Citation: Deren Gong, Xiaoliang Wang, Peng Dong, Shufan Wu, Xiaodan Zhu. Shifted Legendre polynomials-based single and double integral inequalities with arbitrary approximation order: Application to stability of linear systems with time-varying delays[J]. AIMS Mathematics, 2020, 5(5): 4371-4398. doi: 10.3934/math.2020279
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We consider the compressible isentropic Navier-Stokes equations with degenerate viscosities in
{ρt+div(ρu)=0,(ρu)t+div(ρu⊗u)+∇P=divT, | (1) |
where
P=Aργ,γ>1, | (2) |
where
T=μ(ρ)(∇u+(∇u)⊤)+λ(ρ)divuI3, | (3) |
where
μ(ρ)=αρ,λ(ρ)=βρ, | (4) |
where the constants
α>0,2α+3β≥0. |
Here, the initial data are given by
(ρ,u)|t=0=(ρ0,u0)(x),x∈R3, | (5) |
and the far field behavior is given by
(ρ,u)→(0,0)as |x|→∞,t≥0. | (6) |
The aim of this paper is to prove a blow-up criterion for the regular solution to the Cauchy problem (1) with (5)-(6).
Throughout the paper, we adopt the following simplified notations for the standard homogeneous and inhomogeneous Sobolev space:
Dk,r={f∈L1loc(R3):|f|Dk,r=|∇kf|Lr<+∞},Dk=Dk,2(k≥2),D1={f∈L6(R3):|f|D1=|∇f|L2<∞},‖f‖X∩Y=‖f‖X+‖f‖Y,‖f‖s=‖f‖Hs(R3),|f|p=‖f‖Lp(R3),|f|Dk=‖f‖Dk(R3). |
A detailed study of homogeneous Sobolev space can be found in [5].
The compressible isentropic Navier-Stokes system is a well-known mathematical model, which has attracted great attention from the researchers, and some significant processes have been made in the well-posedness for this system.
When
−divT0+∇P(ρ0)=√ρ0g |
for some
When
μ(ρ)=αρδ1,λ(ρ)=βρδ2, | (7) |
where
δ1=1,δ2=0 or 1,α>0,α+β≥0, | (8) |
and (6), where the vacuum cannot appear in any local point. They [12] also prove the same existence result in
(ρ,u)→(ˉρ,0)as|x|→∞, | (9) |
with initial vacuum appearing in some open set or the far field, the constant
1<δ1=δ2≤min(3,γ+12),α>0,α+β≥0. | (10) |
We also refer readers to [3], [6], [10], [13], [18], [26] and references therein for other interesting progress for this compressible degenerate system, corresponding radiation hydrodynamic equations and magnetohydrodynamic equations.
It should be noted that one should not always expect the global existence of solutions with better regularities or general initial data because of Xin's results [23] and Rozanova's results [20]. It was proved that there is no global smooth solutions to (1), if the initial density has nontrivial compact support (
For constant viscosity, Beale-Kato-Majda [1] first proved that the maximum norm of the vorticity controls the blow-up of the smooth solutions to
limT→T∗∫T0|curlu|∞dt=∞, | (11) |
where
limT→T∗∫T0|D(u)|∞dt=∞, | (12) |
where the deformation tensor
When the viscosities depend on density in the form of (4), S. Zhu [25] introduced the regular solutions, which can be defined as
Definition 1.1. [25] Let
(A)(ρ,u) in [0,T]×R3 satisfies the Cauchy problem (1)with (5)−(6)in the sense of distributions;(B)ρ≥0,ργ−12∈C([0,T];H2), (ργ−12)t∈C([0,T];H1);(C)∇logρ∈C([0,T];D1), (∇logρ)t∈C([0,T];L2);(D)u∈C([0,T];H2)∩L2([0,T];D3), ut∈C([0,T];L2)∩L2([0,T];D1). |
The local existence of the regular solutions has been obtained by Zhu [25].
Theorem 1.2. [25] Let
ργ−120≥0,(ργ−120,u0)∈H2,∇logρ0∈D1, | (13) |
then there exist a small time
ρ∈C([0,T∗];H2),ρt∈C([0,T∗];H1). |
Based on Theorem 1.2, we establish the blow-up criterion for the regular solution in terms of
Theorem 1.3. Let
limT↦¯T(sup0≤t≤T|∇logρ|6+∫T0|D(u)|∞ dt)=+∞, | (14) |
and
limsupT↦¯T∫T0‖D(u)‖L∞∩D1,6 dt=+∞. | (15) |
The rest of the paper can be organized as follows. In Section 2, we will give the proof for the criterion (14). Section 3 is an appendix which will present some important lemmas which are frequently used in our proof, and also give the detail derivation for the desired system used in our following proof.
In this section, we give the proof for Theorem 1.3. We use a contradiction argument to prove
limT↦¯T(sup0≤t≤T|∇logρ|6+∫T0|D(u)|∞dt)=C0<+∞ | (16) |
for some constant
Notice that, one can also prove (15) by contradiction argument. Assume that
limsupT↦¯T∫T0‖D(u)‖L∞∩D1,6 dt=C′0<+∞ | (17) |
for some constant
limT↦¯Tsup0≤t≤T|∇logρ|6≤CC′0, |
which implies that under assumption (17), we have (16). Thus, if we prove that (14) holds, then (15) holds immediately.
In the rest part of this section, based on the assumption (16), we will prove that
From the definition of the regular solution, we know for
ϕ=ργ−12,ψ=2γ−1∇logϕ, | (18) |
{ϕt+u⋅∇ϕ+γ−12ϕdivu=0,ψt+∇(u⋅ψ)+∇divu=0,ut+u⋅∇u+2θϕ∇ϕ+Lu=ψ⋅Q(u), | (19) |
where
Lu=−div(α(∇u+(∇u)⊤)+βdivuI3), | (20) |
and terms
Q(u)=α(∇u+(∇u)⊤)+βdivuI3,θ=Aγγ−1. | (21) |
See our appendix for the detailed process of the reformulation.
For
ψt+3∑l=1Al∂lψ+Bψ+∇divu=0. | (22) |
Here
ψ=2γ−1∇ϕϕ=2γ−1∇ργ−12ργ−12=∇ρρ=∇logρ, | (23) |
combing this with
Under (16) and (19), we first show that the density
Lemma 2.1. Let
‖ρ‖L∞([0,T]×R3)+‖ϕ‖L∞([0,T];Lq)≤C,0≤T<¯T, |
where
Proof. First, it is obvious that
ϕ(t,x)=ϕ0(W(0,t,x))exp(−γ−12∫t0divu(s,W(s,t,x))ds), | (24) |
where
{ddtW(t,s,x)=u(t,W(t,s,x)),0≤t≤T,W(s,s,x)=x, 0≤s≤T, x∈R3. |
Then it is clear that
‖ϕ‖L∞([0,T]×R3)≤|ϕ0|∞exp(CC0)≤C. | (25) |
Similarly,
‖ρ‖L∞([0,T]×R3)≤C. | (26) |
Next, multiplying
ddt|ϕ|22≤C|divu|∞|ϕ|22, | (27) |
from (16), (27) and the Gronwall's inequality, we immediately obtain
‖ϕ‖L∞([0,T];L2)≤C. | (28) |
Combing (25)-(28) together, one has
‖ϕ‖L∞([0,T];Lq)≤C,q∈[2,+∞]. |
We complete the proof of this lemma.
Before go further, notice that
|∇ϕ|6=|ϕ∇logϕ|6=2γ−1|ϕ∇logρ|6≤C|ϕ|∞|∇logρ|6≤C, | (29) |
where we have used
Lemma 2.2. Let
sup0≤t≤T|u(t)|22+∫T0|∇u(t)|22dt≤C,0≤T<¯T, |
where
Proof. Multiplying
ddt|u|22+2∫R3(α|∇u|2+(α+β)(divu)2)dx=∫R32(−u⋅∇u⋅u−θ∇ϕ2⋅u+ψ⋅Q(u)⋅u)dx≡:L1+L2+L3. | (30) |
The right-hand side terms can be estimated as follows.
L1=−∫R32u⋅∇u⋅udx≤C|divu|∞|u|22,L2=2∫R3θϕ2divudx≤C|ϕ|22|divu|∞≤C|divu|∞,L3=∫R32ψ⋅Q(u)⋅udx≤C|ψ|6|∇u|2|u|3≤C|∇u|2|u|122|∇u|122≤α2|∇u|22+C|u|2|∇u|2≤α|∇u|22+C|u|22, | (31) |
where we have used (16), (23) and the facts
|u|3≤C|u|122|∇u|122. | (32) |
Thus (30) and (31) yield
ddt|u|22+α|∇u|22≤C(|divu|∞+1)|u|22+C|divu|∞. | (33) |
By the Gronwall's inequality, (16) and (33), we have
|u(t)|22+∫t0|∇u(s)|22ds≤C,0≤t≤T. | (34) |
This completes the proof of this lemma.
The next lemma provides the key estimates on
Lemma 2.3. Let
sup0≤t≤T(|∇u(t)|22+|∇ϕ(t)|22)+∫T0(|∇2u|22+|ut|22)dt≤C,0≤T<¯T, |
where
Proof. Multiplying
12ddt∫R3(α|∇u|2+(α+β)|divu|2)dx+∫R3(−Lu−θ∇ϕ2)2 dx=−α∫R3(u⋅∇u)⋅∇×curlu dx+∫R3(2α+β)(u⋅∇u)⋅∇divu dx+θ∫R3(ψ⋅Q(u))⋅∇ϕ2dx−θ∫R3(u⋅∇u)⋅∇ϕ2 dx+∫R3(ψ⋅Q(u))⋅Lu dx−θ∫R3ut⋅∇ϕ2 dx≡:9∑i=4Li, | (35) |
where we have used the fact that
−△u+∇divu=curl(curlu)=∇×curlu. |
First, from the standard elliptic estimate shown in Lemma 3.3, we have
|∇2u|22−C|θ∇ϕ2|22≤C|div(α(∇u+(∇u)⊤)+βdivuI3)|22−C|θ∇ϕ2|22≤C|div(α(∇u+(∇u)⊤)+βdivuI3)−θ∇ϕ2|22=C∫R3(−Lu−θ∇ϕ2)2dx. | (36) |
Second, we estimate the right-hand side of (35) term by term. According to
{u×curlu=12∇(|u|2)−u⋅∇u,∇×(a×b)=(b⋅∇)a−(a⋅∇)b+(divb)a−(diva)b, |
Hölder's inequality, Young's inequality, (16), (23), (29), Lemma 2.2, Lemma 3.1 and
|L4|=α|∫R3(u⋅∇)u⋅∇×curlu dx|=α|∫R3(curlu⋅∇×((u⋅∇)u))dx|=α|∫R3(curlu⋅∇×(u×curlu))dx|=α|∫R3(12|curlu|2divu−curlu⋅D(u)⋅curlu)dx|≤C|∇u|∞|∇u|22,|L5|=|∫R3(2α+β)(u⋅∇)u⋅∇divu dx|≤|∫R3(2α+β)(−∇u:∇u⊤divu+12(divu)3)dx|+C|∇ϕ|6|u|3|∇u|2|divu|∞≤C(|∇u|22|divu|∞+|u|122|∇u|122|∇u|2|divu|∞)≤C(|∇u|22+|u|2|∇u|2)|divu|∞≤C|divu|∞(|∇u|22+1),L6=θ∫R3(ψ⋅Q(u))⋅∇ϕ2 dx≤C|ψ|6|∇u|3|∇ϕ2|2≤C|∇u|122|∇2u|122|∇ϕ|2|ϕ|∞≤C|∇ϕ|22+C(ϵ)|∇u|22+ϵ|∇2u|22,|L7|=θ|∫R3(u⋅∇u)⋅∇ϕ2dx|=θ|−∫R3∇u:(∇u)⊤ϕ2dx−∫R3ϕ2u⋅∇(divu)dx| | (37) |
=θ|−∫R3∇u:(∇u)⊤ϕ2dx+∫R3(divu)2ϕ2dx+∫R2u⋅∇ϕ2divu dx|≤C(|∇u|22|ϕ2|∞+|u|2|∇ϕ|2|ϕ|∞|divu|∞)≤C(|∇u|22+|divu|∞|∇ϕ|2)≤C(|∇u|22+|divu|∞+|divu|∞|∇ϕ|22),L8=∫R3(ψ⋅Q(u))⋅Lu dx≤C|ψ|6|∇u|3|∇2u|2≤C|∇2u|322|∇u|122≤C(ϵ)|∇u|22+ϵ|∇2u|22,L9=−θ∫R3ut⋅∇ϕ2dx=θ∫R3ϕ2divut dx=θddt∫R3ϕ2divu dx−θ∫R3(ϕ2)tdivu dx=θddt∫R3ϕ2divu dx−θ∫R32ϕϕtdivu dx=θddt∫R3ϕ2divu dx+θ∫R3u⋅∇ϕ2divu dx+θ(γ−1)∫R3ϕ2(divu)2dx≤θddt∫R3ϕ2divu dx+C(|u|2|∇ϕ|2|ϕ|∞|divu|∞+|∇u|22|ϕ2|∞)≤θddt∫R3ϕ2divu dx+C(|∇ϕ|2|divu|∞+|∇u|22)≤θddt∫R3ϕ2divu dx+C(|∇u|22+|divu|∞+|divu|∞|∇ϕ|22), | (38) |
where
12ddt∫R3(α|∇u|2+(α+β)|divu|2−2θϕ2divu)dx+C|∇2u|22≤C((|∇u|22+|∇ϕ|22)(|divu|∞+1)+|divu|∞). | (39) |
Third, applying
(|∇ϕ|2)t+div(|∇ϕ|2u)+(γ−2)|∇ϕ|2divu=−2(∇ϕ)⊤⋅∇u⋅(∇ϕ)−(γ−1)ϕ∇ϕ⋅∇divu=−2(∇ϕ)⊤⋅D(u)⋅(∇ϕ)−(γ−1)ϕ∇ϕ⋅∇divu. | (40) |
Integrating (40) over
ddt|∇ϕ|22≤C(ϵ)(|D(u)|∞+1)|∇ϕ|22+ϵ|∇2u|22. | (41) |
Adding (41) to (39), from the Gronwall's inequality and (16), we immediately obtain
α|∇u(t)|22−2θ∫R3ϕ2divu dx+C|∇ϕ(t)|22+∫t0|∇2u(s)|22ds≤C, |
that is
α|∇u(t)|22+C|∇ϕ(t)|22+∫t0|∇2u(s)|22ds≤C+2θ∫R3ϕ2divu dx≤C(1+|∇u|2|ϕ|2|ϕ|∞)≤C+α4|∇u(t)|22, |
which implies
|∇u(t)|22+|∇ϕ(t)|22+∫t0|∇2u(s)|22ds≤C,0≤t≤T. |
Finally, due to
∫t0|ut|22ds≤C∫t0(|∇2u|22+|∇u|23|u|26+|ϕ|2∞|∇ϕ|22+|∇u|23|ψ|26)ds≤C. |
Thus we complete the proof of this lemma.
Next, we proceed to improve the regularity of
Lemma 2.4. Let
sup0≤t≤T(|ut(t)|22+|u(t)|D2)+∫T0|∇ut|22dt≤C,0≤T<¯T, | (42) |
where
Proof. From the standard elliptic estimate shown in Lemma 3.3 and
Lu=−ut−u⋅∇u−2θϕ∇ϕ+ψ⋅Q(u), | (43) |
one has
|u|D2≤C(|ut|2+|u⋅∇u|2+|ϕ∇ϕ|2+|ψ⋅Q(u)|2)≤C(|ut|2+|u|6|∇u|3+|ϕ|3|∇ϕ|6+|ψ|6|∇u|3)≤C(1+|ut|2+|u|6|∇u|122|∇2u|122+|∇u|122|∇2u|122)≤C(1+|ut|2+|∇2u|122)≤C(1+|ut|2)+12|u|D2, | (44) |
where we have used Sobolev inequalities, (16), (23), (29) and Lemmas 2.1-2.3. Then we immediately obtain that
|u|D2≤C(1+|ut|2). | (45) |
Next, differentiating
utt+Lut=−(u⋅∇u)t−2θ(ϕ∇ϕ)t+(ψ⋅Q(u))t. | (46) |
Multiplying (46) by
12ddt|ut|22+α|∇ut|22≤12ddt|ut|22+∫R3(α|∇ut|2+(α+β)|divut|2)dx=∫R3(−(u⋅∇u)t⋅ut+(ψ⋅Q(u))t⋅ut−2θ(ϕ∇ϕ)t⋅ut)dx≡:12∑i=10Li. | (47) |
Similarly, based on (16), (23),
L10=−∫R3(u⋅∇u)t⋅ut dx=−∫R3((ut⋅∇u)⋅ut+(u⋅∇ut)⋅ut)dx=−∫R3(ut⋅D(u)⋅ut−12(ut)2divu)dx≤C|D(u)|∞|ut|22,L11=∫R3(ψ⋅Q(u))t⋅ut dx=∫R3ψ⋅Q(u)t⋅ut dx+∫R3ψt⋅Q(u)⋅ut dx=∫R3ψ⋅Q(u)t⋅ut dx−∫R3∇divu⋅Q(u)⋅ut dx+∫R3u⋅ψdiv(Q(u)⋅ut)dx≤C(|ψ|6|∇ut|2|ut|3+|∇2u|2|Q(u)|∞|ut|2+|ψ|6|u|6|∇2u|2|ut|6+|ψ|6|u|6|Q(u)|6|∇ut|2)≤C(|∇ut|2|ut|122|∇ut|122+|∇2u|2|Q(u)|∞|ut|2+|∇u|6|∇ut|2+|∇2u|2|∇ut|2)≤α8|∇ut|22+C(1+|D(u)|∞)(|ut|22+|u|2D2),L12=−∫R32θ(ϕ∇ϕ)t⋅ut dx=θ∫R3(ϕ2)tdivutdx=2θ∫R3ϕϕtdivutdx=−2θ∫R3ϕ(u⋅∇ϕ+γ−12ϕdivu)divutdx=−θ∫R3(u⋅∇ϕ2+(γ−1)ϕ2divu)divutdx | (48) |
\begin{equation} \begin{split} = &-\frac{\theta(\gamma-1)}{2}\int_{\mathbb{R}^3} \phi^2 (\text{div} u)^2_t dx-\theta\int_{\mathbb{R}^3} u\cdot\nabla \phi^2\text{div}u_t dx\\ = &-\frac{\theta(\gamma-1)}{2}\frac{d}{dt}\int_{\mathbb{R}^3} \phi^2 (\text{div}u)^2 dx +\theta(\gamma-1)\int_{\mathbb{R}^3} u\phi \phi_t (\text{div} u)^2 dx\\ &-\theta\int_{\mathbb{R}^3} u\cdot \nabla \phi^2 \text{div}u_t dx\\ = &-\frac{\theta(\gamma-1)}{2}\frac{d}{dt}\int_{\mathbb{R}^3} \phi^2 (\text{div}u)^2 dx -\theta(\gamma-1)\int_{\mathbb{R}^3} u\phi (u\cdot\nabla\phi) (\text{div} u)^2 dx\\ &-\frac{\theta(\gamma-1)^2}{2}\int_{\mathbb{R}^3} u\phi^2 (\text{div} u)^3 dx-\theta\int_{\mathbb{R}^3} u\cdot \nabla \phi^2 \text{div}u_t dx\\ = &-\frac{\theta(\gamma-1)}{2}\frac{d}{dt}\int_{\mathbb{R}^3} \phi^2 (\text{div}u)^2 dx +\frac{\theta(\gamma-1)}{2}\int_{\mathbb{R}^3} u\phi^2 \nabla(\text{div}u)^2 dx\\ &+\frac{\theta(\gamma-1)(3-\gamma)}{2}\int_{\mathbb{R}^3} \phi^2 (\text{div}u)^3 dx -\theta\int_{\mathbb{R}^3} u\cdot \nabla \phi^2 \text{div}u_t dx\\ \le&-\frac{\theta(\gamma-1)}{2}\frac{d}{dt}\int_{\mathbb{R}^3} \phi^2 (\text{div}u)^2 dx + C\big(|u|_\infty| \phi|^2_\infty|\nabla u|_2|\nabla^2 u|_2\\ &+|\phi|^2_\infty|D( u)|_\infty|\nabla u|^2_2+|\phi|_\infty|\nabla \phi|_2|u|_\infty|\nabla u_t|_2\big) \\ \le&-\frac{\theta(\gamma-1)}{2}\frac{d}{dt}\int_{\mathbb{R}^3} \phi^2 (\text{div}u)^2 dx\\ &+ C\big(|u|_\infty| |\nabla^2 u|_2+|D( u)|_\infty+|u|_\infty|\nabla u_t|_2\big) \\\\ \end{split} \end{equation} | (49) |
\begin{equation} \begin{split} \le&-\frac{\theta(\gamma-1)}{2}\frac{d}{dt}\int_{\mathbb{R}^3} \phi^2 (\text{div}u)^2 dx + \frac{\alpha}{4}|\nabla u_t|^2_2\\ &+ C(1+|D( u)|_\infty+|u|^2_{D^2}),\\ \end{split} \end{equation} | (50) |
where we also used Hölder's inequality, Young's inequality and
\begin{equation} |u|_{\infty}\le C|u|_{W^{1,3}}\le C(|u|_2^{\frac12}|\nabla u|_2^{\frac12}+|\nabla u|^{\frac12}|\nabla^2 u|^{\frac12}). \end{equation} | (51) |
It is clear from (47)-(50) and (45) that
\begin{equation} \begin{split} & \frac{d}{dt}(|u_t|^2_2+|\phi \text{div}u|^2_2)+|\nabla u_t|^2_2\le C(1+|\nabla^2 u|_2+ |D( u)|_\infty)|u_t|^2_2. \end{split} \end{equation} | (52) |
Integrating (52) over
\begin{equation} \begin{split} &|u_t(t)|^2_2+|\phi \text{div}u(t)|^2_2+\int_\tau^t|\nabla u_t(s)|^2_2 ds\\ \le & |u_t(\tau)|^2_2+|\phi \text{div}u(\tau)|^2_2+C\int_{\tau}^t \Big(\big(1+|\nabla^2 u|_2 +|D( u)|_\infty\big)|u_t|^2_2\Big)(s) ds. \end{split} \end{equation} | (53) |
From the momentum equations
\begin{equation} \begin{split} |u_t(\tau)|_2 \le & C\big(|u\cdot \nabla u|_2+|\phi\nabla\phi|_2+|L u|_2+|\psi\cdot Q(u)|_2\big)(\tau)\\ \le & C\big( |u|_\infty |\nabla u|_2+|\phi|_\infty|\nabla \phi|_2+|u|_{D^2}+|\psi|_6|\nabla u|_3\big)(\tau), \end{split} \end{equation} | (54) |
which, together with the definition of regular solution, gives
\begin{equation} \begin{split} &\lim \sup\limits_{\tau\rightarrow 0}|u_t(\tau)|_2\\ \le& C\big( |u_0|_\infty |\nabla u_0|_2+|\phi_0|_\infty|\nabla \phi_0|_2+|u_0|_{D^2}+|\psi_0|_6|\nabla u_0|_3\big)\le C_0. \end{split} \end{equation} | (55) |
Letting
\begin{equation} |u_t(t)|^2_2+|u(t)|_{D^2}^2+\int_0^t|\nabla u_t(s)|^2_2 ds\le C, \quad 0\le t\le T. \end{equation} | (56) |
This completes the proof of this lemma.
The following lemma gives bounds of
Lemma 2.5. Let
\begin{equation} \begin{split} &\sup\limits_{0\leq t\leq T}\big(\|\phi(t)\|_{W^{1,6}}+|\phi_t(t)|_6\big)+\int_0^T|u(t)|^2_{D^{2,6}} dt\leq C, \quad 0\leq T < \overline{T}, \end{split} \end{equation} | (57) |
where
Proof. First, taking
\begin{equation*} \sup\limits_{0\leq t\leq T} \|\phi(t)\|_{W^{1,6}}\le C,\quad 0\leq T < \overline{T}. \end{equation*} |
Second, one has
\begin{equation} \begin{split} |\phi_t|_6 = & \big|u\cdot\nabla\phi+\frac{\gamma-1}{2}\phi\text{div}u\big|_6\\ \le& C(|\nabla\phi|_6|u|_{\infty}+|\phi|_{\infty}|\text{div}u|_6)\le C, \end{split} \end{equation} | (58) |
where we have used Lemmas 2.3-2.4 and
Third, according to
\begin{equation} Lu = -u_t-u\cdot\nabla u -2\theta\phi\nabla \phi+\psi\cdot Q(u), \end{equation} | (59) |
and the standard elliptic estimate shown in Lemma 3.3, one has
\begin{equation} \begin{split} |\nabla ^2 u|_6 \le& C(|u_t|_6+|u\cdot \nabla u|_6+|\phi\nabla\phi|_6+|\psi \cdot Q( u)|_6)\\ \le & C(|\nabla u_t|_2+|u|_{\infty} |\nabla u|_6+|\phi|_{\infty}|\nabla\phi|_6+|\psi|_{6} |Q( u)|_{\infty})\\ \le & C(1+|\nabla u_t|_2+ |D( u)|^{\frac14}_2 |\nabla D(u)|^{\frac34}_6)\\ \le & C(1+|\nabla u_t|_2+|\nabla^2 u|^{\frac34}_6)\\ \le & C(1+|\nabla u_t|_2)+\frac12|\nabla^2 u|_6, \end{split} \end{equation} | (60) |
where we have used (16), (23), (29), Lemmas 2.1-2.4 and
\begin{equation} \begin{split} & |\text{div}u|_{\infty}\le C|D( u)|_\infty,\\ & |D( u)|_\infty\leq C|D( u)|^{\frac14}_2 |\nabla D(u)|^{\frac34}_6. \end{split} \end{equation} | (61) |
Thus, (60) implies that
\begin{equation} |\nabla^2 u|_6 \le C(1+|\nabla u_t|_2). \end{equation} | (62) |
Combing (62) with Lemma 2.4, one has
\begin{equation} \begin{split} \int_0^t|u(s)|^2_{D^{2,6}} ds \le& C\int_0^t(1+ |\nabla u_t(s)|^2_2) ds\le C,\quad 0\le t\leq T. \end{split} \end{equation} | (63) |
The proof of this lemma is completed.
Lemma 2.5 implies that
\begin{equation} \int_0^t|\nabla u(\cdot, s)|_\infty ds\le C, \end{equation} | (64) |
for any
Lemma 2.6. Let
\begin{equation*} \begin{split} \sup\limits_{0\leq t\leq T}\big(|\phi(t)|^2_{D^2}&+|\psi(t)|^2_{D^1}+\|\phi_t(t)\|^2_1+|\psi_t(t)|^2_{2}\big)\\ &+\int_{0}^{T}\Big(|u(t)|^2_{D^{3}}+|\phi_{tt}(t)|^2_{2}\Big)\mathit{\text{d}}t\leq C, \quad 0\leq T < \overline{T}, \end{split} \end{equation*} |
where
Proof. From
\begin{equation} \begin{split} |u|_{D^3} \le& C\big(|u_t|_{D^1}+|u\cdot \nabla u|_{D^1}+|\phi\nabla\phi|_{D^1}+|\psi \cdot Q(u)|_{D^1}\big)\\ \le & C\big(|u_t|_{D^1}+|u|_{\infty} |\nabla^2 u|_{2}+|\nabla u|_6|\nabla u|_3+|\psi|_6 |\nabla^2 u|_3\\ &+|\nabla\phi|_6|\nabla\phi|_3+|\phi|_{\infty}|\nabla^2\phi|_2+|\nabla\psi|_2|D(u)|_{\infty}\big)\\ \le & C\big(1+|u_t|_{D^1}+| \phi|_{D^2} +| u|^{\frac12}_{D^3}+|\psi|_{D^1} |D( u)|_\infty\big)\\ \le & C\big(1+|u_t|_{D^1}+| \phi|_{D^2}+|\psi|_{D^1} |D( u)|_\infty\big)+\frac12 | u|_{D^3}, \end{split} \end{equation} | (65) |
where we have used Young's inequality, Lemma 2.5, (16), (23), (29) and (61). Thus (65) offers that
\begin{equation} |u|_{D^3} \le C\big(1+|u_t|_{D^1}+| \phi|_{D^2}+|D( u)|_\infty|\psi|_{D^1}\big). \end{equation} | (66) |
Next, applying
\begin{equation} \begin{split} &(\partial_i \psi)_t+\sum\limits_{l = 1}^3 A_l \partial_l\partial_i \psi+B\partial_i \psi+\partial_i \nabla \text{div} u \\ = &\big(-\partial_i(B\psi)+B\partial_i \psi\big)+\sum\limits_{l = 1}^3 \big(-\partial_i(A_l) \partial_l \psi\big). \end{split} \end{equation} | (67) |
Multiplying (67) by
\begin{equation} \begin{split} \frac{d}{dt}|\nabla \psi|^2_2 \le & C\int_{\mathbb{R}^3}\Big( |\text{div} A| |\nabla\psi|^2 +|\nabla^3 u| |\nabla\psi| +|\nabla\psi|^2 |\nabla u|\\ &+|\partial_i(B\psi)-B\partial_i \psi | |\nabla\psi|\Big) dx\\ \le& C\big(\big|\text{div}A\big|_\infty|\nabla \psi|^2_2+|\nabla^3 u|_2|\nabla \psi|_2+|\nabla\psi|^2_2 |\nabla u|_{\infty}\\ &+|\partial_i(B\psi)-B\partial_i \psi |_2|\nabla\psi|_2\big), \end{split} \end{equation} | (68) |
where
\begin{equation} \begin{split} |\partial_i(B\psi)-B\partial_i \psi |_2& = |D^\zeta(B\psi)-BD^\zeta \psi|_2\leq C|\nabla^2 u|_3|\psi|_6\\ &\le C|\nabla^2 u|_2^{\frac12}|\nabla^3 u|_2^{\frac12}\le C|\nabla^3 u|_2^{\frac12}. \end{split} \end{equation} | (69) |
Thus
\begin{equation} \begin{split} \frac{d}{dt}|\nabla \psi|^2_2\leq& C\big(|\nabla u|_\infty|\nabla \psi|^2_2 +|\nabla^3 u|_2|\nabla \psi|_2+|\nabla^3 u |^{\frac12}_2|\nabla\psi|_2\big). \end{split} \end{equation} | (70) |
Combining (70) with (66) and Lemma 3.1, we have
\begin{equation} \begin{split} \frac{d}{dt}|\psi|^2_{D^1}\le & C(1+|\nabla u|_\infty)|\psi|^2_{D^1}+C(1+|\nabla^3 u|_2)|\psi|_{D^1}\\ \leq& C(1+|\nabla u|_\infty)|\psi|^2_{D^1}+C(1+|\phi|^2_{D^2}+|\nabla u_t|^2_2). \end{split} \end{equation} | (71) |
On the other hand, let
\begin{equation} \begin{split} 0 = &(\nabla G)_t+\nabla(\nabla u\cdot G)+\nabla (\nabla G \cdot u)+\frac{\gamma-1}{2}\nabla(G\text{div}u +\phi \nabla \text{div}u\big), \end{split} \end{equation} | (72) |
similarly to the previous step, we multiply (72) by
\begin{equation} \begin{split} \frac{d}{dt}|G|^2_{D^1} \le & C\int_{\mathbb{R}^3} \big(|\nabla^2 u| |G|+ |\nabla u| |\nabla G| +|\nabla \phi| |\nabla^2 u|+|\phi| |\nabla^3 u| \big)|\nabla G| dx\\ \le & C\big(|G|_6|\nabla^2 u|_3+|\nabla u|_\infty |\nabla G|_2+|\phi|_\infty|\nabla^3 u|_2\big)|\nabla G|_2\\ \le & C\big(|\nabla^2 u|_2^{\frac12}|\nabla^3 u|_2^{\frac12}+|\nabla u|_\infty |G|_{D^1}+| u|_{D^3}\big)| G|_{D^1}\\ \le & C\big(|u|_{D^3}^{\frac12}+|u|_{D^3}\big)| G|_{D^1}+C|\nabla u|_\infty | G|_{D^1}^2\\ \le & C(1+| u|_{D^3})| G|_{D^1}+C|\nabla u|_\infty | G|_{D^1}^2\\ \le & C(1+|u_t|_{D^1}+| \phi|_{D^2}+|D( u)|_\infty|\nabla\psi|_2)| G|_{D^1}+C|\nabla u|_\infty | G|_{D^1}^2\\ \le & C(1+|u_t|_{D^1}+|\nabla u|_\infty|\psi|_{D^1})| G|_{D^1}+C(1+|\nabla u|_\infty)| G|_{D^1}^2\\ \le & C(1+|\nabla u|_\infty)(|G|^2_{D^1}+|\psi|^2_{D^1})+C(1+|\nabla u_t|^2_2), \end{split} \end{equation} | (73) |
where we have used the Young's inequality, (29) and (66). This estimate, together with (71), gives that
\begin{equation} \begin{split} \frac{d}{dt}(|G|^2_{D^1}+|\psi|^2_{D^1}) \leq& C(1+|\nabla u|_\infty)(|G|^2_{D^1}+|\psi|^2_{D^1})+C(1+|\nabla u_t|^2_2). \end{split} \end{equation} | (74) |
Then the Gronwall's inequality, (42), (64) and (74) imply
\begin{equation} \begin{split} |\phi(t)|^2_{D^2}+|\psi(t)|^2_{D^1}\le C, \quad 0\leq t\leq T. \end{split} \end{equation} | (75) |
Combing (75) with (66) and Lemma 2.4, one has
\begin{equation} \int_{0}^{t}|u(s)|^2_{D^{3}}ds\le C\int_{0}^{t}(1+|\nabla u_t(s)|^2_2) ds\leq C, \quad 0\leq t\leq T. \end{equation} | (76) |
Finally, using the following relations
\begin{equation} \begin{split} \psi_t = &-\nabla (u \cdot \psi)-\nabla \text{div} u,\ \ \ \phi_t = -u\cdot \nabla \phi-\frac{\gamma-1}{2}\phi\text{div} u,\\ \phi_{tt} = &-u_t\cdot \nabla \phi-u\cdot \nabla \phi_t-\frac{\gamma-1}{2}\phi_t\text{div} u-\frac{\gamma-1}{2}\phi\text{div} u_t, \end{split} \end{equation} | (77) |
according to Hölder's inequality, (16), (29), Lemmas 2.1-2.5, one has
\begin{equation} \begin{split} |\psi_t|_2\le& C(|\nabla u \cdot \psi|_2+|u\cdot \nabla\psi|_2+|\nabla \text{div} u|_2)\\ \le& C(|\nabla u|_3|\psi|_6+|u|_{\infty}|\nabla\psi|_2+|\nabla^2 u|_2)\le C,\\ |\phi_t|_2\le& C(|u\cdot \nabla \phi|_2+|\phi\text{div} u|_2)\\ \le& C(|u|_{\infty}|\nabla\phi|_2+|\phi|_{\infty}|\nabla u|_2)\le C,\\ |\nabla\phi_t|_2 \le & C(|\nabla(u\cdot \nabla\phi)|_2+|\nabla(\phi\text{div} u)|_2)\\ \le& C(|\nabla u\cdot \nabla \phi|_2+| \nabla^2 \phi\cdot u|_2+|\nabla\phi\text{div} u|_2+|\phi\nabla\text{div} u|_2)\\ \le & C(|\nabla u|_3 |\nabla \phi|_6+|u|_{\infty} |\nabla^2\phi|_2+|\nabla\phi|_6|\nabla u|_3+|\phi|_{\infty}|\nabla^2 u|_2)\\ \le& C,\\ |\phi_{tt}|_2\le& C(|u_t\cdot \nabla \phi|_2+|u\cdot \nabla \phi_t|_2+|\phi_t\text{div} u|_2+|\phi\text{div} u_t|_2)\\ \le & C(|u_t|_{6}|\nabla\phi|_3+|u|_{\infty}|\nabla\phi_t|_2+|\phi_t|_6|\nabla u|_3+|\phi|_{\infty}|\nabla u_t|_2)\\ \le & C(1+|\nabla u_t|_2). \end{split} \end{equation} | (78) |
Thus
\begin{equation*} \sup\limits_{0\leq t\leq T}\big(\|\phi_t(t)\|^2_1+|\psi_t(t)|^2_{2}\big)\leq C, \end{equation*} |
and according to (42), one has
\begin{equation*} \int_{0}^{T}|\phi_{tt}(t)|^2_{2} dt \le \int_{0}^{T}\big(1+|\nabla u_{t}(t)|^2_{2}\big) dt \leq C. \end{equation*} |
The proof of this lemma is completed.
Now we know from Lemmas 2.1-2.6 that, if the regular solution
\begin{equation*} (\rho^{\frac{\gamma-1}{2}},\nabla \log\rho,u)|_{t = \overline{T}} = \lim\limits_{t\rightarrow \overline{T}}(\rho^{\frac{\gamma-1}{2}},\nabla \log\rho,u) \end{equation*} |
satisfies the conditions imposed on the initial data
In this subsection, we present some important lemmas which are frequently used in our previous proof. The first one is the well-known Gagliardo-Nirenberg inequality, which can be found in [9].
Lemma 3.1. [9] Let
\begin{equation} |h|_q\leq C|\nabla h|^c_p |h|^{1-c}_r, \end{equation} | (79) |
where
\begin{equation} c = \big(\frac{1}{r}-\frac{1}{q}\big)\big(\frac{1}{r}-\frac{1}{p}+\frac{1}{3}\big)^{-1}, \quad 0\le c\le 1. \end{equation} | (80) |
If
Some common versions of this inequality can be written as
\begin{equation} \begin{split} |f|_3\leq C|f|^{\frac12}_{2}|\nabla f|^{\frac12}_{2},\quad |f|_6\leq C|\nabla f|_{2},\quad |f|_\infty\leq C|f|^{\frac14}_{2}|\nabla f|^{\frac34}_{6}, \end{split} \end{equation} | (81) |
which have be used frequently in our previous proof.
The second one can be found in Majda [17], and we omit its proof.
Lemma 3.2. [17] Let positive constants
\frac{1}{r} = \frac{1}{a}+\frac{1}{b} |
and
\begin{equation} |D^s(fg)-f D^s g|_r\leq C_s\big(|\nabla f|_a |D^{s-1}g|_b+|D^s f|_b|g|_a\big), \end{equation} | (82) |
\begin{equation} |D^s(fg)-f D^s g|_r\leq C_s\big(|\nabla f|_a |D^{s-1}g|_b+|D^s f|_a|g|_b\big), \end{equation} | (83) |
where
The third one is on the regularity estimates for Lam
\begin{equation} \begin{cases} -\alpha\Delta u-(\alpha+\beta)\nabla \text{div} u = f, \\[10pt] u\rightarrow 0 \quad \text{as} \ |x|\rightarrow +\infty, \end{cases} \end{equation} | (84) |
one has
Lemma 3.3. [21] If
\begin{equation} |u|_{D^{k+2,q}} \leq C |f|_{D^{k,q}}, \end{equation} | (85) |
where k is an integer and the constant
\begin{equation} -\Delta u = f,\quad u\to 0\quad \mathit{\text{as}} \ |x|\rightarrow +\infty, \end{equation} | (86) |
then (85) holds and if
\begin{equation} |u|_{D^{1,q}} \leq C |g|_{L^q}. \end{equation} | (87) |
The proof can be obtained via the classical estimates from harmonic analysis, which can be found in [21] or [22]. We omit it here.
Now we show that, via introducing new variables
\begin{equation} \phi = \rho^{\frac{\gamma-1}{2}}, \; \psi = \nabla \log \rho = \frac{2}{\gamma-1}\nabla \phi/\phi, \end{equation} | (88) |
the system (1) can be rewritten as
\begin{equation} \begin{cases} \phi_t+\frac{\gamma-1}{2}\phi\text{div} u+u\cdot\nabla\phi = 0,\\[10pt] \psi_t+\nabla(u\cdot\psi)+\nabla\text{div} u = 0,\\[10pt] u_t+u\cdot\nabla u+2\theta\phi\nabla\phi+Lu = \psi\cdot Q(u). \end{cases} \end{equation} | (89) |
Proof. First, from the momentum equation, one has
\begin{equation*} \begin{split} \rho u_t+\rho u\cdot\nabla u+\nabla P-\rho\text{div}\big(\alpha(\nabla u+&\nabla u^{\top}) +\beta\text{div} u \mathbb{I}_3\big)\\ = &\nabla\rho\cdot [\alpha(\nabla u+\nabla u^{\top})+\beta\text{div} u\mathbb{I}_3], \end{split} \end{equation*} |
where
\begin{equation*} \begin{split} u_t+u\cdot\nabla u+A\gamma\rho^{\gamma-2}\nabla\rho-\text{div}\big(\alpha(\nabla u&+\nabla u^{\top}) +\beta\text{div} u \mathbb{I}_3\big)\\ = &\frac{\nabla\rho}{\rho}\cdot [\alpha(\nabla u+\nabla u^{\top})+\beta\text{div} u\mathbb{I}_3]. \end{split} \end{equation*} |
Denote
\begin{align*} Lu& = -\text{div}\big(\alpha(\nabla u+\nabla u^{\top})+\beta\text{div} u \mathbb{I}_3\big),\\ Q(u)& = \alpha(\nabla u+\nabla u^{\top})+\beta\text{div} u\mathbb{I}_3, \quad \theta = \frac{A\gamma}{\gamma-1}, \end{align*} |
we have
\begin{equation} u_t+u\cdot\nabla u+2\theta\phi\nabla\phi+Lu = \psi \cdot Q(u). \end{equation} | (90) |
Second, for
\begin{equation} \begin{split} \psi_t & = (\nabla\log\rho)_t = \nabla(\log\rho)_t = \nabla\Big(\frac{\rho_t}{\rho}\Big) = \nabla\Big(\frac{-\text{div}(\rho u)}{\rho}\Big)\\ & = \nabla\big(\frac{-\nabla\rho\cdot u-\rho\text{div} u}{\rho}\big) = -\nabla(\nabla\log\rho\cdot u+\text{div} u)\\ & = -\nabla\text{div} u-u\cdot\nabla(\nabla\log\rho)-\nabla\log\rho\cdot\nabla u^{\top}\\ & = -\nabla\text{div} u-u\cdot\nabla\psi-\psi\cdot\nabla u^{\top}\\ & = -\nabla\text{div} u-\nabla(u\cdot\psi). \end{split} \end{equation} | (91) |
Third, for
\begin{equation} \begin{split} \phi_t & = (\rho^{\frac{\gamma-1}{2}})_t = \frac{\gamma-1}{2}\rho^{\frac{\gamma-3}{2}}\rho_t\\ & = \frac{\gamma-1}{2}\rho^{\frac{\gamma-1}{2}}\frac{\rho_t}{\rho} = \frac{\gamma-1}{2}\rho^{\frac{\gamma-1}{2}}\frac{-\text{div}(\rho u)}{\rho}\\ & = \frac{\gamma-1}{2}\phi\frac{-\rho\text{div} u-\nabla\rho\cdot u}{\rho}\\ & = -\frac{\gamma-1}{2}\phi\text{div} u-u\cdot\nabla \phi. \end{split} \end{equation} | (92) |
Combing (90)-(92) together, we complete the proof of the transformation.
The author sincerely appreciates Dr. Shengguo Zhu for his very helpful suggestions and discussions on the problem solved in this paper. The research of Y. Cao was supported in part by China Scholarship Council 201806230126 and National Natural Science Foundation of China under Grants 11571232.
Conflict of Interest: The authors declare that they have no conflict of interest.
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