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Stability of steady-state solutions to Navier–Stokes–Poisson systems
Accepted Manuscript Stability of steady-state solutions to Navier-Stokes-Poisson systems
Yue-Hong Feng, Cun-Ming Liu
PII: DOI: Reference:
S0022-247X(18)30198-7 https://doi.org/10.1016/j.jmaa.2018.03.001 YJMAA 22077
To appear in:
Journal of Mathematical Analysis and Applications
Received date:
29 November 2017
Please cite this article in press as: Y.-H. Feng, C.-M. Liu, Stability of steady-state solutions to Navier-Stokes-Poisson systems, J. Math. Anal. Appl. (2018), https://doi.org/10.1016/j.jmaa.2018.03.001
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STABILITY OF STEADY-STATE SOLUTIONS TO NAVIER-STOKES-POISSON SYSTEMS YUE-HONG FENG1 , CUN-MING LIU2∗ 1
College of Applied Sciences, Beijing University of Technology, 100022 Beijing, China
2 Department
of Mathematics, Qufu Normal University, Qufu 273165, Shandong, China
Email : [email protected], [email protected],
Abstract. This paper is concerned with a stability problem for compressible Navier-Stokes-Poisson systems. It arises in the modeling of semiconductors with a viscosity term in momentum equations. We prove that smooth solutions exist globally in time near the steady-state solution, and converge to the steady state for large time. In this stability result, we don’t give any special assumptions on the given doping profile. The proof is based on the techniques of anti-symmetric matrix and an induction argument on the order of the space derivatives of solutions in energy estimates. Keywords: Navier-Stokes-Poisson system, steady-state, global smooth solution, energy estimate AMS Subject Classification (2000) : 35Q35, 76N10 1. Introduction and main results 1.1. Introduction. We consider smooth solutions of compressible Navier-Stokes-Poisson systems on R3 . This system describes the dynamic of electrons in semiconductors with a viscosity term in momentum equations. The symbols n, p, u = (u1 , u2 , u3 )T and φ stand for the density, pressure, the velocity and the electric potential of the electrons. The scaled system is written as (see [1, 13, 17]) : ⎧ ∂t n + ∇ · (nu) = 0, ⎪ ⎪ ⎨ ∂t (nu) + ∇ · (nu ⊗ u) + ∇p(n) = n∇φ + Δu, (1.1) ⎪ ⎪ ⎩−Δφ = b(x) − n, lim φ = 0, |x|→∞
for t > 0 and x ∈ R3 , where b(x) is the doping profile for semiconductors. We assume b is sufficiently smooth and satisfies b ≥ const. > 0 on R3 , and pressure p is smooth and strictly increasing on (0, +∞). The system is supplemented by the following initial condition (1.2)
t = 0 : (n, u) = (n0 , u0 ),
x ∈ R3 .
For smooth solutions in any non-vacuum field, the momentum equations in (1.1) can be written as Δu , ∂t u + (u · ∇) u + ∇h(n) = ∇φ + n *Corresponding author 1
2
Y.H. Feng and C.M. Liu
where h is the enthalpy function defined by h (n) = increasing on (0, +∞), so is h. Let (1.3)
p (n) . Since p is smooth and strictly n
E = ∇φ.
By the density equation together with Poisson equation in system (1.1), the variable E satisfies (1.4)
∂t E = −∇Δ−1 (∇ · (nu)).
Thus, the Navier-Stokes-Poisson system for variables (n, u, E) can be regarded as a quasilinear hyperbolic-parabolic system with a zero-order term −∇Δ−1 (∇ · (nu)) of (n, u). The local wellposedness of smooth solutions to Cauchy problem (1.1)-(1.2) can be obtained by using standard methods as in [18, 14, 9, 12]. Let s ≥ 3 be an integer, and (n0 − b, u0 , ∇φ0 ) ∈ H s (R3 ) satisfying the compatible condition −Δφ0 = b − n0 and n0 ≥ const. > 0. Then there exists T∗ > 0 such that the Cauchy problem admits a unique solution (n, u, φ) on the domain [0, T∗ ] × R3 , and n − b, ∇φ ∈ C [0, T∗ ]; H s (R3 ) ∩ C 1 [0, T∗ ]; H s−1 (R3 ) , n ≥ const. > 0, u ∈ C [0, T∗ ]; H s (R3 ) ∩ C 1 [0, T∗ ]; H s−2 (R3 ) , ∇u ∈ L2 [0, T∗ ]; H s (R3 ) . Now we consider the steady-state solutions of (1.1) with zero velocity. Let n ¯ , u¯, φ¯ be such a solution of variable x with u¯ = 0. We get ¯ ∇h (¯ n) = ∇φ, (1.5) −Δφ¯ = b − n ¯. This implies n ¯ satisfies an elliptic equation : (1.6)
−Δh (¯ n) = b − n ¯,
in R3 .
In a bounded domain Ω ⊂ R3 with periodic, Dirichlet or Neumann boundary conditions or in the whole space R3 , (1.6) admits a unique solution (see [2, 5, 7, 11]) by using the classical fixedpoint theorem or a variational method. Moreover, b ≥ const. > 0 implies that n ¯ ≥ const. > 0. The existence and uniqueness of solutions to this elliptic equation is given as follows Proposition 1.1. ([7, 11]) Let s1 ≥ 1. Assume b ∈ L∞ (R3 ), ∇b ∈ H s1 −1 (R3 ) and b ≥ const. > 0 ¯=n ¯ (x) satisfying n ¯ − b ∈ H s1 (R3 ), a.e. x ∈ R3 . Then problem (1.5) admits a unique solution n n ¯ ≥ const. > 0. When n ¯ is given by Proposition 1.1, we obtain φ¯ from the first equation of (1.5). In order to ¯ we may assume it is sufficiently smooth which implies study the stability of steady-state (¯ n, 0, φ), p n is bounded. We point out that the the L norm with 2 ≤ p ≤ ∞ of any order derivatives of ∇¯ stability problem here is to investigate the global existence of smooth solutions when the initial ¯ Many research results can be found on data (n0 , u0 , φ0 ) are close to the steady-state (¯ n, 0, φ). such a interesting problem of Navier-Stokes-Poisson systems. When b is a positive constant or is a small perturbation of a constant, this problem is solved in [6, 10, 22, 23, 24, 25] in the Soblev spaces or Besove spaces framework. This is extended to non-isentropic Navier-Stokes-Poisson systems, see [19, 20]. In this situation, since the solution is close to the constant equilibrium state, the classical energy method and linearized method are valid. We also mentioned the global well-posedness result for the one-dimensional model with large initial data, see [21]. ¯ We find all the techniques used When b is large, so is the steady-state solution (¯ n, 0, φ). for small b or constant b no longer work due to the appearance of lower order terms which will make essential difficulties in the energy estimates, and there is no result for Navier-StokesPoisson systems. In this paper, we solve this problem by using an anti-symmetric matrix
Stability of steady-state solutions to NSP systems
3
technique and employing an induction argument on the order of the derivatives of solutions in energy estimates. We point out that the technique of anti-symmetric matrix was first used by Guo-Strauss when they considered the stability problem for the isentropic Euler-Poisson systems with boundary conditions in [5], namely, for the system (1.1) where the equation for the velocity u is replaced by (1.7)
∂t u + (u · ∇) u + ∇h(n) = ∇φ − u.
Next, Peng employed an induction argument on the order of the mixed time-space derivatives of solutions in energy estimates when he studied the global existence of solutions for Euler-Maxwell system in [16]. This technique was also applied to the two-fluid isentropic Euler-Maxwell system, see [4]. In our problems, since the high order derivative terms with respect to t of (n, u) depends on the higher order derivative terms with respect to x of u from the momentum equations, it is impossible to completely follow the same technique of induction argument employed in [16]. To overcome it, we obtain refined estimates by use of classical Sobolev embedding theorems and integrations by parts. It allows us to use a slightly different induction argument together with anti-symmetric matrix technique to complete the proof. In this paper, we prove the global existence of solutions to Cauchy problem (1.1)-(1.2), and the solution tends to the steady-state for large time. It is stated in Theorem 1.1. Remark that the result in Theorem 1.1 still holds for system (1.1) where the momentum equations are replaced by (1.8)
∂t (nu) + ∇ · (nu ⊗ u) + ∇p(n) = n∇φ + Δu + ∇ div u.
Since the term ∇ div u brings us better dissipations of div u, and doesn’t produce any obstacle in the energy estimates, so we omit it in momentum equations. 1.2. Main results. Let us introduce some notations. For all integer s ∈ N, we denote by H s , L2 and L∞ the usual Sobolev spaces H s (R3 ), L2 (R3 ) and L∞ (R3 ), and by · s , · and
· ∞ the corresponding norms, respectively. ·, · stands for the inner product of L2 (R3 ). We also denote by C, C0 , C1 > 0 constants independent of any time, but may be dependent of the ¯ For a multi-index α = (α1 , α2 , α3 ) ∈ N3 , we denote steady-state (¯ n, 0, φ). ∂ α = ∂xα11 ∂xα22 ∂xα33 = ∂1α1 ∂2α2 ∂3α3 , with |α| = α1 + α2 + α3 . For α = (α1 , α2 , α3 ) and β = (β1 , β2 , β3 ) ∈ N3 , β ≤ α stands for βj ≤ αj for j = 1, 2, 3, and β < α stands for β ≤ α and β = α. We also recall the Leibniz formula Cαβ ∂ α−β u∂ β v, ∀ α ∈ N3 , ∂ α (uv) = β≤α
where Cαβ are positive constants. The main results of this paper can be stated as follows. Theorem 1.1. Let s ≥ 3 be an integer. Then there exist constants δ0 > 0, C > 0 such that if
n0 − n ¯ , u0 , ∇φ0 − ∇φ¯ s ≤ δ0 , Cauchy problem (1.1)-(1.2) has a unique global smooth solution (n, u, φ) satisfying t
2 2 2 ¯
n(t) − n
n(t ) − n ¯ s + ∇u(t ) s dt ¯ , u(t), ∇φ(t) − ∇φ s + 0
2 (1.9) ≤ C n0 − n ¯ , u0 , ∇φ0 − ∇φ¯ s , ∀ t ≥ 0,
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Y.H. Feng and C.M. Liu
which implies ¯ s−1 = 0, lim n(t) − n
(1.10)
t→∞
and lim ∇u(t) s−3 = 0.
(1.11)
t→∞
This paper is organized as follows. In the next section, we focus on the symmetrization of the Navier-Stokes equations with perturbation variables and show an useful lemma. Section 3 is devoted to the energy estimates. Finally, we give the proof of the above theorem in section 4. 2. Symmetrization of Navier-Stokes equations Let N =n−n ¯,
(2.1) and (2.2)
U=
N u
,
U0 =
N0 u0
¯ Φ = φ − φ,
,
W =
U ∇Φ
,
W0 =
U0 ∇Φ0
where ¯ N0 = n0 − n ¯ , ∇Φ0 = ∇φ0 − ∇φ. By (2.1) and (1.5), system (1.1) can be written as ⎧ ∂t N + u · ∇N + n∇ · u + ∇¯ n · u = 0, ⎪ ⎪ ⎨ Δu (2.3) ∂t u + (u · ∇) u + ∇h (n) − ∇h (¯ n) − ∇Φ − = 0, ⎪ n ⎪ ⎩ ΔΦ = N. A straightforward computation implies n) N + r (¯ n, N ) , ∇h (n) − ∇h (¯ n) = h (n) ∇N + ∇h (¯ where
n) − h (¯ n) N ) ∇¯ n = O N2 . r (¯ n, N ) = (h (n) − h (¯ The Navier-Stokes equations in (2.3) can be rewritten as (2.4)
∂t U +
3
Aj (n, u) ∂j U + L (x) U + M (W ) = f (r) ,
j=1
with (2.5)
(2.6) and (2.7)
uj
neTj
, j = 1, 2, 3, h (n) ej uj I3 0 (∇¯ n )T L (x) = , ∇h (¯ n) 0
Aj (n, u) =
⎛ M (W ) = ⎝
0
⎞
Δu ⎠ , f (r) = −∇Φ − n
0 −r
.
,
Stability of steady-state solutions to NSP systems
5
Here, (e1 , e2 , e3 ) denotes the canonical basis of R3 , I3 denotes the 3 × 3 unit matrix and we use eTj to denote the transpose of ej . It is clear that the matrix A0 (n) defined by h (n) 0 , (2.8) A0 (n) = 0 nI3 is symmetric and uniformly positive definite for n > 0, and h (n)uj h (n)neTj j (n, u) = A0 (n)Aj (n, u) = (2.9) A h (n)nej nuj I3 is symmetric. These expressions of A0 and A˜j are useful in energy estimates. Furthermore, the 3 j (n, u) − 2A0 (n)L(x) defined by matrix B(n, u, x) = ∂j A j=1
(2.10)
(∇p (n)−2h (n)∇¯ n )T ∇·(h (n)u) B(n, u, x) = ∇p (n)−2n∇h (¯ n) ∇·(nu)I3
is anti-symmetric at (n, u) = (¯ n, 0). Finally, we list some useful inequalities in Sobolev space. Lemma 2.1. (Moser-type calculus inequalities, see [8, 12].) Let s ≥ 3 be an integer. Suppose u ∈ H s , ∇u ∈ L∞ , v ∈ H s−1 ∩ L∞ and f is a smooth function. Then for all multi-index α with 1 ≤ |α| ≤ s, one has ∂ α (uv) − u∂ α v ∈ L2 and
∂ α (uv) − u∂ α v ≤ C ∇u ∞ D|α|−1 v + D|α| u
v ∞ ,
∂ α f (u) ≤ C(1 + u s )s−1 u s , where the constant C may depend on u ∞ and s, and
∂ α u .
Ds u = |α|=s
3. Energy estimates According to [15], the global existence of smooth solutions follows from the local existence and uniform estimates of solutions with respect to t. In what follows, we focus on uniform estimates for proving Theorem 1.1. Let T > 0 and W be smooth solution of (2.3) defined on time interval [0, T ] with initial value W0 . Let (3.1)
UT = sup ||U (t)||s . 0≤t≤T
We assume that s ≥ 3, and UT is uniformly sufficiently small with respect to T . Then from the continuous embedding H s−1 → L∞ (see [3]), it implies (3.2)
q0 ≤ n ≤ q1 , h0 ≤ h (n),
where q0 , q1 and h0 are positive constants independent of any time. The following lemma is easily obtained by use of the density equation and the antisymmetry of the matrix B(¯ n, 0, x).
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Y.H. Feng and C.M. Liu
Lemma 3.1. For all t ∈ [0, T ], it holds
∂t N + ∂t N ∞ ≤ C||∇u||2 ,
(3.3)
∂t A0 (n) ∞ ≤ C||∇u||2 ,
and | B(n, u, x)U, U | ≤ C||U ||s (||N ||2s + ||∇u||2s ).
(3.4)
Proof. By the density equation ∂t N = −n∇ · u − ∇N · u, and the smallness of UT , we have
∂t N ≤ C( n L∞ ∇u + ∇N
u ∞ ) ≤ C ∇u 2 ,
∂t N L∞ ≤ n ∞ ∇u ∞ + u ∞ ∇N ∞ ≤ C ∇u 2 , and then
∂t A0 (n) ∞ = A0 (n) ∂t N ∞ ≤ C ∇u 2 .
Thus, we get (3.3). Next, we prove (3.4). Noting the expression (2.10), we have B(n, u, x)U, U = B1 U, U + B2 U, U , where
n )T 0 (∇p (n)−2h (n)∇¯ B1 = ∇p (n)−2n∇h (¯ n) 0
,
0 ∇·(h (n)u) B2 = 0 ∇·(nu)I3
.
¯ , we obtain Since the matrix B1 is anti-symmetric at n = n B1 U, U ≤ C N 2 u ∞ ≤ C N 2 ∇u 1 . For the term B2 U, U , using an integration by parts gives | B2 U, U | = | ∇ · (h (n)u) N, N + ∇ · (nu) u, u | =2 | h (n)N u, ∇N + nu, (u · ∇) u | ≤C ∇u 1 ( N ∇N + u ∇u ) . Therefore, we get (3.4). Firstly, we obtain an L2 estimate with respect to variable (N, u, ∇Φ).
2
Proposition 3.1. For all t ∈ [0, T ], it holds d (3.5) A0 (n) U, U + ∇Φ 2 + 2 ∇u 2 ≤ C||U ||s (||N ||2s + ||∇u||2s ). dt Proof. Taking the inner product of (2.4) with 2A0 (n) U in L2 yields the classical energy equality for U : d A0 (n) U, U = divA (n, u) U, U + 2 A0 (n) (f − LU − M (W )) , U , dt where 3 divA (n, u) = ∂t A0 (n) + ∂j A˜j (n, u) . j=1
Then d A0 (n) U, U = ∂t A0 (n) U, U + B(n, u, x)U, U + 2 A0 (n) (f − M (W )) , U . dt Applying Lemma 3.1 yields ∂t A0 (n) U, U = h (n) ∂t N, |N |2 + ∂t N, |u|2 ≤ C U s (||N ||2s + ||∇u||2s )
Stability of steady-state solutions to NSP systems
7
and | B(n, u, x)U, U | ≤ C||U ||s (||N ||2s + ||∇u||2s ). In view of the expression (2.7) and r (¯ n, N ) = O (N 2 ), we have A0 (n) M (W ) , U = ∇u 2 − nu, ∇Φ , and | A0 (n) f, U | ≤ C N 2 u ∞ ≤ C U s (||N ||2s + ||∇u||2s ). These formulas yield d A0 (n) U, U + 2 ∇u 2 − 2 nu, ∇Φ ≤ C U s (||N ||2s + ||∇u||2s ). dt By the Poisson equation in (2.3), we get −2 nu, ∇Φ =
(3.6)
d
∇Φ 2 . dt
Thus, (3.5) follows from the last two inequalities.
2
3.1. Higher order energy estimates. Let α ∈ N3 with 1 ≤ |α| ≤ s. Applying ∂ α on (2.4) gives (3.7)
∂t ∂ α U +
3
Aj ∂j ∂ α U + L∂ α U + ∂ α M = ∂ α f + g α ,
j=1
where (3.8)
gα =
3
(Aj ∂j ∂ α U − ∂ α (Aj ∂j U )) + L∂ α U − ∂ α (LU ) .
j=1
Lemma 3.2. For all t ∈ [0, T ] and α ∈ N3 with 1 ≤ |α| ≤ s, it holds (3.9)
d A0 ∂ α U, ∂ α U + ∂ α ∇Φ 2 + ∂ α∇u 2 dt ≤C N 2|α|−1 + C ∇u 2|α|−1 + C U s (||N ||2s + ||∇u||2s ).
Proof. Taking the inner product of (3.7) with 2A0 (n) ∂ α U in L2 , we obtain
(3.10)
d A0 ∂ α U, ∂ α U = ∂t A0 (n) ∂ αU, ∂ αU + B(n, u, x)∂ αU, ∂ αU dt + 2 A0 ∂ α f, ∂ α U − 2 A0 ∂ α M, ∂ α U + 2 A0 g α , ∂ α U =I1α + I2α + I3α + I4α + I5α ,
with the natural correspondence for I1α , I2α , · · · , I5α . Let us estimate them in the following. Estimates of I1α , I2α and I3α . Applying Lemmas 2.1, 3.1 yields (3.11)
|I1α | = | ∂t A0 (n) ∂ α U, ∂ α U | ≤C U s (||N ||2s + ||∇u||2s ),
(3.12)
|I2α | = | B(n, u, x)∂ α U, ∂ α U | ≤ C U s (||N ||2s + ||∇u||2s ),
and (3.13)
|I3α | = | A0 ∂ α f, ∂ α U | ≤ C U s (||N ||2s + ||∇u||2s ).
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Y.H. Feng and C.M. Liu
Estimate of I4α . In view of the expression of (2.7), using the Leibniz formula and an integration by parts gives d α 2 2 αβ α α β αβ I + 2 ∂ ∇u
+ ∇Φ
= 2 C I − 2 Cαβ I42 ,
∂ 41 4 α (3.14) dt β<α
where (3.15)
αβ I41
=
n∂
α−β
1 β α ∂ Δu, ∂ u , n
β<α
αβ I42 = ∂ α−β n∂ β u, ∂ α ∇Φ .
αβ In order to estimate I41 , we write
1 N 1 = − . n n ¯ n¯ n When |β| = |α| − 1, noting n ¯ is sufficiently smooth and bounded (see Proposition 1.1), applying Lemma 2.1 and an integration by parts, we get αβ I41 |β|=|α|−1 β<α
≤
n∂ α−β 1 ∂ β Δu, ∂ α u + n∂ α−β N ∂ β Δu, ∂ α u n n¯ n |β|=|α|−1 β<α
(3.16)
n∂ α−β 1 ∂ β ∇u, ∂ α ∇u + ∇ n∂ α−β 1 · ∂ β ∇u, ∂ α u ≤ n n ¯ |β|=|α|−1 β<α
+ C U s (||N ||2s + ||∇u||2s ) ≤C ∂ α ∇u
∇u |α|−1 + ∇u 2|α|−1 + C U s (||N ||2s + ||∇u||2s ) 1 ≤ ∂ α ∇u 2 + C ∇u 2|α|−1 + C U s (||N ||2s + ||∇u||2s ). 2 When |β| < |α| − 1, applying Lemma 2.1, we have αβ I41 ≤ C ∇u 2|α|−1 + C U s (||N ||2s + ||∇u||2s ). (3.17) |β|<|α|−1 β<α
αβ , applying Lemma 2.1 and the Poisson equations ΔΦ = N yields Similarly, for the term I42 αβ ¯ ∂ β u, ∂ α ∇Φ + C ∂ α−β N ∂ β u, ∂ α ∇Φ I42 ≤C ∂ α−β n (3.18) ≤C ∇u 2|α|−1 + C N 2|α|−1 + C U s (||N ||2s + ||∇u||2s ).
Therefore, combining (3.14)-(3.18), we obtain (3.19)
d 3 I4α + ∂ α ∇φ 2 + ∂ α ∇u 2 ≤ C ∇u 2|α|−1 +C N 2|α|−1 +C U s (||N ||2s +||∇u||2s ). dt 2
Estimate of I5α . From (3.8), we have (3.20)
I5α = 2 A0 g α , ∂ αU = 2
3
A0 (Aj ∂j ∂ αU −∂ α (Aj ∂j U)) , ∂ αU
j=1
+ 2 A0 (L∂ αU −∂ α (LU)) , ∂ αU .
Stability of steady-state solutions to NSP systems
9
For the first term on the right-hand side of (3.20), by using the Leibniz formula, we have αβ αβ αβ αβ α α α A (A ∂ ∂ U − ∂ (A ∂ U )) , ∂ U = − Cαβ (I51 + I52 + I53 + I54 ), 0 j j j j (3.21) β<α
where
αβ αβ = h (n) ∂ α−β uj ∂ β ∂j N, ∂ α N , I52 = ∂ α−β n∂ β ∂j u, ∂ α N , I51 αβ αβ = ∂ α−β (h (n)) ∂ β ∂j N, ∂ α u , I54 = ∂ α−β uj ∂ β ∂j u, ∂ α u . I53
αβ αβ and I54 yields Applying Lemma 2.1 to the terms I51 αβ (3.22) I51 = h (n) ∂ α−β uj ∂ β ∂j N, ∂ α N ≤ C U s (||N ||2s + ||∇u||2s )
and
αβ α−β uj ∂ β ∂j u, ∂ α u ≤ C U s (||N ||2s + ||∇u||2s ). I54 = ∂
(3.23)
αβ For the term I52 , when |β| = |α| − 1, let ∂ α = ∂ β ∂x . By an integration by parts and Lemma 2.1, we have αβ ∂ α−β n ¯ ∂ β ∂j u, ∂ α N + ∂ α−β N ∂ β ∂j u, ∂ α N I52 ≤ |β|=|α|−1 β<α
|β|=|α|−1 β<α
≤
(3.24)
∂x ∂ α−β n ¯ ∂ β ∂j u , ∂ β N + ∂ α−β N ∂ β ∂j u, ∂ α N
|β|=|α|−1 β<α
≤ε ∂ α ∇u 2 + C ∇u 2|α|−1 + C N 2|α|−1 + C U s (||N ||2s + ||∇u||2s ), where ε > 0 is a small constant to be chosen. When |β| < |α| − 1, we easily get αβ 2 2 I52 ≤C ∇u |α|−1 + C N |α|−1 + C U s (||N ||2s + ||∇u||2s ). (3.25) |β|<|α|−1 β<α
αβ , applying Lemma 2.1, we obtain Similarly, for the term I53 αβ 2 2 2 I53 ≤ε ∂ α ∇u + C ∇u |α|−1 + C N |α|−1 + C U s (||N ||2s + ||∇u||2s ). (3.26) β<α
Next we estimate the second term on the right-hand side of (3.20). The Leibniz formula gives A0 (L∂ α U − ∂ α (LU )) , ∂ α U β α−β (3.27) C β h (n) ∂ α−β (∇¯ n) ∂ β u, ∂ α N − C n∂ (h (n)) ∂ β N, ∂ α u . =− α
β<α
α
β<α
Applying an integration by parts and Lemma 2.1 to the first term on the right-hand side of (3.27) yields h (n) ∂ α−β (∇¯ n) ∂ β u, ∂ α N = ∂x h (n) ∂ α−β ∇¯ n ∂ β u, ∂ α1 N + h (n) ∂ α−β ∇¯ n∂x ∂ β u, ∂ α1 N (3.28)
≤ C ∇u 2|α|−1 + C N 2|α|−1 + C U s (||N ||2s + ||∇u||2s ),
where α1 ∈ N3 with ∂ α1 ∂x = ∂ α . Similarly, for the second term, by Lemma 2.1, we get 2 2 (3.29) n∂ α−β (h (n)) ∂ β N, ∂ α u ≤ C ∇u |α|−1 + C N |α|−1 + C U s (||N ||2s + ||∇u||2s ).
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Y.H. Feng and C.M. Liu
The combining of (3.20)-(3.29) gives (3.30)
|I5α | ≤ 2ε ∂ α ∇u 2 + C ∇u 2|α|−1 + C N 2|α|−1 + C U s (||N ||2s + ||∇u||2s ).
Choosing 0 < ε < 14 , then (3.9) follows from (3.11)-(3.13), (3.19) and (3.30).
2
Now, from Proposition 3.1 and Lemma 3.2, we obtain Proposition 3.2. For all t ∈ [0, T ], α ∈ N3 with 1 ≤ |α| ≤ s, it holds
2
d A0 ∂ β U, ∂ β U + ∂ β ∇Φ + ∇u 2|α| dt β≤α
2 2 (3.31) ≤C ∇u |α|−1 + N |α|−1 + C U s (||N ||2s + ||∇u||2s ). Proof. Summing (3.5) and (3.9) (in which α = β) for all indexes with |β| ≤ |α| yields (3.31). 2 3.2. Dissipation estimate for N . Estimate (3.31) contains a recurrence relation on the time dissipation of ∇u. It is clear that this estimate is not sufficient to control the higher order term in (3.31) and the dissipation estimates of N is necessary. Proposition 3.3. For all t ∈ [0, T ], α ∈ N3 with 1 ≤ |α| ≤ s, it holds d β β ∂ u, ∂ ∇N +C0 N 2|α| ≤ C ∇u 2|α| +C N 2|α|−1 +C U s (||N ||2s + ||∇u||2s ), (3.32) dt |β|<|α|
(3.33) and
d u, ∇ (h (¯ n) N ) + C0 N 21 ≤ C ∇u 21 + C U s (||N ||2s + ||∇u||2s ), dt d A0 ∂ α U, ∂ α U + ∂ α ∇Φ 2 + ∇u 21 dt |α|≤1
(3.34)
≤ ε0 N 21 + C ∇u 2 +C U s (||N ||2s + ||∇u||2s ),
where C0 is a positive constant independent of any time and the positive constant ε0 > 0 is determined in the next section. Proof. For β ∈ N3 with |β| ≤ |α| − 1, applying ∂ β to the second equation in (2.3) gives 1 β β β Δu − ∂t ∂ β u − I6β , (3.35) n) N ) − ∂ ∇Φ = ∂ ∂ ∇ (h (¯ n ¯ where N Δu I6β = ∂ β (u · ∇) u + (h (¯ . n + N ) − h (¯ n)) ∇N + r + n¯ n By Lemma 2.1, it is easy to see that
β
I6 ≤ C U s (||N ||s + ||∇u||s ). Let
I7β = ∂ β ∇ (h (¯ n) N ) − h (¯ n) ∂ β ∇N.
Since the highest derivatives order of N in I7β is |β|, it follows that
β
I7 ≤ C N |β| .
Stability of steady-state solutions to NSP systems
11
Taking the inner product of (3.35) with ∂ β ∇N in L2 , using integration by parts and Lemma 2.1 yields
2
n)∂ β ∇N − ∂ β ∇Φ, ∂ β ∇N
h (¯
(3.36)
1 = ∂ β Δu , ∂ β ∇N − ∂t ∂ β u, ∂ β ∇N − I6β + I7β , ∂ β ∇N n ¯
β
2 ≤δ ∂ ∇N + C N 2|β| + C ∇u 2|α| − ∂t ∂ β u, ∂ β ∇N + C U s (||N ||2s + ||∇u||2s ),
where δ > 0 is small enough. In view of the density equation ∂t N = −∇ · (nu), for the fourth term on the right-hand side of (3.36), we have − ∂t ∂ β u, ∂ β ∇N d β ∂ u, ∂ β ∇N + ∂ β ∇ · u, ∂ β ∇ · (¯ nu) + ∂ β ∇ · u, ∂ β ∇ · (N u) =− (3.37) dt d β ∂ u, ∂ β ∇N + C ∇u 2|β| + C U s (||N ||2s + ||∇u||2s ). ≤− dt On the other hand, the Poisson equation ∂ β ΔΦ = ∂ β N gives
2
(3.38) − ∂ β ∇Φ, ∂ β ∇N = ∂ β N . Then, combining (3.36)-(3.38), and taking 0 < δ <
h0 , 2
we get
h0 β d β
∂ ∇N 2 + ∂ β N 2 ∂ u, ∂ β ∇N + dt 2 ≤C ∇u 2|α| + C N 2|β| + C U s (||N ||2s + ||∇u||2s ).
(3.39) Remark that
0≤|β|≤|α|−1
h0
∂ β ∇N 2 + ∂ β N 2 2
N 2|α| .
is equivalent to Summing (3.39) for all indexes β with |β| ≤ |α| − 1 yields (3.32). Next, we estimate (3.33). Taking the inner product of (3.35) (in which β = 0) with ∇(h (¯ n)N ) in L2 yields 2
(3.40)
∇ (h (¯ n) N ) − ∇Φ, ∇ (h (¯ n) N ) 1 = Δu, ∇ (h (¯ n) N ) − ∂t u, ∇ (h (¯ n) N ) − I60 , ∇ (h (¯ n) N ) n ¯ 1 d 2 ≤ ∇ (h (¯ n) N ) + C ∇u 21 − u, ∇ (h (¯ n) N ) + u, ∇∂t (h (¯ n) N ) 2 dt + C U s (||N ||2s + ||∇u||2s ).
From the density equation ∂t N = − div(nu) and the Poisson equation ΔΦ = N , it gives − ∇Φ, ∇ (h (¯ (3.41) n) N ) = h (¯ n)N 2 , and (3.42)
n) N ) = ∇ · u, h (¯ n)∇ · (nu) u, ∇∂t (h (¯ ≤C ∇u 2 + C U s (||N ||2s + ||∇u||2s ).
12
Y.H. Feng and C.M. Liu
From the above inequalities, we get d 1 2 u, ∇ (h (¯ n) N ) + ∇ (h (¯ n) N ) + h (¯ n)N 2 dt 2 (3.43) ≤C ∇u 21 + C U s (||N ||2s + ||∇u||2s ). Noting h (n) ≥ h0 > 0, we obtain (3.33). Finally, we estimate (3.34), which is a different version of (3.31) for |α| = 1. Following the αβ αβ αβ steps of proof of Lemma 3.2, we only need to re-estimate the terms of I42 , I52 , I53 and the two terms on the right-hand side of (3.27). Indeed, it is easy to get ε αβ 0 2 2 (3.44) I42 ≤ N + C ∇u + C U s (||N ||2s + ||∇u||2s ), 3 ε αβ αβ 0 2 2 (3.45) I52 + I53 ≤ ∇N + C ∇u + C U s (||N ||2s + ||∇u||2s ), 3 and ε0 | h (n) ∂ α−β (∇¯ (3.46) n) ∂ β u, ∂ α N | ≤ ∇N 2 + C ∇u 2 , 6 α−β ε0 β α n∂ (3.47) (h (n)) ∂ N, ∂ u ≤ ∇N 2 + C ∇u 2 + C U s ( N 2s + ∇u 2s ). 6 Then, by (3.14)-(3.16) and (3.44) we get d 3 I4α + ∂ α ∇φ 2 + ∂ α ∇u 2 ≤ C ∇u 2 +ε0 N 2 +C U s (||N ||2s +||∇u||2s ). dt 2 On the other hand, by (3.20)-(3.23), (3.45), (3.27) and (3.46)-(3.47), we obtain 1 (3.49) |I5α | ≤ ∂ α ∇u 2 + C ∇u 2 + ε0 ∇N 2 + C U s (||N ||2s + ||∇u||2s ). 2 Then, by (3.11)-(3.13), (3.48) and (3.49), we have d A0 ∂ α U, ∂ α U + ∂ α ∇Φ 2 + ∂ α∇u 2 dt (3.50) ≤ε0 N 21 + C ∇u 2 + C U s (||N ||2s + ||∇u||2s ). (3.48)
Therefore, summing (3.5) and (3.50) for all indexes α with |α| = 1 yields (3.34).
2
Proposition 3.4. There exists small positive constants π and C1 such that, for all t ∈ [0, T ], α ∈ N3 with 1 ≤ |α| ≤ s, it holds ⎛ ⎞
d ⎝ 2 ∂ γ u, ∂ γ ∇N ⎠ A0 ∂ β U, ∂ β U + ∂ β ∇Φ + π dt β≤α |γ|≤|α|−1
(3.51) + C1 N 2|α| + ∇u 2|α|
≤C ∇u 2|α|−1 + N 2|α|−1 + C U s (||N ||2s + ||∇u||2s ). Proof. Choose π > 0 sufficiently small such that 1 . π≤ 2C Let πC0 1 − Cπ , , C1 = min 2 2
Stability of steady-state solutions to NSP systems
13
where C0 is given by Proposition 3.2. The summation of (3.31) and (3.32) multiplying π, and then summing the resulting equation yields (3.51). 2 4. Proof of Theorem 1.1 We carry on the induction on |α|(1 ≤ |α| ≤ s) of space derivatives for (3.51) and (3.33)(3.34). The step of the induction is increasing from |α| = 1 to |α| = s. More precisely, for |α| = 1, we first multiplies π on both sides of (3.33). Remark that the term Cπ ∇u 21 can be controlled by ∇u 21 on the left-hand side of (3.34), and the term ε0 N 21 on the right-hand side of (3.34) can be controlled by N 21 when ε0 > 0 is sufficiently small. Thus, there exists a positive constant a1 such that
d n) N ) + (N, ∇u) 21 A0 ∂ αU, ∂ αU + ∂ α ∇Φ 2 + π u, ∇ (h (¯ a1 dt (4.1) |α|≤1
≤C ∇u 2 + C U s (||N ||2s + ||∇u||2s ). In the same way, for |α| ≥ 2, C( ∇u 2|α|−1 + N 2|α|−1 ) on the right-hand side of (3.51) can be controlled by ∇u 2|α| + N 2|α| in the preceding step on the left-hand side of (3.51) multiplying an appropriate large positive constant. So, we have s d ak ∂ γ u, ∂ γ ∇N + (||N ||2s + ||∇u||2s ) A0 ∂ α U, ∂ α U + ∂ α ∇Φ 2 + π dt k=1 |α|≤k
|γ|≤|α|−1
(4.2) ≤ C ∇u 2 + C U s (||N ||2s + ||∇u||2s ), where ak > 0(k = 1, · · · , s) are some constants. By use of formulas (3.5) and (4.2) and noting
U is small, we obtain (4.3) s d ak A0 ∂ α U, ∂ α U + ∂ α ∇Φ 2 + π dt k=1 |α|≤k
∂ γ u, ∂ γ ∇N
+ (||N ||2s + ||∇u||2s ) ≤ 0,
|γ|≤|α|−1
where the constant ak > 0(k = 1, · · · , s) may be amended again. When π > 0 is small enough, s ak ∂ γ u, ∂ γ ∇N A0 ∂ α U, ∂ α U + ∂ α ∇Φ 2 + π k=1
U 2s
|α|≤k
|γ|≤|α|−1
∇Φ 2s .
is equivalent to + Integrating (4.3) on from 0 to t, we get (1.9). Moreover, (1.9) implies that, for all |β| ≤ s − 1 and |γ| ≤ s − 3, ∂ β N ∈ L∞ R+ , L2 , ∂ γ ∇u ∈ L∞ R+ , L2 , and
∂t ∂ β N ∈ L∞ R+ , L2 ,
which implies (1.10)-(1.11).
∂t ∂ γ ∇u ∈ L∞ R+ , L2 , 2
Acknowledgments : The authors would like to thank Professor Yue-Jun.Peng for his precious suggestions and fruitful discussions. This research was supported in part by the the BNSF (1164010, 1132006), NSFC (11671295,11401421,11371042), the general project of scientific research project of the Beijing education committee of China, NSF of Qinghai Province, the 2016
14
Y.H. Feng and C.M. Liu
Beijing project of scientific activities for the excellent students studying abroad and the Beijing University of Technology foundation funded project. References [1] F. Chen, Introduction to Plasma Physics and Controlled Fusion, Vol. 1, Plenum Press, New York, 1984. [2] P. Degond and P. Markowich, A steady state potential flow model for semiconductors, Ann. Mat. Pura Anal. 52 (1993), 87-98. [3] L. C. Evans, Partial differential equations, Graduate Studies in Mathematics, 19, American Mathe matical Society, Providence, RI, 1998. [4] Y.H. Feng, Y.J. Peng and S. Wang, Stability of non-constant equilibrium solutions for two-fluid EulerMaxwell systems, Nonlinear Anal. Real World, 26 (2015), 372-390. [5] Y. Guo and W. Strauss, Stability of semiconductor states with insulating and contact boundary conditions, Arch. Rational Mech. Anal. 179 (2005), 1-30. [6] C. Hao and H.L. Li, Global existence for compressible Navier-Stokes-Poisson equations in three and higher dimensions, J. Differential Equations, 246 (2009), 4791-4812. [7] L. Hsiao, Q. Ju and S. Wang, The asymptotic behavior of global smooth solutions to the multi-dimensional hydrodynamic model for semiconductors, Math. Meth. Appl. Sci. 26 (2003), 1187-1210. [8] S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math. 34 (1981), 481-524. [9] T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal. 58 (1975), 181-205. [10] H.L. Li, A. Matsumura and G. Zhang, Optimal decay rate of the compressible Navier-Stokes-Poisson system in R3 , Arch. Ration. Mech. Anal. 196 (2010), 681-713. [11] C.M. Liu, Z.J. Guo and Y.J. Peng, Global Stability of large steady-states for an isentropic Euler-Maxwell system in R3 , preprint, 2017. [12] A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer-Verlag, New York, 1984. [13] P. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor equations, Springer, 1990. [14] J. Nash, Le probl`eme de Cauchy pour les ´equations diff´erentielles d’un fluide g´en´eral, Bull. Soc. Math. France, 90 (1962), 487-497. [15] T. Nishida, Nonlinear hyperbolic equations and related topics in fluids dynamics, Publications Math´ematiques d’Orsay, Universit´e Paris-Sud, Orsay, No. 78-02, 1978. [16] Y.J. Peng, Stability of non-constant equilibrium solutions for Euler-Maxwell equations, J. Math. Pures Appl. 103 (2015), 39-67. [17] H. Rishbeth and O.K. Garriott, Introduction to Ionospheric Physics, Academic Press, 1969. [18] J. Serrin, On the uniqueness of compressible fluid motion, Arch. Rational. Mech. Anal. 3 (1959), 271-288. [19] Z. Tan, Y. Wang and X. Zhang, Large time behavior of solutions to the non-isentropic compressible NavierStokes-Poisson system in R3 , Kinetic and Related Models, 5 (2017), 615-638. [20] Z. Tan and G. Wu, Global existence for the non-isentropic compressible Navier-Stokes-Poisson system in three and higher dimensions, Nonlinear Anal. Real World, 13 (2012), 650-664. [21] Z. Tan, T. Yang, H. Zhao and Q. Zou, Global solutions to the one-dimensional compressible Navier-StokesPoisson Equations with large data, SIAM J. Math. Anal. 45 (2013), 547-571. [22] W. Wang and Z. Wu, Pointwise estimates of solution for the Navier-Stokes-Poisson equations in multidimensions, J. Differential Equations, 248 (2010), 1617-1636. [23] Y.Z. Wang and K. Wang, Asymptotic behavior of classical solutions to the compressible Navier-StokesPoisson equations in three and higher dimensions, J. Differential Equations, 259 (2015), 25-47. [24] G. Zhang, H.L. Li and C. Zhu, Optimal decay rate of the non-isentropic compressible Navier-Stokes-Poisson system in R3 , J. Differential Equations, 250 (2011), 866-891. [25] X. Zheng, Global well-posedness for the compressible Navier-Stokes-Poisson system in the Lp framework, Nonlinear Anal. T. M. A. 75 (2012), 4156-4175.
Yue-Hong Feng, Cun-Ming Liu
PII: DOI: Reference:
S0022-247X(18)30198-7 https://doi.org/10.1016/j.jmaa.2018.03.001 YJMAA 22077
To appear in:
Journal of Mathematical Analysis and Applications
Received date:
29 November 2017
Please cite this article in press as: Y.-H. Feng, C.-M. Liu, Stability of steady-state solutions to Navier-Stokes-Poisson systems, J. Math. Anal. Appl. (2018), https://doi.org/10.1016/j.jmaa.2018.03.001
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STABILITY OF STEADY-STATE SOLUTIONS TO NAVIER-STOKES-POISSON SYSTEMS YUE-HONG FENG1 , CUN-MING LIU2∗ 1
College of Applied Sciences, Beijing University of Technology, 100022 Beijing, China
2 Department
of Mathematics, Qufu Normal University, Qufu 273165, Shandong, China
Email : [email protected], [email protected],
Abstract. This paper is concerned with a stability problem for compressible Navier-Stokes-Poisson systems. It arises in the modeling of semiconductors with a viscosity term in momentum equations. We prove that smooth solutions exist globally in time near the steady-state solution, and converge to the steady state for large time. In this stability result, we don’t give any special assumptions on the given doping profile. The proof is based on the techniques of anti-symmetric matrix and an induction argument on the order of the space derivatives of solutions in energy estimates. Keywords: Navier-Stokes-Poisson system, steady-state, global smooth solution, energy estimate AMS Subject Classification (2000) : 35Q35, 76N10 1. Introduction and main results 1.1. Introduction. We consider smooth solutions of compressible Navier-Stokes-Poisson systems on R3 . This system describes the dynamic of electrons in semiconductors with a viscosity term in momentum equations. The symbols n, p, u = (u1 , u2 , u3 )T and φ stand for the density, pressure, the velocity and the electric potential of the electrons. The scaled system is written as (see [1, 13, 17]) : ⎧ ∂t n + ∇ · (nu) = 0, ⎪ ⎪ ⎨ ∂t (nu) + ∇ · (nu ⊗ u) + ∇p(n) = n∇φ + Δu, (1.1) ⎪ ⎪ ⎩−Δφ = b(x) − n, lim φ = 0, |x|→∞
for t > 0 and x ∈ R3 , where b(x) is the doping profile for semiconductors. We assume b is sufficiently smooth and satisfies b ≥ const. > 0 on R3 , and pressure p is smooth and strictly increasing on (0, +∞). The system is supplemented by the following initial condition (1.2)
t = 0 : (n, u) = (n0 , u0 ),
x ∈ R3 .
For smooth solutions in any non-vacuum field, the momentum equations in (1.1) can be written as Δu , ∂t u + (u · ∇) u + ∇h(n) = ∇φ + n *Corresponding author 1
2
Y.H. Feng and C.M. Liu
where h is the enthalpy function defined by h (n) = increasing on (0, +∞), so is h. Let (1.3)
p (n) . Since p is smooth and strictly n
E = ∇φ.
By the density equation together with Poisson equation in system (1.1), the variable E satisfies (1.4)
∂t E = −∇Δ−1 (∇ · (nu)).
Thus, the Navier-Stokes-Poisson system for variables (n, u, E) can be regarded as a quasilinear hyperbolic-parabolic system with a zero-order term −∇Δ−1 (∇ · (nu)) of (n, u). The local wellposedness of smooth solutions to Cauchy problem (1.1)-(1.2) can be obtained by using standard methods as in [18, 14, 9, 12]. Let s ≥ 3 be an integer, and (n0 − b, u0 , ∇φ0 ) ∈ H s (R3 ) satisfying the compatible condition −Δφ0 = b − n0 and n0 ≥ const. > 0. Then there exists T∗ > 0 such that the Cauchy problem admits a unique solution (n, u, φ) on the domain [0, T∗ ] × R3 , and n − b, ∇φ ∈ C [0, T∗ ]; H s (R3 ) ∩ C 1 [0, T∗ ]; H s−1 (R3 ) , n ≥ const. > 0, u ∈ C [0, T∗ ]; H s (R3 ) ∩ C 1 [0, T∗ ]; H s−2 (R3 ) , ∇u ∈ L2 [0, T∗ ]; H s (R3 ) . Now we consider the steady-state solutions of (1.1) with zero velocity. Let n ¯ , u¯, φ¯ be such a solution of variable x with u¯ = 0. We get ¯ ∇h (¯ n) = ∇φ, (1.5) −Δφ¯ = b − n ¯. This implies n ¯ satisfies an elliptic equation : (1.6)
−Δh (¯ n) = b − n ¯,
in R3 .
In a bounded domain Ω ⊂ R3 with periodic, Dirichlet or Neumann boundary conditions or in the whole space R3 , (1.6) admits a unique solution (see [2, 5, 7, 11]) by using the classical fixedpoint theorem or a variational method. Moreover, b ≥ const. > 0 implies that n ¯ ≥ const. > 0. The existence and uniqueness of solutions to this elliptic equation is given as follows Proposition 1.1. ([7, 11]) Let s1 ≥ 1. Assume b ∈ L∞ (R3 ), ∇b ∈ H s1 −1 (R3 ) and b ≥ const. > 0 ¯=n ¯ (x) satisfying n ¯ − b ∈ H s1 (R3 ), a.e. x ∈ R3 . Then problem (1.5) admits a unique solution n n ¯ ≥ const. > 0. When n ¯ is given by Proposition 1.1, we obtain φ¯ from the first equation of (1.5). In order to ¯ we may assume it is sufficiently smooth which implies study the stability of steady-state (¯ n, 0, φ), p n is bounded. We point out that the the L norm with 2 ≤ p ≤ ∞ of any order derivatives of ∇¯ stability problem here is to investigate the global existence of smooth solutions when the initial ¯ Many research results can be found on data (n0 , u0 , φ0 ) are close to the steady-state (¯ n, 0, φ). such a interesting problem of Navier-Stokes-Poisson systems. When b is a positive constant or is a small perturbation of a constant, this problem is solved in [6, 10, 22, 23, 24, 25] in the Soblev spaces or Besove spaces framework. This is extended to non-isentropic Navier-Stokes-Poisson systems, see [19, 20]. In this situation, since the solution is close to the constant equilibrium state, the classical energy method and linearized method are valid. We also mentioned the global well-posedness result for the one-dimensional model with large initial data, see [21]. ¯ We find all the techniques used When b is large, so is the steady-state solution (¯ n, 0, φ). for small b or constant b no longer work due to the appearance of lower order terms which will make essential difficulties in the energy estimates, and there is no result for Navier-StokesPoisson systems. In this paper, we solve this problem by using an anti-symmetric matrix
Stability of steady-state solutions to NSP systems
3
technique and employing an induction argument on the order of the derivatives of solutions in energy estimates. We point out that the technique of anti-symmetric matrix was first used by Guo-Strauss when they considered the stability problem for the isentropic Euler-Poisson systems with boundary conditions in [5], namely, for the system (1.1) where the equation for the velocity u is replaced by (1.7)
∂t u + (u · ∇) u + ∇h(n) = ∇φ − u.
Next, Peng employed an induction argument on the order of the mixed time-space derivatives of solutions in energy estimates when he studied the global existence of solutions for Euler-Maxwell system in [16]. This technique was also applied to the two-fluid isentropic Euler-Maxwell system, see [4]. In our problems, since the high order derivative terms with respect to t of (n, u) depends on the higher order derivative terms with respect to x of u from the momentum equations, it is impossible to completely follow the same technique of induction argument employed in [16]. To overcome it, we obtain refined estimates by use of classical Sobolev embedding theorems and integrations by parts. It allows us to use a slightly different induction argument together with anti-symmetric matrix technique to complete the proof. In this paper, we prove the global existence of solutions to Cauchy problem (1.1)-(1.2), and the solution tends to the steady-state for large time. It is stated in Theorem 1.1. Remark that the result in Theorem 1.1 still holds for system (1.1) where the momentum equations are replaced by (1.8)
∂t (nu) + ∇ · (nu ⊗ u) + ∇p(n) = n∇φ + Δu + ∇ div u.
Since the term ∇ div u brings us better dissipations of div u, and doesn’t produce any obstacle in the energy estimates, so we omit it in momentum equations. 1.2. Main results. Let us introduce some notations. For all integer s ∈ N, we denote by H s , L2 and L∞ the usual Sobolev spaces H s (R3 ), L2 (R3 ) and L∞ (R3 ), and by · s , · and
· ∞ the corresponding norms, respectively. ·, · stands for the inner product of L2 (R3 ). We also denote by C, C0 , C1 > 0 constants independent of any time, but may be dependent of the ¯ For a multi-index α = (α1 , α2 , α3 ) ∈ N3 , we denote steady-state (¯ n, 0, φ). ∂ α = ∂xα11 ∂xα22 ∂xα33 = ∂1α1 ∂2α2 ∂3α3 , with |α| = α1 + α2 + α3 . For α = (α1 , α2 , α3 ) and β = (β1 , β2 , β3 ) ∈ N3 , β ≤ α stands for βj ≤ αj for j = 1, 2, 3, and β < α stands for β ≤ α and β = α. We also recall the Leibniz formula Cαβ ∂ α−β u∂ β v, ∀ α ∈ N3 , ∂ α (uv) = β≤α
where Cαβ are positive constants. The main results of this paper can be stated as follows. Theorem 1.1. Let s ≥ 3 be an integer. Then there exist constants δ0 > 0, C > 0 such that if
n0 − n ¯ , u0 , ∇φ0 − ∇φ¯ s ≤ δ0 , Cauchy problem (1.1)-(1.2) has a unique global smooth solution (n, u, φ) satisfying t
2 2 2 ¯
n(t) − n
n(t ) − n ¯ s + ∇u(t ) s dt ¯ , u(t), ∇φ(t) − ∇φ s + 0
2 (1.9) ≤ C n0 − n ¯ , u0 , ∇φ0 − ∇φ¯ s , ∀ t ≥ 0,
4
Y.H. Feng and C.M. Liu
which implies ¯ s−1 = 0, lim n(t) − n
(1.10)
t→∞
and lim ∇u(t) s−3 = 0.
(1.11)
t→∞
This paper is organized as follows. In the next section, we focus on the symmetrization of the Navier-Stokes equations with perturbation variables and show an useful lemma. Section 3 is devoted to the energy estimates. Finally, we give the proof of the above theorem in section 4. 2. Symmetrization of Navier-Stokes equations Let N =n−n ¯,
(2.1) and (2.2)
U=
N u
,
U0 =
N0 u0
¯ Φ = φ − φ,
,
W =
U ∇Φ
,
W0 =
U0 ∇Φ0
where ¯ N0 = n0 − n ¯ , ∇Φ0 = ∇φ0 − ∇φ. By (2.1) and (1.5), system (1.1) can be written as ⎧ ∂t N + u · ∇N + n∇ · u + ∇¯ n · u = 0, ⎪ ⎪ ⎨ Δu (2.3) ∂t u + (u · ∇) u + ∇h (n) − ∇h (¯ n) − ∇Φ − = 0, ⎪ n ⎪ ⎩ ΔΦ = N. A straightforward computation implies n) N + r (¯ n, N ) , ∇h (n) − ∇h (¯ n) = h (n) ∇N + ∇h (¯ where
n) − h (¯ n) N ) ∇¯ n = O N2 . r (¯ n, N ) = (h (n) − h (¯ The Navier-Stokes equations in (2.3) can be rewritten as (2.4)
∂t U +
3
Aj (n, u) ∂j U + L (x) U + M (W ) = f (r) ,
j=1
with (2.5)
(2.6) and (2.7)
uj
neTj
, j = 1, 2, 3, h (n) ej uj I3 0 (∇¯ n )T L (x) = , ∇h (¯ n) 0
Aj (n, u) =
⎛ M (W ) = ⎝
0
⎞
Δu ⎠ , f (r) = −∇Φ − n
0 −r
.
,
Stability of steady-state solutions to NSP systems
5
Here, (e1 , e2 , e3 ) denotes the canonical basis of R3 , I3 denotes the 3 × 3 unit matrix and we use eTj to denote the transpose of ej . It is clear that the matrix A0 (n) defined by h (n) 0 , (2.8) A0 (n) = 0 nI3 is symmetric and uniformly positive definite for n > 0, and h (n)uj h (n)neTj j (n, u) = A0 (n)Aj (n, u) = (2.9) A h (n)nej nuj I3 is symmetric. These expressions of A0 and A˜j are useful in energy estimates. Furthermore, the 3 j (n, u) − 2A0 (n)L(x) defined by matrix B(n, u, x) = ∂j A j=1
(2.10)
(∇p (n)−2h (n)∇¯ n )T ∇·(h (n)u) B(n, u, x) = ∇p (n)−2n∇h (¯ n) ∇·(nu)I3
is anti-symmetric at (n, u) = (¯ n, 0). Finally, we list some useful inequalities in Sobolev space. Lemma 2.1. (Moser-type calculus inequalities, see [8, 12].) Let s ≥ 3 be an integer. Suppose u ∈ H s , ∇u ∈ L∞ , v ∈ H s−1 ∩ L∞ and f is a smooth function. Then for all multi-index α with 1 ≤ |α| ≤ s, one has ∂ α (uv) − u∂ α v ∈ L2 and
∂ α (uv) − u∂ α v ≤ C ∇u ∞ D|α|−1 v + D|α| u
v ∞ ,
∂ α f (u) ≤ C(1 + u s )s−1 u s , where the constant C may depend on u ∞ and s, and
∂ α u .
Ds u = |α|=s
3. Energy estimates According to [15], the global existence of smooth solutions follows from the local existence and uniform estimates of solutions with respect to t. In what follows, we focus on uniform estimates for proving Theorem 1.1. Let T > 0 and W be smooth solution of (2.3) defined on time interval [0, T ] with initial value W0 . Let (3.1)
UT = sup ||U (t)||s . 0≤t≤T
We assume that s ≥ 3, and UT is uniformly sufficiently small with respect to T . Then from the continuous embedding H s−1 → L∞ (see [3]), it implies (3.2)
q0 ≤ n ≤ q1 , h0 ≤ h (n),
where q0 , q1 and h0 are positive constants independent of any time. The following lemma is easily obtained by use of the density equation and the antisymmetry of the matrix B(¯ n, 0, x).
6
Y.H. Feng and C.M. Liu
Lemma 3.1. For all t ∈ [0, T ], it holds
∂t N + ∂t N ∞ ≤ C||∇u||2 ,
(3.3)
∂t A0 (n) ∞ ≤ C||∇u||2 ,
and | B(n, u, x)U, U | ≤ C||U ||s (||N ||2s + ||∇u||2s ).
(3.4)
Proof. By the density equation ∂t N = −n∇ · u − ∇N · u, and the smallness of UT , we have
∂t N ≤ C( n L∞ ∇u + ∇N
u ∞ ) ≤ C ∇u 2 ,
∂t N L∞ ≤ n ∞ ∇u ∞ + u ∞ ∇N ∞ ≤ C ∇u 2 , and then
∂t A0 (n) ∞ = A0 (n) ∂t N ∞ ≤ C ∇u 2 .
Thus, we get (3.3). Next, we prove (3.4). Noting the expression (2.10), we have B(n, u, x)U, U = B1 U, U + B2 U, U , where
n )T 0 (∇p (n)−2h (n)∇¯ B1 = ∇p (n)−2n∇h (¯ n) 0
,
0 ∇·(h (n)u) B2 = 0 ∇·(nu)I3
.
¯ , we obtain Since the matrix B1 is anti-symmetric at n = n B1 U, U ≤ C N 2 u ∞ ≤ C N 2 ∇u 1 . For the term B2 U, U , using an integration by parts gives | B2 U, U | = | ∇ · (h (n)u) N, N + ∇ · (nu) u, u | =2 | h (n)N u, ∇N + nu, (u · ∇) u | ≤C ∇u 1 ( N ∇N + u ∇u ) . Therefore, we get (3.4). Firstly, we obtain an L2 estimate with respect to variable (N, u, ∇Φ).
2
Proposition 3.1. For all t ∈ [0, T ], it holds d (3.5) A0 (n) U, U + ∇Φ 2 + 2 ∇u 2 ≤ C||U ||s (||N ||2s + ||∇u||2s ). dt Proof. Taking the inner product of (2.4) with 2A0 (n) U in L2 yields the classical energy equality for U : d A0 (n) U, U = divA (n, u) U, U + 2 A0 (n) (f − LU − M (W )) , U , dt where 3 divA (n, u) = ∂t A0 (n) + ∂j A˜j (n, u) . j=1
Then d A0 (n) U, U = ∂t A0 (n) U, U + B(n, u, x)U, U + 2 A0 (n) (f − M (W )) , U . dt Applying Lemma 3.1 yields ∂t A0 (n) U, U = h (n) ∂t N, |N |2 + ∂t N, |u|2 ≤ C U s (||N ||2s + ||∇u||2s )
Stability of steady-state solutions to NSP systems
7
and | B(n, u, x)U, U | ≤ C||U ||s (||N ||2s + ||∇u||2s ). In view of the expression (2.7) and r (¯ n, N ) = O (N 2 ), we have A0 (n) M (W ) , U = ∇u 2 − nu, ∇Φ , and | A0 (n) f, U | ≤ C N 2 u ∞ ≤ C U s (||N ||2s + ||∇u||2s ). These formulas yield d A0 (n) U, U + 2 ∇u 2 − 2 nu, ∇Φ ≤ C U s (||N ||2s + ||∇u||2s ). dt By the Poisson equation in (2.3), we get −2 nu, ∇Φ =
(3.6)
d
∇Φ 2 . dt
Thus, (3.5) follows from the last two inequalities.
2
3.1. Higher order energy estimates. Let α ∈ N3 with 1 ≤ |α| ≤ s. Applying ∂ α on (2.4) gives (3.7)
∂t ∂ α U +
3
Aj ∂j ∂ α U + L∂ α U + ∂ α M = ∂ α f + g α ,
j=1
where (3.8)
gα =
3
(Aj ∂j ∂ α U − ∂ α (Aj ∂j U )) + L∂ α U − ∂ α (LU ) .
j=1
Lemma 3.2. For all t ∈ [0, T ] and α ∈ N3 with 1 ≤ |α| ≤ s, it holds (3.9)
d A0 ∂ α U, ∂ α U + ∂ α ∇Φ 2 + ∂ α∇u 2 dt ≤C N 2|α|−1 + C ∇u 2|α|−1 + C U s (||N ||2s + ||∇u||2s ).
Proof. Taking the inner product of (3.7) with 2A0 (n) ∂ α U in L2 , we obtain
(3.10)
d A0 ∂ α U, ∂ α U = ∂t A0 (n) ∂ αU, ∂ αU + B(n, u, x)∂ αU, ∂ αU dt + 2 A0 ∂ α f, ∂ α U − 2 A0 ∂ α M, ∂ α U + 2 A0 g α , ∂ α U =I1α + I2α + I3α + I4α + I5α ,
with the natural correspondence for I1α , I2α , · · · , I5α . Let us estimate them in the following. Estimates of I1α , I2α and I3α . Applying Lemmas 2.1, 3.1 yields (3.11)
|I1α | = | ∂t A0 (n) ∂ α U, ∂ α U | ≤C U s (||N ||2s + ||∇u||2s ),
(3.12)
|I2α | = | B(n, u, x)∂ α U, ∂ α U | ≤ C U s (||N ||2s + ||∇u||2s ),
and (3.13)
|I3α | = | A0 ∂ α f, ∂ α U | ≤ C U s (||N ||2s + ||∇u||2s ).
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Y.H. Feng and C.M. Liu
Estimate of I4α . In view of the expression of (2.7), using the Leibniz formula and an integration by parts gives d α 2 2 αβ α α β αβ I + 2 ∂ ∇u
+ ∇Φ
= 2 C I − 2 Cαβ I42 ,
∂ 41 4 α (3.14) dt β<α
where (3.15)
αβ I41
=
n∂
α−β
1 β α ∂ Δu, ∂ u , n
β<α
αβ I42 = ∂ α−β n∂ β u, ∂ α ∇Φ .
αβ In order to estimate I41 , we write
1 N 1 = − . n n ¯ n¯ n When |β| = |α| − 1, noting n ¯ is sufficiently smooth and bounded (see Proposition 1.1), applying Lemma 2.1 and an integration by parts, we get αβ I41 |β|=|α|−1 β<α
≤
n∂ α−β 1 ∂ β Δu, ∂ α u + n∂ α−β N ∂ β Δu, ∂ α u n n¯ n |β|=|α|−1 β<α
(3.16)
n∂ α−β 1 ∂ β ∇u, ∂ α ∇u + ∇ n∂ α−β 1 · ∂ β ∇u, ∂ α u ≤ n n ¯ |β|=|α|−1 β<α
+ C U s (||N ||2s + ||∇u||2s ) ≤C ∂ α ∇u
∇u |α|−1 + ∇u 2|α|−1 + C U s (||N ||2s + ||∇u||2s ) 1 ≤ ∂ α ∇u 2 + C ∇u 2|α|−1 + C U s (||N ||2s + ||∇u||2s ). 2 When |β| < |α| − 1, applying Lemma 2.1, we have αβ I41 ≤ C ∇u 2|α|−1 + C U s (||N ||2s + ||∇u||2s ). (3.17) |β|<|α|−1 β<α
αβ , applying Lemma 2.1 and the Poisson equations ΔΦ = N yields Similarly, for the term I42 αβ ¯ ∂ β u, ∂ α ∇Φ + C ∂ α−β N ∂ β u, ∂ α ∇Φ I42 ≤C ∂ α−β n (3.18) ≤C ∇u 2|α|−1 + C N 2|α|−1 + C U s (||N ||2s + ||∇u||2s ).
Therefore, combining (3.14)-(3.18), we obtain (3.19)
d 3 I4α + ∂ α ∇φ 2 + ∂ α ∇u 2 ≤ C ∇u 2|α|−1 +C N 2|α|−1 +C U s (||N ||2s +||∇u||2s ). dt 2
Estimate of I5α . From (3.8), we have (3.20)
I5α = 2 A0 g α , ∂ αU = 2
3
A0 (Aj ∂j ∂ αU −∂ α (Aj ∂j U)) , ∂ αU
j=1
+ 2 A0 (L∂ αU −∂ α (LU)) , ∂ αU .
Stability of steady-state solutions to NSP systems
9
For the first term on the right-hand side of (3.20), by using the Leibniz formula, we have αβ αβ αβ αβ α α α A (A ∂ ∂ U − ∂ (A ∂ U )) , ∂ U = − Cαβ (I51 + I52 + I53 + I54 ), 0 j j j j (3.21) β<α
where
αβ αβ = h (n) ∂ α−β uj ∂ β ∂j N, ∂ α N , I52 = ∂ α−β n∂ β ∂j u, ∂ α N , I51 αβ αβ = ∂ α−β (h (n)) ∂ β ∂j N, ∂ α u , I54 = ∂ α−β uj ∂ β ∂j u, ∂ α u . I53
αβ αβ and I54 yields Applying Lemma 2.1 to the terms I51 αβ (3.22) I51 = h (n) ∂ α−β uj ∂ β ∂j N, ∂ α N ≤ C U s (||N ||2s + ||∇u||2s )
and
αβ α−β uj ∂ β ∂j u, ∂ α u ≤ C U s (||N ||2s + ||∇u||2s ). I54 = ∂
(3.23)
αβ For the term I52 , when |β| = |α| − 1, let ∂ α = ∂ β ∂x . By an integration by parts and Lemma 2.1, we have αβ ∂ α−β n ¯ ∂ β ∂j u, ∂ α N + ∂ α−β N ∂ β ∂j u, ∂ α N I52 ≤ |β|=|α|−1 β<α
|β|=|α|−1 β<α
≤
(3.24)
∂x ∂ α−β n ¯ ∂ β ∂j u , ∂ β N + ∂ α−β N ∂ β ∂j u, ∂ α N
|β|=|α|−1 β<α
≤ε ∂ α ∇u 2 + C ∇u 2|α|−1 + C N 2|α|−1 + C U s (||N ||2s + ||∇u||2s ), where ε > 0 is a small constant to be chosen. When |β| < |α| − 1, we easily get αβ 2 2 I52 ≤C ∇u |α|−1 + C N |α|−1 + C U s (||N ||2s + ||∇u||2s ). (3.25) |β|<|α|−1 β<α
αβ , applying Lemma 2.1, we obtain Similarly, for the term I53 αβ 2 2 2 I53 ≤ε ∂ α ∇u + C ∇u |α|−1 + C N |α|−1 + C U s (||N ||2s + ||∇u||2s ). (3.26) β<α
Next we estimate the second term on the right-hand side of (3.20). The Leibniz formula gives A0 (L∂ α U − ∂ α (LU )) , ∂ α U β α−β (3.27) C β h (n) ∂ α−β (∇¯ n) ∂ β u, ∂ α N − C n∂ (h (n)) ∂ β N, ∂ α u . =− α
β<α
α
β<α
Applying an integration by parts and Lemma 2.1 to the first term on the right-hand side of (3.27) yields h (n) ∂ α−β (∇¯ n) ∂ β u, ∂ α N = ∂x h (n) ∂ α−β ∇¯ n ∂ β u, ∂ α1 N + h (n) ∂ α−β ∇¯ n∂x ∂ β u, ∂ α1 N (3.28)
≤ C ∇u 2|α|−1 + C N 2|α|−1 + C U s (||N ||2s + ||∇u||2s ),
where α1 ∈ N3 with ∂ α1 ∂x = ∂ α . Similarly, for the second term, by Lemma 2.1, we get 2 2 (3.29) n∂ α−β (h (n)) ∂ β N, ∂ α u ≤ C ∇u |α|−1 + C N |α|−1 + C U s (||N ||2s + ||∇u||2s ).
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Y.H. Feng and C.M. Liu
The combining of (3.20)-(3.29) gives (3.30)
|I5α | ≤ 2ε ∂ α ∇u 2 + C ∇u 2|α|−1 + C N 2|α|−1 + C U s (||N ||2s + ||∇u||2s ).
Choosing 0 < ε < 14 , then (3.9) follows from (3.11)-(3.13), (3.19) and (3.30).
2
Now, from Proposition 3.1 and Lemma 3.2, we obtain Proposition 3.2. For all t ∈ [0, T ], α ∈ N3 with 1 ≤ |α| ≤ s, it holds
2
d A0 ∂ β U, ∂ β U + ∂ β ∇Φ + ∇u 2|α| dt β≤α
2 2 (3.31) ≤C ∇u |α|−1 + N |α|−1 + C U s (||N ||2s + ||∇u||2s ). Proof. Summing (3.5) and (3.9) (in which α = β) for all indexes with |β| ≤ |α| yields (3.31). 2 3.2. Dissipation estimate for N . Estimate (3.31) contains a recurrence relation on the time dissipation of ∇u. It is clear that this estimate is not sufficient to control the higher order term in (3.31) and the dissipation estimates of N is necessary. Proposition 3.3. For all t ∈ [0, T ], α ∈ N3 with 1 ≤ |α| ≤ s, it holds d β β ∂ u, ∂ ∇N +C0 N 2|α| ≤ C ∇u 2|α| +C N 2|α|−1 +C U s (||N ||2s + ||∇u||2s ), (3.32) dt |β|<|α|
(3.33) and
d u, ∇ (h (¯ n) N ) + C0 N 21 ≤ C ∇u 21 + C U s (||N ||2s + ||∇u||2s ), dt d A0 ∂ α U, ∂ α U + ∂ α ∇Φ 2 + ∇u 21 dt |α|≤1
(3.34)
≤ ε0 N 21 + C ∇u 2 +C U s (||N ||2s + ||∇u||2s ),
where C0 is a positive constant independent of any time and the positive constant ε0 > 0 is determined in the next section. Proof. For β ∈ N3 with |β| ≤ |α| − 1, applying ∂ β to the second equation in (2.3) gives 1 β β β Δu − ∂t ∂ β u − I6β , (3.35) n) N ) − ∂ ∇Φ = ∂ ∂ ∇ (h (¯ n ¯ where N Δu I6β = ∂ β (u · ∇) u + (h (¯ . n + N ) − h (¯ n)) ∇N + r + n¯ n By Lemma 2.1, it is easy to see that
β
I6 ≤ C U s (||N ||s + ||∇u||s ). Let
I7β = ∂ β ∇ (h (¯ n) N ) − h (¯ n) ∂ β ∇N.
Since the highest derivatives order of N in I7β is |β|, it follows that
β
I7 ≤ C N |β| .
Stability of steady-state solutions to NSP systems
11
Taking the inner product of (3.35) with ∂ β ∇N in L2 , using integration by parts and Lemma 2.1 yields
2
n)∂ β ∇N − ∂ β ∇Φ, ∂ β ∇N
h (¯
(3.36)
1 = ∂ β Δu , ∂ β ∇N − ∂t ∂ β u, ∂ β ∇N − I6β + I7β , ∂ β ∇N n ¯
β
2 ≤δ ∂ ∇N + C N 2|β| + C ∇u 2|α| − ∂t ∂ β u, ∂ β ∇N + C U s (||N ||2s + ||∇u||2s ),
where δ > 0 is small enough. In view of the density equation ∂t N = −∇ · (nu), for the fourth term on the right-hand side of (3.36), we have − ∂t ∂ β u, ∂ β ∇N d β ∂ u, ∂ β ∇N + ∂ β ∇ · u, ∂ β ∇ · (¯ nu) + ∂ β ∇ · u, ∂ β ∇ · (N u) =− (3.37) dt d β ∂ u, ∂ β ∇N + C ∇u 2|β| + C U s (||N ||2s + ||∇u||2s ). ≤− dt On the other hand, the Poisson equation ∂ β ΔΦ = ∂ β N gives
2
(3.38) − ∂ β ∇Φ, ∂ β ∇N = ∂ β N . Then, combining (3.36)-(3.38), and taking 0 < δ <
h0 , 2
we get
h0 β d β
∂ ∇N 2 + ∂ β N 2 ∂ u, ∂ β ∇N + dt 2 ≤C ∇u 2|α| + C N 2|β| + C U s (||N ||2s + ||∇u||2s ).
(3.39) Remark that
0≤|β|≤|α|−1
h0
∂ β ∇N 2 + ∂ β N 2 2
N 2|α| .
is equivalent to Summing (3.39) for all indexes β with |β| ≤ |α| − 1 yields (3.32). Next, we estimate (3.33). Taking the inner product of (3.35) (in which β = 0) with ∇(h (¯ n)N ) in L2 yields 2
(3.40)
∇ (h (¯ n) N ) − ∇Φ, ∇ (h (¯ n) N ) 1 = Δu, ∇ (h (¯ n) N ) − ∂t u, ∇ (h (¯ n) N ) − I60 , ∇ (h (¯ n) N ) n ¯ 1 d 2 ≤ ∇ (h (¯ n) N ) + C ∇u 21 − u, ∇ (h (¯ n) N ) + u, ∇∂t (h (¯ n) N ) 2 dt + C U s (||N ||2s + ||∇u||2s ).
From the density equation ∂t N = − div(nu) and the Poisson equation ΔΦ = N , it gives − ∇Φ, ∇ (h (¯ (3.41) n) N ) = h (¯ n)N 2 , and (3.42)
n) N ) = ∇ · u, h (¯ n)∇ · (nu) u, ∇∂t (h (¯ ≤C ∇u 2 + C U s (||N ||2s + ||∇u||2s ).
12
Y.H. Feng and C.M. Liu
From the above inequalities, we get d 1 2 u, ∇ (h (¯ n) N ) + ∇ (h (¯ n) N ) + h (¯ n)N 2 dt 2 (3.43) ≤C ∇u 21 + C U s (||N ||2s + ||∇u||2s ). Noting h (n) ≥ h0 > 0, we obtain (3.33). Finally, we estimate (3.34), which is a different version of (3.31) for |α| = 1. Following the αβ αβ αβ steps of proof of Lemma 3.2, we only need to re-estimate the terms of I42 , I52 , I53 and the two terms on the right-hand side of (3.27). Indeed, it is easy to get ε αβ 0 2 2 (3.44) I42 ≤ N + C ∇u + C U s (||N ||2s + ||∇u||2s ), 3 ε αβ αβ 0 2 2 (3.45) I52 + I53 ≤ ∇N + C ∇u + C U s (||N ||2s + ||∇u||2s ), 3 and ε0 | h (n) ∂ α−β (∇¯ (3.46) n) ∂ β u, ∂ α N | ≤ ∇N 2 + C ∇u 2 , 6 α−β ε0 β α n∂ (3.47) (h (n)) ∂ N, ∂ u ≤ ∇N 2 + C ∇u 2 + C U s ( N 2s + ∇u 2s ). 6 Then, by (3.14)-(3.16) and (3.44) we get d 3 I4α + ∂ α ∇φ 2 + ∂ α ∇u 2 ≤ C ∇u 2 +ε0 N 2 +C U s (||N ||2s +||∇u||2s ). dt 2 On the other hand, by (3.20)-(3.23), (3.45), (3.27) and (3.46)-(3.47), we obtain 1 (3.49) |I5α | ≤ ∂ α ∇u 2 + C ∇u 2 + ε0 ∇N 2 + C U s (||N ||2s + ||∇u||2s ). 2 Then, by (3.11)-(3.13), (3.48) and (3.49), we have d A0 ∂ α U, ∂ α U + ∂ α ∇Φ 2 + ∂ α∇u 2 dt (3.50) ≤ε0 N 21 + C ∇u 2 + C U s (||N ||2s + ||∇u||2s ). (3.48)
Therefore, summing (3.5) and (3.50) for all indexes α with |α| = 1 yields (3.34).
2
Proposition 3.4. There exists small positive constants π and C1 such that, for all t ∈ [0, T ], α ∈ N3 with 1 ≤ |α| ≤ s, it holds ⎛ ⎞
d ⎝ 2 ∂ γ u, ∂ γ ∇N ⎠ A0 ∂ β U, ∂ β U + ∂ β ∇Φ + π dt β≤α |γ|≤|α|−1
(3.51) + C1 N 2|α| + ∇u 2|α|
≤C ∇u 2|α|−1 + N 2|α|−1 + C U s (||N ||2s + ||∇u||2s ). Proof. Choose π > 0 sufficiently small such that 1 . π≤ 2C Let πC0 1 − Cπ , , C1 = min 2 2
Stability of steady-state solutions to NSP systems
13
where C0 is given by Proposition 3.2. The summation of (3.31) and (3.32) multiplying π, and then summing the resulting equation yields (3.51). 2 4. Proof of Theorem 1.1 We carry on the induction on |α|(1 ≤ |α| ≤ s) of space derivatives for (3.51) and (3.33)(3.34). The step of the induction is increasing from |α| = 1 to |α| = s. More precisely, for |α| = 1, we first multiplies π on both sides of (3.33). Remark that the term Cπ ∇u 21 can be controlled by ∇u 21 on the left-hand side of (3.34), and the term ε0 N 21 on the right-hand side of (3.34) can be controlled by N 21 when ε0 > 0 is sufficiently small. Thus, there exists a positive constant a1 such that
d n) N ) + (N, ∇u) 21 A0 ∂ αU, ∂ αU + ∂ α ∇Φ 2 + π u, ∇ (h (¯ a1 dt (4.1) |α|≤1
≤C ∇u 2 + C U s (||N ||2s + ||∇u||2s ). In the same way, for |α| ≥ 2, C( ∇u 2|α|−1 + N 2|α|−1 ) on the right-hand side of (3.51) can be controlled by ∇u 2|α| + N 2|α| in the preceding step on the left-hand side of (3.51) multiplying an appropriate large positive constant. So, we have s d ak ∂ γ u, ∂ γ ∇N + (||N ||2s + ||∇u||2s ) A0 ∂ α U, ∂ α U + ∂ α ∇Φ 2 + π dt k=1 |α|≤k
|γ|≤|α|−1
(4.2) ≤ C ∇u 2 + C U s (||N ||2s + ||∇u||2s ), where ak > 0(k = 1, · · · , s) are some constants. By use of formulas (3.5) and (4.2) and noting
U is small, we obtain (4.3) s d ak A0 ∂ α U, ∂ α U + ∂ α ∇Φ 2 + π dt k=1 |α|≤k
∂ γ u, ∂ γ ∇N
+ (||N ||2s + ||∇u||2s ) ≤ 0,
|γ|≤|α|−1
where the constant ak > 0(k = 1, · · · , s) may be amended again. When π > 0 is small enough, s ak ∂ γ u, ∂ γ ∇N A0 ∂ α U, ∂ α U + ∂ α ∇Φ 2 + π k=1
U 2s
|α|≤k
|γ|≤|α|−1
∇Φ 2s .
is equivalent to + Integrating (4.3) on from 0 to t, we get (1.9). Moreover, (1.9) implies that, for all |β| ≤ s − 1 and |γ| ≤ s − 3, ∂ β N ∈ L∞ R+ , L2 , ∂ γ ∇u ∈ L∞ R+ , L2 , and
∂t ∂ β N ∈ L∞ R+ , L2 ,
which implies (1.10)-(1.11).
∂t ∂ γ ∇u ∈ L∞ R+ , L2 , 2
Acknowledgments : The authors would like to thank Professor Yue-Jun.Peng for his precious suggestions and fruitful discussions. This research was supported in part by the the BNSF (1164010, 1132006), NSFC (11671295,11401421,11371042), the general project of scientific research project of the Beijing education committee of China, NSF of Qinghai Province, the 2016
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Y.H. Feng and C.M. Liu
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