Solvability of heat equations coupled with Navier–Stokes equations

136 Pages·2016·0.8 MB·English

Solvability of heat equations coupled with
Navier–Stokes equations
(cid:674)(cid:3300)(cid:3661)(cid:3092)(cid:2268)(cid:881)Navier–Stokes(cid:3661)(cid:3092)(cid:2268)(cid:887)(cid:4040)(cid:3953)(cid:1829)(cid:887)(cid:1348)(cid:1394)(cid:2641)(cid:675)
Yutaka TSUZUKI
(cid:3150)(cid:2969) (cid:1518)
Department of Mathematics
Tokyo University of Science
February 5, 2016
Acknowledgements
First of all I would like to declare that this thesis is anything but a fruit of only my efforts. I would
like to express my sincere and profound gratitude to dear supervisor Professor Tomomi Yokota for
providingmeconsiderableguidanceinmathematics, officeworksandalsomylife. Hisvaluableand
pertinent direction made me correct own mistakes and take appropriate actions. His enthusiastic
and gracious leading stimulated me continue studying mathematics. His irreplaceable guide has
brought me the results in this thesis and what I am today.
I am deeply thankful to Professor Motohiro Sobajima for providing me much instruction and
many advices in mathematics and my life. He particularly gave the mathematical idea that a
nonlinear term can be regarded as a Lipschitz perturbation. That led the result in Chapter 3.
I am in cordial appreciation for Professor Takeshi Fukao’s kind guidance and advices for the
studyonthisthesis. EspeciallyhisideagreatlyhelpedmetoaccomplishthemainresultinChapter
6. Indeed, loss of regularity of a convective term was made up for by using the p-Laplace operator
and solvability was established by the key tool called “Time Returned Method”. Moreover he took
a time for argument on the study in Chapter 6 at his laboratory all the day. Furthermore, his
introduction to the concept of hysteresis was a great motive for the work in Part II.
Moreover I feel heartfelt grateful to Professor Nobuyuki Kenmochi, Professor Toyohiko Aiki,
Professor Ken Shirakawa, Professor Kota Kumazaki and Professor Hiroshi Watanabe for encour-
aging me to study and enjoy mathematics not only in conferences and workshops but also while
eating meals. In particular, Professor Toyohiko Aiki gave me a chance to talk on the 10th AIMS
conference in Madrid, Spain. My gratitude expands to Professor Masahiro Kubo for giving me
some gracious comments each time I reported present situation on my mathematical research by
sending my paper. He also sent me so many papers for further development of my paper [56].
Professor Kota Kumazaki kindly taught me a calculation method on his work related to hysteresis
when I was in deadlock to obtain uniqueness for a problem with hysteresis as in Part II.
I would like to express my thanks to Professor Noboru Okazawa. He listened my talk based on
the result in Chapter 8 which deals with fractional power of Stokes operators, and after that he
gave me some papers with comments which deal with logarithmic power of operators and which
can derive a new work.
I also thank Chief examiner, Professor Masahito Ohta and Associate examiners, Professor
Emiko Ishiwata, Professor Keiichi Kato, Professor Hiroshi Kaneko, Professor Toyohiko Aiki and
Professor Tomomi Yokota for sparing their precious time for thesis defense.
My gratitude goes to Professor Kentarou Yoshii and Professor Toshiyuki Suzuki for giving me
worthwhile advices in a weekly seminar. Professor Kentarou Yoshii, also in other occasions, always
provide affluent knowledge which is useful in life. Professor Toshiyuki Suzuki gave me an idea of
another application of the Gagliardo–Nirenberg inequality which made Lemma 3.4.3.
I am much grateful to the financial support by Grant-in-Aid for JSPS Fellows, No. 15J01987.
Finally, I am very thankful to all seminar members, my friends and my family for being anchor.
Yutaka TSUZUKI
i
ii
Contents
1 Introduction 1
1.1 Heat equations with constraints and Navier–Stokes equations . . . . . . . . . . . . . 2
1.2 Heat equations with hysteresis and Navier–Stokes equations . . . . . . . . . . . . . 8
2 Mathematical tools 11
2.1 Interpolation inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 The Lions–Aubin compactness theorem . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Subdifferential operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Fixed point theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 Evolution equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
I Heat equations with constraints and Navier–Stokes equations 19
3 Existence and uniqueness for heat equations with nonlinear reactions coupled
with Navier–Stokes equations in 2D 21
3.1 Problem and purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Formulation and definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 Main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 Auxiliary facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.5 Solvability of heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.6 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4 Existence and uniqueness for heat equations with p-Laplace diffusions coupled
with Navier–Stokes equations in 2D 39
4.1 Problem and purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 Formulation and definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.3 Main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.4 Auxiliary facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.5 Solvability of heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.6 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5 Existenceanduniquenessforheatequationswithx-dependentreactionscoupled
with Navier–Stokes equations in 2D 55
5.1 Problem and purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.2 Formulation and definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.3 Main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
iii
5.4 Auxiliary facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.5 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6 Existence for heat equations with p-Laplace diffusions coupled with Navier–
Stokes equations in 3D 67
6.1 Problem and purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.2 Formulation and definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.3 Main theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.4 Solvability of approximate heat equation . . . . . . . . . . . . . . . . . . . . . . . . 71
6.5 Futher estimate toward Theorem 6.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.6 Proof of the main theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
II Heat equations with hysteresis and Navier–Stokes equations 81
7 Existence for heat equations with hysteresis coupled with Navier–Stokes equa-
tions in 2D 83
7.1 Problem and purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.2 Formulation and definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7.3 Main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.4 Solvability of equation for hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
7.5 Solvability of heat equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
7.6 Solvability of Navier–Stokes equation . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.7 Solvability of heat equation with hysteresis . . . . . . . . . . . . . . . . . . . . . . . 94
7.8 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
8 Existence and uniqueness for heat equations with hysteresis coupled with
Navier–Stokes equations in 2D and 3D 103
8.1 Problem and purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
8.2 Formulation and definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
8.3 Main theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
8.4 Estimates for the convective terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
8.5 Solvability of heat equation with hysteresis . . . . . . . . . . . . . . . . . . . . . . . 109
8.6 Solvability of Navier–Stokes equation . . . . . . . . . . . . . . . . . . . . . . . . . . 112
8.7 Proof of the main theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
References 127
List of original papers 131
iv
Chapter 1
Introduction
In this thesis we consider thermohydraulic models for heat equations coupled with Navier–Stokes
equations with some effect on the temperature in a water tank with thermostat devices. In more
detail, we consider the following problem:
⎧
⎪∂θ
⎪⎨ −Δ θ+v ·∇θ+|θ|q−1θ−αθ+w = f in Q := (0,T)×Ω,
p
∂t
⎪
⎪∂v
⎩ −Δv +(v ·∇)v = g(θ)−∇π, divv = 0 in Q.
∂t
Here T > 0, Ω ⊂ RN (N = 2,3) be a bounded domain with smooth boundary Γ. θ : Q → R,
v : Q → RN and π : Q → R stand for the temperature, the velocity and the pressure, respectively,
and are unknown functions; f : Q → R, g : R → RN, p ≥ 2, q ≥ 1 and α ∈ R are given; Δ is
p
the so-called p-Laplace operator defined as Δ θ := div(|∇θ|p−2∇θ). Moreover w means the effect
p
mentioned above. Indeed, we can restrict the temperature θ by changing largeness of w (the bigger
w is, the smaller ∂θ/∂t is; the smaller w is, the bigger ∂θ/∂t is). After Chapter 2 as mathematical
tools this thesis consists of two parts. In Part I, which is divided into Chapters 3, 4, 5 and 6, we
consider time-dependent constraints on heat equations directly. More precisely, the restriction w
in Part I satisfies the condition with given functions ψ ,ψ : Q → R,
1 2
⎧
⎪⎪⎪ψ1 ≤ θ ≤ ψ2 in Q,
⎨
w = 0 in Q[ψ < θ < ψ ],
1 2
⎪⎪w ≤ 0 in Q[θ = ψ ],
⎪ 1
⎩
w ≥ 0 in Q[θ = ψ ]
2
(see Figure 1.1). Here Q[∗] means the set of elements in Q satisfying ∗. Next, in Part II, which
w
ψ
1
O ψ θ
2
Figure 1.1: Restriction in Part I
1
is divided into Chapters 7 and 8, we deal with hysteresis on heat equations which is indirect
constraints. In more detail, the restriction w in Part II satisfies the condition
⎧
⎪⎪⎪ψ1(θ) ≤ w ≤ ψ2(θ) in Q,
⎪
⎪∂w
⎪
⎪⎪⎨ ∂t = 0 in Q[ψ1(θ) < w < ψ2(θ)],
∂w
⎪⎪⎪⎪⎪ ∂t ≥ 0 in Q[w = ψ1(θ)],
⎪
⎪
⎪⎩∂w
≤ 0 in Q[w = ψ (θ)],
2
∂t
with a given function ψ : θ ∈ R (cid:9)→ ψ (θ) ∈ R for i = 1,2 (see Figure 1.2). Problems concerned
i i
w
ψ ψ
2 1
O θ
Figure 1.2: Restriction in Part II
with heat equations, Navier–Stokes equations and system of them have been dealt with in various
works. On the other hand, heat equations with the restriction w is also considered by many
authors (see Sections 1.1.1 and 1.2.1). However there is very few studies on heat equations with
the restriction w coupled with Navier–Stokes equations. In this thesis we study for such problem
and mainly deal with solvability of the problem.
1.1 Heat equations with constraints and Navier–Stokes
equations
Let T > 0. Let Ω ⊂ RN (N = 2,3) be a bounded domain with smooth boundary Γ. In Part I
we consider a system of heat equations with constraints and Navier–Stokes equations such as the
following problem (P ):
0
⎧
⎪⎪⎪ψ1 ≤ θ ≤ ψ2 in Q = (0,T)×Ω,
⎪
⎪⎪∂θ
⎪⎪⎪ −Δpθ+v ·∇θ+|θ|q−1θ−αθ = f in Q[ψ1 < θ < ψ2],
⎪∂t
⎪
⎪
⎪
⎪⎪⎪⎪⎪⎪∂∂θt −Δpθ+v ·∇θ+|θ|q−1θ−αθ ≥ f in Q[θ = ψ1],
⎪
⎨
∂θ
(P ) −Δ θ+v ·∇θ+|θ|q−1θ−αθ ≤ f in Q[θ = ψ ],
0 ⎪⎪∂t p 2
⎪
⎪
⎪
⎪∂v
⎪⎪⎪ −Δv +(v ·∇)v = g(θ)−∇π in Q,
⎪⎪∂t
⎪
⎪⎪divv = 0 in Q,
⎪
⎪
⎪
⎪⎪θ = 0, v = 0 on Σ := (0,T)×Γ,
⎪
⎩
θ(0,·) = θ , v(0,·) = v in Ω.
0 0
2
Here θ : Q → R, v : Q → RN and π : Q → R stand for the temperature, the velocity and the
pressure, respectively, and are unknown functions; ψ ,ψ : Q → R are given obstacle functions
1 2
with ψ ≤ ψ ; f : Q → R, g : R → RN, θ : Ω → R, v : Ω → RN, p ≥ 2, q ≥ 1 and α ∈ R are also
1 2 0 0
given; Δ is the so-called p-Laplace operator defined as Δ θ := div(|∇θ|p−2∇θ). Moreover Q[∗]
p p
means the set of elements in Q satisfying ∗.
From a viewpoint of physics, the problem (P ) (practically in the case p = 2) represents the
0
model describing temperature θ = θ(t,x) and velocity v = v(t,x) of incompressible fluid at each
place x on the domain Ω at each time t ∈ [0,T]. Artificially, we impose constraint on temperature.
In more detail, the equation
∂θ
−Δ θ+v ·∇θ+|θ|q−1θ−αθ = f
p
∂t
holds when ψ < θ < ψ . On another hand if θ = ψ , then the left-hand side of the above equation
1 2 1
including the time derivative ∂θ/∂t bigger than the right-hand side so that the constraint keeps.
On the other hand if θ = ψ , similarly, then the left-hand side is smaller than the right-hand side.
2
1.1.1 Related works on the problem (P )
0
Problems associated with systems of heat equations and Navier–Stokes equations (Boussinesq sys-
tem) are studied in many papers. Morimoto [41] showed existence for a Boussinesq system under
a mixed boundary condition. D´ıaz–Galiano [11] studied the Boussinesq system with nonlinear
thermal diffusion. Lorca–Boldrini [38] proved well-posedness for a generalized Boussinesq system.
Fukao–Kenmochi [15] derived an existence result for a Stefan problem in a non-cylindrical domain.
Kubo [34] established solvability of a heat equation with a nonlinear heat flux coupled with an
incompressible Navier–Stokes equation under the Dirichlet boundary condition. Fukao–Kubo [17]
also considered it under the Neumann boundary condition in a three-dimensional domain. Larios–
Lunasin–Titi [36] proved well-posedness for an anisotropic Boussinesq equation. Li–Xu [37] estab-
lished existence and uniqueness for an inviscid Boussinesq equation with temperature-dependent
thermal diffusivity. Miao–Zheng [39] gave an existence result for a Boussinesq equation with hori-
zontal dissipation. Fukao–Kenmochi [16] showed existence for a three-dimensional Navier–Stokes
equation with a temperature-dependent constraint coupled with a heat equation. The mathemat-
ical theory of 2D and 3D Navier–Stokes equations is discussed in detail in Temam [52], [53] and
Constantin–Foias [9].
Concerning problems with obstacle functions such as ψ , ψ in (P ), we can find some mathe-
1 2 0
matical studies. Brezis–Crandall–Pazy [5] originally dealt with an obstacle problem in the frame-
work of evolution equations in the case where obstacle functions are independent of t ∈ [0,T]. In
the case where obstacle functions depend on t ∈ [0,T], Yamazaki–Ito–Kenmochi [60] first studied
such obstacle problem under strict conditions on obstacle functions. After that, Fukao–Kubo [18]
dealt with such time-dependent obstacle problem without strict conditions by employing a similar
argument in [5].
In particular, Fukao–Kubo [18, Theorem 1] asserts that in the case where N = 2, p = 2, q = 1
and α = 1, namely, the heat equation is the form
dθ
−Δθ+v ·∇θ = f,
dt
3
there exists a unique solution (θ,v) to (P ) without the nonlinear term |θ|q−1θ such that
0
θ ∈ H1(0,T;L2(Ω))∩L∞(0,T;H1(Ω))∩L2(0,T;H2(Ω)),
0
v ∈ H1(0,T;(H1(Ω))∗)∩L∞(0,T;L2(Ω))∩L2(0,T;H1(Ω))
σ σ σ
under the following conditions:
• ψ ,ψ ∈ H1(0,T;L2(Ω))∩L2(0,T;H2(Ω))∩L∞(0,T;L∞(Ω)),
1 2
ψ ≤ ψ in [0,T]×Ω, ψ ≤ 0 ≤ ψ on [0,T]×Γ;
1 2 1 2
• f ∈ L2(0,T;L2(Ω)), g ∈ Lip(R;R2);
• θ ∈ H1(Ω) with ψ (0) ≤ θ ≤ ψ (0), v ∈ L2(Ω),
0 0 1 0 2 0 σ
2 1
where L (Ω) and H (Ω) are the Lebesgue and Sobolev spaces of functions v satisfying divv = 0
σ σ
and Lip(R;R2) is the set of Lipschitz continuous functions defined on R to R2.
However, in view of physics, there are several subjects on the problem dealt with in [18]. One is
that the heat equation does not have nonlinear terms while natural phenomena can have nonlinear
effect. Another is that the problem was considered in 2D domain while the natural world is in 3D
domains. Therefore the purposes of Part I are as follows:
(I) To nonlinearize the heat equation.
(II) To deal with the problem in 3D domains.
1.1.2 Problem in Chapter 3
In Chapter 3 we consider a nonlinear term
|θ|q−1θ
and add it to the heat equation, that is to say, we deal with the following problem (P ):
1
⎧
⎪⎪⎪ψ1 ≤ θ ≤ ψ2 in Q = (0,T)×Ω,
⎪
⎪⎪∂θ
⎪⎪⎪ −Δθ+v ·∇θ+|θ|q−1θ = f in Q[ψ1 < θ < ψ2],
⎪∂t
⎪
⎪
⎪
⎪⎪⎪⎪⎪⎪∂∂θt −Δθ+v ·∇θ+|θ|q−1θ ≥ f in Q[θ = ψ1],
⎪
⎨
∂θ
(P ) −Δθ+v ·∇θ+|θ|q−1θ ≤ f in Q[θ = ψ ],
1 ⎪⎪∂t 2
⎪
⎪
⎪
⎪∂v
⎪⎪⎪ −Δv +(v ·∇)v = g(θ)−∇π in Q,
⎪⎪∂t
⎪
⎪⎪divv = 0 in Q,
⎪
⎪
⎪
⎪⎪θ = 0, v = 0 on Σ = (0,T)×Γ,
⎪
⎩
θ(0,·) = θ , v(0,·) = v in Ω.
0 0
Here T > 0 and Ω ⊂ R2 is a bounded domain with smooth boundary Γ; θ : Q → R, v : Q → R2
and π : Q → R stand for the temperature, the velocity and the pressure, respectively, and are
4
unknown functions; ψ ,ψ : Q → R are given obstacle functions with ψ ≤ ψ ; f : Q → R,
1 2 1 2
g : R → R2, θ : Ω → R, v : Ω → R2 and q ≥ 1 are also given.
0 0
Existence and uniqueness for the problem (P ) without the nonlinear term |θ|q−1θ have been
1
obtained by [18] under the assumption ψ ,ψ ∈ L∞(0,T;L∞(Ω)) mentioned above. However,
1 2
solvability is not obtained in [18] when ψ ,ψ (cid:11)∈ L∞(0,T;L∞(Ω)). In this case the nonlinear term
1 2
|θ|q−1θ has a meaning because it is not regarded as a Lipschitz perturbation. So these problems
are still open as in Table 1.1 below.
Solvability ψ ,ψ ∈ L∞(Q) ψ ,ψ (cid:11)∈ L∞(Q)
1 2 1 2
(P) without |θ|q−1θ Fukao-Kubo [18] Unknown (i)
(P) (with |θ|q−1θ) Lipschitz perturbation of [18] Unknown (ii)
Table 1.1: Solvability for (P)
Our purpose in Chapter 3 (based on Sobajima–Tsuzuki–Yokota [50]) is to solve the unknown
problems (i) and (ii) in Table 1.1, that is, to establish existence and uniqueness of solutions to the
problem (P ) which includes the nonlinear term |θ|q−1θ. Indeed, (i) is affirmatively answered and
1
(ii) is solved under the mild condition ψ ,ψ ∈ L∞(0,T;L∞(Ω)).
1 2
1.1.3 Problem in Chapter 4
In Chapter 4 we consider a nonlinear diffusion
Δ θ = div(|∇θ|p−2∇θ)
p
in the heat equation with a logistic term |θ|q−1θ −αθ, that is to say, we deal with the following
problem (P ):
2
⎧
⎪⎪⎪ψ1 ≤ θ ≤ ψ2 in Q = (0,T)×Ω,
⎪
⎪⎪∂θ
⎪⎪⎪ −Δpθ+v ·∇θ+|θ|q−1θ−αθ = f in Q[ψ1 < θ < ψ2],
⎪∂t
⎪
⎪
⎪
⎪⎪⎪⎪⎪⎪∂∂θt −Δpθ+v ·∇θ+|θ|q−1θ−αθ ≥ f in Q[θ = ψ1],
⎪
⎨
∂θ
(P ) −Δ θ+v ·∇θ+|θ|q−1θ−αθ ≤ f in Q[θ = ψ ],
2 ⎪⎪∂t p 2
⎪
⎪
⎪
⎪∂v
⎪⎪⎪ −Δv +(v ·∇)v = g(θ)−∇π in Q,
⎪⎪∂t
⎪
⎪⎪divv = 0 in Q,
⎪
⎪
⎪
⎪⎪θ = 0, v = 0 on Σ = (0,T)×Γ,
⎪
⎩
θ(0,·) = θ , v(0,·) = v in Ω.
0 0
Here θ : Q → R, v : Q → R2 and π : Q → R stand for the temperature, the velocity and the
pressure, respectively, and are unknown functions; ψ ,ψ : Q → R are given obstacle functions
1 2
with ψ ≤ ψ ; f : Q → R, g : R → R2, θ : Ω → R, v : Ω → R2, q ≥ 1 and α ∈ R are also given.
1 2 0 0
The p-Laplacian Δ θ implies the effect of highlighting extreme values in the temperature dis-
p
tribution, and such effect becomes stronger depending on largeness of p. Accordingly, convective
5