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Large-time behavior of solutions to a thermo-diffusion system with Smoluchowski interactions. Journal of Differential Equations
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Large-time behavior of solutions to a thermo-diffusion system
with Smoluchowski interactions
Toyohiko Aiki
Department of Mathematical and Physical Sciences, Faculty of Science, Japan Women’s University
2-8-1 Mejirodai, Bunkyo-ku, Tokyo, 112-8681 Japan (aikit@fc.jwu.ac.jp)
Adrian Muntean
Department of Mathematics and Computer Science, Karlstad University SE-651 88 Karlstad, Sweden
(adrian.muntean@kau.se)
Abstract. We prove the large time behavior of solutions to a coupled thermo-diffusion arising in the modelling of the motion of hot colloidal particles in porous media. Addition-ally, we also ensure the uniqueness of solutions of the target problem. The main mathe-matical difficulty is due to the presence in the right-hand side of the equations of products between temperature and concentration gradients. Such terms mimic the so-called ther-modynamic Soret and Dufour effects. These are cross-coupling terms emphasizing in this context a strong interplay between heat conduction and molecular diffusion.
Keywords. Thermo-diffusion; gradient estimates; large-time behavior; Sorret and Du-four effects
MSC 2010: 35Q79; 35K55; 35B45; 35B40
1
Introduction
Populations of colloids can be driven into motion by gradients in chemical, electrostatic, or thermal fields that may exist externally to the colloids; see [4, 6], e.g. This paper is concerned with the mathematical analysis of a scenario involving the joint effect of gradi-ents in chemical and thermal fields that we refer to here as thermo-diffusion. Particularly, we study the large-time behavior of the following class of thermo-diffusion systems – a nonlinear coupled system of partial differential equations with homogeneous Neumann boundary conditions described as follows:
Let Ω ⊂ R3 be a bounded domain with the smooth boundary Γ := ∂Ω, and Q(T ) = (0, T ) × Ω and S(T ) := (0, T ) × Γ for T > 0. The problem is to find a pair of functions
(θ, u), with u = (u1, u2, · · · , uN), satisfying θt− κ∆θ − τ N X i=1 ∇δ0u i · ∇θ = 0 in Q(T ), (1.1)
uit− κi∆ui− τi∇θ · ∇ui = Ri(u) in Q(T ) for each i, (1.2)
− κ∇θ · ν = 0, −κi∇ui· ν = 0 for each i on S(T ), (1.3)
θ(0, x) = θ0(x), ui(0, x) = u0i(x) for each i on Ω, (1.4)
where κ and κi(i = 1, 2 · · · , N ) are the diffusion constants, δ0, τ and τi(i = 1, 2 · · · , N )
are positive constants, ν is the outward normal vector to Γ, and Ri : RN → R is given as
Ri(u) = 1 2 X k+j=i βkju+ku+j − N X j=1 βiju+i u + j for each i,
where r+indicates its positive part for r ∈ R, βkj are positive constants (discrete values of
aggregation and fragmentation kernels) such that βkj = βjk (j, k = 1, 2 · · · , N ). Moreover,
for a given choice of δ > 0, we use following notation:
J (x) = Cmexp(−
1
1−|x|2) if |x| < 1, 0 otherwise,
where Cm is a positive constant chosen such that
R
R3J (x)dx = 1. Here, we put Jδ(x) =
δ−3J (x/δ) and, for a measurable function f on Ω, we employ
(∇δf )(x) = ∇(Jδ∗ f )(x) = ∇(
Z
R3
Jδ(x − y) ˜f (y)dy),
where ˜f = f on Ω and ˜f = 0, otherwise.
We denote by P := P(θ0, u0) the above system (1.1) ∼ (1.4).
The pair (θ, u) refers to the unknowns in the system, i.e. θ is the temperature field, while u = (u1, u2, · · · , uN) is the vector of N interacting colloidal populations. The
reaction term R(·) models the classical Smoluchovski interaction production (see, for instance, chapter 2 in [7]). By means of this production term, cluster sizes (i.e. up to the maximum size N ) grow or decrease. From the mathematical point of view, the most interesting feature of our system is the presence of the coupling terms between the evolution of the colloids ui and the evolution of the temperature θ. Such setting is
sometimes referred to as thermophoresis or thermo-migration, thermo-diffusion, or the Soret effect, in the context of mixtures of mobile particles, where the different particle sizes have distinct responses to the presence of a temperature gradient.
The precise structure of this system has been proposed in [7] as a mathematical model supposed to describe simultaneous effects between heat conduction and moelcular diffusion arising when populations of hot colloids like to ”diffuse” inside porous materials. To fix ideas, this process is called here thermo-diffusion. It is worth noting that considerable phenomenological understanding is available around this topic; compare for instance [11] or the more recent accounts by Wojnar [14, 15]. Regarding the presence in the right-hand side of the model equations of the products between temperature and concentration
gradients – mimicking thermodynamic Soret and Dufour effects pointing out a strong interplay between heat conduction and molecular diffusion – we refer the reader to [8, 9, 13]. In these settings, such strongly nonlinear structures arising in the model equations play a decisive role in capturing the expected evolution of the physical system.
In the framework of this paper, we are interested in understanding the large time behavior of a given porous material exposed to thermo-diffusive infiltrations, very much in the spirit of related mathematical work done for a conceptually different problem referring to the chemical corrosion of concrete; see e.g. [2, 10].
The major mathematical difficulty encountered here is the presence of nonlinear terms of the type ∇θ · ∇ui. A careful look at our estimates will discover that the presence of
terms like ∇δ0u
i· ∇θ is essential to ensure ultimately a good (time independent) control
on the L∞bounds on the gradients of both temperature and colloidal concentrations. The regularization parameter δ0 arising in ∇δ0ui can be removed only in one space dimension
using a suitable combination of compactness arguments for strong solutions to problem P and the Leray-Schauder fixed point principle; see for instance the line of thought in [3]. Using a couple of approximating problems and a suitable grip on the gradient of con-centrations and of the gradient of temperature, we prove that, for sufficiently large time, all transport terms in problem P disappear, the limiting evolution of the concentration being simply governed by the ordinary differential equations governing the Smoluchowski dynamics.
2
Main result
We begin with the definition of our concept of solution to problem P.
Definition 2.1. Let θ and ui(i = 1, 2, · · · , N ) be non-negative functions on Q(T ) for
T > 0 and u = (u1, u2, · · · , uN). We call that a pair (θ, u) is a solution of P on [0, T ] if
the conditions (S1) and (S2) hold:
(S1) (θ, u) ∈ X(T )N +1, where X(T ) = L∞(Q(T ))∩W1,2(0, T ; L2(Ω))∩L∞(0, T ; H1(Ω))∩ L2(0, T ; H2(Ω)).
(S2) (1.1) ∼ (1.4) hold in the usual sense.
Moreover, we say that (θ, u) is a solution of P on [0, ∞), if it is a solution of P on [0, T ] for any T > 0.
For simplicity, we put
L∞+(Ω) = {z ∈ L∞(Ω) : z ≥ 0 a.e. on Ω},
and write
|u|L2(Ω) := |u|L2(Ω)N, |∇θ|L2(Ω):= |∇θ|L2(Ω)3 and so on.
The first theorem of this paper guarantees the existence and the large time behavior of the solution to the problem P.
Theorem 2.1. Let δ0 > 0. If (θ0, u) ∈ (H1(Ω) ∩ L∞+(Ω))N +1, then P(θ0, u0) has a
solution (θ, u) on [0, ∞) satisfying 0 ≤ θ(t) ≤ |θ0|L∞(Ω) a.e. on Ω for t ≥ 0, and for each i = 1, 2 · · · , N there exists a function yi ∈ L∞(0, ∞) ∩ L2(0, ∞) such that
0 ≤ ui(t) ≤ yi(t) a.e. on Ω for t > 0. (2.1)
The inequality (2.1) conclude that ui(t) → 0 as t → ∞ for i.
The second theorem is concerned with the uniqueness of solutions to problem P.
Theorem 2.2. Under the same assumptions as in Theorem 2.1 if (θ0, u0) ∈ W1,∞(Ω)N +1,
then there exists a solution (θ, u) of P(θ0, u0) on [0, ∞) satisfying (θ, u) ∈ L∞(0, T ; W1,∞(Ω))N +1.
Moreover, let T > 0 and (θ(k), u(k)) be a solution of P(θ
0, u0) on [0, T ] for k = 1, 2.
If (θ(k), u(k)) ∈ L∞(0, T ; W1,∞(Ω))N +1, k = 1, 2, then θ(1) = θ(2) and u(1) = u(2) a.e. on
Q(T ).
First, we prove Theorem 2.2 in this paper. To prove Theorem 2.2 we consider the approximation problem Pn(θ0, u0) := {(1.1), (2.2), (1.3), (1.4)} of the problem P for n > 0:
uit− κi∆ui− τi∇θ · ∇ui = Rin(u) in Q(T ) for each i, (2.2)
where Rin(s1, s2, · · · , sN) := Ri(σn(s1), σn(s2), · · · , σn(sN)) and σn(r) = n if r > n, r if 0 ≤ r ≤ n, 0 otherwise, for r ∈ R.
In Section 3 we show the existence of a unique solution (θ(n), u(n)) to P
n(θ0, u0) on
[0, T ] for n > 0, by using the fixed point theory, when θ0 and u0 are sufficiently smooth.
After we get the uniform estimate of (θ(n), u(n)) in L∞(Ω) with respect to n, we say that
for large n (θ(n), u(n)) is a solution of P. Thus, we shall prove Theorem 2.2.
Next, in order to prove Theorem 2.1 we approximate initial values θ0 and u0by smooth
functions {(θ0(ε), u(ε)0 )}ε∈(0,1]. Also, we give some uniform estimates for solutions (θ(ε), u(ε))
of P(θ0(ε), u(ε)0 ) with respect to ε in Section 4. Finally, after controlling in terms of uniform estimates the approximate solutions, we give in Section 5 the proof of Theorem 2.1.
Throughout this paper we assume that the boundary Ω is sufficiently smooth such that
|f |H2(Ω) ≤ CΩ(|∆f |L2(Ω)+ |f |H1(Ω)) for f ∈ H2(Ω) with ∇f · ν = 0 on Γ, (2.3) where CΩ is a positive constant (see Theorem 25.3 in Chapter of [5]).
At the end of this section we give here a inequality concerned with ∇δ, (see for example, [1]). For all 1 ≤ p ≤ ∞ and δ > 0 it holds that
|∇δf |
Lp(Ω) ≤ cδ,p|f |L2(Ω) for f ∈ L2(Ω), (2.4) where cδ,p is a positive constant.
3
Proof of Theorem 2.2
In order to prove Theorem 2.2 we solve the approximate problem Pn(θ0, u0) in the following
way:
Proposition 3.1. Let T > 0, δ0 > 0 and n > 0. If (θ0, u0) ∈ (W1,∞(Ω)∩L∞+(Ω))N +1, then
Pn(θ0, u0) has a unique solution (θ, u) on [0, T ] satisfying (θ, u) ∈ (X(T )∩L∞(0, T ; W1,∞(Ω)))N +1
and that θ and ui (1 ≤ i ≤ N ) are non-negative a.e. on Q(T ).
Before the proof of Proposition 3.1, we consider the following initial boundary value problem for a linear parabolic equation:
vt− κ0∆v + ∇a · ∇v = f in Q(T ), (3.1)
v(0, x) = v0(x) for x ∈ Ω, −κ0∇v · ν = 0 on S(T ), (3.2)
where κ0 is a positive constant, a and f are given functions on Q(T ), and v0 is an initial
function.
For this initial boundary value problem the next classical lemma is valid.
Lemma 3.1. [12, Chapter 3] Let T > 0. If v0 ∈ W1,∞(Ω), a ∈ L∞(0, T ; W1,∞(Ω)) and
f ∈ L∞(0, T ; L∞(Ω)), then the above problem (3.1) and (3.2) has a unique solution v such that v ∈ X(T ) ∩ L∞(0, T ; W1,∞(Ω)). Moreover, for r > 0 there exists a positive constant C∗(r) such that
if |v0|W1,∞(Ω)+ |a|L∞(0,T ;W1,∞(Ω))+ |f |L∞(0,T ;L∞(Ω))≤ r, then |v|L∞(0,T ;W1,∞(Ω)) ≤ C∗(r).
First, for a given function ˆu = (ˆu1, ˆu2, · · · , ˆuN) on Q(T ) we consider the following
problem APn(ˆu) for n > 0: θt− κ∆θ − τ N X i=1 ∇δ0uˆ i· ∇θ = 0 in Q(T ), (3.3) − κ∇θ · ν = 0 on S(T ), θ(0, x) = θ0(x) for x ∈ Ω, (3.4)
uit− κi∆ui− τi∇θ · ∇ui = Rin(ˆu) in Q(T ) for each i, (3.5)
− κi∇ui· ν = 0 on S(T ), ui(0, x) = u0i(x) for x ∈ Ω and each i. (3.6)
Here, for T > 0 and M > 0 we put
K(T, M ) = {ˆu = (ˆu1, ˆu2, · · · , ˆuN) ∈ L2(0, T ; L2(Ω))N : 0 ≤ ˆui ≤ M a.e. on Q(T )}.
On the problem APn(ˆu) for n > 0 we give the following lemma.
Lemma 3.2. Let T > 0, δ0 > 0, n > 0 and M > 0. If (θ0, u0) ∈ (W1,∞(Ω) ∩ L∞+(Ω))N +1
and ˆu ∈ K(T, M ), then APn(ˆu) has a unique solution (θ, u) on [0, T ] such that (θ, u) ∈
X(T )N +1 ∩ L∞(0, T ; W1,∞(Ω))N +1, and θ and u
i (1 ≤ i ≤ N ) are nonnegative a.e on
Proof. Since by using (2.4) we see that |∇δ0uˆ i(t)|L∞(Ω) ≤ cδ 0,∞M |Ω| 1/2 for a.e. t ∈ [0, T ], (3.7) where |Ω| = R
Ωdx, Lemma 3.1 implies the existence of a function θ ∈ X(T ) satisfying
(3.3) and (3.4) with θ ∈ L∞(0, T ; W1,∞(Ω)).
Next, we show that θ is non-negative on Q(T ). In fact, we put ¯θ = −θ. Then, it holds that ¯ θt− κ∆¯θ = τ N X i=1 ∇δ0uˆ i· ∇¯θ a.e. on Q(T ). (3.8)
Here, by multiplying (3.8) by [¯θ]+ and then integrating it over Ω we obtain
1 2 d dt|[¯θ] +|2 L2(Ω)+ κ|∇[¯θ]+|2L2(Ω) = τ N X i=1 Z Ω (∇δ0uˆ i· ∇¯θ)[˜θ]+dx ≤ τ N X i=1 |∇δ0uˆ i|L∞(Q(T ))|∇[¯θ]+|L2(Ω)|[¯θ]+|L2(Ω) a.e. on [0, T ] so that 1 2 d dt|[¯θ] +|2 L2(Ω)+ κ 2|∇[¯θ] +|2 L2(Ω) ≤ τ2N 2κ N X i=1 |∇δ0uˆ i|2L∞(Q(T ))|[¯θ]+|2L2(Ω) a.e. on [0, T ]. (3.9) By applying Gronwall’s inequality to the above inequality, we get |[¯θ]+(t)|2
L2(Ω) = 0 for t ∈ [0, T ], namely, θ is non-negative.
From Lemma 3.1 together with the definition of Rin and θ ∈ L∞(0, T ; W1,∞(Ω))
it follows that there exists a function u ∈ X(T )N such that (3.5) and (3.6) hold, and
u ∈ L∞(0, T ; W1,∞(Ω))N. Because of the positivity of Rinwe can show the non-negativity
of ui for each i in a similar way.
On account of Lemma 3.2 we can define the solution operator Λ : K(T, M ) → L2(0, T ; L2(Ω))N by Λˆu = u for M > 0 and T > 0.
Proof of Proposition 3.1. Let T > 0 and n > 0. As a first step of this proof, we show that
Λ : K(T, M ) → K(T, M ) for large M > 0. (3.10)
By the definition of Rin, we take a positive constant Cn independent of M such that
|Rin(ˆu)| ≤ Cn a.e. on Q(T ) for any ˆu ∈ K(T, M ) and M > 0.
Here, for ˆu ∈ K(T, M ) let (θ, u) be a solution of APn(ˆu) on [0, T ] and put u = (u1, · · · , uN).
constant and see that Z Ω uit(t)[ui(t) − M0(t + 1)]+dx + κi Z Ω ∇ui(t) · ∇[ui(t) − M0(t + 1)]+dx =τi Z Ω ∇θ · ∇ui[ui(t) − M0(t + 1)]+dx + Z Ω Rin(ˆu(t))ui[ui(t) − M0(t + 1)]+dx
for a.e. t ∈ [0, T ] and i. It easy to see that d dt|[ui(t) − M0(t + 1)] +|2 L2(Ω)+ κi|∇[ui(t) − M0(t + 1)]+|2L2(Ω) ≤ τi|∇θ|L∞(Q(T ))|∇[ui(t) − M0(t + 1)]+|L2(Ω)|[ui(t) − M0(t + 1)]+|L2(Ω) + Z Ω
(Rin(ˆu(t)) − M0)|[ui(t) − M0(t + 1)]+dx for a.e. t ∈ [0, T ] and i.
By choosing M0 with M0 = max{Cn, |u0|L∞(Ω)}, we have d dt|[ui(t) − M0(t + 1)] +|2 L2(Ω)+ κi 2|∇[ui(t) − M0(t + 1)] +|2 L2(Ω) ≤ τ 3 i 2κi |∇θ|2
L∞(Q(T ))|[ui(t) − M0(t + 1)]+|L2(Ω) for a.e. t ∈ [0, T ] and i. Hence, thanks to the Gronwall inequality, we infer that
ui ≤ M0(T + 1) a.e. on Q(T ).
Accordingly, for M ≥ M0(T + 1) (3.10) holds.
Moreover, by (3.7) Lemma 3.1 implies
|∇θ|L∞(Q(T )) ≤ C∗(r1) and |∇u|L∞(Q(T )) ≤ C∗(r2), (3.11) where r1 = cδ0,∞M |Ω|
1/2+ |θ
0|W1,∞(Ω) and r2 = C∗(r1) + Cn+ |u0|W1,∞(Ω).
Next, we show Λm is a contraction mapping on K(T, M ) for a large positive integer
m. Indeed, let ˆu(j) = (ˆu(j)
1 , · · · , ˆu (j)
N ) ∈ K(T, M ), (θ(j), u(j)) be a solution of APn(ˆu(j))
on [0, T ] for each j = 1, 2, and put θ = θ(1) − θ(2), ˆu = ˆu(1) − ˆu(2) = (ˆu
1, · · · , ˆuN), and
u = u(1)− u(2) = (u
1, · · · , uN). Then, it is clear that
θt− κ∆θ = τ N X i=1 (∇δ0uˆ i· ∇θ(1)+ ∇δ0uˆ (2) i · ∇θ) on Q(T ), (3.12) uit− κi∆ui = τi(∇θ · ∇u (1) i + ∇θ (2)· ∇u i) + Rin(ˆu(1)) − Rin(ˆu(2)) on Q(T ) for each i. (3.13) By multiplying (3.12) with θ, it follows from (2.4) that
1 2 d dt|θ| 2 L2(Ω)+ κ|∇θ|2L2(Ω) = τ N X i=1 Z Ω (∇δ0uˆ i· ∇θ(1)+ ∇δ0uˆ (2) i · ∇θ)θdx ≤τ C∗(r1)cδ0,2 N X i=1 |ˆui|L2(Ω)|θ|L2(Ω)+ τ N M |Ω|1/2|∇θ|L2(Ω)|θ|L2(Ω) a.e. on [0, T ].
Easily, we get 1 2 d dt|θ| 2 L2(Ω)+ κ 2|∇θ| 2 L2(Ω) ≤τ C∗(r1)cδ0,2 2 |ˆu| 2 L2(Ω)+ ( τ C∗(r1)cδ0,2N 2 + τ2N2M2|Ω| 2κ )|θ| 2 L2(Ω) a.e. on [0, T ]. (3.14) Similarly to (3.14), we observe that
1 2 d dt|ui| 2 L2(Ω)+ κi|∇ui|2L2(Ω)
≤(τiC∗(r2)|∇θ|L2(Ω)+ τiC∗(r1)|∇ui|L2(Ω)+ Cn0| ˆui|L2(Ω))|ui|L2(Ω) a.e. on [0, T ] for i, where Cn0 = max{Lip(Rin) : 1 ≤ i ≤ N } and Lip(Rin) is the Lipschitz constant of Rin.
Then we have 1 2 d dt|ui| 2 L2(Ω)+ κi 2|∇ui| 2 L2(Ω) ≤ κ 4|∇θ| 2 L2(Ω)+ Cn0 2 | ˆui| 2 L2(Ω) + (τ 2 iC∗(r2)2 2κ + τi2C∗(r1)2 2κi +C 0 n 2 )|ui| 2 L2(Ω) a.e. on [0, T ] for i. (3.15) By adding (3.14) and (3.15), we see that
d dt(|θ|
2
L2(Ω)+ |u|2L2(Ω)) + µ(|∇θ|2L2(Ω)+ |∇u|2L2(Ω))
≤C1(|ˆu|2L2(Ω)+ |θ|L22(Ω)+ |u|2L2(Ω)) a.e. on [0, T ], (3.16) where µ and C1 are positive constants. On account of the Gronwall inequality it yields
that
|θ(t1)|2L2(Ω)+ |u(t1)|2L2(Ω)≤ C1eC1t1
Z t1
0
|ˆu(t)|2L2(Ω)dt for any t1 ∈ [0, T ].
Hence, it holds that
|Λ2uˆ(1)(t) − Λ2uˆ(2)(t)|2L2(Ω) ≤ t1(C1eC1t1)2|ˆu|2L2(0,t
1;L2(Ω)) for 0 < t1 ≤ T. Then, recursively, we get
|Λmuˆ(1)(t) − Λmuˆ(2)(t)|2 L2(Ω) ≤ tm−11 (C1eC1t1)m (m − 1)! |ˆu| 2 L2(0,t 1;L2(Ω)) for 0 < t1 ≤ T and m, and |Λmuˆ(1)− Λmuˆ(2)|2 L2(0,T ;L2(Ω)) ≤ (C1eC1TT )m m! |ˆu| 2 L2(0,T ;L2(Ω)) for m = 1, 2, · · · . (3.17)
Here, we can take m such that (C1eC1TT )m
m! < 1 so that Λ
m is contraction on K(T, M ).
Therefore, the Banach fixed point guarantees that Λm has a unique fixed point u ∗ ∈
K(T, M ). Clearly, u∗ is also the fixed point of Λ. Thus, we have proved the existence of
a solution Pn(θ0, u0) on [0, T ].
Now, we shall prove Theorem 2.2.
Proof of Theorem 2.2. As discussed in Proposition 3.1, for n = 1, 2, · · · , Pn(θ0, u0) has a
solution (θ(n), u(n)) on [0, T ]. In order to prove Theorem 2.2 it is sufficient to show that
there exists positive constants M and n0 such that
0 ≤ u(n)i ≤ M a.e. on Q(T ) for any i = 1, 2, · · · , N and n ≥ n0, (3.18)
where u(n)= (u(n)
1 , · · · , u (n) N ).
The non-negativity of u(n)i is already proved in Lemma 3.2.
Next, let y1 be a solution of the following initial value problem for the ordinary
differ-ential equation:
y10 = −β11y21 on [0, ∞), y1(0) = |u01|L∞(Ω). Clearly, y1 ∈ L2(0, ∞) ∩ L∞(0, ∞). Then, from (2.2), we see that
u(n)1t − y1t− κ1∆(u (n) 1 − y1) =τ1∇θ(n)· ∇(u (n) 1 − y1) + R1n(u(n)) + β11y12 ≤τ1∇θ(n)· ∇(u(n)1 − y1) − β11σn(u(n)1 ) 2 + β11y12 a.e. on Q(T ). (3.19) Multiply (3.19) by [u(n)1 − y1]+. Since (−σn(u (n) 1 )2+ y21)[u (n) 1 − y1]+ ≤ 0 a.e. on Q(T ) for n ≥ |y1|L∞(0,∞), we have 1 2 d dt|[u (n) 1 − y1]+|2L2(Ω)+ κ1|∇[u (n) 1 − y1]+|2L2(Ω) ≤ τ1 Z Ω (∇θ(n)· ∇[u(n)1 − y1]+)[u (n) 1 − y1]+dx ≤ κ1 2|∇[u (n) 1 − y1]+|2L2(Ω)+ τ2 1 2κ1 |∇θ(n)|2 L∞(Q(T ))|[u (n) 1 − y1]+|2L2(Ω) a.e. on [0, T ]. The Gronwall’s inequality yields that u(n)1 ≤ y1 a.e. on Q(T ).
Here, we assume that for i = 1, 2, · · · , i0 there exists yi ∈ L∞(0, ∞) ∩ L2(0, ∞) ∩
W1,1(0, ∞) ∩ C1([0, ∞)) satisfying that for n ≥ max{|y
i|L∞(0,∞) : i = 1, 2, · · · , i0} 0 ≤ u(n)i ≤ yi a.e. on Q(T ).
Let i = i0+ 1 and yi be a solution of
y0i = β0 2 i−1 X k=1 yk2− βiiyi2 on [0, ∞), yi(0) = |u0i|L∞(Ω), where β0 = max{βkj : 1 ≤ k ≤ N, 1 ≤ j ≤ N }.
Then, by choosing n ≥ max{|yi|L∞(0,∞) : i = 1, 2, · · · , i0 + 1} it is easy to see that u(n)it − yit− κi∆(u(n)i − yi) = τi∇θ(n)· ∇(u (n) i − yi) + Rin(u(n)) − β0 2 i−1 X k=1 y2k+ βiiyi2 ≤ τi∇θ(n)· ∇(u (n) i − yi) + β0 2 i−1 X k=1 |σn(u (n) k )| 2 − β ii|σn(u (n) i )| 2− β0 2 i−1 X k=1 yk2+ βiiyi2 ≤ τi∇θ(n)ε · ∇(u (n) i − yi) − βii(|σn(u (n) i )| 2 − y2 i) a.e. on Q(T ). Similarly to (??), we obtain 0 ≤ u(n)i ≤ yi a.e. Q(T ). (3.20)
Thus, by taking n ≥ max{|yi|L∞(0,∞) : i = 1, 2, · · · , N } we have
0 ≤ u(n)i ≤ M := max{|yi|L∞(0,∞) : i = 1, 2, · · · , N } a.e. on Q(T ) for any T > 0, that is, Rin(u(n)) = Ri(u(n)) a.e on (0, T ) × Ω for T > 0. Now, we have proved this
theorem.
4
Uniform estimates for approximate solutions
Let (θ0, u0) ∈ (H1(Ω) ∩ L∞+(Ω))N +1. Then we can take a sequence {(θ (ε) 0 , u
(ε)
0 )}ε∈(0,1] ⊂
(W1,∞(Ω) ∩ L∞+(Ω))N +1 such that (θ0(ε), u(ε)0 ) → (θ0, u0) in H1(Ω)N +1 as ε ↓ 0 and
{(θ(ε)0 , u(ε)0 )}ε∈(0,1] is bounded in L∞(Ω)N +1. Also, by Theorem 2.2 P(θ (ε) 0 , u
(ε)
0 ) has a
unique solution (θ(ε), u(ε)) on [0, T ] for any ε ∈ (0, 1].
In this section we assume the above conditions for initial functions, put u(ε)= (u(ε)
1 , · · · , u (ε) N ),
and give several auxiliary lemmas dealing with the derivation of uniform estimates of ap-proximate solutions (θ(ε), u(ε)).
Lemma 4.1. There exists a positive constant K1 independent of ε ∈ (0, 1] such that
0 ≤ θ(ε)≤ K1 a.e. on Q(T ) for ε ∈ (0, 1],
0 ≤ u(ε)i ≤ K1 a.e. on Q(T ) for ε ∈ (0, 1] and i.
We can prove this lemma in a similar way to the proof of Lemma 3.2 and (3.18) so that we omit its proof.
Lemma 4.2. Then there exists a positive constant K2 independent of ε ∈ (0, 1] such that
|θ(ε)|W1,2(0,T ;L2(Ω))+ |θ(ε)|L∞(0,T ;H1(Ω))+ |θ(ε)|L2(0,T ;H2(Ω)) ≤ K2 for ε ∈ (0, 1], (4.1) |u(ε)|
Proof. First, we multiply (1.1) by θ(ε)t and integrate it over Ω. Then on account of (2.4) we see that |θt(ε)|2L2(Ω)+ κ 2 d dt|∇θ (ε)|2 L2(Ω) = τ N X i=1 Z Ω (∇δ0u(ε) i · ∇θ (ε))θ(ε) t dx ≤ 1 2|θ (ε) t |2L2(Ω)+ τ2 2 N X i=1 |∇δ0u(ε) i | 2 L∞(Ω)|∇θ(ε)|L2(Ω) ≤ 1 2|θ (ε) t |2L2(Ω)+ τ2c2 δ0,∞ 2 N X i=1 |u(ε)i |2 L2(Ω)|∇θ(ε)|2L2(Ω) a.e. on [0, T ] for ε ∈ (0, 1]. Obviously, we get 1 2|θ (ε) t |2L2(Ω)+ κ 2 d dt|∇θ (ε)|2 L2(Ω)≤ τ2c2 δ0,∞ 2 K 2 1|Ω|N2|∇θ(ε)|2L2(Ω) a.e. on [0, T ] for ε ∈ (0, 1]. By applying the Grownwall’s inequality, (2.3) and (1.1) imply (4.1).
Next, we multiply (2.2) by u(ε)i and integrate it over Ω. Then for each i we get 1 2 d dt|u (ε) i | 2 L2(Ω)+ κi Z Ω |∇u(ε)i |2dx = τi Z Ω (∇θ(ε)· ∇u(ε)i )u(ε)i dx + Z Ω Ri(u(ε))u (ε) i dx ≤ τi|∇θ(ε)|L2(Ω)|∇u(ε)i |L2(Ω)|u(ε)i |L∞(Ω)+ Z Ω CRK1dx ≤ τiK1|∇θ(ε)|L2(Ω)|∇u(ε)i |L2(Ω)+ CRK1|Ω| ≤ κi 2 Z Ω |∇u(ε)i |2dx + τi2K12 2κi |∇θ(ε)|2 L2(Ω)+ CRK1|Ω| a.e. on [0, T ] for ε ∈ (0, 1], (4.3) where CR = sup{Ri(q1, · · · , qN) : |qj| ≤ K1, i, j = 1, 2, · · · , N }.
By (4.3) and (4.1) we can get (4.2).
The following two lemmas are concerned with the essential estimates for the proof of existence part of Theorem 2.1.
Lemma 4.3. The set {∇θ(ε) : ε ∈ (0, 1]} is bounded in L4(Q(T ))3.
Proof. Let ε ∈ (0, 1]. Because of θ(ε)(t) ∈ W1,∞(Ω) for a.e. t ∈ [0, T ], θ(ε)|∇θ(ε)|2 ∈
H1(Ω)3 for a.e. t ∈ [0, T ]. Then we can multiply it to (1.1) and get
Z Ω θ(ε)t (θ(ε)|∇θ(ε)|2)dx + κ Z Ω ∇θ(ε)· ∇(θ(ε)|∇θ(ε)|2)dx = τ N X i=1 Z Ω (∇δ0u(ε) i · ∇θ (ε) )θ(ε)|∇θ(ε)|2dx a.e. on [0, T ].
We note that ∇θ(ε)· ∇(θ(ε)|∇θ(ε)|2) =|∇θ(ε)|4+ 2θ(ε) 3 X k,j=1 ∂2θ(ε) ∂xk∂xj ∂θ(ε) ∂xk ∂θ(ε) ∂xj so that κ Z Ω |∇θ(ε)|4dx = − Z Ω θt(ε)(θ(ε)|∇θ(ε)|2)dx − 2κ Z Ω θ(ε) 3 X k,j=1 ∂2θ(ε) ∂xk∂xj ∂θ(ε) ∂xk ∂θ(ε) ∂xj dx + τ N X i=1 Z Ω (∇δ0u(ε) i · ∇θ (ε))θ(ε)|∇θ(ε)|2dx (=: I 1+ I2 + I3) a.e. on [0, T ]. (4.4) It is easy to obtain I1 ≤ K1|θ (ε) t |L2(Ω)|∇θ(ε)|2L4(Ω) ≤ κ 4|∇θ (ε)|4 L4(Ω)+ 1 κK 2 1|θ (ε) t | 2 L2(Ω), (4.5) I2 ≤ 2κK1 3 X k,j=1 Z Ω | ∂ 2θ(ε) ∂xk∂xj ||∇θ(ε)|2dx ≤ κ 4|∇θ (ε)|4 L4(Ω)+ 36κK12 3 X k,j=1 Z Ω | ∂ 2θ(ε) ∂xk∂xj |2dx, (4.6) I3 ≤ τ K1 N X i=1 Z Ω |∇δ0u(ε) i ||∇θ (ε)|3dx ≤ κ 4|∇θ (ε)|4 L4(Ω)+ CκK14 N X i=1 Z Ω |∇δ0u(ε) i | 4dx ≤ κ 4|∇θ (ε)|4 L4(Ω)+ Cκcδ0,4K14 N X i=1 |u(ε)i |4L2(Ω) a.e. on [0, T ], (4.7) where Cκ is a positive constant depending only on κ. By combining (4.4) ∼ (4.7) we have
κ 8|∇θ (ε)|4 L4(Ω)≤ K2 1 κ |θ (ε) t | 2 L2(Ω)+ 36κK12|θ(ε)|H22(Ω)+ Cκcδ0,4K 4 1|u (ε) i | 4 L2(Ω) a.e. on [0, T ].
Hence, thanks to Lemma 4.2 we have proved the conclusion of this lemma.
Lemma 4.4. The set {u(ε)|ε ∈ (0, 1]} is bounded in W1,2(0, T ; L2(Ω)N), L∞(0, T ; H1(Ω)N)
and L2(0, T ; H2(Ω)N), and {∇u(ε)
Proof. Let ε ∈ (0, 1]. First, we multiply (1.2) by u(ε)it and then get |u(ε)it |2 L2(Ω)+ κi 2 d dt|∇u (ε) i | 2 L2(Ω) = τi Z Ω (∇θ(ε)· ∇u(ε)i )u(ε)it dx + Z Ω Ri(u(ε))u (ε) it dx ≤1 2|u (ε) it | 2 L2(Ω)+ τi2 Z Ω |∇θ(ε)|2|∇u(ε) i | 2dx + Z Ω |Ri(u(ε))|2dx a.e. on [0, T ] for i.
Here, for an arbitrary positive number η we can easily get 1 2|u (ε) it | 2 L2(Ω)+ κi 2 d dt|∇u (ε) i | 2 L2(Ω) ≤ η|∇u(ε)i |4 L4(Ω)+ Cητi4|∇θ (ε)|4 L4(Ω)+ CR2|Ω| a.e. on [0, T ] for i, (4.8) where Cη is a positive constant depending only on η and CR is already defined in the
proof of Lemma 4.2.
Next, by multiplying (1.2) by −∆u(ε)i we can see that 1 2 d dt|∇u (ε) i | 2 L2(Ω)+ κi|∆u (ε) i | 2 L2(Ω) ≤ τi Z Ω |∇θ(ε)||∇u(ε) i ||∆u (ε) i | + Z Ω |Ri(u(ε))||∆u (ε) i |dx ≤κi 2|∆u (ε) i | 2 L2(Ω)+ τi2 κi Z Ω |∇θ(ε)|2|∇u(ε)i |2dx +C 2 R κi |Ω| a.e. on [0, T ] for i. Similarly to (4.8), we have 1 2 d dt|∇u (ε) i | 2 L2(Ω)+ κi 2|∆u (ε) i | 2 L2(Ω) ≤η|∇u(ε)i |4 L4(Ω)+ τ4 iCη κ2 i |∇θ(ε)|4 L4(Ω)+ C2 R κi |Ω| a.e. on [0, T ] for i. (4.9) Moreover, we multiply (2.2) by u(ε)i |∇u(ε)i |2 and in the similar way to that of (4.4) we
observe that κi Z Ω |∇u(ε)i |4dx = − Z Ω u(ε)it u(ε)i |∇u(ε)i |2dx − κ i Z Ω u(ε)i ∇u(ε)i · ∇(|∇u(ε)i |2)dx + τ i Z Ω
(∇θ(ε)· ∇u(ε)i )(u(ε)i |∇u(ε)i |2)dx
+ Z Ω Ri(u(ε))u (ε) i |∇u (ε) i | 2dx (=: J 1+ J2+ J3 + J4) a.e. on [0, T ] for i. (4.10)
By elementary calculations we infer that
J1 ≤ |u (ε) i |L∞(Ω) Z Ω |u(ε)it ||∇u(ε)i |2dx ≤ κi 4|∇u (ε) i | 4 L4(Ω)+ K2 1 κi |u(ε)it |2 L2(Ω), (4.11)
J2 ≤ κi|u (ε) i |L∞(Ω)
Z
Ω
|∇u(ε)i ||∇(|∇uiε|2)|dx
≤ 2κiK1 Z Ω |∇u(ε)i |2| 3 X k,j=1 ∂2u(ε)i ∂xk∂xj |2dx ≤ κi 4|∇u (ε) i | 4 L4(Ω)+ 36κiK12|u (ε) i | 2 H2(Ω), (4.12) J3 ≤ τi|u (ε) i |L∞(Ω) Z Ω |∇θ(ε)||∇u(ε) i | 3dx ≤ κi 4|∇u (ε) i | 4 L4(Ω)+ K14Cκiτi4|∇θ(ε)|4L4(Ω), (4.13) J4 ≤ CRK1|∇u (ε) i | 2 L2(Ω) a.e. on [0, T ] for i, (4.14) where Cκi is a positive constant depending only on κi for each i. From (4.10) ∼ (4.14) it follows that |∇u(ε)i |4L4(Ω) ≤ C2(|u (ε) it | 2 L2(Ω)+ |∆u (ε) i | 2 L2(Ω)+ |∇θ(ε)|4L4(Ω)+ |∇u (ε) i | 2 L2(Ω)+ 1) (4.15) a.e. on [0, T ] for i, where C2 is a positive constant independent of ε.
Furthermore, by combining (4.8), (4.9) and (4.15), and using (2.3) we see that
d dt N X i=1 (1 2+ κi 2)|∇u (ε) i | 2 L2(Ω)+ 1 2|u (ε) t |2L2(Ω)+ N X i=1 κi 2|∆u (ε) i | 2 L2(Ω) ≤2η N X i=1 |∇u(ε)i |4 L4(Ω)+ C 0 η|∇θ (ε)|4 L4(Ω)+ C3 ≤2ηC2 N X i=1 (|u(ε)it |2 L2(Ω)+ |∆u (ε) i | 2 L2(Ω)+ |∇θ(ε)|4L4(Ω)+ |∇u (ε) i | 2 L2(Ω)+ 1|) + Cη0|∇θ(ε)|4 L4(Ω)+ C3 a.e. on [0, T ],
where Cη0 and C3 are positive constants independent of ε. Since we can take sufficiently
small η in the above inequality, there exist positive constants C4 and µ1 independent of
ε ∈ (0, 1] such that d dt N X i=1 (1 2+ κi 2)|∇u (ε) i | 2 L2(Ω)+ µ1|u (ε) t |2L2(Ω)+ µ1 N X i=1 |∆u(ε)i |2 L2(Ω) ≤C4(|∇θ(ε)|4L4(Ω)+ 1) a.e. on [0, T ]. (4.16) Therefore, the assertion of this lemma is a direct consequence of Lemma 4.3.
5
Proof of Theorem 2.1
Proof of Theorem 2.1. For ε ∈ (0, 1] let (θ(ε), u(ε)) be a solution of P(θ(ε)0 , u(ε)0 ) on [0, T ]. By Lemmas 4.1 ∼ 4.4 the sets {θ(ε)|ε ∈ (0, 1]} and {u(ε)i |ε ∈ (0, 1]} (i = 1, · · · , N ) are
bounded in L∞(Q(T )), W1,2(0, T ; L2(Ω)), L∞(0, T ; H1(Ω)), L2(0, T ; H2(Ω)) and L4(0, T ; W1,4(Ω)).
Then, there exists a subsequence {εj} such that θ(j):= θ(εj) → θ and u (j) i := u
(εj)
i → ui in
L2(0, T ; L2(Ω)), weakly in W1,2(0, T ; L2(Ω)), L2(0, T ; H2(Ω)) and in L4(0, T ; W1,4(Ω)), and weakly* in L∞(Q(T )) and L∞(0, T ; H1(Ω)) as j → ∞, where θ, u
i ∈ X(T ) ∩
L4(0, T ; W1,4(Ω)) for i = 1, 2, · · · , N .
First, we show that
∇δ0u(j)
i · ∇θ
(j)→ ∇δ0u
i· ∇θ weakly in L2(Q(T )) as j → ∞ for each i. (5.1)
In fact, for each i and any η ∈ L2(Q(T )) we have
| Z Q(T ) (∇δ0u(j) i · ∇θ (j)− ∇δ0u i· ∇θ)ηdxdt| ≤| Z Q(T ) (∇δ0u(j) i − ∇ δ0u i) · ∇θ(j)ηdxdt| + | Z Q(T ) ∇δ0u i· (∇θ(j)− ∇θ)ηdxdt|
=:I1j + Ij2 for each j.
Since ∇δ0u
i ∈ L∞(Q(T ))3, namely, η∇δ0ui ∈ L2(Q(T ))3, it is easy to see that I2j → 0 as
j → ∞. Also, by (2.4) we have I1j ≤( Z T 0 |∇δ0(u(j) i − ui)|4L4(Ω)dt)1/4|∇θ(j)|L4(Q(T ))|η|L2(Q(T )) ≤cδ0,4( Z T 0 |u(j)i − ui|4L2(Ω)dt)1/4|∇θ(j)|L4(Q(T ))|η|L2(Q(T ))→ 0 as j → ∞. Thus (5.1) holds.
As a next step, we prove
∇θ(j)→ ∇θ in L2(Q(T ))3 as j → ∞. (5.2)
Indeed, for j it is holds that
θ(j)t − κ∆θ(j)− τ N X i=1 ∇δ0u(j) i · ∇θ (j) = 0 in Q(T ).
The convergences as above and (5.1) imply that
θt− κ∆θ − τ N X i=1 ∇δ0u i· ∇θ = 0 in Q(T ).
Accordingly, we see that
(θ(j)− θ)t− κ∆(θ(j)− θ) = τ N X i=1 (∇δ0u(j) i · ∇θ (j)− ∇δ0u i· ∇θ) in Q(T ) for j. (5.3)
and multiply (5.3) by θ(j)− θ. Then we obtain 1 2 d dt|θ (j)− θ|2 L2(Ω)+ κ|∇(θ(j)− θ)|2L2(Ω) =τ N X i=1 Z Ω (∇δ0u(j) i · ∇θ (j)− ∇δ0u i· ∇θ)(θ(j)− θ)dx a.e. on [0, T ] for j and 1 2|θ (j)(T ) − θ(T )|2 L2(Ω)+ κ Z T 0 |∇(θ(j)− θ)|2 L2(Ω)dt =τ N X i=1 Z Q(T ) (∇δ0u(j) i · ∇θ (j)− ∇δ0u i· ∇θ)(θ(j)− θ)dxdt + 1 2|θ (j) 0 − θ0|2L2(Ω) ≤τ N X i=1 (|∇δ0u(j) i |L4(Q(T ))|∇θ(j)|L4(Q(T ))+ |∇δ0ui|L4(Q(T ))|∇θ|L4(Q(T )))|θ(j)− θ|L2(Q(T )) +1 2|θ (j) 0 − θ0|2L2(Ω) for j.
By letting j → ∞ in this inequality (5.2) is proved. The third step of this proof is to get
∇θ(j)· ∇u(j)
i → ∇θ · ∇ui weakly in L2(Q(T )) as j → ∞ for each i. (5.4)
On account of the boundedness in L4(0, T ; W1,4(Ω)) for approximate solutions it is suffi-cient to show that
Z
Q(T )
(∇θ(j)· ∇u(j)i − ∇θ · ∇ui)ξdxdt → 0 as j → ∞ for ξ ∈ C∞(Q(T )).
Then for ξ ∈ C∞(Q(T )) and each j it holds that
| Z Q(T ) (∇θ(j)· ∇u(j)i − ∇θ · ∇ui)ξdxdt| ≤| Z Q(T ) (∇θ(j)− ∇θ) · ∇u(j)i ξdxdt| + | Z Q(T ) ∇θ · (∇u(j)i − ∇ui)ξdxdt| =: ˆI1j + ˆI2j.
Immediately, for each j we have ˆ
I2j =
Z
Q(T )
∇θ · (∇u(j)i − ∇ui)ξdxdt → 0 as j → ∞,
since ξ(∇θ) ∈ L2(Q(T ))3 and ∇u(j)
i → ∇ui weakly in L2(Q(T ))3. Also, we infer that
ˆ
I1j ≤|∇θ(j)− ∇θ|L2(Q(T ))|∇ui(j)|L2(Q(T ))|ξ|L∞(Q(T )) → 0 as j → ∞. The above arguments guarantee (5.4).
From (5.4) it follows
uit− κi∆ui− τi∇θ · ∇ui = Ri(u) a.e. on Q(T ) for i.
Therefore, (θ, u) is a solution of P(θ0, u0) on [0, T ] for any T > 0 and then is also a solution
of P(θ0, u0) on [0, ∞).
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