J. Math. Anal. Appl. 463 (2018) 198–221
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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
On the Rayleigh–Taylor instability in compressible viscoelastic fluids Weiwei Wang, Youyi Zhao ∗ College of Mathematics and Computer Science, Fuzhou University, Fuzhou, China
a r t i c l e
i n f o
Article history: Received 30 December 2017 Available online 14 March 2018 Submitted by D. Wang Keywords: Viscoelastic fluid Rayleigh–Taylor instability Bootstrap instability method
a b s t r a c t We mathematically investigate the Rayleigh–Taylor (abbr. RT) instability in the compressible viscoelastic fluid in the presence of a uniform gravitational field in a bounded domain based on Oldroyd-B model. We first analyze the linearized equations around the viscoelastic RT equilibrium solution, and obtain an instability condition. Then we construct solutions of the linearized viscoelastic RT problem that grow in time in the Sobolev space H 3 under an instability condition, thus leading to the linear instability. Finally, with the help of the constructed unstable solutions of the linearized viscoelastic RT problem and a local well-posedness result of smooth solutions to the nonlinear viscoelastic RT problem, we mathematically prove the instability of viscoelastic RT (abbr. VRT) problem in the sense of Hadamard. © 2018 Elsevier Inc. All rights reserved.
1. Introduction The equilibrium of the heavier fluid on top of the lighter one under the gravity is unstable, and such instability is called the Rayleigh–Taylor (abbr. RT) instability. In this case, the equilibrium state is unstable to sustain small disturbances, and this unstable disturbance will grow and lead to a release of potential energy, as the heavier fluid moves down under the gravitational force, and the lighter one is displaced upwards. This phenomenon was first studied by Rayleigh [28] and then Taylor [29], and is called therefore the RT instability. The mathematical proof of RT instability have been extensively established in the sense of Hadamard, see [14,15,20,21]. It has been also widely investigated how the RT instability evolves under the effects of other physical factors, such as rotation [3], internal surface tension [6,30,32], magnetic fields [2,16,18,19,22,23,31] and so on. Recently, the RT instability in the incompressible viscoelastic fluid have been investigated, see [13,24,25]. In this article, we further mathematically prove the RT instability in compressible viscoelastic fluid based on the following idea Oldroyd-B model in the presence of a uniform gravitational field: * Corresponding author. E-mail addresses:
[email protected] (W. Wang),
[email protected] (Y. Zhao). https://doi.org/10.1016/j.jmaa.2018.03.018 0022-247X/© 2018 Elsevier Inc. All rights reserved.
W. Wang, Y. Zhao / J. Math. Anal. Appl. 463 (2018) 198–221
⎧ ρ + div ρv = 0, ⎪ ⎪ ⎨ t ρvt + ρv · ∇v + ∇P − μ1 Δv − μ2 ∇divv = κdiv (ρU U T ) − ρge3 , ⎪ ⎪ ⎩ Ut + v · ∇U − ∇vU = 0.
199
(1.1)
Here the unknowns ρ := ρ(t, x), v := v(t, x), and U := U (t, x) denote the density, velocity, and deformation tensor (a 3 × 3 matrix valued function), respectively. κ > 0, and g > 0 stand for the elastic coefficient and gravitational constant, respectively. e3 := (0, 0, 1)T is the vertical unit vector, and −ρge3 represents the gravitational force. μ1 > 0 is the coefficient of shear viscosity and μ2 := ν + μ1 /3 with ν being the positive bulk viscosity. In this article, the pressure function P := P (ρ) is always assumed to be smooth, positive, and strictly increasing with respect to the density ρ. In the system (1.1), the equation (1.1)1 is a continuity equation, (1.1)2 describes the balance law of momentum, while (1.1)3 is called the deformation equation. The well-posedness problem of the equations (1.1) without the gravity has been widely investigated by many authors, see [9–12] for examples. To investigate the RT instability of the above equations, we shall construct an equilibrium state to the equations (1.1). To begin with, we choose a density profile ρ¯ := ρ¯(x3 ), which is independent of (x1 , x2 ) and satisfies ¯ ρ¯ ∈ C 4 (Ω),
inf ρ¯ > 0,
x∈Ω
(1.2)
and the RT condition ρ¯ (x03 )|x3 =x03 > 0
for some x03 ∈ {x3 | (x1 , x2 , x3 )T ∈ Ω}.
(1.3)
The RT condition assures that there is at least a region in which the RT density has larger density with increasing height x3 , thus leading to the classical RT instability [17]. Then, for given ρ¯ and g, we define ⎛
u ¯ ⎜ ¯ U := ⎝ 0 0
⎞ 0 0 ⎟ u ¯ 0⎠ 0 u ¯
and u ¯≡u ¯(x3 ) := ±
F (P (¯ ρ)¯ ρ + g ρ¯) + C , κ¯ ρ
(1.4)
where F (P (¯ ρ)¯ ρ + g ρ¯) denotes a primitive function of P (¯ ρ)¯ ρ + g ρ¯ and C is a positive constant satisfying inf {F (P (¯ ρ)¯ ρ + g ρ¯) + C} > 0.
¯ x∈Ω
It is easy to see that (1.4) makes sense for a bounded domain Ω, and P (¯ ρ)¯ ρ = κ(¯ ρu ¯2 ) − g ρ¯
(1.5)
¯ ) is an equilibrium where P (¯ ρ) = P (s)|s=ρ¯ and ρ¯ (x3 ) := d/dx3 . Thus, we immediately see that (¯ ρ, 0, U solution of (1.1). Now, we denote the perturbation around the equilibrium state by = ρ − ρ¯,
v = v − 0,
¯ V =U −U
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then the equations (1.1) can be written as the following perturbation equations: ⎧ ⎪ t + div ( + ρ¯)v = 0, ⎪ ⎪ ⎪ ⎪ ⎨ ( + ρ¯)vt + ( + ρ¯)v · ∇v + ∇(P ( + ρ¯) − P (¯ ρ)) − μ1 Δv − μ2 ∇divv ¯T + U ¯ V T + V V T ) + κdiv (U ¯U ¯ T ) − ge3 , ⎪ = κdiv ( + ρ¯)(V U ⎪ ⎪ ⎪ ⎪ ¯ ) − ∇v(V + U ¯ ) = 0. ⎩ Vt + v · ∇(V + U
(1.6)
In addition, we shall impose the initial-boundary value conditions: (, v, V )|t=0 = (0 , v0 , V0 ) in Ω, v ∂Ω = 0 for any t > 0,
(1.7) (1.8)
where Ω is a C 4 -smooth bounded domain. We call the initial-boundary value problem (1.6)–(1.8) the VRT (i.e., viscoelastic RT) problem. It is well-known that the elasticity has the stabilizing effect in the RT instability. However, if the elasticity is sufficiently small, then the above VRT problem may be unstable. ¯ ), it seems to be convenient of analyzing the linearized To study the instability of equilibrium state (¯ ρ, 0, U VRT problem, because it gives us an insight into the physical and mathematical mechanisms, and also provide a starting point towards the nonlinear theory. Thus, we linearize the equations (1.6) around the ¯ ), then the resulting linearized viscoelastic equations as steady state (¯ ρ, 0, U ⎧ ρv) = 0, t + div(¯ ⎪ ⎪ ⎪ ⎪ ⎨ ρ¯v + ∇(P (¯ ρ)) − μ1 Δv − μ2 ∇divv t ¯T + U ¯ V T ) + κdiv (U ¯U ¯ T ) − ge3 , ⎪ = κdiv ρ¯(V U ⎪ ⎪ ⎪ ⎩ ¯ − v · ∇U ¯. Vt = ∇v U
(1.9)
We call the initial-boundary value problem (1.7)–(1.9) the linearized VRT problem. Based on the linearized equations, we indeed obtain an sufficient condition for the instability of the above VRT problem. Before further stating our result, we shall introduce some notations used thoughtout this paper. We denote Lp := Lp (Ω) = W 0,p (Ω) for 1 p ∞, := , Ω
:= W01,2 (Ω), H k := W k,2 (Ω), · k := · H k (Ω) for k 0, Ψ(w) := ρ¯u ¯2 (∇w)T : ∇w + |∇w|2 − |divw|2 dx, E(w) := g(¯ ρ w3 + 2¯ ρdivw)w3 dx − P (¯ ρ)¯ ρ|divw|2 dx − κΨ(w), H01
κC := sup w∈H01
E(w) , Ψ(w)
a b means that a cb for some constant c > 0.
The letter c will denote a generic constant which may depend on the domain Ω and the physical parameters, such as g, μ1 , μ2 and ρ¯ in the perturbation equations (1.1). Next we introduce our main result: ¯ satisfies (1.2) and (1.3). Theorem 1.1. Let Ω be a C 4 -smooth bounded domain, density profile ρ¯ ∈ C 4 (Ω) If the instability condition κC > 0 holds, then the VRT problem (1.6)–(1.8) is unstable, that is, there exist
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positive constants ε and ι, and a quaternion (˜ 0 , v˜0 , V˜0 , vr ) ∈ H 3 , such that for any δ ∈ (0, ι) and the initial 2 ˜ data (0 , v0 , V0 ) := δ(˜ 0 , v˜0 , V0 ) +(0, δ vr , 0), there is a unique classical solution (, v, V ) of the VRT problem on [0, T max ), but (, v, V )(T )3 ε for some time T ∈ (0, T max ), where (0 , v0 , V0 ) satisfies the compatibility condition ∂ti v(x, 0)∂Ω = 0 (i = 0, 1), and T max denotes the maximal time of existence of the solution (, v, V ), and ε, ι and (˜ 0 , v˜0 , V˜0 , vr ) depend on Ω and the physical parameters in the VRT problem. Remark 1.1. We mention that Ψ(w) 0,
(1.10)
for all w ∈ H01 . In fact, by the integration by part, we have
2
ρ¯u ¯ ∂1 w2 ∂2 w1 dx =
ρ¯u ¯2 ∂1 w1 ∂2 w2 dx.
Thus, we can derive that
ρ¯u ¯2 (∇w)T : ∇w + |∇w|2 − |divw|2 dx = ρ¯u ¯2 (∂1 w2 − ∂2 w1 )2 + (∂1 w3 + ∂3 w1 )2 + (∂2 w3 + ∂3 w2 )2 + (∂1 w1 + ∂1 w2 − ∂3 w3 )2 dx
Hence (1.10) holds. Moreover, we can easily see that, if Ψ(w) = 0 for some w ∈ H01 , then w = 0. This means that the instability condition κC > 0 is equivalent to the following condition: there exists a function w0 ∈ H01 such that E(w0 ) > 0. Obviously, the instability condition κC > 0 is also equivalent to the following condition: κ<κ ˜ C := sup w∈H01
ρdivw)w3 dx − g(¯ ρ w3 + 2¯ Ψ(w)
P (¯ ρ)¯ ρ|divw|2 dx
.
Theorem 1.1 presents that the RT instability will occur in the VRT problem for κC > 0. The RT instability arising from viscoelastic flows is often called viscoelastic RT instability, which has been mathematically verified in [24] for incompressible viscoelastic flows based on the method of bootstrap instability from a linear instability to a dynamical nonlinear instability. The method of bootstrap instability was introduced by Guo and Strauss in [7,8]. Later, various versions of bootstrap approaches were presented in the proof of dynamical instability of various physical models, but no general theory has been established so far. Recently, Jiang and Jiang development a new version of bootstrap instability method to construct an unstable strong solution to the incompressible viscoelastic RT problem [19]. Thus, applying [5, Lemma 1.1] to our compressible problem, the proof procedure shall be divided into five steps. Firstly, we shall construct unstable solutions to the linearized VRT problem, this can be achieved by the modified variational method as in [6,17] due to the presence of viscosity, see Proposition 2.1. Secondly, we introduce a local well-posedness of the solutions to the VRT problem (1.6)–(1.8), see Lemma 2.2. Thirdly, we want to use the initial data for the linearized VRT problem to construct the initial data for the corresponding nonlinear problem in [19], but unfortunately, due to the presence of an interface, the initial data of the linearized and corresponding
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nonlinear problems have to satisfy the different compatibility condition on boundary ∂Ω. To circumvent this difficulty, we use the elliptic regularity theory to modify the initial data of the linearized VRT problem, so that the obtained modified initial data satisfy the condition (1.8), and at the same time, are close to the initial data of the linearized VRT problem, see Lemma 3.1. Fourthly, we deduce the error estimates between the solutions of the linearized and nonlinear problems. Due to the compressible case, we need more techniques to estimate the terms involving the compressible term, thus leading to a complicated derivation of the error estimates. Finally, we show the existence of escape times and thus obtain Theorem 1.1. Finally, we mention that, in the proof of Theorem 1.1, we will repeatedly use Cauchy–Schwarz’s inequality, Hölder’s inequality, and the embedding inequality (see [1, 4.12 Theorem]) f Lp f 1 for 2 p 6,
(1.11)
f L∞ f 2 .
(1.12)
And the interpolation inequality in H j (see [1, 5.2 Theorem]) 1− ji
f j f 0
j
f ii C f 0 + f i
(1.13)
for any 0 j < i and for any constant > 0, and the constant C depends on Ω and . In addition, we shall also repeatedly use the following estimate: f 0 ∂j f 0 for f ∈ H01 .
(1.14)
The rest of this paper can be organized as follows. In section 2, we establish the instability of the linearized VRT problem (1.7)–(1.9) and then we introduce the local well-posedness of the VRT problem (1.6)–(1.8). Finally, we will show the instability of the VRT problem (1.6)–(1.8) under the instability condition in Section 3. 2. Preliminaries This section is devoted to establishing some preliminary results for the proof of Theorem 1.1. 2.1. Linear instability To begin with, we use the modified variational method to construct unstable solutions of the linearized problem (1.7)–(1.9). The modified variational method was firstly used by Guo and Tice to construct unstable solutions to a class of ordinary differential equations arising from a linearized RT instability problem [6]. Later, Jiang and Jiang [17,18] further extend the modified variational method to construct unstable solutions to the partial differential equations (PDEs) arising from a linearized RT instability problem. Next we construct unstable solutions of the linearized problem by exploiting the modified variational method of PDE in [18] and a regularity theory of elliptic equations. To begin with, we make the following ansatz of growing mode solutions to the linearized viscoelastic equations (1.9): (x, t) = ˜(x)eΛt ,
v(x, t) = v˜(x)eΛt ,
V (x, t) = V˜ (x)eΛt
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for some Λ. Substituting this ansatz into (1.9), we get ⎧ ⎪ v = 0, Λ˜ + ρ¯ v˜3 + ρ¯div˜ ⎪ ⎪ ⎪ ⎪ ⎨ Λ¯ ρv˜ + ∇(P (¯ ρ)˜ ) − μ1 Δ˜ v − μ2 ∇div˜ v T T ˜ ¯ ¯ ˜ ¯U ¯ T ) − g ˜e3 , ⎪ = κdiv ρ¯(V U + U V ) + κdiv (˜ U ⎪ ⎪ ⎪ ⎪ ¯ − v˜ · ∇U ¯, ⎩ ΛV˜ = ∇˜ vU
(2.1)
and then eliminating ˜ and V˜ by using (2.1)1 and (2.1)3 , we arrive at a time-independent boundary problem ⎧ 2 v + Λμ2 ∇div˜ v + ∇(P (¯ ρ)(¯ ρ v˜3 + ρ¯div˜ v )) + g(¯ ρ v˜3 + ρ¯div˜ v )e3 ⎪ ⎨ Λ ρ¯v˜ =Λμ1 Δ˜ T T ¯ − v˜ · ∇U ¯ )U ¯ +U ¯ (∇˜ ¯ − v˜ · ∇U ¯ ) ) − κdiv ((¯ ¯U ¯T ) + κdiv ρ¯((∇˜ vU vU ρ v˜3 + ρ¯div˜ v )U ⎪ ⎩ v˜ = 0.
(2.2)
∂Ω
Now we apply a modified variational method to construct a solution of the boundary value problem (2.2), so we modify (2.2) as follows: ⎧ v + sμ2 ∇div˜ v + ∇(P (¯ ρ)(¯ ρ v˜3 + ρ¯div˜ v )) + g(¯ ρ v˜3 + ρ¯div˜ v )e3 ρv˜ = sμ1 Δ˜ ⎪ ⎨ α¯ T T ¯ − v˜ · ∇U ¯ )U ¯ +U ¯ (∇˜ ¯ − v˜ · ∇U ¯ ) ) − κdiv ((¯ ¯U ¯T ) + κdiv ρ¯((∇˜ vU vU ρ v˜3 + ρ¯div˜ v )U ⎪ ⎩ v˜ = 0,
(2.3)
∂Ω
where s > 0 is a parameter, and α := α(s) depends on s. Then, the standard energy functional for (2.3) is given by Θ(˜ v , s) := E(˜ v ) − s(μ1 ∇˜ v 20 + μ2 div˜ v 20 ) with an associated admissible set A :=
w ∈ H01
2 ρ ¯ w dx = 1 .
Thus, we can find an α by maximizing α := sup Θ(w, s).
(2.4)
w∈A
More precisely, we have the following results: Lemma 2.1. Assume that the density profile ρ¯ satisfies (1.2), then for any but fixed s > 0, the following assertions are valid. (1) Θ(w, s) achieves its supremum on A. (2) Let v˜ be a maximizer and α is defined by (2.4), then v˜ ∈ H01 is a weak solution to the boundary value problem (2.3). (3) If α > 0, then the maximizer v˜ further satisfies v˜3 = 0,
(2.5)
v˜12 + v˜22 = 0,
(2.6)
where v˜i denotes the i-th component of v˜. In addition, div(¯ ρv˜) = 0,
provided ρ¯ 0.
(2.7)
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Proof. (1) Let {wn }∞ n=1 ⊂ A be a maximizing sequence, using Cauchy–Schwarz’s inequality, one see that Θ(wn , s) = E(wn ) − s(μ1 ∇wn 20 + μ2 divwn 20 ) (g ρ¯ |w3n |2 + 2g ρ¯divwn w3n − P (¯ ρ)¯ ρ|divwn |2 )dx − s(μ1 ∇wn 20 + μ2 divwn 20 )
g ρ¯ |w3n |2 dx +
g 2 ρ¯|w3n |2 dx − s(μ1 ∇wn 20 + μ2 divwn 20 ), P (¯ ρ)
which yields s(μ1 ∇wn 20 + μ2 divwn 20 ) + Θ(wn , s)
g ρ¯ |w3n |2 dx +
g 2 ρ¯|w3n |2 dx. P (¯ ρ)
We easily see from the above inequality that Θ(wn , s) is bounded due to {wn }∞ n=1 ⊂ A, and consequently, wn is bounded in H01 . Thus, there exists a w ˜ ∈ A and a sequence (still denoted by wn for simplicity), such that wn → w ˜ weakly in H01 and strongly in L2 . Moreover, by the lower semi-continuity, one has sup Θ(w, s) = lim sup Θ(wn , s) w∈A
n→∞
(g ρ¯ |w3n |2 + 2g ρ¯divwn w3n )dx ρ)¯ ρ|divwn |2 + κ¯ ρu ¯2 ((∇wn )T : ∇wn + |∇wn |2 − |divwn |2 ) P (¯ − lim inf
= lim
n→∞
n→∞
+ s(μ1 |∇wn |2 + μ2 |divwn |2 ) dx
Θ(w, ˜ s) sup Θ(w, s), w∈A
which shows that Θ(w, s) achieves its supremum on A. (2) To show the second assertion, we write (2.4) as follows.
α = sup w∈H01
Θ(w, s) , J(w)
(2.8)
where we have defined J(w) := ρ¯|w|2 dx. For any τ ∈ R and ψ ∈ H01 , we take v˜(τ ) := v˜ + τ ψ. Then, (2.8) implies Θ(˜ v (τ ), s) − αJ(˜ v (τ )) 0. Let I(τ ) = Θ(˜ v (τ ), s) − αJ(˜ v (τ )), then we see that I(τ ) ∈ C 1 (R), I(τ ) 0 for all τ ∈ R, I(0) = 0 and dI(τ ) = 0. dτ τ =0 Notice that (∇˜ v )T : ∇ψ = (∇ψ)T : ∇˜ v.
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Hence, a direct computation leads to α
ρ¯v˜ · ψdx = (g ρ¯ v˜3 e3 + g ρ¯div˜ v e3 ) · ψ + g ρ¯divψe3 · v˜ dx v − P (¯ ρ)¯ ρdiv˜ v divψdx − κ¯ ρu ¯2 (∇˜ v + (∇˜ v )T ) : ∇ψdx + κ¯ ρu ¯2 div˜ v : ∇ψ + μ2 div˜ v divψ)dx, − s (μ1 ∇˜
and then using the condition (1.5) to the term g ρ¯divψe3 · v˜dx, we can change the above weak form as follows α ρ¯v˜ · ψdx = − P (¯ ρ)(¯ ρ v˜3 + ρ¯div˜ v ) − κ¯ u2 (¯ ρ v˜3 + ρ¯div˜ v ) − 2κ¯ ρu ¯u ¯ v˜3 divψdx (2.9) v + (∇˜ v )T ) : ∇ψdx − s (μ1 ∇˜ v : ∇ψ + μ2 div˜ v divψ)dx − κ ρ¯u ¯2 (∇˜ v e3 ) · ψ dx, + (g ρ¯ v˜3 e3 + g ρ¯div˜ which implies that v˜ is a weak solution to (2.3). (3) Next, we turn to prove (2.5)–(2.7) by contradiction. By the second assertion, we know the maximizer v˜ ∈ A satisfies (2.9), thus α = Θ(˜ v , s). Suppose v˜3 = 0, 2 then α = Θ(˜ v , s) < 0 due to ρ¯|˜ v | dx = 1, which contradict with the condition α > 0. Hence, v˜3 = 0. Suppose that v˜12 + v˜22 = 0 or div(¯ ρv˜) = 0, then
2 g ρ¯ v˜3 + 2g ρ¯v˜3 ∂3 v˜3 − P (¯ ρ)¯ ρ|∂3 v˜3 |2 dx 2 2 2 2 v |2 + μ2 |div˜ v |2 )dx − κ ρ¯u ¯ |∂1 v˜3 | + |∂2 v˜3 | + |∂3 v˜3 | dx − s (μ1 |∇˜ 2 ρ)¯ ρ|∂3 v˜3 | dx − κ ρ¯u ¯2 |∂1 v˜3 |2 + |∂2 v˜3 |2 + |∂3 v˜3 |2 dx = − P (¯ v |2 + μ2 |div˜ v |2 )dx < 0, − s (μ1 |∇˜
0<α=
or
2 g ρ¯ v˜3 + (2g ρ¯v˜3 − P (¯ ρ)¯ ρdiv˜ v )div˜ v dx v )T : ∇˜ v + |∇˜ v |2 − |div˜ v |2 dx − s (μ1 |∇wn |2 + μ2 |divwn |2 )dx − κ ρ¯u ¯2 (∇˜ 2 ρ)¯ ρ|div˜ v |2 dx − κ ρ¯u ¯2 (∇˜ v )T : ∇˜ v + |∇˜ v |2 − |div˜ v |2 dx =− g ρ¯ v˜3 + P (¯ v |2 + μ2 |div˜ v |2 )dx < 0, − s (μ1 |∇˜
0<α=
which is a contradiction. Therefore, (2.6) and (2.7) hold. This completes the proof. 2 Exploiting the regularity theory of elliptic equations, we can further establish the following result: Lemma 2.2. Let v˜ ∈ H01 be a weak solution of the boundary value problem (2.2), then v˜ ∈ H 4 .
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Proof. Firstly, we write (2.3) as follows. ⎧ ρu ¯2 + P (¯ ρ)¯ ρ)∂12 v˜1 + (sμ1 + κ¯ ρu ¯2 )∂22 v˜1 + (sμ1 + κ¯ ρu ¯2 )∂32 v˜1 (sμ1 + sμ2 + κ¯ ⎪ ⎪ ⎪ ⎪ ⎪ +(sμ2 + P (¯ ρ)¯ ρ)∂1 ∂2 v˜2 + (sμ2 + P (¯ ρ)¯ ρ)∂1 ∂3 v˜3 ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ = α¯ ρv˜1 + g ρ¯∂1 v˜3 − κ(¯ ρu ¯ ) (∂1 v˜3 + ∂3 v˜1 ), ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ (sμ1 + κ¯ ρu ¯ )∂1 v˜2 + (sμ1 + sμ2 + κ¯ ρu ¯2 + P (¯ ρ)¯ ρ)∂22 v˜2 + (sμ1 + κ¯ ρu ¯2 )∂32 v˜2 ⎪ ⎪ ⎪ ⎪ ⎨ + (sμ2 + P (¯ ρ)¯ ρ)∂2 ∂1 v˜1 + (sμ2 + P (¯ ρ)¯ ρ)∂2 ∂3 v˜3 2 ⎪ = α¯ ρv˜2 + g ρ¯∂2 v˜3 − κ(¯ ρu ¯ ) (∂2 v˜3 + ∂3 v˜2 ), ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ (sμ1 + κ¯ ρu ¯ )∂1 v˜3 + (sμ1 + κ¯ ρu ¯2 )∂22 v˜3 ⎪ ⎪ ⎪ ⎪ ⎪ ρ)¯ ρ)∂1 v˜1 + (sμ2 + P (¯ ρ)¯ ρ)∂2 v˜2 + (sμ1 + sμ2 + κ¯ ρu ¯2 + P (¯ ρ)¯ ρ)∂3 v˜3 ] ⎪ + ∂3 [(sμ2 + P (¯ ⎪ ⎪ ⎪ 2 ⎪ = α¯ ρv˜3 − gdiv(¯ ρv˜) + ∂3 (g ρ¯v˜3 ) + κ(¯ ρu ¯ ) (∂1 v˜1 + ∂2 v˜2 ), ⎪ ⎪ ⎪ ⎩ v˜|∂Ω = 0.
(2.10)
Let f = − α¯ ρv˜1 + g ρ¯∂1 v˜3 − κ(¯ ρu ¯2 ) (∂1 v˜3 + ∂3 v˜1 ), α¯ ρv˜2 + g ρ¯∂2 v˜3 − κ(¯ ρu ¯2 ) (∂2 v˜3 + ∂3 v˜2 ), α¯ ρv˜3 − gdiv(¯ ρv˜) + T αβ 1α,β3 2 ∂3 (g ρ¯v˜3 ) + κ(¯ ρu ¯ ) (∂1 v˜1 + ∂2 v˜2 ) and (Aij )1i,j3 be the matrix of coefficients of the linear elliptic equations (2.10), then (2.10) can be written as −∂α (Aαβ ˜j ) = fi , ij ∂β v where we have used the Einstein convention of summing over repeated indices, and the non-zero coefficients are A11 ρu ¯2 + P (¯ ρ)¯ ρ, 11 = sμ1 + sμ2 + κ¯
33 A22 ρu ¯2 , 11 = A11 = sμ1 + κ¯
13 A12 ρ)¯ ρ. 12 = A13 = sμ2 + P (¯
33 A11 ρu ¯2 , 22 = A22 = sμ1 + κ¯
A22 ρu ¯2 + P (¯ ρ)¯ ρ, 22 = sμ1 + sμ2 + κ¯
23 A21 ρ)¯ ρ. 21 = A23 = sμ2 + P (¯
22 A11 ρu ¯2 , 33 = A33 = sμ1 + κ¯
32 A31 ρ)¯ ρ, 31 = A32 = sμ2 + P (¯
A33 ρu ¯2 + P (¯ ρ)¯ ρ. 33 = sμ1 + sμ2 + κ¯
Noting that, for any ξ, η ∈ R3 , Aαβ ρu ¯2 )|ξ|2 |η|2 + (sμ2 + P (¯ ρ)¯ ρ)(ξ1 η1 + ξ2 η2 + ξ3 η3 )2 ij ξα ξβ ηi ηj = (sμ1 + κ¯ (sμ1 + κ¯ ρu ¯2 )|ξ|2 |η|2 , αβ 4 0,1 hence Aαβ (Ω) and ij satisfies the strong elliptic condition. On the other hand, ∂Ω is C -smooth, Aij ∈ C f ∈ L2 . Thus, if we apply [4, Theorem 4.11] to the weak form (2.9), we get v˜ ∈ H 2 . Moreover, in view of the definition of f , if we repeat the above improving regularity method with v˜ ∈ H 2 , we have v˜ ∈ H 4 . 2
Next, we further show that there exists a fixed point s = Λ > 0 such that shall give some properties of α(s) as a function of s > 0.
α(Λ) = Λ. To this end, we
Lemma 2.3. Under the assumptions of Lemma 2.1, the function α(s) defined on (0, ∞) enjoys the following properties: 0,1 (0, ∞) is nonincreasing. (1) α(s) ∈ Cloc (2) If κC > 0, then there are constants c1 , c2 > 0, which depend on g, ρ¯, u ¯, μ1 , μ2 , and κ, such that α(s) c1 − sc2 .
(2.11)
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s
Proof. (1) Let {wnj }∞ n=1 ⊂ A be a maximizing sequence of supw∈A Θ(w, sj ) for j = 1, 2. Then α(s1 ) lim sup Θ(wns2 , s1 ) lim inf Θ(wns2 , s2 ) = α(s2 ) n→∞
n→∞
for any 0 < s1 < s2 < ∞. Hence α(s) is nonincreasing on (0, ∞). Next we use this fact to show the continuity of α(s). Let [a, b] ⊂ (0, ∞) be a bounded interval. For any w ∈ A, by Cauchy–Schwarz’s inequality,
(g ρ¯ |w3 |2 + 2g ρ¯divww3 − P (¯ ρ)¯ ρ|divw|2 )dx g ρ¯ . + g ρ¯ ∞ P (¯ ρ) L∞ L
Θ(w, s)
Hence, from the monotonicity of α(s) we get ρ¯ g := L < ∞ |α(s)| max |α(a)|, |α(b)|, g + ρ¯ ∞ P (¯ ρ) L∞ L
for any s ∈ [a, b].
(2.12)
On the other hand, for any given s ∈ [a, b], it’s easy to know that there exists a maximizing sequence {wsn } ⊂ A of supw∈A Θ(w, s), such that |α(s) − Θ(wns , s)| < 1.
(2.13)
Using (2.12) and (2.13), we infer from the definition of Θ(w, s) that 0
(μ1 |∇wns |2 + μ2 |divwns |2 )dx +
1 s
κ s
ρ¯u ¯2 ((∇wns T ) : ∇wns + |∇wns |2 − |divwns |2 )dx
s 2 s (g ρ¯ |wn3 | + 2g ρ¯divwns wn3 − P (¯ ρ)¯ ρ|divwns |2 )dx −
Θ(wns , s) s
s 2 g 2 ρ¯|wn3 Θ(wns , s) | dx − P (¯ ρ) s g g 1 + 2L 1+L ρ¯ := K, + + a ρ¯ L∞ (R) P (¯ ρ) L∞ a a
1 s
s 2 g ρ¯ |wn3 | dx +
1 s
s where wn3 denotes the third component of wns . Thus, for sj ∈ I(j = 1, 2), we further find that
α(s1 ) =
lim sup Θ(wns1 , s1 ) n→∞
lim sup Θ(wns1 , s2 ) n→∞
+ |s1 − s2 | lim sup n→∞
α(s2 ) + K|s1 − s2 |, which yields α(s1 ) − α(s2 ) K|s1 − s2 |. Similarly, we also have α(s2 ) − α(s1 ) K|s1 − s2 |.
(μ1 |∇wns |2 + μ2 |divwns |2 )dx
(2.14)
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The above two inequalities implies |α(s1 ) − α(s2 )| K|s1 − s2 |, 0,1 which yields α(s) ∈ Cloc (0, ∞). (2) We turn to the proof of (2.11). Noting that κC > 0, we can deduce from the definition of κC that there is a v ∈ H01 , such that E(v) = g(¯ ρ v3 + 2¯ ρdivv)v3 dx − P (¯ ρ)¯ ρ|divv|2 dx − κΨ(v) > 0
Thus, one has Θ(w, s) J(w) w∈A (μ1 |∇v|2 dx + μ2 |divv|2 )dx Θ(v, s) E(v) = 2 −s := c1 − sc2 J(v) ρ¯v dx ρ¯v 2 dx
α(s) = sup Θ(w, s) = sup
w∈H01
for two positive constant c1 := c1 (g, ρ¯, u ¯, κ) and c2 := c2 (μ1 , μ2 , ρ¯). This completes the proof of Lemma 2.3. 2 Next we show that there exists a function v˜ satisfying (2.2) with a growth rate Λ > 0. Let S := sup{s | α(τ ) > 0 for any τ ∈ (0, s)}. By virtue of Lemma 2.3, S > 0; and moreover, α(s) > 0 for any s < S. Since α(s) = supw∈A Θ(w, s) < ∞, using the monotonicity of α(s), we see that lim α(s) exists and the limit is a positive constant.
s→0
(2.15)
On the other hand, by virtue of Poincaré’s inequality, there is a constant c3 dependent of g, ρ¯, Ω such that
g ρ¯ w32 dx +
g 2 ρ¯ 2 w dx c3 P (¯ ρ) 3
|∇w|2 dx
for any w ∈ A.
Thus, if s > c3 /μ1 , then
g ρ¯ w32 dx +
g 2 ρ¯ 2 w dx − s P (¯ ρ) 3
(μ1 |∇w|2 dx + μ2 |divw|2 )dx < 0 for any w ∈ A,
which implies that α(s) 0
for any s > c3 /μ1 .
Hence S < ∞. Moreover, lim α(s) = 0.
s→S
(2.16)
Now, exploiting (2.15) and (2.16) and the continuity of α(s) on (0, S), we find by a fixed-point argument on (0, S) that there is a unique Λ ∈ (0, S) satisfying Λ=
α(Λ) =
sup Θ(w, Λ) > 0. w∈A
(2.17)
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Thus, by virtue of Lemma 2.1 and Lemma 2.2, there is a solution v˜ ∈ H 4 to the boundary value problem (2.2) with Λ constructed in (2.17), where v˜ satisfies (2.5) and (2.7). Thus, we conclude the following Proposition, which gives Theorem 2.1. Lemma 2.4. Under the assumptions of Theorem 1.1, then there exists a v˜ ∈ H 4 ∩ A which solve the boundary value problem (2.2) with a finite growth rate Λ > 0 satisfying (2.17). Moreover, v˜ satisfies (2.5)–(2.7). In ¯ −˜ ¯ )/Λ), then (˜ particular, let (˜ ρ, V˜ ) = (−div(¯ ρv˜)(x)/Λ, (∇˜ vU v ·∇U ρ, v˜, V˜ ) ∈ H 3 ×H 4 ×H 3 solves (1.7)–(1.9). In addition, ρ˜ = 0 provided ρ¯ 0. In view of Lemma 2.4, we obtain the following instability result of the linearized VRT problem. ¯ ) of the linearized ρ, 0, U Proposition 2.1. Under the assumptions of Theorem 1.1, the equilibrium state (¯ problem (1.7)–(1.9) is unstable, that is, there is an unstable solution in the form ¯ − v˜ · ∇U ¯ )(x)/Λ) (, v, V )(t, x) := eΛt (−div(¯ ρv˜)(x)/Λ, v˜(x), (∇˜ vU to the linearized VRT problem, where v˜ ∈ H 4 ∩ A solve the boundary value problem (2.2) with Λ > 0 being a constant satisfying Λ2 = sup Θ(w, Λ) = Θ(˜ v , Λ),
(2.18)
w∈A
where A :=
w∈
H01
2 ρ¯w dx = 1 ,
and
Θ(w, Λ) = E(w) − Λ(μ1 ∇w20 + μ2 divw20 ).
Moreover, v˜3 = 0.
(2.19)
2.2. Local well-posedness Now we further introduce the local well-posedness of the solutions to the VRT problem (1.6)–(1.8). Proposition 2.2. Under the assumptions of Theorem 1.1, for any given initial data (0 , u0 , V0 ) ∈ H 3 , satisfying inf x∈Ω {(0 + ρ¯)(x)} > 0, and the compatibility condition ∂ti v(·, 0)|∂Ω = 0 for i = 0, 1, there exist a T max and a unique classical solution (, v, V ) ∈ C 0 ([0, T max ), H 3 ) to the VRT problem, where (, v, V ) enjoys the following properties: vt ∈ C 0 ([0, T max ), H 1 ) ∩ L2 ((0, T max ), H 2 ) and 0 <
inf
{(x, t) + ρ¯},
(x,t)∈Ω×(0,T max )
where T max denotes the maximal time of existence of the solution (, v, V ). In addition, if inf {¯ ρ(x)}/2 ( + ρ¯)(x, t) 2 sup{¯ ρ(x)} on Ω × (0, T max ),
x∈Ω
(2.20)
x∈Ω
then S((t), v(t), V (t)) := v(t)21 + (t), V (t), t , vt , Vt 20 C1 (, V )(t)21 + v(t)22 + (, v, V )(t)21 (, v, V )(t)22 1 + V 21
(2.21)
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for any t ∈ (0, T max ), where the constant C1 1, only depends on Ω and the known physical parameters in the perturbation equations. Proof. Proposition 2.2 can be established by a standard iteration scheme as in [22, Proposition 3.1], and hence, we omit its proof here which involves only tedious calculations. Here we only show (2.21). In view of (1.5) and the relation P ( + ρ¯) = P¯ + P (¯ ρ) +
( − z)P (z + ρ¯)dz,
0
we can write (1.6)2 as follows ( + ρ¯)vt + ( + ρ¯)v · ∇v + ∇ P (¯ ρ) + ( − z)P (z + ρ¯)dz 0
¯T + U ¯ V T + V V T ) + κdiv (U ¯U ¯ T ) − ge3 . = μ1 Δv + μ2 ∇divv + κdiv ( + ρ¯)(V U Multiplying the above identity by vt in L2 and using (2.20) to the term ( + ρ¯)v · ∇v, then we deduce that ( + ρ¯)|vt |2 dx vt 20 2 2 2 2 2 + C (, V )(t)1 + v(t)2 + (, v, V )(t)1 (, v, V )(t)2 1 + V 1 for any > 0, where the constant C depends on . Thus, one can use (2.20) again to get vt 20 v(t)22 + (, V )(t)21 + (, v, V )(t)21 (, v, V )(t)22 (1 + V 21 ). Using (1.6)1 and (1.6)3 , we have t 0 (1 + 2 )v1 ,
Vt 0 (1 + V 2 )v1 .
From the above three estimates we can get (2.21) immediately.
2
3. Proof of Theorem 1.1 In this section, we will give the detailed results of instability for the VRT problem (1.6)–(1.8). We will use Propositions 2.1–2.2 to construct unstable solutions to the VRT problem. To begin with, we use Proposition 2.1 to give a solution of the linearized perturbation equations (1.9) in the form:
l , v l , V l = eΛt ˜0 , v˜0 , V˜0 ∈ H 3 ,
(3.1)
where ¯ − v˜ · ∇U ¯ )/Λ). (˜ 0 , v˜0 , V˜0 ) := (−div(¯ ρv˜)(x)/Λ, v˜(x), (∇˜ vU Now, denote (a , v a , V a ) := δ(l , v l , V l ),
δ > 0, a constant.
(3.2)
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Then, (a , v a , V a ) solves the linearized perturbation equations (1.9) with initial data δ(˜ 0 , v˜0 , V˜0 ), and satisfies (a , v a , V a )3 δeΛt .
(3.3)
In the rest of this paper, we call (3.2) an approximate solution of the VRT problem for fixed δ. Now, we shall further modify the initial data δ(˜ 0 , v˜0 , V˜0 ) to construct a solution of the VRT problem for sufficiently small δ > 0. Lemma 3.1. Let (˜ 0 , v˜0 , V˜0 ) be the same as in (3.1). Then there are an error function vr and a constant δ1 ∈ (0, 1) depending on (˜ 0 , v˜0 , V˜0 ), such that for any δ ∈ (0, δ1 ), (1) The modified initial data (δ0 , v0δ , V0δ ) := δ(˜ 0 , v˜0 , V˜0 ) + (0, δ 2 vr , 0)
(3.4)
satisfies the compatibility conditions on boundary of the VRT problem (1.6)–(1.8). (2) vr satisfies the following estimate: vr 3 C2 , where the constant C2 1 depends on (˜ 0 , v˜0 , V˜0 )3 and other physical parameters, but is independent of δ. 0 , v˜0 , V˜0 ) satisfies v˜0 |∂Ω = 0 and Proof. Note that (˜
T T T ˜ ¯ ¯ ˜ ¯ ¯ μ1 Δ˜ v0 + μ2 ∇div˜ v0 − ∇(P (¯ ρ)˜ 0 ) + κdiv ρ¯(V0 U + U V0 ) + κdiv (˜ 0 U U ) − g ˜0 e3
= 0. ∂Ω
Hence, if the modified initial data satisfy (3.4), then we expect to vr satisfy the following problem: ⎧ v0 μ1 Δvr + μ2 ∇divvr − δ 2 (δ ˜0 + ρ¯)vr · ∇vr − δ(δ ˜0 + ρ¯)vr · ∇˜ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ˜0 ⎪ ⎪ ⎪ ¯T + U ¯ V˜ T ) − κdiv (δ ˜0 + ρ¯)V˜0 V˜ T ⎨ =∇ (˜ 0 − z)P (δz + ρ¯)dz − κdiv ˜0 (V˜0 U 0 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ =: F (˜ 0 , v˜0 , V˜0 ), ⎪ ⎪ ⎪ ⎩ v | = 0. r ∂Ω
(3.5)
Thus the modified initial data naturally satisfy the compatibility conditions on boundary. Next we shall look for a solution vr to the boundary problem (3.5) when δ is sufficiently small. We begin with the linearization of (3.5) which reads as μ1 Δvr + μ2 ∇divvr = F (˜ 0 , v˜0 , V˜0 ) + δ 2 (δ ˜0 + ρ¯)w · ∇w + (δ ˜0 + ρ¯)w · ∇˜ v0
(3.6)
with boundary condition vr |∂Ω = 0.
(3.7)
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Let w ∈ H 3 , then it follows from the elliptic theory that there is a solution vr of (3.6)–(3.7) satisfying vr 3 F (˜ 0 , v˜0 , V˜0 ) + δ 2 (δ ˜0 + ρ¯)w · ∇w + (δ ˜0 + ρ¯)w · ∇˜ v0 1 C3 1 + δ 2 w22 , δ ∈ (0, 1), where C3 depends on (˜ 0 , v˜0 , V˜0 )3 and other physical parameters. Now, taking δ such that δ −1 min{(2C3 ) , 1}, we have for any w3 2C3 that vr 3 2C3 . Therefore, one can construct an approximate function sequence vrn , such that μΔvrn+1 + μ2 ∇divvrn+1 − δ 2 (δ ˜0 + ρ¯)vrn · ∇vrn − (δ ˜0 + ρ¯)vrn · ∇˜ v0 = F (˜ 0 , v˜0 , V˜0 ) for any n, vrn 3 2C3 and vrn+1 − vrn 3 C4 vrn − vrn−1 3 for some constant C4 independent of δ and n. Finally, we choose a sufficiently small δ so that C4 δ < 1, and use then a compactness argument to get a limit function which solves the nonlinear boundary problem (3.5). Moreover vr 3 2C3 . Thus we have proved Lemma 3.1. 2 ρ(x)} > 0, and the embedding inequality (1.12), we can choose a Now, in view of the condition inf x∈Ω {¯ sufficiently small τ ∈ (0, δ1 ), such that inf x∈Ω {¯ ρ(x)} inf (δ0 (x) + ρ¯(x)) 2 sup{¯ ρ(x)} x∈Ω 2 x∈Ω
for any δ ∈ (0, τ ).
(3.8)
Thus, by virtue of Proposition 2.2, for any given δ < τ , there exists a unique local-in-time classical solution (δ , v δ , V δ ) ∈ C 0 ([0, T max ), H 3 ) to the VRT problem, emanating from the initial data (δ0 , v0δ , V0δ ). We estimate the error between the approximation solution (a , v a , V a ) and (δ , v δ , V δ ). Denote d d ( , v , V d ) = (δ , v δ , V δ ) − (a , v a , V a ). Then (d , v d , V d ) satisfies the following error equations: ⎧ d ρv d ) = F δ , t + div(¯ ⎪ ⎪ ⎪ ⎪ ⎨ (δ + ρ¯)v d + ∇(P (¯ ρ)d ) − μ1 Δv d − μ2 ∇divv d t ¯T + U ¯ (V d )T )) − κdiv (d U ¯U ¯ T ) + gd e3 = Gδ , ⎪ − κdiv (¯ ρ(V d U ⎪ ⎪ ⎪ ⎩ d ¯ + Hδ, ¯ − v d · ∇U Vt = ∇v d U
(3.9)
where F δ := −div(δ v δ ), ¯ (V δ )T ) ¯T + U Gδ := κdiv ((δ + ρ¯)(V δ (V δ )T )) + κdiv (δ (V δ U −∇ (δ − z)P (δz + ρ¯)dz − (δ + ρ¯)v δ · ∇v δ − δ vta , δ
0
H δ := ∇v δ V δ − v δ · ∇V δ , with initial conditions (d (0), v d (0), V d (0)) = (0, δ 2 vr , 0).
(3.10)
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Next, we can deduce the following error estimate from the error equations (3.9), which plays a critical role in the proof of Theorem 1.1. Lemma 3.2. Let ∈ (0, 1) and δ ∈ (0, τ ) ⊂ (0, 1). Denote C5 := 2(1 + Λ)((˜ 0 , v˜0 , V˜0 )2 + 2C2 )2 ,
T ∗ := sup t ∈ (0, T max ) S(δ (τ ), v δ (τ ), V δ (τ )) + vsδ L2 ((0,τ ),H 1 ) 2C1 C5 δeΛτ for any τ ∈ [0, t] If T max = ∞, and (δ , v δ , V δ ) satisfies (2.20) and (δ , v δ , V δ )3 on [0, ∞),
(3.11)
then there is a constant C6 , such that S(d (t), v d (t), V d (t)) + vtd L2 ((0,t),H 1 ) C6
f ( , δ, t) for any t ∈ (0, T ∗ ], 2
7
where S(d (t), v d (t), V d (t)) is defined by (2.21) and f ( , δ, t) := δ 3 e3Λt + 3 δ 3 e
7Λt 3
3
9
+ 4 δ 4 e
(3.12) 9Λt 4
.
Remark 3.1. It should be remarked that (2.20) automatically holds for sufficiently small . In fact, by virtue of (1.12), there is a positive constant 0 , such that (2.20) holds for any δ satisfying δ 3 0 . Proof. From Lemma 3.1 and (3.11) one gets C5 δ . 1 + (δ0 , v0δ , V0δ )21 (δ0 , v0δ , V0δ )2 δ 1 + ((˜ 0 , v˜0 , V˜0 )2 + C2 )2 (˜ 0 , v˜0 , V˜0 )2 + C2 ) 2 Thus, in view of the regularity of (δ , v δ , V δ ) and the estimate (2.21), we see that T ∗ > 0. Moreover, by definition of T ∗ S(δ (t), v δ (t), V δ (t)) + vτδ L2 ((0,t),H 1 ) 2C1 C5 δeΛt on [0, T ∗ ],
(3.13)
or S(δ (T ∗ ), v δ (T ∗ ), V δ (T ∗ )) + vτδ L2 ((0,T ∗ ),H 1 ) = 2C1 C5 δeΛT
∗
in the case of T ∗ < ∞.
(3.14)
We next show (3.12). By (3.9)2 we find that
δ d2 δ d d + ρ¯ |vt | dx = 2 ( + ρ¯)vt t · vt dx − δt |vtd |2 dx d 2 d d d ρ)t − κ¯ u t + 2κ¯ ρu ¯u ¯ v3 divvt dx − 2 gdt e3 · vtd dx =2 P (¯ δ δ δ T d Gt − δt vtd · vtd dx − 2 κ¯ ρu ¯ H + (H ) : ∇vt dx + 2 2 d d T d − 2 κ¯ ρu ¯ (∇v + (∇v ) ) : ∇vt dx − 2 (μ1 |∇vtd |2 dx + μ2 |divvtd |2 )dx
d dt
˜ 1 (t) + R ˜ 2 (t) + R3 (t) + R4 (t) =: R − 2 κ¯ ρu ¯2 (∇v d + (∇v d )T ) : ∇vtd dx − 2 (μ1 |∇vtd |2 dx + μ2 |divvtd |2 )dx,
(3.15)
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˜ 1 (t), where the first four integrals on the right-hand side of the second equality in (3.15) are denoted by R ˜ R2 (t), R3 (t) and R4 (t), respectively. On the other hand, since (∇vtd )T : ∇v d = (∇v d )T : ∇vtd , we can use (3.9)1 , (3.9)3 , and (1.5) to infer that ˜ 2 (t) − 2 κ¯ ˜ 1 (t) + R ρu ¯2 (∇v d + (∇v d )T ) : ∇vtd dx R ρ)(¯ ρ v3d + ρ¯divv d ) + κ¯ u2 (¯ ρ v3d + ρ¯divv d ) + 2κ¯ ρu ¯u ¯ v3d divvtd dx =2 −P (¯ d ρu ¯2 (∇v d + (∇v d )T ) : ∇vtd dx + R2 (t) +2 g ρ¯ u3 + g ρ¯divud ∂t v3d dx − 2 κ¯ ρ)¯ ρ v3d + κ¯ u2 ρ¯ v3d + 2κ¯ ρu ¯u ¯ v3d divvtd dx =2 −P (¯ d ρ)¯ ρdivv d divvtd dx + 2 g ρ¯ v3 + g ρ¯divv d ∂t v3d dx − 2 P (¯ ρu ¯2 (∇v d + (∇v d )T ) : ∇vtd dx + R2 (t) + 2 κ¯ ρu ¯2 divv d divvtd dx − 2 κ¯ =
d E(v d (t)) + R2 (t), dt
where R2 (t) := 2
F
δ
ρ) − κ¯ u P (¯
2
divvtd dx
− 2g
F δ ∂t v3d dx.
Thus the equality (3.15) can be written as d dt
d2 d + ρ¯ |vt | dx − E(v (t)) + 2 (μ1 |∇vtd |2 dx + μ2 |divvtd |2 )dx =
4
δ
Ri (t).
(3.16)
i ,
(3.17)
i=2
Recalling that v d (0) = δ 2 vr , we can integrate (3.16) in time from 0 to t to get d 2 δ + ρ¯vt (t)0 + 2 (μ1 ∇vτd 20 dx + μ2 divvτd 20 )dx = E(v d (t)) + t
4
i=1
0
where 1 :=
δ d2 + ρ¯ |vt | dx
− E(δ 2 vr )
t=0
t i :=
Ri (τ )dτ, i = 2, 3, 4, 0
which can be bounded from below. Multiplying (3.9)2 by vtd in L2 , one gets
δ + ρ¯ |vtd |2 dx =
¯ (V d )T )) + ¯T + U μ1 Δv d + μ2 ∇divv d + κdiv (¯ ρ(V d U ¯U ¯ T ) − gd e3 − ∇(P (¯ κdiv ( U ρ)d ) + Gδ d
· vtd dx.
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Applying (2.20) and Cauchy–Schwarz’s inequality, we obtain δ + ρ¯vtd (t)20 (d , V d )21 + v d 22 + Gδ 20 .
(3.18)
From the definition of vta it follows that ∂tj v a k = Λj δeΛt ˜ v0 k Λj δeΛt
for 0 k, j 3.
(3.19)
So, using (3.19), (3.13), (3.11), (2.20), the interpolation inequality (1.13), the Nirenberg interpolation inequality (see [1, 5.8 Theorem]) 1
1
wL∞ w02 w32 ,
(3.20)
and Nirenberg’s interpolation inequality (see [26, Theorem]) 3
1
3
1
wL∞ w04 ∇2 w04 + w0 w04 w24 ,
(3.21)
we arrive at Gδ 20 δ + ρ¯21 V δ 4L∞ + δ + ρ¯2L∞ V δ 21 V δ 2L∞ + v δ 2L∞ v δ 21 + δ 20 vta 22 + (δ , V δ )2L∞ (δ , V δ )21 3
1
5
3
(1 + δ 1 )2 (V δ 04 V δ 24 )4 + (V δ 04 V δ 24 )2 7 3
(3.22)
5 3
+ v δ 0 v δ 3 v δ 21 + (δ , V δ )0 (δ , V δ )3 + δ 4 e4Λt 3
5
δ 3 e3Λt + 2 δ 2 e
5Λt 2
5
7
+ 3 δ 3 e
7Λt 3
+ δ 4 e4Λt .
Chaining the estimates (3.18) and (3.22) together and taking the limit as t → 0. We apply Lemma 3.1 and (3.10) to get 1 = lim δ + ρ¯vtd (t)20 − E(δ 2 vr ) t→0 ! " 3 5 5Λt 5 7 7Λt lim (d , V d )(t)21 + v d 22 + δ 3 e3Λt + 2 δ 2 e 2 + 3 δ 3 e 3 + δ 4 e4Λt + δ 4
(3.23)
t→0
5
7
1
5
2
7
1
5
δ 4 + δ 3 + 3 δ 3 + 2 δ 2 δ 3 + 3 δ 3 + 2 δ 2 . To bound 2 (t) and 3 (t), recalling the definition of 2 (t) and 3 (t), we have t 2 (t) + 3 (t)
(F δ 0 + H δ 0 )∂t v d 1 dτ.
(3.24)
0
By the interpolation inequality [27, Theorem 1.49] in Lp , (1.13) in H k and the embedding inequality (1.11), we see that 1
1
1
1
1
1
∇wL3 ∇w02 ∇wL2 6 w12 w22 w02 w32 ,
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from which, (3.20) and the embedding inequality it follows that F δ 20 + H δ 0 v δ 1 ∇(δ , V δ )L3 + (δ , V δ )L∞ ∇v δ 0 1
1
1
1
v δ 1 (δ , V δ )02 (δ , V δ )32 + (δ , V δ )02 (δ , V δ )32 v δ 1 1 2
3 2
δ e
3Λt 2
(3.25)
.
Putting (3.25) into (3.24), we make use of (3.19), (3.13) and Hölder’s inequality to conclude ⎛ t ⎞ 12 ⎛ t ⎞ 12 t 1 5 5Λt 2 (t) + 3 (t) ⎝ δ 3 e3Λt dτ ⎠ ⎝ vτa 21 dτ + vτδ 21 dτ ⎠ 2 δ 2 e 2 . 0
0
(3.26)
0
Next, to control 4 (t), we integrate by parts to find that t
κ δτ (V δ (V δ )T ) + (δ + ρ¯) Vτδ (V δ )T + (δ + ρ¯) V δ (Vτδ )T : ∇vτd dxdτ
4 (t) = −2 0
t −2
κ¯ u δτ S(V δ ) + δ S(Vτδ ) : ∇vτd dxdτ
0
t −2 0
δ ( + ρ¯)vτδ · ∇v δ + (δ + ρ¯)v δ · ∇vτδ + δ vτaτ · vτd dxdτ ⎛
t
⎞
δ
⎜ δτ ⎝(vτa + vτδ + 2v δ · ∇v δ ) · vτd − 2
−
0
⎟ P (δz + ρ¯)dzdivvτd ⎠ dxdτ
0
:= 4,1 + 4,2 + 4,3 + 4,4 , where S(V δ ) = V δ + (V δ )T . And the terms on the right hand side can be estimated as follows, employing (2.20), (3.11), (3.13), (3.19), (3.20) and (3.21), we obtain t 4,1 + 4,2 = −2κ
δ ¯V + u ¯(V δ )T + V δ (V δ )T + (δ + ρ¯) Vτδ (V δ )T + V δ (V δ )Tτ δτ u
0
δ + δ u ¯ Vτ + u ¯(Vτδ )T : ∇vτd dxdτ t
δτ 0 V δ L∞ 1 + V δ 2 + Vτδ 0 (δ , V δ )L∞ δ + ρ¯L∞ + 1 vτd 1 dτ
0
⎛ t ⎞ 12 ⎛ t ⎞ 12 t 3 1 ⎝ (1 + )2 δ 2 e2Λτ (δ , V δ )02 (δ , V δ )22 dτ ⎠ ⎝ vτa 21 dτ + vτδ 21 dτ ⎠ 0
1
0
7
4 (1 + )δ 4 e
7Λt 4
⎞ 12 ⎛ t t 11Λt ⎝ vτa 21 dτ + vτδ 21 dτ ⎠ 14 δ 11 4 e 4 . 0
0
0
(3.27)
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While one can utilize (3.19), (3.13), (2.20) and (1.11) to deduce that t 4,3
δ 0 vτaτ 1 + v δ 1 vτδ 1 vτd 1 dτ
0
t
2 2Λτ Λτ
δ e (δe + vτδ 1 ) + δeΛτ vτδ 21 dτ
(3.28)
0
t
δeΛτ δ 2 e2Λτ + vτδ 21 dτ
0
δ 3 e3Λt . Finally, similar to the derivation of (3.28), and use (3.20), one has t 4,4
δτ 0 vτa L3 + vτδ L3 + δ L∞ + v δ 1 vτδ 2 vτd 1 dτ
0
t
! " 1 1 δeΛτ δ 02 δ 32 vτd 1 + δ 2 e2Λτ + vτδ 21 dτ
(3.29)
0 5
δ2e
5Λt 2
1
1
( 2 + δ 2 e
Λt 2
).
Thus, summing up the estimates (3.27)–(3.29), (3.26) and (3.23), we use Young’s inequality to infer 4 1
5
i δ 3 e3Λt + 2 δ 2 e
5Λt 2
2
7
1
+ 3 δ 3 + 4 δ
11 4
e
11Λt 4
f ( , δ, t).
(3.30)
i=1
Combining (3.17) with (3.30), one obtains δ + ρ¯vtd (t)20 + 2 μ1 ∇vτd 20 + μ2 divvτd 20 dτ E(v d (t) + cf ( , δ, t). t
0
Thanks to (2.18), we have E(v d (t)) Λ2 = Λ2
ρ¯|v d |2 dx + Λ μ1 ∇v d 20 + μ2 divv d 20 (δ + ρ¯)|v d |2 dx + Λ μ1 ∇v d 20 + μ2 divv d 20 − Λ2
δ |v d |2 dx.
On the one hand,
1
1
1
5
δ |v d |2 dx δ L∞ v d 20 δ 02 δ 32 δ 2 e2Λt 2 δ 2 e
5Λt 2
.
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218
Combining the above three estimates together, we arrive at d 2 δ + ρ¯vt (t)0 + 2 μ1 ∇vτd 20 + μ2 divvτd 20 dτ t
Λ2
0
(3.31)
δ + ρ¯v d (t)20 + Λ μ1 ∇v d 20 + μ2 divv d 20 + cf ( , δ, t).
Moreover, recalling that v d ∈ C 0 ([0, T max ), H 3 ) and v d (0) = δ 2 vr , we apply Newton–Leibniz’s formula and Cauchy–Schwarz’s inequality to find that Λ μ1 ∇v d 20 + μ2 divv d 20 t
t ∇v : d
= 2Λμ1
∇vτd dxdτ
0
+δ
4
divv d : divvτd dxdτ
+ 2Λμ2 0
Λ(μ1 ∇vr 20
+
(3.32)
μ2 divvr 20 )
t Λ
t (μ1 ∇v d 20 dτ
2
+
μ2 divv d 20 )dτ
(μ1 ∇vτd 20 + μ2 divvτd 20 )dτ + cf ( , δ, t).
+
0
0
Combining (3.31) with (3.32), one gets 1 δ + ρ¯vtd (t)20 + μ1 ∇v d 20 + μ2 divv d 20 Λ t d 2 δ Λ + ρ¯v (t)0 + 2Λ μ1 ∇v d 20 + μ2 divv d 20 dτ + cf ( , δ, t).
(3.33)
0
On the other hand,
d dt
(δ
+
ρ¯)v d 20
=2
( + ρ¯)v · δ
d
vtd dx
+
δt |v d |2 dx
1 d 2 d 2 δ δ ( + ρ¯)vt 0 + Λ ( + ρ¯)v 0 + δt |v d |2 dx Λ
and by virtue of (3.13) and (3.3), we obtain
δt |v d |2 dx = −
div(¯ ρv δ + δ v δ )|v d |2 dx
v δ 1 + δ 2 v δ 1 v d 21 δ 3 e3Λt , for any t ∈ (0, T ∗ ). If we put the previous three estimates together, we get the differential inequality d (δ + ρ¯)v d 20 + μ1 ∇v d 20 + μ2 divv d 20 dt ⎛ ⎞ t 2Λ ⎝ δ + ρ¯v d (t)20 + μ1 ∇v d 20 + μ2 divv d 20 dτ ⎠ + cf ( , δ, t). 0
(3.34)
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Recalling v d (0) = δ 2 vr , one can apply Gronwall’s inequality to (3.34) to conclude t t d 2 d 2 δ ( + ρ¯)v 0 + μ1 ∇v 0 dτ + μ2 divv d 20 dτ 0
0
⎛ t ⎞ e2Λt ⎝ cf ( , δ, τ )e−2Λτ dτ + δ 4 (δ0 + ρ¯)vr 20 ⎠ f ( , δ, t), for any t ∈ (0, T ∗ ).
(3.35)
0
Moreover, using (1.14) and (2.20), we can further from (3.35), (3.33) and (3.31) that v d (t)21 + vtd (t)20 + vτd (t)2L2 ((0,t),H 1 ) f ( , δ, t).
(3.36)
Next we turn to the derivation of d and V d . First, it follow from the equation (3.9)1 that 1 d d 2 0 = − 2 dt
div(¯ ρv d + δ v δ )d dx
=−
div(¯ ρv d + a v δ ) + d divv δ /2 d dx
v d 1 + a 2 v δ 1 + d L∞ v δ 1 d 0 . Therefore, from (3.36), (3.20), (3.13) and (3.11), it follows that
0 d
t !
" 1 1 v d 1 + a 2 v δ 1 + d 02 d 32 v δ 1 dτ f ( , δ, t).
(3.37)
0
Using (3.9)1 again, we can argue analogously (3.25) to deduce dt 20 = div(¯ ρv d + δ v δ )20 f ( , δ, t).
(3.38)
Similarly to (3.37) and (3.38), we get from (3.9)3 and (3.25) that V d 20 + Vt d 20 f ( , δ, t).
(3.39)
Now, (3.12) follows from the estimate (3.36)–(3.39) immediately. This completes the proof of Lemma 3.2. 2 Finally, let Tδ =
1
ln > 0, Λ δ
δ
i.e. = δeΛT ,
(3.40)
where (C1 C5 )2 ˜ v0 20
= min 0 , , , 1 > 0, 4C62 16C62
(3.41)
and 0 is the same as in Remark 3.1. Then we have the following conclusion concerning the relation between T δ and T ∗ . Lemma 3.3. Under the assumptions of Lemma 3.2, T δ < T ∗ .
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Proof. It suffices to show that T ∗ < ∞, and we prove it by contradiction. Suppose that T δ T ∗ . By virtue of the definition of (a , ua , V a ), C5 , f ( , δ, t) and in (3.40), it is to verify that S(a (t), v a (t), V a (t)) + vτa L2 ((0,t),H 1 ) Λ ˜ v0 1 δeΛt ˜ v0 21 + (˜ 0 , V˜0 , Λ˜ 0 , Λ˜ v0 , ΛV˜0 )20 + 2 2 (1 + Λ)˜ 0 , v˜0 , V˜0 1 δeΛt C5 δeΛt C1 C5 δeΛt and f ( , δ, t) 3 δ 2 e2Λt ,
for any t T ∗ .
(3.42)
Consequently, S(δ (T ∗ ), v δ (T ∗ ), V δ (T ∗ )) + vτδ L2 ((0,T ∗ ),H 1 ) S(a (T ∗ ), v a (T ∗ ), V a (T ∗ )) + vτa L2 ((0,T ∗ ),H 1 ) + S(d (T ∗ ), v d (T ∗ ), V d (T ∗ )) + vτd L2 ((0,T ∗ ),H 1 ) √ ∗ ∗ ∗ C1 C5 δeΛT + C6 f ( , δ, T ∗ ) < δeΛT (C1 C5 + 2C6 ) < 2C1 C5 δeΛT , which contradicts (3.14). Hence, T δ < T ∗ . The proof is complete. 2 Now, we are in position to show Theorem 1.1. Let be given by (3.41), and ε := min{1,
˜ v0 0 }, 2
then ε > 0 by (2.19) and the fact that v˜0 = v˜. For any given δ ∈ (0, ι), (1) if the solution of the VRT problem (δ , v δ , V δ ) satisfies (δ , v δ , V δ )(T )3 > for some T ∈ (0, T max ),
(3.43)
then Theorem 1.1 automatically holds by virtue of ε. (2) if (3.43) fails, then one has T max = ∞ by Proposition 2.2 and Remark 3.1, and (δ , v δ , V δ )(t)3
for any t > 0.
Thus, we can use (3.40)–(3.41) and Lemma 3.2–3.3 to deduce that v δ (T δ )0 v a (T δ )0 − v d (T δ )0 = δv l (T δ )0 − v d (T δ )0 √ δ δ > ˜ v0 0 δeΛT − 2C6 δeΛT √ = (˜ v0 0 − 2C6 ) > ˜ v0 0 /2 ε,
(3.44)
which proves Theorem 1.1. Acknowledgments The research of Weiwei Wang was supported by NSFC (Grant No. 11501116) and the Education Department of Fujian Province (Grant No. JA15063).
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