ 2−k . 0 Lemma 4.1. Denote by R1 ≡ V00 2H s+1 + n00 2H s+2 , then, there are T0 ∈ (0, T ] and positive constant C(R1 , C0 , T0 ), such that when T ≤ T0 , U k = V k − V00 , η k = nk − n00 (k = 0, 1, 2, ...) satisfy T max |U
0≤t≤T
k
|2s
+ max |η 0≤t≤T
k
|2s
|∇η k |2s dt ≤
+ 0
max
0≤t≤T
Δη k 2H s−1
≤ C(R1 , C0 , T0 ).
3 C0 , 4
(4.1)
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Proof. Assume that (4.1) is satisfied at order k, this implies A0 (V k ) satisfy (2.1). According to Theorem 3.1, Theorem 3.2, Corollary 3.1, we can get T max |U
0≤t≤T
k+1
|2s
+ max |η
k+1
0≤t≤T
|2s
|∇η k+1 |2s dt
+ 0
C B 34 C0 +R1 +1 T
≤e
1 3 C0 + R1 + B C0 + R1 T , 2 4
3 C0 + R1 + R1 max Δnk+1 2H s−1 ≤ C B 0≤t≤T 4 3 1 3 C B 4 C0 +R1 +1 T C0 + R1 + B C0 + R1 +e T . 2 4 Therefore, there is a T0 > 0, such that for any T ≤ T0 , we have T max |U
k+1
0≤t≤T
|2s
+ max |η 0≤t≤T
k+1
|2s
|∇η k+1 |2s dt ≤
+
3 C0 , 4
0
and there exist constant C(R1 , C0 , T0 ) > 0 which depends on R1 , C0 , T0 such that when T ≤ T0 max Δη k+1 2H s−1 ≤ C(R1 , C0 , T0 ).
0≤t≤T
2
∞ Lemma 4.2. There exist T∗ ∈ (0, T ], 0 < τ < 1, {αi }∞ i=1 and {βi }i=1 with Σi |αi | < ∞, Σi |βi | < ∞, such that for k = 1, 2, · · · , when T ≤ T∗ ,
T max V k+1 − V k 2L2 + max nk+1 − nk 2L2 +
0≤t≤T
∇(nk+1 − nk )2L2 dt
0≤t≤T
≤τ
0
max V k − V k−1 2L2 + max nk − nk−1 2L2 + αk + βk .
0≤t≤T
0≤t≤T
Proof. In order to prove the result, we consider the following problem for V k+1 − V k and nk+1 − nk , ⎧ 3 k+1 k k−1 ⎪ − V k) ) k ∂(V k ∂(V − V ⎪ ⎪ + (V ) A (V ) = F (V k , I k ) − F (V k−1 , I k−1 ) + I, A 0 j ⎪ ⎨ ∂t ∂xj j=1
⎪ (nk+1 − nk )t − Δ(nk+1 − nk ) = (θk )4 − (θk−1 )4 − (nk − nk−1 ), ⎪ ⎪ ⎪ ⎩ k+1 − V k )|t=0 = V0k+1 − V0k , (nk+1 − nk )|t=0 = nk+1 − nk0 , (V 0 ∂V k ∂V k − (Aj (V k ) − Aj (V k−1 )) . ∂t ∂xj j=1 3
where I = −(A0 (V k ) − A0 (V k−1 ))
(4.2)
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Applying the energy estimates (3.2), (2.1) and Lemma 4.1 in the above problem, it follows: T max V
0≤t≤T
k+1
−V
k 2 L2
+ max n
k+1
0≤t≤T
−
nk 2L2
∇(nk+1 − nk )2L2 dt
+ 0
⎛
T
≤ C1 (R1 , C0 , T0 ) ⎝
T V k − V k−1 2L2 dt +
0
⎞ nk − nk−1 2L2 dt⎠
(4.3)
0
+ C V0k+1 − V0k 2L2 + nk+1 − nk0 2L2 0 ≤ C1 (R1 , C0 , T0 ) max V k − V k−1 2L2 + nk − nk−1 2L2 T + αk + βk , 0≤t≤T
where C1 (R1 , C0 , T0 ) is a positive constant depending on R1 , C0 and T0 . Next, we choose T∗ ∈ (0, T ], such that 0 < C1 (R1 , C0 , T0 )T∗ < 1. Thus, for all T ≤ T∗ , according to (4.3), we can find a τ , 0 < τ < 1 such that (4.2) holds. 2 Then, by interpolation inequality that for any
5 2
< s < s, we have 1−s /s
max V k+1 − V k H s ≤ C max V k+1 − V k L2
0≤t≤T
0≤t≤T
s /s
max V k+1 − V k H s ,
0≤t≤T
and 1−s /s
max nk+1 − nk H s ≤ C max nk+1 − nk L2
0≤t≤T
0≤t≤T
s /s
max nk+1 − nk H s .
0≤t≤T
Thus, by Lemmas 4.1, 4.2 and Sobolev’s inequality, we can obtain the local existence of (2.3). Finally, we prove uniqueness. Lemma 4.3. The solutions of (2.3) obtained at above are unique. Proof. Suppose that (V , n ) and (V , n ) are two solutions of (2.3). Denote by ω1 = V − V , ω2 = n − n . Thus, we obtain the following equations ⎧ 3 ⎪ ∂ω1 ⎪ ∂ω1 ⎪ + (V ) Aj (V ) = A ⎪ 0 ⎪ ⎪ ∂t ∂xj ⎪ j=1 ⎪ ⎪ ⎪ 3 ⎨ ∂V ∂V − (Aj (V ) − Aj (V )) + F (V , n ) − F (V , n ), −(A0 (V ) − A0 (V )) ∂t ∂x ⎪ j ⎪ j=1 ⎪ ⎪ ⎪ 4 4 ⎪ ⎪ ω − Δω = (θ − θ ) − ω , 2,t 2 2 1 2 ⎪ ⎪ ⎪ ⎩ω | = 0, ω | = 0. 1 t=0
2 t=0
Utilizing Theorem 3.1, Theorem 3.2, Lemma 4.1, Hölder’s inequality and Young’s inequality, if t ∈ (0, T ], we deduce that d |ω1 (t)|2s ≤ C(|V |2s , |V |2s , |n |2s , |n |2s )(|ω1 |2s + |ω2 |2s ) + |∇ω2 |2s , dt d |ω2 (t)|2s + |∇ω2 |2s ≤ C(|V |2s , |V |2s , |n |2s , |n |2s )(|ω1 |2s + |ω2 |2s ), dt where > 0 is a sufficiently small constant. According to (4.4) we obtain
(4.4)
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d |ω1 (t)|2s + |ω2 (t)|2s + |∇ω2 |2s dt ≤ C |V |2s , |V |2s , |n |2s , |n |2s |ω1 |2s + |ω2 |2s , which follows ω1 = ω2 = 0 immediately by using Gronwall’s inequality. 2 So the proof of Theorem 2.1 is completed. Acknowledgment This work is partially supported by the Fundamental Research Funds for the Central Universities (Grant No. 2016B07214). References [1] J.W. Bond, K.M. Watson, J.A. Welch, Atomic Theory of Gas Dynamics, Addison-Wesley, Reading, Massachusetts, 1965. [2] C. Buet, B. Després, Asymptotic analysis of fluid models for the coupling of radiation and hydrodynamics, J. Quant. Spectrosc. Radiat. Transf. 85 (2004) 385–418. [3] Th. Goudon, P. Lafitte, A coupled model for radiative transfer: Doppler effects, equilibrium and non equilibrium diffusion asymptotics, Multiscale Model. Simul. 4 (2005) 1245–1279. [4] P. Jiang, Unique global solution of an initial-boundary value problem to a diffusion approximation model in radiation hydrodynamics, Discrete Contin. Dyn. Syst. Ser. A 35 (7) (2015) 3015–3037. [5] P. Jiang, Global well-posedness and large time behavior of classical solutions to the diffusion approximation model in radiation hydrodynamics, Discrete Contin. Dyn. Syst. Ser. A 37 (4) (2017) 2045–2063. [6] S. Jiang, F.C. Li, F. Xie, Nonrelativistic limits of the compressible Navier–Stokes–Fourier–P1 approximation model arising in radiation hydrodynamics, SIAM J. Math. Anal. 47 (5) (2015) 3726–3746. [7] S. Jiang, F. Xie, J.W. Zhang, A Global Existence Result in Radiation Hydrodynamics, Industrial and Applied Mathematics in China, Series in Contemporary Applied Mathematics, High Edu. Press and World Scientific, Beijing, Singapore, 2009. [8] S. Kawashima, S. Nishibata, Shock waves for a model system of the radiating gases, SIAM J. Math. Anal. 30 (1) (1999) 95–117. [9] S. Kawashima, S. Nishibata, A singular limit for hyperbolic–elliptic coupled systems in radiation hydrodynamics, Indiana Univ. Math. J. 50 (2001) 567–589. [10] R. Kippenhahn, A. Weigert, Stellar Structure and Evolution, Springer Verlag, Berlin–Heidelberg, 1994. [11] C. Lin, Asymptotic stability of rarefaction waves in radiation hydrodynamics, Commun. Math. Sci. 9 (2011) 207–223. [12] C. Lin, J.F. Coulombel, Th. Goudon, Shock profiles for non equilibrium radiating gases, Phys. D 218 (2006) 83–94. [13] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, vol. 53, Springer-Verlag, New York, Berlin, Heidelberg, 1986. [14] S.S. Penner, D.B. Olfe, Radiation and Reentry, Academic Press, New York, 1968. [15] G.C. Pomraning, The Equations of Radiation Hydrodynamics, Pergamon Press, 1973. [16] C. Rohde, F. Xie, Global existence and blowup phenomenon for a 1D radiation hydrodynamics model problem, Math. Methods Appl. Sci. 35 (5) (2012) 564–573. [17] Y.B. Zeldovich, Y.P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomenon, Academic Press, 1966. [18] X. Zhong, S. Jiang, Local existence and finite time blow-up in multidimensional radiation hydrodynamics, J. Math. Fluid Mech. 9 (2007) 543–564.
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J. Math. Anal. Appl. ••• (••••) •••–•••
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Smooth solutions to diffusion approximation radiation hydrodynamics equations Peng Jiang ∗ , Yulei Zhou Department of Mathematics, College of Science, Hohai University, Nanjing 210098, PR China
a r t i c l e
i n f o
Article history: Received 18 July 2017 Available online xxxx Submitted by W. Layton MSC: 35K05 47J40 26A39
a b s t r a c t In this paper, we consider the smooth solutions to the Cauchy problem for the diffusion approximation radiation hydrodynamics equations in R3 . The model describes the interaction of an inviscid gas with photons. The local existence and uniqueness of smooth solutions is obtained by using the energy method with more subtle energy estimates. © 2018 Elsevier Inc. All rights reserved.
Keywords: Radiation hydrodynamics Diffusion approximation Smooth solutions
1. Introduction The key aim of the radiation hydrodynamics is to include the radiation effects into hydrodynamics. The importance of thermal radiation in physical problems increases as the temperature is raised. At moderate temperatures (say, thousands of degrees Kelvin), the role of the radiation is transporting energy by radiative processes. At higher temperatures (say, millions of degrees Kelvin), the energy and momentum densities of the radiation field may become dominate the corresponding fluid quantities. In this case, the radiation field significantly affects the dynamics of the fluid. In this paper, we are interested diffusion approximation (also called the Eddington approximation) model in radiation hydrodynamics, which is the energy flow due to radiative process in a semi-quantitative sense, and is particularly accurate if the specific intensity of radiation is almost isotropic (cf. [15]). Such systems can be used to simulate, for instance, nonlinear stellar pulsation, supernova explosions and stellar winds in astrophysics and so on (cf. [10,14,17]). The diffusion approximation is valid for optically thick regions where the photons emitted by the gas have a high probability of reabsorption within the region (cf. [1]). Based on the standard hydrodynamics, the governing * Corresponding author. E-mail addresses: [email protected] (P. Jiang), [email protected] (Y. Zhou). https://doi.org/10.1016/j.jmaa.2018.05.071 0022-247X/© 2018 Elsevier Inc. All rights reserved.
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equations of the diffusion approximation in radiation hydrodynamics for 3-D flow of a inviscid gas, can be written in terms of Euler coordinates as follows: ρt + div(ρu) = 0,
(1.1)
ρ(ut + u · ∇u) + ∇P = −∇n,
(1.2)
cν ρ(θt + u · ∇θ) + P divu = n − θ4 + u · ∇n,
(1.3)
nt − Δn = θ4 − n.
(1.4)
Here, the unknowns are (ρ, u, θ, n), where ρ = ρ(x, t) > 0, θ = θ(x, t), u = u(x, t) = (u1 (x, t), u2 (x, t), u3 (x, t)), for t > 0, x ∈ R3 denote the mass density, temperature and velocity field of the fluid respectively, and n = n(x, t) > 0 for t > 0, x ∈ R3 denote the radiation field, P = Rρθ is the material pressure, cν > 0 being the heat conductivity, R = cν (γ − 1), and γ > 1 being the specific heat ratio. More discuss of this system can be seen [1,5,15]. Due to its physical importance, complexity, rich phenomena, and mathematical challenges, there is large of literature on the studies of radiation hydrodynamics from the mathematical/physical point of view, see, for example [2,3,6,18]. Specially, for diffusion approximation model, in [7], Jiang et al. showed the global existence of smooth solutions for one-dimensional case with viscosity. However, the effect of radiation on the momentum (i.e. ∇n in (1.2)) is omitted for technical reason, and for full model, the global existence results can be seen [4]. The global well-posedness and large time behavior of classical solutions for multi-dimensional case see [5]. For inviscid case, Lin, Coulombel and Goudon consider a simplified model as follows (see [12]): ρt + (ρu)x = 0, (ρu)t + (ρu2 + P )x = 0, 1 2 1 2 ρθ + ρu + ρθ + ρu u + P u = n − θ4 , 2 2 t x
(1.5)
− nxx = θ4 − n, which describes the interaction between an inviscid gas and photons. They showed the existence of smooth traveling waves, called “shock profiles”, when the strength of shock is small. While, system (1.5) can be seen as a stationary radiation field case and excluded the effect of radiative pressure from the momentum equation of (1.1)–(1.4). As a deep simplified model of (1.5), the “radiating gases” model reads as follows: ut + (
u2 )x = −qx , 2
(1.6)
−qxx + q = −ux . The thorough study on (1.6) motivated a lot of works, see, for example, [8,9,11,16] and the references cited therein. In this paper, we are interested in the local existence of smooth solutions to system (1.1)–(1.4) with the initial conditions: (ρ, u, θ, n)|t=0 = (ρ0 , u0 , θ0 , n0 )(x),
x ∈ R3 .
(1.7)
To obtain this result, we use an iteration and the Banach contraction mapping principle, a standard procedure see, e.g., [13]. However, we should point out here that the main difficulties in the proof lie in dealing with the nonlinear and non-local terms in (1.1)–(1.4). Especially for the nonlinear term θ4 and radiation
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3
effect term ∇n, which requires appropriate iteration scheme and delicate energy estimates to control the terms and some new ideas and new ingredients in the proof. The rest of this paper is organized as follows. In Section 2, we reformulate the problem and state the main result. In Section 3, we consider the corresponding linearized problem of (1.1)–(1.4) and give some useful estimates. Finally, in section 4 we prove the main result of this paper. Through the paper we use · H s , · W 1,∞ to denote norm of H s (R3 ), W 1,∞ ([0, T ] × R3 ) respectively, B(·) denotes the smooth functions with respect to its argument and C denotes a positive constant which may take different values in different place. 2. Main results and reformulation Firstly, we introduce some notations. For positive number T and s, we define space X s (T ) = C 0 ([0, T ], H s (R3 )) ∩ C 1 ([0, T ], H s−1 (R3 )), and Y s (T ) = L2 ([0, T ], H s+1 (R3 )) ∩ H 1 ([0, T ], H s (R3 )). We also use the notation |u(t)|s = u(t)H s + ∂t u(t)H s−1 . Denote by V = (ρ, u1 , u2 , u3 , θ)t , then, we state the main result of this paper: Theorem 2.1. Assume that there is a positive constant C0 such that 0 < C0−1 ≤ ρ0 , θ0 ≤ C0 , x ∈ R3 and V0 (x) ∈ H s (R3 ), n0 (x) ∈ H s+1 (R3 ), where s > 32 + 1. Then, there exists a T > 0, such that the problem (1.1)–(1.4), (1.7) have unique smooth solutions (V, n) on [0, T ] satisfying V ∈ C 1 ([0, T ] × R3 ), n ∈ C 1 ([0, T ] × R3 ) ∩
H j ([0, T ]; H s+1−j (R3 ))
with
Δn ∈ C 0 ([0, T ] × R3 ),
j=0,1
and
0
−1
≤ ρ, θ ≤ C
for any
(x, t) ∈ R3 × [0, T ].
Remark 2.1. Moreover, under the same conditions in Theorem 2.1, we can prove the solutions which establish above have V ∈
s−1
C j ([0, T ]; H s−j (R3 )),
j=0
n∈
s−1
C j ([0, T ]; H s−j (R3 )) ∩
j=0
s−1
H j ([0, T ]; H s+1−j (R3 )).
j=0
For the solutions (V, n) defined in Theorem 2.1, there is a positive definite symmetric matrix A0 (V ) defined by ⎛
ρ−1 ⎜ 0 ⎜ ⎜ A0 (V ) = ⎜ 0 ⎜ ⎝ 0 0 with A0 (V ) is a smooth function of V satisfying
0 ρ Rθ
0 0 0
0 0 ρ Rθ
0 0
0 0 0 ρ Rθ
0
0 0 0 0 cν ρ Rθ 2
⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎠
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CId5 ≤ A0 (V ) ≤ C −1 Id5
(2.1)
for some positive constant C. Now, we rewrite system (1.1)–(1.3) to a symmetric hyperbolic system of first order. Firstly, (1.1)–(1.3) can be rewritten as ∂V j (V ) ∂V = H(V, n), + A ∂t ∂xj j=1 3
(2.2)
j (V ) = { where A amn }5×5 , aii = uj , a1(j+1) = ρ, a(j+1)1 = Rθ a(j+1)5 = R, a5(j+1) = ρ , 1, 2, 3, 4, 5 and the remaining elements of aij equal to 0, and
Rθ cν
for j = 1, 2, 3, i =
H(V, n) = (0, g1 , g2 , g3 , g4 )t , 1 1 (n − θ4 + u · ∇n). Then, we symmetrize (2.2) by multiplying with gj = − nxj for j = 1, 2, 3, and g4 = ρ cν ρ A0 (V ). Therefore, (2.2) can be represented by the quasilinear symmetric system ∂V ∂V + Aj (V ) = F (V, n), ∂t ∂xj j=1 3
A0 (V ) j (V ), with where Aj (V ) = A0 (V )A ⎛
u1 ρ
⎜ 1 ⎜ ⎜ A1 (V ) = ⎜ 0 ⎜ ⎝ 0 0
1 ρu1 Rθ
0 0 ρ θ
0 0 ρu1 Rθ
0 0
0 0 0
0 ρ θ
0 0
ρu1 Rθ
0
⎛
⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎠
cν ρu1 Rθ 2 u3 ρ
⎜ 0 ⎜ ⎜ A3 (V ) = ⎜ 0 ⎜ ⎝ 1 0
0 ρu3 Rθ
0 0 0
⎛
u2 ρ
⎜ 0 ⎜ ⎜ A2 (V ) = ⎜ 1 ⎜ ⎝ 0 0 0 0 ρu3 Rθ
0 0
1 0 0
0 0 0
ρu3 Rθ ρ θ
ρ θ cν ρu3 Rθ 2
0 ρu2 Rθ
0 0 0 ⎞
1 0 ρu2 Rθ
0 ρ θ
0 0 0 ρu2 Rθ
0
0 0 ρ θ
0
⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎠
cν ρu2 Rθ 2
⎟ ⎟ ⎟ ⎟, ⎟ ⎠
F (V, n) = A0 (V )H(V, n) = (0, f1 , f2 , f3 , f4 )t with fj = −
1 ∂n , Rθ ∂xj
for j = 1, 2, 3,
and f4 =
1 (n − θ4 + u · ∇n). Rθ2
So the problem we consider in this paper can be written as follows: ⎧ 3 ⎪ ∂V ∂V ⎪ ⎪ + (V ) Aj (V ) = F (V, n), A 0 ⎪ ⎨ ∂t ∂xj j=1
⎪ nt − Δn = θ4 − n, ⎪ ⎪ ⎪ ⎩ (ρ, u, θ, n)|t=0 = (ρ0 , u0 , θ0 , n0 )(x),
x ∈ R3 .
(2.3)
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[m3L; v1.238; Prn:6/06/2018; 11:41] P.5 (1-14)
P. Jiang, Y. Zhou / J. Math. Anal. Appl. ••• (••••) •••–•••
5
3. Energy estimates of linearized problems In this section, we consider the corresponding linearized problem of (2.3) as follows: ⎧ 3 ⎪ ∂U ∂U ⎪ ⎪ + (h) Aj (h) = F (h, l, ∇η) + F (h, l, ∇n00 ) + G, A ⎪ ⎨ 0 ∂t ∂xj j=1
⎪ ηt − Δη = h45 − l + Δn00 , ⎪ ⎪ ⎪ ⎩ (U, η)|t=0 = (0, 0),
(3.1)
and then give some useful energy estimates. Here U = V − V00 , η = n − n00 , V00 = j0 ∗ V0 , n00 = j0 ∗ n0 x with j(x) ∈ C0∞ (R3 ), suppj ⊆ {x : |x| ≤ 1}, j ≥ 0 and j(x)dx = 1, j0 = −3 0 j( 0 ), where 0 > 0 will be R3
˜ h5 )t , l = l(x, t) are the smooth functions of (x, t), and A0 (h) chosen later, h = (h1 , h2 , h3 , h4 , h5 )t = (h1 , h, ˜ f˜ )t with f˜ = − 1 ∇η, f˜ = 1 (l − h4 + h ˜ · ∇η), satisfy (2.1), F (h, l, ∇η) = (0, f˜1 , f˜2 , f˜3 , f˜4 )t = (0, f, 4 4 5 Rh5 Rh25 3 ∂V 0 G=− Aj (h) 0 . First, we introduce a useful lemma ∂xj j=1 Lemma 3.1. If s > 32 , then for f, g ∈ H s and |α| ≤ s, we have Dα (f g)L2 ≤ Cs f H s gH s . 3.1. L2 estimate Theorem 3.1. Given h ∈ W 1,∞ ([0, T ]×R3 )∩L∞ ([0, T ]; L2 (R3 )), l ∈ L∞ ([0, T ]; L2 (R3 ))∩L2 ([0, T ]; H 1 (R3 )). If U ∈ L∞ ([0, T ]; L2 (R3 )), η ∈ L∞ ([0, T ]; L2 (R3 )) ∩ L2 ([0, T ]; H 1 (R3 )) are solutions of (3.1), then, the following inequality holds: T max (U 2L2 0≤t≤T
+
η2L2 )
+
∇η2L2 dt
0 C B(hW 1,∞ )+1 T
≤e
(3.2)
0 2 V0 H 1 + η00 2H 2 + B (h, l)L∞ ([0,T ];L2 ) , hW 1,∞ T .
Proof. Firstly, taking the dot product between the first equation of (3.1) and U , integrating over R3 , we deduce that d (U, A0 (h)U )dx dt R3
3 ≤ |(U, A0 (h)t U )|dx + U, Aj (h)xj U dx j=1 R3 R3 0 + 2 |U · F (h, l, ∇η)|dx + 2 |U · F (h, l, ∇n0 )|dx + 2 |U · G|dx
R3
≤ A0 (h)t L∞ U 2L2
R3
3 + A (h) j xj j=1
R3
U 2L2 + B((h, l)L∞ )n00 H 1 U L2 L∞
(3.3)
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Doctopic: Partial Differential Equations
[m3L; v1.238; Prn:6/06/2018; 11:41] P.6 (1-14)
P. Jiang, Y. Zhou / J. Math. Anal. Appl. ••• (••••) •••–•••
6
3 ˜ + |(θ − θ0 )f˜ | dx + 0 − u +2 ) · f A (h) (u j xj 0 0 4 j=1 3 R
V00 H 1 U L2 . L∞
Next, we estimate the fourth term of (3.3). By Hölder’s and Young’s inequality, we have u − u00 2 2 2 2 (u − u00 ) · f˜ dx ≤ C h5 2 + ∇ηL2 ≤ CB(hL∞ )U L2 + ∇ηL2 , L
R3
|(θ − θ00 )f˜4 |dx ≤ CB(hL∞ )(h2L2 + l2L2 + U 2L2 ) + ∇η2L2 , R3
with > 0 a sufficiently small constant. By (3.3), it follows immediately that d dt
(U, A0 (h)U )dx ≤ CB(hW 1,∞ )(V00 2H 1 + n00 2H 1 + h2L2 + l2L2 + U 2L2 ) + ∇η2L2 .
(3.4)
R3
Then, multiplying the second equation of (3.1) by η, and integrating over R3 we get d η2L2 + dt
|∇η|2 dx ≤ CηL2 h45 L2 + lL2 + n00 H 2
R3
≤C
η2L2
+
n00 2H 2
+
h45 2L2
+
l2L2
(3.5) .
For h45 2L2 , we have h45 2L2 =
h85 dx ≤ h5 6L∞
R3
h25 dx ≤ CB(hL∞ )h2L2 .
(3.6)
R3
Substituting (3.6) into (3.5), we get d η2L2 + dt
|∇η|2 dx ≤ C η2L2 + n00 2H 2 + B(hL∞ )h2L2 + l2L2 .
(3.7)
R3
According to (3.4), (3.7), we have d dt
(U, A0 (h)U )dx + R3
d η2L2 + dt
≤ C(B(hW 1,∞ ) +
|∇η|2 dx (3.8)
R3
1)(U 2L2
+
η2L2
+
V00 2H 1
+
n00 2H 2
+
h2L2
+
l2L2 ),
applying Gronwall’s inequality to (3.8), and use (2.1), we get (3.2). 2 3.2. H s estimate Theorem 3.2. Given h ∈ X s (T ), l ∈ X s (T ) ∩ Y s (T ), s > solutions of (3.1), then, the following inequality holds:
3 2
+ 1. If U ∈ X s (T ), η ∈ X s (T ) ∩ Y s (T ) are
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AID:22303 /FLA
Doctopic: Partial Differential Equations
[m3L; v1.238; Prn:6/06/2018; 11:41] P.7 (1-14)
P. Jiang, Y. Zhou / J. Math. Anal. Appl. ••• (••••) •••–•••
T max (|U |2s + |η|2s ) +
C B( max |(h,l)|s )+1 T
|∇η|2s dt ≤ e
0≤t≤T
7
0≤t≤T
0
0 2 0 2 0 2 0 2 × V0 H s + n0 H s + V0 H s+1 + n0 H s+2 + B( max |(h, l)|s ) T .
(3.9)
0≤t≤T
Proof. First we estimate (U, η)H s . Differentiating the second equation of (3.1) α (|α| ≤ s) times with respect to x, multiplying the resulting equation by Dα η and integrating over R3 , we deduce that d Dα η2L2 + dt
|Dα (∇η)|2 dx ≤ CDα ηL2 Dα h45 L2 + Dα lL2 + Dα Δn00 L2
R3
≤ C Dα η2L2 + Dα Δn00 2L2 + Dα h45 2L2 + Dα l2L2 .
Then, we give the estimate of Dα h45 2L2 , take |α| = 3 for example, 2 Dα h45 2L2 ≤ C h5 (Dh5 )3 + h25 D2 h5 Dh5 + h35 D3 h5 L2 2 ≤ C h5 L∞ Dh5 3L∞ + h5 2L∞ h5 H s Dh5 L∞ + h5 3L∞ h5 H s ≤ CB(h5 H s ). For other case of |α| > 1, we can obtain similar conclusion. Thus d η2H s + ∇η2H s ≤ C(η2H s + n00 2H s+2 + B((h, l)H s )). dt
(3.10)
Multiplying the first equation of (3.1) by A−1 0 (h), differentiating α times with respect to x, and multiplying the resulting equation by A0 (h), we have ∂(Dα U ) ∂(Dα U ) + A0 (h) Aj (h) ∂t ∂xj j=1 3
−1 −1 0 = A0 (h)Dα (A−1 0 (h)F (h, l, ∇η) + A0 (h)F (h, l, ∇n0 ) + A0 (h)G) + Fα ,
with Fα is defined by the commutator terms as
Fα =
∂(Dα U ) ∂U −1 α A . A0 (h) A−1 (h)A (h) − D (h)A (h) j j 0 0 ∂xj ∂xj j=1
3
Now, multiplying (3.11) by Dα U , and then integrating over R3 , by Lemma 3.1, we deduce that d dt
(Dα U, A0 (h)Dα U )dx R3
3 α α α α ≤ |(D U, A0 (h)t D U )| dx + (D U, Aj (h)xj D U ) dx j=1 R3 R3 α + 2 (A0 (h)Dα (A−1 0 (h)F (h, l, ∇η)), D U ) dx
R3
(3.11)
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AID:22303 /FLA
Doctopic: Partial Differential Equations
[m3L; v1.238; Prn:6/06/2018; 11:41] P.8 (1-14)
P. Jiang, Y. Zhou / J. Math. Anal. Appl. ••• (••••) •••–•••
8
+2
(A0 (h)Dα (A−1 (h)F (h, l, ∇n00 )), Dα U ) dx 0
R3
+2
(A0 (h)Dα (A−1 (h)G), Dα U ) dx + 2
0
R3
(3.12)
|(Fα , Dα U )| dx
R3
≤ A0 (h)t L∞ + Aj (h)xj L∞ Dα U 2L2 + CDα U 2L2 + CFα 2L2 + B((h, l)H s )n00 H s+1 Dα U L2 + B(hH s )V00 H s+1 Dα U L2 h1 α cν h1 α 0 α 0 α + Rh5 D (u − u0 ) · D g˜ + Rh2 D (θ − θ0 )D g˜4 dx, 5 R3
where g˜ = − 1 ∇η, h1
g˜4 =
1 ˜ · ∇η). (l − h45 + h cν h1
For the last term of (3.12), h1 α 0 α dx ≤ C D (u − u ) · D g ˜ 0 Rh5
R3
R3
0≤|β|≤|α|
h1 α−β 1 β α 0 − D ∇η · D (u − u0 ) dx. Rh5 D h1
When |β| = |α|, α α 0 − 1 D ∇η · D (u − u0 ) dx ≤ CB(hL∞ )U 2H s + ∇η2H s ; Rh5
R3
when |β| = |α| − 1, h1 1 α−1 α 0 2 2 D D − ∇η · D (u − u ) 0 dx ≤ CB(hW 1,∞ (R3 ) )U H s + CηH s ; Rh5 h1
R3
when 0 < |β| < |α| − 1, using Sobolev’s inequality, we have h1 α−β 1 β α 0 − D D ∇η · D (u − u ) 0 dx Rh5 h1
R3
h1 α−β 1 D Dβ ∇ηL4 Dα U L2 ≤ Rh5 ∞ 4 h 1 L L ≤ CU 2H s + CB(hH s )η2H s ;
when |β| = 0, h1 h1 α 1 α 0 dx ≤ − ∇η · D D (u − u ) 0 Rh5 Rh5 h1
R3
α 1 ∇ηL∞ Dα U L2 D 2 h ∞ 1 L L
≤ CU 2H s + CB(hH s )η2H s . According to (3.13), we obtain
(3.13)
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[m3L; v1.238; Prn:6/06/2018; 11:41] P.9 (1-14)
P. Jiang, Y. Zhou / J. Math. Anal. Appl. ••• (••••) •••–•••
9
h1 α 0 α 2 2 2 Rh5 D (u − u0 ) · D g˜ dx ≤ C (B(hH s ) + 1) (U H s + ηH s ) + ∇ηH s .
(3.14)
R3
h1 α 0 α Similar to the discuss of D (u − u0 ) · D g˜ dx and by Lemma 3.1, we have Rh5 R3
cν h1 α 0 α Rh2 D (θ − θ0 )D g˜4 dx 5
R3
≤C
cν h1 α 0 α 1 4 D (θ − θ )D (l − h ) 0 5 dx Rh2 cν h1 5
R3
+C
R3 0≤|β|≤|α|
˜ cν h1 α h 0 α−β β · D ∇η dx Rh2 D (θ − θ0 )D c h ν 1 5
(3.15)
≤ CB((h, l)H s )U 2H s + C (B(hH s ) + 1) (U 2H s + η2H s ) + ∇η2H s ≤ C (B((h, l)H s ) + 1) (U 2H s + η2H s ) + ∇η2H s . For Fα 2L2 , we have
Fα 2L2
2 3 ∂U −1 α−β β = −A0 (h) Cα,β D (A0 Aj (h))D ∂x j 2 j=1 0<|β|≤|α| ≤C
3
j=1 0<|β|≤|α|
2 A0 (h)Dα−β (A−1 Aj (h))Dβ ∂U . 0 ∂xj L2
L
We also divide three cases to discuss. When |β| = 1, 2 2 A0 (h)D(A−1 Aj (h))Dα−1 ∂U ≤ CA0 (h)2 2 D(A−1 Aj (h))2L∞ Dα−1 ∂U L 0 0 ∂xj 2 ∂xj 2 L
L
≤ CB(hH s )U 2H s ; when 1 < |β| < |α|, 2 2 A0 (h)Dβ (A−1 Aj (h))Dα−β ∂U ≤ CA0 (h)2L∞ Dβ (A−1 Aj (h))2 4 Dα−β ∂U L 0 0 ∂xj L2 ∂xj L4 2 −1 2 β 2 α−β ∂U ≤ CA0 (h)L∞ D (A0 Aj (h))H 1 D ∂xj H 1 ≤ CB(hH s )U 2H s ; when |β| = |α|, 2 2 A0 (h)Dα (A−1 Aj (h)) ∂U ≤ CA0 (h)2 2 Dα (A−1 Aj (h))2 2 ∂U L L 0 0 ∂xj 2 ∂xj ∞ L
L
≤ Therefore, we can conclude that
CB(hH s )U 2H s .
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Doctopic: Partial Differential Equations
[m3L; v1.238; Prn:6/06/2018; 11:41] P.10 (1-14)
P. Jiang, Y. Zhou / J. Math. Anal. Appl. ••• (••••) •••–•••
10
Fα 2L2 ≤ CB(hH s )U 2H s .
(3.16)
Utilizing (3.12), (3.14)–(3.16), we deduce that d dt
(Dα U, A0 (h)Dα U )dx ≤ C (B(|(h, l)|s ) + 1) (U 2H s + η2H s ) + ∇η2H s .
(3.17)
R3
Applying Gronwall’s inequality to (3.10) and (2.1), (3.17) we have T max (U 2H s 0≤t≤T
+
η2H s )
+
∇η2H s dt
0 C B( max |(h,l)|s )+1 T
≤e
0≤t≤T
(3.18)
V00 2H s+1 + n00 2H s+2 + B( max (h, l)H s ) T . 0≤t≤T
Now we estimate (Ut , ηt )H s−1 . Acting the operator Dα−1 ∂t on the second equation of (3.1), multiplying the resulting equation by Dα−1 ∂t η, and then integrating over R3 , we obtain d Dα−1 ηt 2L2 + dt
|Dα−1 (∇ηt )|2 dx ≤ CDα−1 ηt L2 Dα−1 (h45 )t L2 + Dα lt L2
R3
≤ C Dα−1 ηt 2L2 + Dα−1 (h45 )t 2L2 + Dα−1 lt 2L2 ≤ C Dα−1 ηt 2L2 + CB(|h|s ) + Dα−1 lt 2L2 .
Therefore, d ηt 2H s−1 + ∇ηt 2H s−1 ≤ C(ηt 2H s−1 + B(|(h, l)|s )). dt
(3.19)
α−1 Next, multiplying the second equation of (3.1) by A−1 ∂t , and multiplying the 0 (h), acting the operator D resulting equation by A0 (h), we deduce that
∂(Dα−1 Ut ) ∂(Dα−1 Ut ) + Aj (h) = A0 (h)Dα−1 ∂t (A−1 0 (h)F (h, l, ∇η)) ∂t ∂x j j=1 3
A0 (h)
(3.20)
0 α−1 ∂t (A−1 + A0 (h)Dα−1 ∂t (A−1 0 (h)F (h, l, ∇n0 )) + A0 (h)D 0 (h)G) + Fα ,
with Fα =
∂(Dα−1 Ut ) ∂U −1 α−1 A . A0 (h) A−1 (h)A (h) − D ∂ (h)A (h) j t j 0 0 ∂xj ∂xj j=1
3
Multiply (3.20) by Dα−1 ∂t , similarly to the proof of U H s , we get d dt
(Dα−1 Ut , A0 (h)Dα−1 Ut )dx (3.21)
R3
≤ C (B(|(h, l)|s ) + 1) (Ut 2H s−1 + ηt 2H s−1 + V00 2H s + n00 2H s ) + ∇ηt 2H s−1 . By (2.1), (3.19), (3.21) and Gronwall’s inequality we have
JID:YJMAA
AID:22303 /FLA
Doctopic: Partial Differential Equations
[m3L; v1.238; Prn:6/06/2018; 11:41] P.11 (1-14)
P. Jiang, Y. Zhou / J. Math. Anal. Appl. ••• (••••) •••–•••
T max (Ut 2H s−1 + ηt 2H s−1 ) +
C B( max |(h,l)|s )+1 T
∇ηt 2H s−1 dt ≤ e
0≤t≤T
11
0≤t≤T
0
(3.22)
× V00 2H s + n00 2H s + V00 2H s + n00 2H s + B( max |(h, l)|s ) T . 0≤t≤T
2
Together (3.18) with (3.22) we obtain (3.9). At last, we give the estimate of Δη.
Corollary 3.1. Under the assumptions of Theorem 3.2, and assume (V, η) are solutions of (3.1), we have following estimate:
max Δη2H s−1 ≤ C B( max |(h, l)|s ) + n00 2H s+1
0≤t≤T
0≤t≤T
×
+e
C B( max |(h,l)|s )+1 T 0≤t≤T
V00 2H s
+
n00 2H s
+
V00 2H s+1
+
n00 2H s+2
+ B( max |(h, l)|s ) T 0≤t≤T
(3.23)
.
Proof. By ηt − Δη = h45 − l + Δn00 , we can get that Dα−1 Δη = Dα−1 ηt − Dα−1 h45 + Dα−1 l − Dα−1 Δn00 . Therefore max Δη2H s−1 ≤ max ηt 2H s−1 + B(|h|s ) + l2H s−1 + n00 2H s+1 ,
0≤t≤T
0≤t≤T
by (3.9), we can deduce (3.23). 2 4. Proof of the main result In this section, we will give the proof of Theorem 2.1. Firstly, we construct an approximate solution sequence to (2.3) through the following iteration scheme. First we set V 0 = V00 , n0 = n00 and define V k+1 (x, t) and nk+1 (x, t) inductively as the solution of the following linearized problem: ⎧ 3 k+1 k+1 ⎪ k ∂V k ∂V ⎪ ⎪ + (V ) A (V ) = F (V k , nk , ∇nk+1 ), A 0 j ⎪ ⎨ ∂t ∂xj j=1
⎪ nk+1 − Δnk+1 = (θk )4 − nk , ⎪ t ⎪ ⎪ ⎩ k+1 |t=0 = V0k+1 (x), nk+1 |t=0 = nk+1 (x), V 0
x ∈ R3 ,
where V0k = jk ∗ V0 , nk0 = jk ∗ n0 with k = 2−k 0 . For a positive constant C, we can choose appropriate 0 such that (V0 − V0k , n0 − nk0 )H s ≤ C4 , k = 0, 1, 2, · · · and (V0k+1 − V0k , nk+1 − nk0 )L2 ≤ 2−k . 0 Lemma 4.1. Denote by R1 ≡ V00 2H s+1 + n00 2H s+2 , then, there are T0 ∈ (0, T ] and positive constant C(R1 , C0 , T0 ), such that when T ≤ T0 , U k = V k − V00 , η k = nk − n00 (k = 0, 1, 2, ...) satisfy T max |U
0≤t≤T
k
|2s
+ max |η 0≤t≤T
k
|2s
|∇η k |2s dt ≤
+ 0
max
0≤t≤T
Δη k 2H s−1
≤ C(R1 , C0 , T0 ).
3 C0 , 4
(4.1)
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[m3L; v1.238; Prn:6/06/2018; 11:41] P.12 (1-14)
P. Jiang, Y. Zhou / J. Math. Anal. Appl. ••• (••••) •••–•••
12
Proof. Assume that (4.1) is satisfied at order k, this implies A0 (V k ) satisfy (2.1). According to Theorem 3.1, Theorem 3.2, Corollary 3.1, we can get T max |U
0≤t≤T
k+1
|2s
+ max |η
k+1
0≤t≤T
|2s
|∇η k+1 |2s dt
+ 0
C B 34 C0 +R1 +1 T
≤e
1 3 C0 + R1 + B C0 + R1 T , 2 4
3 C0 + R1 + R1 max Δnk+1 2H s−1 ≤ C B 0≤t≤T 4 3 1 3 C B 4 C0 +R1 +1 T C0 + R1 + B C0 + R1 +e T . 2 4 Therefore, there is a T0 > 0, such that for any T ≤ T0 , we have T max |U
k+1
0≤t≤T
|2s
+ max |η 0≤t≤T
k+1
|2s
|∇η k+1 |2s dt ≤
+
3 C0 , 4
0
and there exist constant C(R1 , C0 , T0 ) > 0 which depends on R1 , C0 , T0 such that when T ≤ T0 max Δη k+1 2H s−1 ≤ C(R1 , C0 , T0 ).
0≤t≤T
2
∞ Lemma 4.2. There exist T∗ ∈ (0, T ], 0 < τ < 1, {αi }∞ i=1 and {βi }i=1 with Σi |αi | < ∞, Σi |βi | < ∞, such that for k = 1, 2, · · · , when T ≤ T∗ ,
T max V k+1 − V k 2L2 + max nk+1 − nk 2L2 +
0≤t≤T
∇(nk+1 − nk )2L2 dt
0≤t≤T
≤τ
0
max V k − V k−1 2L2 + max nk − nk−1 2L2 + αk + βk .
0≤t≤T
0≤t≤T
Proof. In order to prove the result, we consider the following problem for V k+1 − V k and nk+1 − nk , ⎧ 3 k+1 k k−1 ⎪ − V k) ) k ∂(V k ∂(V − V ⎪ ⎪ + (V ) A (V ) = F (V k , I k ) − F (V k−1 , I k−1 ) + I, A 0 j ⎪ ⎨ ∂t ∂xj j=1
⎪ (nk+1 − nk )t − Δ(nk+1 − nk ) = (θk )4 − (θk−1 )4 − (nk − nk−1 ), ⎪ ⎪ ⎪ ⎩ k+1 − V k )|t=0 = V0k+1 − V0k , (nk+1 − nk )|t=0 = nk+1 − nk0 , (V 0 ∂V k ∂V k − (Aj (V k ) − Aj (V k−1 )) . ∂t ∂xj j=1 3
where I = −(A0 (V k ) − A0 (V k−1 ))
(4.2)
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AID:22303 /FLA
Doctopic: Partial Differential Equations
[m3L; v1.238; Prn:6/06/2018; 11:41] P.13 (1-14)
P. Jiang, Y. Zhou / J. Math. Anal. Appl. ••• (••••) •••–•••
13
Applying the energy estimates (3.2), (2.1) and Lemma 4.1 in the above problem, it follows: T max V
0≤t≤T
k+1
−V
k 2 L2
+ max n
k+1
0≤t≤T
−
nk 2L2
∇(nk+1 − nk )2L2 dt
+ 0
⎛
T
≤ C1 (R1 , C0 , T0 ) ⎝
T V k − V k−1 2L2 dt +
0
⎞ nk − nk−1 2L2 dt⎠
(4.3)
0
+ C V0k+1 − V0k 2L2 + nk+1 − nk0 2L2 0 ≤ C1 (R1 , C0 , T0 ) max V k − V k−1 2L2 + nk − nk−1 2L2 T + αk + βk , 0≤t≤T
where C1 (R1 , C0 , T0 ) is a positive constant depending on R1 , C0 and T0 . Next, we choose T∗ ∈ (0, T ], such that 0 < C1 (R1 , C0 , T0 )T∗ < 1. Thus, for all T ≤ T∗ , according to (4.3), we can find a τ , 0 < τ < 1 such that (4.2) holds. 2 Then, by interpolation inequality that for any
5 2
< s < s, we have 1−s /s
max V k+1 − V k H s ≤ C max V k+1 − V k L2
0≤t≤T
0≤t≤T
s /s
max V k+1 − V k H s ,
0≤t≤T
and 1−s /s
max nk+1 − nk H s ≤ C max nk+1 − nk L2
0≤t≤T
0≤t≤T
s /s
max nk+1 − nk H s .
0≤t≤T
Thus, by Lemmas 4.1, 4.2 and Sobolev’s inequality, we can obtain the local existence of (2.3). Finally, we prove uniqueness. Lemma 4.3. The solutions of (2.3) obtained at above are unique. Proof. Suppose that (V , n ) and (V , n ) are two solutions of (2.3). Denote by ω1 = V − V , ω2 = n − n . Thus, we obtain the following equations ⎧ 3 ⎪ ∂ω1 ⎪ ∂ω1 ⎪ + (V ) Aj (V ) = A ⎪ 0 ⎪ ⎪ ∂t ∂xj ⎪ j=1 ⎪ ⎪ ⎪ 3 ⎨ ∂V ∂V − (Aj (V ) − Aj (V )) + F (V , n ) − F (V , n ), −(A0 (V ) − A0 (V )) ∂t ∂x ⎪ j ⎪ j=1 ⎪ ⎪ ⎪ 4 4 ⎪ ⎪ ω − Δω = (θ − θ ) − ω , 2,t 2 2 1 2 ⎪ ⎪ ⎪ ⎩ω | = 0, ω | = 0. 1 t=0
2 t=0
Utilizing Theorem 3.1, Theorem 3.2, Lemma 4.1, Hölder’s inequality and Young’s inequality, if t ∈ (0, T ], we deduce that d |ω1 (t)|2s ≤ C(|V |2s , |V |2s , |n |2s , |n |2s )(|ω1 |2s + |ω2 |2s ) + |∇ω2 |2s , dt d |ω2 (t)|2s + |∇ω2 |2s ≤ C(|V |2s , |V |2s , |n |2s , |n |2s )(|ω1 |2s + |ω2 |2s ), dt where > 0 is a sufficiently small constant. According to (4.4) we obtain
(4.4)
JID:YJMAA
AID:22303 /FLA
Doctopic: Partial Differential Equations
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P. Jiang, Y. Zhou / J. Math. Anal. Appl. ••• (••••) •••–•••
14
d |ω1 (t)|2s + |ω2 (t)|2s + |∇ω2 |2s dt ≤ C |V |2s , |V |2s , |n |2s , |n |2s |ω1 |2s + |ω2 |2s , which follows ω1 = ω2 = 0 immediately by using Gronwall’s inequality. 2 So the proof of Theorem 2.1 is completed. Acknowledgment This work is partially supported by the Fundamental Research Funds for the Central Universities (Grant No. 2016B07214). References [1] J.W. Bond, K.M. Watson, J.A. Welch, Atomic Theory of Gas Dynamics, Addison-Wesley, Reading, Massachusetts, 1965. [2] C. Buet, B. Després, Asymptotic analysis of fluid models for the coupling of radiation and hydrodynamics, J. Quant. Spectrosc. Radiat. Transf. 85 (2004) 385–418. [3] Th. Goudon, P. Lafitte, A coupled model for radiative transfer: Doppler effects, equilibrium and non equilibrium diffusion asymptotics, Multiscale Model. Simul. 4 (2005) 1245–1279. [4] P. Jiang, Unique global solution of an initial-boundary value problem to a diffusion approximation model in radiation hydrodynamics, Discrete Contin. Dyn. Syst. Ser. A 35 (7) (2015) 3015–3037. [5] P. Jiang, Global well-posedness and large time behavior of classical solutions to the diffusion approximation model in radiation hydrodynamics, Discrete Contin. Dyn. Syst. Ser. A 37 (4) (2017) 2045–2063. [6] S. Jiang, F.C. Li, F. Xie, Nonrelativistic limits of the compressible Navier–Stokes–Fourier–P1 approximation model arising in radiation hydrodynamics, SIAM J. Math. Anal. 47 (5) (2015) 3726–3746. [7] S. Jiang, F. Xie, J.W. Zhang, A Global Existence Result in Radiation Hydrodynamics, Industrial and Applied Mathematics in China, Series in Contemporary Applied Mathematics, High Edu. Press and World Scientific, Beijing, Singapore, 2009. [8] S. Kawashima, S. Nishibata, Shock waves for a model system of the radiating gases, SIAM J. Math. Anal. 30 (1) (1999) 95–117. [9] S. Kawashima, S. Nishibata, A singular limit for hyperbolic–elliptic coupled systems in radiation hydrodynamics, Indiana Univ. Math. J. 50 (2001) 567–589. [10] R. Kippenhahn, A. Weigert, Stellar Structure and Evolution, Springer Verlag, Berlin–Heidelberg, 1994. [11] C. Lin, Asymptotic stability of rarefaction waves in radiation hydrodynamics, Commun. Math. Sci. 9 (2011) 207–223. [12] C. Lin, J.F. Coulombel, Th. Goudon, Shock profiles for non equilibrium radiating gases, Phys. D 218 (2006) 83–94. [13] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, vol. 53, Springer-Verlag, New York, Berlin, Heidelberg, 1986. [14] S.S. Penner, D.B. Olfe, Radiation and Reentry, Academic Press, New York, 1968. [15] G.C. Pomraning, The Equations of Radiation Hydrodynamics, Pergamon Press, 1973. [16] C. Rohde, F. Xie, Global existence and blowup phenomenon for a 1D radiation hydrodynamics model problem, Math. Methods Appl. Sci. 35 (5) (2012) 564–573. [17] Y.B. Zeldovich, Y.P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomenon, Academic Press, 1966. [18] X. Zhong, S. Jiang, Local existence and finite time blow-up in multidimensional radiation hydrodynamics, J. Math. Fluid Mech. 9 (2007) 543–564.