Hindawi Publishing Corporation
International Journal of Differential Equations
Volume 2011, Article ID 329014, 19 pages
doi:10.1155/2011/329014
Research Article
Global Existence and Asymptotic Behavior of
Self-Similar Solutions for the
Navier-Stokes-Nernst-Planck-Poisson System in R3
Jihong Zhao,1 Chao Deng,2 and Shangbin Cui3
1
Institute of Applied Mathematics, College of Science, Northwest A&F University, Shaanxi,
Yangling 712100, China
2
Department of Mathematics, Xuzhou Normal University, Jiangsu, Xuzhou 221009, China
3
Department of Mathematics, Sun Yat-sen University, Guangdong, Guangzhou 510275, China
Correspondence should be addressed to Jihong Zhao, zhaojihong2007@yahoo.com.cn
Received 5 May 2011; Accepted 5 September 2011
Academic Editor: Chérif Amrouche
Copyright q 2011 Jihong Zhao et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We study the Navier-Stokes-Nernst-Planck-Poisson system modeling the flow of electrohydrodynamics. For small initial data, the global existence, uniqueness, and asymptotic stability as time
goes to infinity of self-similar solutions to the Cauchy problem of this system posed in the whole
three dimensional space are proved in the function spaces of pseudomeasure type.
1. Introduction
In this paper, we consider the Cauchy problem for the normalized Navier-Stokes-NernstPlanck-Poisson system which governs the hydrodynamic transport of binary diffusion charge
densities as follows see 1:
∂t u − Δu u · ∇u ∇p Δφ∇φ in R3 × 0, ∞,
1.1
∇ · u 0 in R3 × 0, ∞,
∂t v u · ∇v ∇ · ∇v − v∇φ
in R3 × 0, ∞,
∂t w u · ∇w ∇ · ∇w w∇φ
in R3 × 0, ∞,
1.2
Δφ v − w
ux, 0 u0 x,
in R3 × 0, ∞,
vx, 0 v0 x,
wx, 0 w0 x
1.3
1.4
1.5
in R3 .
1.6
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Equations 1.1 and 1.2 are the momentum conservation and the mass conservation equations of incompressible flow. u ux, t u1 x, t, u2 x, t, u3 x, t ∈ R3 , p px, t ∈ R
and φ φx, t ∈ R denote, respectively, the velocity field, the pressure of the fluid, and
the electrostatic potential, and the right-hand side term in 1.1 is the Lorentz force caused by
the charged particles. Equations 1.3 and 1.4 model the balance between diffusion and convective transport of charge densities by the flow and the electric fields. v vx, t ∈ R and
w wx, t ∈ R denote the charge densities of the negatively and positively charged species,
respectively, hence the sign difference in front of the convective term in either equation. Equation 1.5 is the Poisson equation for the electrostatic potential φ, and the right-hand side is the
net charge density. For simplicity, we have chosen the fluid density, viscosity, charge mobility and dielectric constant to be unit.
To start with, let us recall two special cases of 1.1–1.6. In the case that the flow is
charge free, that is, v w φ 0, the system 1.1–1.6 reduces into the well-known NavierStokes equations:
∂t u − Δu u · ∇u ∇P 0
in R3 × 0, ∞,
1.7
∇ · u 0 in R3 × 0, ∞,
ux, 0 u0 x
in R3 .
After the pioneering work 2, the Navier-Stokes equations 1.7 has drawn great attention of
researchers for many years and a huge number of works can be found from the literature, for
example, 3–8 and the references therein. If, on the other hand, the velocity field u is identically vanishing, then 1.1–1.6 reduces into the following Nernst-Planck-Poisson equations
which was formulated by W. Nernst and M. Planck at the end of the nineteenth century as a
basic model for the diffusion of ions in an electrolytes cf. 9:
in R3 × 0, ∞,
∂t v ∇ · ∇v − v∇φ
in R3 × 0, ∞,
∂t w ∇ · ∇w w∇φ
Δφ v − w
vx, 0 v0 x,
in R3 × 0, ∞,
wx, 0 w0 x
1.8
in R3 .
In some literatures, it is also called the Debye-Hückel system cf. 10. It has drawn much
attention of analysts during the past twenty years, and some works concerning existence of
large weak solutions, small mild solutions, convergence rate estimates to stationary solutions of time-dependent solutions and other related topics can be found from the literature,
cf., for example, 10–16 and the references therein.
In 2002, Jerome 17 proved that the system 1.1–1.6 has a unique local smooth solution for smooth initial data where he verified the local existence in Kato’s semigroup framework. In 18, by using the energy inequalities and the Schauder fixed point theorem,
Schmuck established global existence of weak solutions to the system 1.1–1.6 in a bounded domain Ω with homogeneous Neumann boundary conditions with initial data u0 ∈
L2 Ωn and v0 , w0 ∈ L∞ Ω for n 2, 3. In 19, Ryham studied existence, uniqueness, and
regularity of weak solutions of 1.1–1.6 in a bounded domain with no-flux boundary
conditions for general L2 initial data in n 2 and for small initial data in n 3. The convergence to the stationary solution with a rate is also established in 19. In our recent work 20,
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3
by using the Lp -Lq estimates of the heat semigroup and the classical Hardy-Littlewood-Sobolev inequality, we established local well-posedness of 1.1–1.6 in critical and subcritical
Lebesgue spaces i.e., u0 ∈ Lq Rn n , and v0 , w0 ∈ Lp Rn , n ≤ q < ∞, n/2 ≤ p < n and global
well-posedness for small initial data in critical Lebesgue spaces i.e., u0 ∈ Ln Rn n , v0 , w0 ∈
Ln/2 Rn and u0 Ln n v0 Ln/2 w0 Ln/2 is sufficiently small. For computational simulations
of the problem 1.1–1.6, see 21–23.
The most important results stated in this paper are theorems on the global existence,
uniqueness and asymptotic stability as time goes to infinity of self-similar solutions to the system 1.1–1.6 in the functional spaces of pseudomeasure-type. Let us recall that the solution
u, p, v, w, φ of the system 1.1–1.6 is called a self-similar solution if it satisfies the following scaling invariant property: ut, x uλ x, t, pt, x pλ x, t, vt, x vλ x, t, wt, x wλ x, t and φt, x φλ x, t for all λ > 0, x ∈ R3 , and t ≥ 0, where
uλ x, t λu λx, λ2 t ,
pλ x, t λ2 p λx, λ2 t ,
vλ x, t λ2 v λx, λ2 t ,
wλ x, t λ2 w λx, λ2 t ,
φλ x, t φ λx, λ2 t .
1.9
It is clear that if u, p, v, w, φ is a solution of 1.1–1.5 with initial data 1.6, then, for each
λ > 0, uλ , pλ , vλ , wλ , φλ also solves 1.1–1.5 with initial data
u0,λ x λu0 λx,
v0,λ x λ2 v0 λx,
w0,λ x λ2 w0 λx.
1.10
Apparently such initial data do not belong to any Lebesgue and Sobolev spaces due to their
strong singularity at x 0 as well as slow decay as |x| → ∞. In 24, the authors found explicit formulas for a one-parameter family of stationary solutions of the three-dimensional
Navier-Stokes equations with zero external force; these solutions are global but not smooth,
more precisely, they are singular at the origin with a singularity of the kind ∼1/|x| for all time.
Note that 1/|x| ∈
/ L3 R3 but 1/|x| ∈ PM2 see 1.16 below for definition of this functional
space. Similar phenomenon also appeared for the Nernst-Planck-Poisson equations; see 25.
This is the reason why we consider the system 1.1–1.6 in the pseudomeasure-type spaces.
By a standard contraction argument, we establish global existence of solution for small initial
data. It is worth pointing out that this solution is unique in a ball of the functional spaces
in which the existence of solutions is going to be obtained. To overcome this restrictive
condition, we establish a stability result in terms of a perturbation of initial data, which allow
us to give a complete answer to the uniqueness problem of solution. Moreover, we establish
the asymptotic stability of self-similar solutions as time goes to infinity. Here, we refer the
reader to see 26–28 and the references cited there for more details related to the NavierStokes equations with measures as initial data.
The self-similar solution is related to an asymptotic behavior, for large time, of global
solution to the system 1.1–1.6, and we could characterize the self-similar condition in
the following way. Here, we disregard the functions p and φ because when u, v, and w are
determined, p and φ can be easily obtained from 1.2 and 1.5. A vector function u, v, w
has the self-similar property to the system 1.1–1.6
if and only if √
there exists a vector
√ √
function
√ U, V, W such that ux, t Ux/ t/ t, vx, t V x/ t/t and wx, t Wx/ t/t for all x ∈ R3 and t > 0. In fact, when U, V, W exists, these last equalities give the
definition for u, v, w, and it is straightforward to see that it is self-similar. Conversely, when
the self-similar solution u, v, w is given, we define Ux ux, 1, V x vx, 1 and
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Wx wx, 1 for all x ∈ R3 . Then, the self-similar condition on u,
√ v, w turns out the expected equality between u, v, w and U, V, W by choosing λ 1/ t.
Now, as a standard practice, we can reformulate the problem 1.1–1.6 into a system
of integral equations. To this end, we first solve 1.5 to get φ as a functional of w − v:
φx −Δ−1 w − vx F−1 |ξ|−2 Fwξ − Fvξ x
1.11
in the distributional sense, where F and F−1 denote the Fourier transform and the inverse
Fourier transform, respectively. Next, it is convenient to eliminate the pressure p by applying
the Leray projector P to both sides of 1.1, by 1.11, 1.1 and 1.2 can be transformed into
the following equations:
∂u
− Δu Pu · ∇u Pv − w∇ −Δ−1 w − v .
∂t
1.12
Recalling that P is given formally by the formula P I ∇−Δ−1 div; that is, P is the 3 × 3
matrix pseudo-differential operator in R3 with the symbol FPξ δjk − ξj ξk /|ξ|2 3j,k1 ,
where I denotes the unit operator and δjk is the Kronecker symbol. It is obvious that all these
components are bounded, that is,
sup |FPξ| < ∞.
ξ∈R3 \{0}
1.13
Finally, by the well-known Duhamel principle, we see that the problem 1.1–1.6 can be further reduced into the following system of integral equations:
u e u0 tΔ
t
et−τΔ G1 uτ, vτ, wτdτ,
0
v etΔ v0 t
et−τΔ G2 uτ, vτ, wτdτ,
1.14
0
w e w0 tΔ
t
et−τΔ G3 uτ, vτ, wτdτ,
0
where etΔ is the heat operator which can be regarded as the convolution with the heat kernel
Gx, t 4πt−3/2 exp−|x|2 /4t, and
G1 u, v, w −P∇ · u ⊗ u Pv − w∇ −Δ−1 w − v ,
G2 u, v, w −∇ · uv − ∇ · v∇−Δ−1 w − v ,
G3 u, v, w −∇ · uw ∇ · w∇−Δ−1 w − v .
Later on we will work on this system of integral equations instead of 1.1–1.6.
1.15
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Before giving the explicit meaning of solutions to the system 1.1–1.6, we define the
functional spaces relevant to the existence of solutions of 1.1–1.6. Let us first define
PM a
f ∈S R
3
: Ff ∈
L1loc
a
R , f PMa ess sup|ξ| Ffξ < ∞ ,
3
1.16
ξ∈R3
where a ≥ 0 is a given parameter. Since PMa is not separable and the heat semigroup {etΔ }t≥0
is not strongly continuous on this space but only weakly continuous, we will use the notation Cw to denote by the space of functions which are weakly continuous as distributions with
respect to t. Next, we construct the solution u, v, w of the system 1.1–1.6 with the velocity u in the space
3 X Cw 0, ∞, PM2
1.17
equipped with the norm uX supt≥0 utPM2 3 , and the components v and w in the space
Ya v ∈ Cw 0, ∞, PM
1
: sup t
a−1/2
t>0
vtPMa < ∞
1.18
equipped with norm vYa supt≥0 vtPM1 supt>0 ta−1/2 vtPMa , where a is a given
parameter satisfying 1 < a < 2 in the whole paper. For simplicity, we denote · Ya ,1 and
· Ya ,2 for each part in the norm of Ya , and the product of Banach spaces X × Y × Z will be
equipped with the norm u, v, wX×Y×Z max{uX , vY , wZ }.
Remark 1.1. Let f ∈ S R3 ∩L1loc R3 ; for a positive parameter λ, we denote fλ x fλx. It is
easy to verify that, for each λ > 0,
F fλ ξ λ−3 F f λ−1 ξ ,
fλ·
PMa
λa−3 f PMa .
1.19
Hence, the norms in X and Ya are invariant under the scaling 1.9.
Definition 1.2. The solution of 1.1–1.6 is a vector function u, v, w with components satisfying, for some 0 < T ≤ ∞,
3 ,
u ∈ Cw 0, T , PM2
v, w ∈ Cw 0, T , PM1 ,
and the following equalities hold for all 0 ≤ t < T :
2
Fuξ, t e−t|ξ| Fu0 ξ −
t
0
t
2
e−t−τ|ξ| FPξ iξ · Fu ⊗ uξ, τdτ
0
2
e−t−τ|ξ| FPξF v − w∇ −Δ−1 w − v ξ, τdτ,
1.20
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2
Fvξ, t e−t|ξ| Fv0 ξ −
−
t
t
0
2
e−t−τ|ξ| iξ · F v∇ −Δ−1 w − v ξ, τdτ,
0
2
Fwξ, t e−t|ξ| Fw0 ξ −
t
2
e−t−τ|ξ| iξ · Fuvξ, τdτ
t
2
e−t−τ|ξ| iξ · Fuwξ, τdτ
0
2
e−t−τ|ξ| iξ · F w∇ −Δ−1 w − v ξ, τdτ.
0
1.21
Now, we state the main results of this paper as follows.
Theorem 1.3. Let u0 ∈ PM2 3 , ∇·u0 0, and v0 , w0 ∈ PM1 . There exists a constant ε such that if
u0 , v0 , w0 PM2 3 ×PM1 2 ≤ ε, then the system 1.1–1.6 has a global solution u, v, w in the
space X × Ya 2 , and this is the unique solution under the condition
u, v, wX×Ya 2 ≤ 2ηε,
1.22
where 0 < ε < 1/4η2 and the constant η is defined by 2.23. Moreover, the solution depends con 0 PM2 3 ×PM1 2 ≤ ε, then one denotes by
tinuously on initial data in the following sense: if u0 , v0 , w
0 , and u, v , w
X×Ya 2 ≤
u, v , w
the unique solution of 1.1–1.6 with initial data u0 , v0 , w
2ηε, then, one has
0 , v0 − v0 , w0 − w
, v − v , w − w
0 PM2 3 ×PM1 2 ,
1.23
X×Ya 2 ≤ C ε, η u0 − u
u − u
where Cε, η η/1 − 4εη2 .
Remark 1.4. In Theorem 1.3, we obtained only a partial answer to the uniqueness problem of
solution; that is, under the restrictive condition 1.22, the solution of 1.1–1.6 is unique.
For a complete answer to this problem, see Corollary 1.8 below.
Based on the uniqueness of solution in Theorem 1.3, by a standard way, we can deduce
the existence of self-similar solution to the system 1.1–1.6.
Corollary 1.5. Assume that u0 , v0 , and w0 satisfy the assumptions of Theorem 1.3. Assume that,
moreover,
u0 x λu0 λx,
v0 x λ2 v0 λx,
w0 x λ2 w0 λx.
1.24
Then, the solution u, v, w constructed in Theorem 1.3 is a self-similar solution.
In order to give a complete answer to the uniqueness problem of solutions to the system 1.1–1.6, we will establish the following stability result.
0 , v0 −
u0 , v0 , w
0 belong to PM2 3 ×PM1 2 such that u0 − u
Theorem 1.6. Let u0 , v0 , w0 and 2
0 PM2 3 ×PM1 2 < 1/8η , where η is a constant defined by 2.23, and let u, v, w and
v0 , w0 − w
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u0 , v0 , w
0 , respec
u, v , w
be two solutions of 1.1–1.6 with initial conditions u0 , v0 , w0 and tively. Then, one has
0 , v0 − v0 , w0 − w
, v − v , w − w
0 PM2 3 ×PM1 2 .
X×Ya 2 ≤ 2ηu0 − u
u − u
1.25
Remark 1.7. Theorem 1.6 implies that we can measure the difference of two solutions in terms
of the difference of their initial data provided that the difference between these initial data is
small enough.
The direct consequence of Theorem 1.6 is the following corollary.
Corollary 1.8. Assume that u0 ∈ PM2 3 , ∇ · u0 0 in the distributional sense, and v0 , w0 ∈ PM1 .
Then, there exists at most one solution of 1.1–1.6 with initial data u0 , v0 , w0 .
Finally, we study the asymptotic stability of solutions in the sense proposed in 13
and developed in 27.
Theorem 1.9. Let ε be a sufficiently small number such that ε < min{1/4η2 , 1/η},
where η and
u0 , v0 , w
0 satisfy the
η are defined by 2.23 and 4.19, respectively. Assume that u0 , v0 , w0 and assumptions of Theorem 1.3, and let u, v, w and u, v , w
be two global solutions of 1.1–1.6
u0 , v0 , w
0 , respectively. Then, the following two conditions
with initial conditions u0 , v0 , w0 and are equivalent:
0 , v0 − v0 , w0 − w
0 lim etΔ u0 − u
t→∞
3
PM2 ×PM1 ×etΔ v0 − v0 , w0 − w
0 PMa 2
2
ta−1/2
0,
, v − v , w − w
PMa 2 0.
PM2 3 ×PM1 2 ta−1/2 v − v , w − w
lim u − u
t→∞
1.26
1.27
Remark 1.10. As an interesting application of Theorem 1.9, we get the asymptotic stability
of self-similar solutions to the system 1.1–1.6, namely, under the assumptions of
Theorem 1.9, besides, assume that u0 , v0 , w0 satisfies 1.24. Then, we know that the mild
solution u, v , w
tends to the self-similar solution u, v, w as time goes to infinity as long as
0 satisfies the condition 1.26.
u0 , v0 , w
Notations
Let A and B be two real numbers; we denote A B if there is a universal constant C, which
does not depend on varying parameters of the problem, such that A ≤ CB. We denote A ∼ B
if A B and B A. In the rest part of this paper, we will use “sup” instead of “ess sup” for
convenience.
Structure of the Paper
In Section 2, we prove Theorem 1.3 and Corollary 1.5. The purpose of Section 3 is to prove
Theorem 1.6. In the last section, we present the proof of Theorem 1.9.
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2. Global-in-Time Solutions
In this section, we give the proofs of Theorem 1.3 and Corollary 1.5. Thus, throughout this
section, we assume that u0 ∈ PM2 3 , ∇ · u0 0, and v0 , w0 ∈ PM1 .
2.1. The Proof of Theorem 1.3
Let a be a fixed number such that 1 < a < 2, and let X X × Ya 2 . Given u, v, w ∈ X, we
define Fu, v, w u, v, w, where
t
2.1
u etΔ u0 et−τΔ G1 uτ, vτ, wτdτ,
0
v etΔ v0 t
et−τΔ G2 uτ, vτ, wτdτ,
2.2
0
w e w0 tΔ
t
et−τΔ G3 uτ, vτ, wτdτ.
2.3
0
We fulfil the proof of Theorem 1.3 through the following two lemmas.
Lemma 2.1. The map F is well defined and maps X into itself.
Proof. Note that although some parts of the proof were given in 27, we would rather give it
for completeness. We first prove that u is well defined and u ∈ X. From 2.1, we can denote
u u1 u2 u3 , where
u1 t etΔ u0 ,
u2 t u3 t t
t
et−τΔ P−∇ · u ⊗ uτdτ,
0
2.4
et−τΔ P v − w∇ −Δ−1 w − v τdτ.
0
For u1 , since u0 ∈ PM2 3 , it is easy to see that
2
tΔ e u0 sup sup|ξ|2 e−t|ξ| Fu0 ξ ≤ u0 PM2 3 .
X
2.5
t≥0 ξ∈R3
Thus, etΔ u0 ∈ L∞ 0, ∞, PM2 3 . To prove the weak continuity of u1 with respect to t, due to
the properties of heat semigroup etΔ , it suffices to prove this for t 0. For every ϕ ∈ SR3 3 ,
by applying the Plancherel formula, we obtain
2
tΔ
e−t|ξ| − 1 Fu0 ξF ϕ ξdξ
e u0 − u0 , ϕ R3
−t|ξ|2
e
−1 ≤ t sup u0 PM2 3 F ϕ L1 R3 3 −→ 0 as t −→ 0.
2
ξ∈R3 t|ξ|
2.6
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This implies that
u1 ∈ X,
u1 X ≤ u0 PM2 3 .
2.7
For u2 , using 1.13 and the properties of the Fourier transform, we get
u2 PM2 3
t
−t−τ|ξ|2
sup|ξ| e
FPξiξ · Fu ⊗ uξ, τdτ 3
0
ξ∈R
t
2
sup |ξ|3 e−t−τ|ξ| |ξ|−2 ∗ |ξ|−2 dτu2X
2
ξ∈R3
sup
0
t
0
ξ∈R3
2.8
2
|ξ|2 e−t−τ|ξ| dτu2X
u2X .
Here, we have used the fact that |ξ|−2 ∗ |ξ|−2 ∼ |ξ|−1 see 29, Chapter 5, Section 1, 8. For
the weak continuity of u2 with respect to t, we can prove it by a standard argument cf. 30.
Hence,
u2 ∈ X,
u2 X u2X .
2.9
For u3 , using similar calculations,
u3 PM2 3
t
−1
−t−τ|ξ|2
sup|ξ| e
FPξF v − w∇ −Δ w − v ξ, τdτ 3
0
ξ∈R
t
2
sup |ξ|2 e−t−τ|ξ| |ξ|−a ∗ |ξ|−a−1 τ −a−1 dτv, w2Ya ,2
2
ξ∈R3
sup
ξ∈R3
sup
ξ∈R3
0
t
0
2
|ξ|4−2a e−t−τ|ξ| τ −a−1 dτv, w2Ya ,2
t t − τ|ξ|2
2−a
0
2.10
e−t−τ|ξ| t − τ−2−a τ −a−1 dτv, w2Ya ,2
2
v, w2Ya ,2 .
Here, we have used the fact |ξ|−a ∗ |ξ|−a−1 ∼ |ξ|2−2a and the assumption 1 < a < 2 to ensure that
t
the integral 0 t − τ−2−a τ −a−1 dτ is finite and independent of t. It remains to show the weak
continuity of u3 , but this is a standard argument as we mentioned before. Hence,
u3 ∈ X,
u3 X v, w2Ya ,2 .
2.11
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Combining the above estimates 2.7–2.11, we see that
uX u0 PM2 3 u2X v, w2Ya ,2 .
u ∈ X,
2.12
Next, we prove that v is well defined and v ∈ Ya . Note that, from 2.2, we can denote
v v1 v2 v3 , where
v1 t etΔ v0 ,
t
v2 t et−τΔ −∇ · uvτdτ,
2.13
0
t
v3 t et−τΔ −∇ · v∇ −Δ−1 w − v
τdτ.
0
Since v0 ∈ PM1 , as in the proof of u1 , it can be easily seen that
v1 ∈ Ya ,
v1 Ya v0 PM1 .
2.14
Indeed, it suffices to estimate the second term in the norm of Ya as follows:
2
sup ta−1/2 v1 PMa sup sup ta−1/2 |ξ|a e−t|ξ| |Fv0 ξ|
t>0
t>0 ξ∈R3
2.15
a−1/2
2
≤ sup sup t|ξ|2
e−t|ξ| |ξ||Fv0 ξ| v0 PM1 .
t>0 ξ∈R3
For v2 , we can do the same calculations to deal with the first term in the norm of Ya , and one
obtains that
v2 Ya ,1 uX vYa ,2 .
2.16
To deal with the second term in the norm of Ya , we need to calculate more. Note first the
following two elementary inequalities:
|ξ|
2−a
t/2
2
2
e−t−τ|ξ| τ −a−1/2 dτ ≤ |ξ|2−a e−t|ξ| /2
0
|ξ|
2−a
t/2
τ −a−1/2 dτ |ξ|−a t−a−1/2 ,
0
t
e
2
−t−τ|ξ|
τ
−a−1/2
dτ |ξ|
2−a −a−1/2
2.17
t
t
t/2
e
−t−τ|ξ|2
−a −a−1/2
dτ |ξ| t
.
t/2
Hence, taking the Fourier transform to v2 , we get
|Fv2 | t
0
2
|ξ|e−t−τ|ξ| |ξ|−2 ∗ |ξ|−a τ −a−1/2 dτuX vYa ,2
uX vYa ,2 |ξ|
−a −a−1/2
|ξ| t
2−a
t/2 t 2
e−t−τ|ξ| τ −a−1/2 dτ
0
uX vYa ,2 .
t/2
2.18
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This implies that v2 Ya ,2 uX vYa ,2 . By a standard argument, we can prove that v2 is
weakly continuous with respect to t. Hence,
v2 ∈ Ya ,
v2 Ya uX vYa ,2 .
2.19
For v3 , we can do the same calculations as for v2 and obtain
v3 ∈ Ya ,
v3 Ya v, w2Ya ,2 .
2.20
Concluding the above estimates 2.14–2.20, we have already proved that
v ∈ Ya ,
vYa v0 PM1 uX v, wYa ,2 v, wYa ,2 .
2.21
Similarly, for w, we can prove that
w ∈ Ya ,
wYa w0 PM1 uX v, wYa ,2 v, wYa ,2 .
2.22
The proof of Lemma 2.1 is complete by 2.12, 2.21, and 2.22.
From Lemma 2.1, there exists a constant η > 0 such that, for any u, v, w ∈ X and
u, v, w Fu, v, w, one has
u, v, wX ≤ η u0 , v0 , w0 PM2 3 ×PM1 2 u, v, w2X .
2.23
Let ε > 0 be sufficiently small so that 4η2 ε < 1. If u0 , v0 , w0 PM2 3 ×PM1 2 ≤ ε, then from
2.23 one has
u, v, wX ≤ ηε ηu, v, w2X .
2.24
Now, let B be a closed ball in X with radius 2ηε, that is,
B u, v, w ∈ X : u, v, wX ≤ 2ηε .
2.25
For any u, v, w ∈ B, from 2.24, we see that
2 u, v, wX ≤ ηε η 2ηε 1 4η2 ε ηε ≤ 2ηε.
2.26
This implies that F maps B into itself.
Lemma 2.2. Let ε be as before ε < 1/4η2 . If u0 , v0 , w0 PM2 3 ×PM1 2 ≤ ε, then F is a contraction
mapping.
12
International Journal of Differential Equations
Proof. Let u1 , v1 , w1 , u2 , v2 , w2 ∈ B, and let uj , vj , wj Fuj , vj , wj , j 1, 2. Then by a
similar argument as in the proof of Lemma 2.1, we obtain the following estimate:
u1 − u2 , v1 − v2 , w1 − w2 X ≤ ηu1 , v1 , w1 X u2 , v2 , w2 X ×u1 − u2 , v1 − v2 , w1 − w2 X
2.27
≤ 4η2 εu1 − u2 , v1 − v2 , w1 − w2 X .
Since 4η2 ε < 1, we see that F is a contraction mapping. By using Lemmas 2.1 and 2.2 and the
Banach fixed point theorem, we know there exists a global solution u, v, w of 1.1–1.6 in
the space X×Ya 2 , and this is the unique solution satisfying the condition u, v, wX×Ya 2 ≤
2ηε. It remains to show that the solution depends continuously on initial data. Let u, v, w
and u, v , w
be two solutions of 1.1–1.6 corresponding to initial conditions u0 , v0 , w0 0 , respectively, and
and u0 , v0 , w
u0 , v0 , w0 PM2 3 ×PM1 2 ≤ ε,
u0 , v0 , w
0 PM2 3 ×PM1 2 ≤ ε.
2.28
Then, proceeding as in 2.27, we get
0 , v0 − v0 , w0 − w
, v − v , w − w
0 PM2 3 ×PM1 2
X×Ya 2 ≤ ηu0 − u
u − u
, v − v , w − w
4η2 εu − u
X.
2.29
Since 4η2 ε < 1, 2.29 yields that
0 , v0 − v0 , w0 − w
, v − v , w − w
0 PM2 3 ×PM1 2 ,
X×Ya 2 ≤ C ε, η u0 − u
u − u
2.30
where Cε, η η/1 − 4εη2 . This proves Theorem 1.3.
2.2. The Proof of Corollary 1.5
Proof. On the one hand, from Theorem 1.3, we know, the system 1.1–1.6 admits a unique
global solution u, v, w with initial data u0 , v0 , w0 . Moreover, u, v, wX×Ya 2 ≤ 2ηε. On
the other hand, since u0 , v0 , and w0 satisfy the condition 1.24, by the scaling invariance of
1.1–1.6, for each λ > 0, the function uλ , vλ , wλ see 1.9 is also a solution with the same
initial data. Note that the norm of X is invariant under the scaling 1.9, that is,
uλ , vλ , wλ X×Ya 2 u, v, wX×Ya 2 .
2.31
Hence, by the uniqueness result of Theorem 1.3, the solution u, v, w of 1.1–1.6 is selfsimilar.
International Journal of Differential Equations
13
3. Stability of Solutions
In this short section, we prove Theorem 1.6. Let us pick up any two solutions u, v, w and
u0 , v0 , w
0 , respectively, lying in
u, v , w
associated with initial conditions u0 , v0 , w0 and PM2 3 × PM1 2 . As in the proof of Lemma 2.1, one has
t
t−τΔ
uτ, v τ, wτdτ
G1 uτ, vτ, wτ − G1 e
0
X
X uX ≤ η u − u
uX v − v , w − w
v, w
Ya
Ya v, wYa 3.1
≤ 2ηM2 ,
, v − v , w − w
u, v , w
X , u − u
X }. Similarly,
where M max{u, v, wX , t
t−τΔ
G2 uτ, vτ, wτ − G2 uτ, v τ, wτdτ
e
≤ 2ηM2 ,
0
Ya
t
uτ, v τ, wτdτ
≤ 2ηM2 .
et−τΔ G3 uτ, vτ, wτ − G3 0
3.2
Ya
Now, we subtract the integral equation 1.14 for u, v , w
from the analogous expression for
u, v, w, using the definition of the norm of X and 3.1-3.2, we obtain
0 , v0 − v0 , w0 − w
, v − v , w − w
0 2ηM2
X ≤ etΔ u0 − u
u − u
X
3.3
0 , v0 − v0 , w0 − w
≤ ηu0 − u
0 PM2 3 ×PM1 2 2ηM2 .
Applying the definition of M, 3.3 yields that
0 , v0 − v0 , w0 − w
0 PM2 3 ×PM1 2 2ηM2 .
M ≤ ηu0 − u
3.4
0 , v0 − v0 , w0 − w
Since we have assumed that u0 − u
0 PM2 3 ×PM1 2 < 1/8η2 , the continuity
argument implies that M ≤ M1 , where M1 is the smallest root corresponding to the following
quadratic equation:
0 , v0 − v0 , w0 − w
0 PM2 3 ×PM1 2 0.
2ηM2 − M ηu0 − u
3.5
From 3.4 we know that this root satisfies
0 , v0 − v0 , w0 − w
0 PM2 3 ×PM1 2 .
M1 ≤ 2ηu0 − u
3.6
Since M ≤ M1 , this last inequality yields that
0 , v0 − v0 , w0 − w
, v − v , w − w
0 PM2 3 ×PM1 2 .
X ≤ 2ηu0 − u
u − u
The proof of Theorem 1.6 is complete.
3.7
14
International Journal of Differential Equations
4. Large Time Behavior of Solutions
We are now in a position to show Theorem 1.9 on the large time behavior of solutions to the
system 1.1–1.6. Let u, v, w and u, v , w
be two solutions of 1.1–1.6 constructed in
u0 , v0 , w
0 , respectively.
Theorem 1.3 which correspond to initial conditions u0 , v0 , w0 and Let us recall that, by Lemma 2.1, there exists a constant 2ηε, the radius of B, such that
u, v, wX ≤ 2ηε,
u, v , w
X ≤ 2ηε.
4.1
Now, to simplify the notations, we introduce the following two auxiliary functions:
0 , v0 − v0 , w0 − w
ht etΔ u0 − u
0 3
PM2 ×PM1 ta−1/2 etΔ v0 − v0 , w0 − w
0 PMa 2
2
,
4.2
, v − v , w − w
lt u − u
2
3
PM2 ×PM1 ta−1/2 v − v , w − w
PMa 2 .
We first assume that 1.26 holds. It follows immediately from Lemma 2.1 that
ht ∈ L∞ 0, ∞,
lim ht 0.
4.3
t→∞
, we can easily get
By calculating the norm · PM2 3 of u − u
PM2 3
u − u
0 ≤ etΔ u0 − u
PM2 t
et−τΔ
0
3
t
t−τΔ
⊗u
dτ e
P∇u ⊗ u − u
0
PM2 3
−1
−1
v − w
∇−Δ w
× P v − w ∇−Δ w − v − − v dτ PM2 3
I0 I1 I2 .
4.4
For I1 , let δ ∈ 0, 1 be a constant to be chosen later, we decompose the integral
δt
t
· · · dτ δt · · · dτ and estimate each term separately:
0
t
0
· · · dτ into
δt t 2
PM2 3 dτ I11 I12 .
uPM2 3 u − u
I1 sup
|ξ|2 e−t−τ|ξ| uPM2 3 ξ∈R3
0
δt
4.5
International Journal of Differential Equations
15
For I11 , we change the variables τ ts and use the following identity:
2
sup |ξ|2 e−1−st|ξ| ξ∈R3
2
1
1
· sup |σ|2 e−|σ| .
1 − st σ∈R3
1 − ste
4.6
Thus, from 4.1, one has
I11 4ηε
δ
0
tsPM2 3 ds.
1 − s−1 uts − u
4.7
For I12 , we can estimate it directly that
τPM2 3 .
I12 4ηε sup uτ − u
4.8
δt≤τ≤t
Hence, it follows immediately from 4.7 and 4.8 that
I1 4ηε
δ
0
−1
tsPM2 3 ds sup uτ − u
τPM2 3
1 − s uts − u
.
4.9
δt≤τ≤t
Now, we compute I2 , by using the same argument, we obtain
δt t 2
I2 sup
|ξ|4−2a e−t−τ|ξ| τ −a−1
ξ∈R3
0
δt
× τ a−1/2 v, wPMa 2 τ a−1/2 v, w
PMa 2
4.10
× τ a−1/2 v − v , w − w
PMa 2 dτ
I21 I22 .
For I21 , we change the variables τ ts, and from 4.1,
I21 4ηε
δ
0
1 − s−2−a s−a−1 tsa−1/2 vts − v ts, wts − wts
PMa 2 ds.
4.11
For I22 , we estimate it directly that
I22 4ηε sup vτ − v τ, wτ − wτ
PMa 2 .
δt≤τ≤t
4.12
16
International Journal of Differential Equations
From 4.11 and 4.12, we get
I2 4ηε
δ
1 − s−2−a s−a−1 tsa−1/2 vts − v ts, wts − wts
PMa 2 ds
0
sup vτ − v τ, wτ − wτ
PMa 2
4.13
.
δt≤τ≤t
Combining the above estimates 4.9 and 4.13, we have already proved that
PM2 3 etΔ u0 − u
0 u − u
×
δ
0
PM2 3
4ηε
tsPM2 3 ds sup uτ − u
τPM2 3
1 − s−1 uts − u
δt≤τ≤t
δ
0
1 − s−2−a s−a−1 tsa−1/2 vts − v ts, wts − wts
PMa 2 ds
sup vτ − v τ, wτ − wτ
PMa 2 .
δt≤τ≤t
4.14
By using the analogous argument above, we can estimate v − v and obtain the following estimate:
v − v PM1 ta−1/2 v − v PMa etΔ v0 − v0 ×
δ
PM1
ta−1/2 etΔ v0 − v0 PMa
4ηε
1 − s−3−a/2 s−a−1/2
0
tsPM2 3
× uts − u
tsa−1/2 vts − v tsPMa ds
δ
1 − s−2−a s−a−1
0
× tsa−1/2 vts − v ts, wts − wts
PMa 2 ds
δ
0
1 − s−1 s−a−1/2
tsPM2 3
× uts − u
International Journal of Differential Equations
17
tsa−1/2 vts − v tsPMa ds
δ
1 − s−3−a/2 s−a−1
0
× tsa−1/2 vts − v ts, wts − wts
PMa 2 ds
τPM2 3
sup uτ − u
δt≤τ≤t
τ
a−1/2
vτ − v τ, wτ − wτ
PMa 2
.
4.15
The estimate of w − w
has exactly the same form as 4.15. Now, let
M lim sup lt t→∞
lim sup lt.
k∈N,k → ∞ t≥k
4.16
In order to prove 1.27, it suffices to prove M 0. Note that from 4.1 we know that M is
nonnegative and finite. Hence, by applying the Lebesgue dominated convergence theorem to
4.14, 4.15, and the same estimate as 4.15 for w − w
and using the assumption 1.26 we
obtain
M 4ηεFδ 1M,
4.17
where Fδ is defined by
δ
1
Fδ log
1 − s−2−a s−a−1 ds
1 − δ
0
δ
1 − s−3−a/2 s−a−1/2 ds
4.18
0
δ
0
1 − s−1 s−a−1/2 ds δ
1 − s−3−a/2 s−a−1 ds.
0
Thus, there exists a universal constant η which may depend on η such that
M ≤ ηεFδ
1M.
4.19
Since we have assumed that εη < 1, we can choose δ sufficiently small such that ηεFδ1
<
1 by the fact that limδ → 0 Fδ 0. This implies M 0 by 4.19. We complete the proof of
1.27.
18
International Journal of Differential Equations
Conversely, we assume that 1.27 holds. Note that from 4.1 one has
lt ∈ L∞ 0, ∞,
lim lt 0.
t→∞
4.20
We need to prove 1.26. Repeat calculations similar to the proofs of 4.14 and 4.15, and,
from the boundedness of u, v, w and u, v , w
in 4.1, we can obtain the following estimate:
ht ≤ ηεFδ
1lt.
4.21
It is obvious that ηεFδ
1 is bounded and independent of t, so 1.26 follows immediately
from 1.27 and 4.21. This proves Theorem 1.9.
Acknowledgments
This work is supported by the China National Natural Science Foundation under the Grant
no. 11171357. The authors would like to thank the anonymous referee for invaluable suggestions.
References
1 I. Rubinstein, Electro-Diffusion of Ions, vol. 11 of SIAM Studies in Applied Mathematics, Society for
Industrial and Applied Mathematics, Philadelphia, Pa, USA, 1990.
2 J. Leray, “Sur le mouvement d’un liquide visqueux emplissant l’espace,” Acta Mathematica, vol. 63,
no. 1, pp. 193–248, 1934.
3 J. Bourgain and N. Pavlović, “Ill-posedness of the Navier-Stokes equations in a critical space in 3D,”
Journal of Functional Analysis, vol. 255, no. 9, pp. 2233–2247, 2008.
4 M. Cannone, “Harmonic analysis tools for solving the incompressible Navier-Stokes equations,”
in Handbook of Mathematical Fluid Dynamics, S. Friedlander and D. Serre, Eds., vol. 3, pp. 161–244,
Elsevier, Amsterdam, The Netherlands, 2004.
5 H. Fujita and T. Kato, “On the Navier-Stokes initial value problem. I,” Archive for Rational Mechanics
and Analysis, vol. 16, pp. 269–315, 1964.
6 T. Kato, “Strong Lp -solutions of the Navier-Stokes equation in Rm , with applications to weak solutions,” Mathematische Zeitschrift, vol. 187, no. 4, pp. 471–480, 1984.
7 H. Koch and D. Tataru, “Well-posedness for the Navier-Stokes equations,” Advances in Mathematics,
vol. 157, no. 1, pp. 22–35, 2001.
8 P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, vol. 431 of Chapman & Hall/
CRC Research Notes in Mathematics, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2002.
9 P. Debye and E. Hückel, “Zur theorie der elektrolyte, II: das grenzgesetz für die elektrische leitfähigkeit,” Physik Zeitschrift, vol. 24, pp. 305–325, 1923.
10 P. Biler, W. Hebisch, and T. Nadzieja, “The Debye system: existence and large time behavior of solutions,” Nonlinear Analysis, vol. 23, no. 9, pp. 1189–1209, 1994.
11 P. Biler and J. Dolbeault, “Long time behavior of solutions of Nernst-Planck and Debye-Hückel driftdiffusion systems,” Annales Henri Poincaré, vol. 1, no. 3, pp. 461–472, 2000.
12 H. Gajewski and K. Gröger, “On the basic equations for carrier transport in semiconductors,” Journal
of Mathematical Analysis and Applications, vol. 113, no. 1, pp. 12–35, 1986.
13 G. Karch, “Scaling in nonlinear parabolic equations,” Journal of Mathematical Analysis and Applications,
vol. 234, no. 2, pp. 534–558, 1999.
14 M. Kurokiba and T. Ogawa, “Well-posedness for the drift-diffusion system in Lp arising from the
semiconductor device simulation,” Journal of Mathematical Analysis and Applications, vol. 342, no. 2,
pp. 1052–1067, 2008.
International Journal of Differential Equations
19
15 T. Ogawa and S. Shimizu, “The drift-diffusion system in two-dimensional critical Hardy space,”
Journal of Functional Analysis, vol. 255, no. 5, pp. 1107–1138, 2008.
16 T. Ogawa and M. Yamamoto, “Asymptotic behavior of solutions to drift-diffusion system with
generalized dissipation,” Mathematical Models & Methods in Applied Sciences, vol. 19, no. 6, pp. 939–
967, 2009.
17 J. W. Jerome, “Analytical approaches to charge transport in a moving medium,” Transport Theory and
Statistical Physics, vol. 31, no. 4-6, pp. 333–366, 2002.
18 M. Schmuck, “Analysis of the Navier-Stokes-Nernst-Planck-Poisson system,” Mathematical Models &
Methods in Applied Sciences, vol. 19, no. 6, pp. 993–1015, 2009.
19 R. J. Ryham, “Existence, uniqueness, regularity and long-term behavior for dissipative systems
modeling electrohydrodynamics,” submitted to Analysis of PDEs, http://arxiv.org/abs/0910.4973v1.
20 J. Zhao, C. Deng, and S. Cui, “Well-posedness of a dissipative system modeling electrohydrodynamics
in Lebesgue spaces,” Differential Equations & Applications, vol. 3, no. 3, pp. 427–448, 2011.
21 M. Longaretti, B. Chini, J. W. Jerome, and R. Sacco, “Electrochemical modeling and characterization
of voltage operated channels in nano-bio-electronics,” Sensor Letters, vol. 6, no. 1, pp. 49–56, 2008.
22 M. Longaretti, B. Chini, J. W. Jerome, and R. Sacco, “Computational modeling and simulation of
complex systems in bio-electronics,” Journal of Computational Electronics, vol. 7, no. 1, pp. 10–13, 2008.
23 M. Longaretti, G. Marino, B. Chini, J. W. Jerome, and R. Sacco, “Computational models in nanobioelectronics: simulation of ionic transport in voltage operated channels,” Journal of Nanoscience and
Nanotechnology, vol. 8, no. 7, pp. 3686–3694, 2008.
24 G. Tian and Z. Xin, “One-point singular solutions to the Navier-Stokes equations,” Topological Methods
in Nonlinear Analysis, vol. 11, no. 1, pp. 135–145, 1998.
25 P. Biler, M. Cannone, I. A. Guerra, and G. Karch, “Global regular and singular solutions for a model
of gravitating particles,” Mathematische Annalen, vol. 330, no. 4, pp. 693–708, 2004.
26 G. Benfatto, R. Esposito, and M. Pulvirenti, “Planar Navier-Stokes flow for singular initial data,”
Nonlinear Analysis, vol. 9, no. 6, pp. 533–545, 1985.
27 M. Cannone and G. Karch, “Smooth or singular solutions to the Navier-Stokes system?” Journal of
Differential Equations, vol. 197, no. 2, pp. 247–274, 2004.
28 Y. Le Jan and A. S. Sznitman, “Stochastic cascades and 3-dimensional Navier-Stokes equations,”
Probability Theory and Related Fields, vol. 109, no. 3, pp. 343–366, 1997.
29 E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series,
Princeton University Press, Princeton, NJ, USA, 1970.
30 M. Yamazaki, “The Navier-Stokes equations in the weak-Ln space with time-dependent external
force,” Mathematische Annalen, vol. 317, no. 4, pp. 635–675, 2000.
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