Lecture 4. The parabolic equations and time dependent Stokes problem Ching-hsiao (Arthur) Cheng Department of Mathematics National Central University Taiwan, ROC The National Center for Theoretic Sciences, Summer 2012 Ching-hsiao Cheng Lecture 4. The parabolic theory and Stokes equations
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Lecture 4. The parabolic equations and timedependent Stokes problem
Ching-hsiao (Arthur) Cheng
Department of MathematicsNational Central University
Taiwan, ROC
The National Center for Theoretic Sciences, Summer 2012
Ching-hsiao Cheng Lecture 4. The parabolic theory and Stokes equations
Parabolic equations
Let Ω ⊆ Rn be a bounded and smooth domain. We consider
ut + Lu = f in Ω× (0,T ),
u = u0 on Ω× t = 0,boundary conditions on ∂Ω× (0,T ),
where Lu is a (time-dependent) uniformly elliptic operator
defined byLu = − ∂
∂xi
(aij ∂u∂xj
)+ bi ∂u
∂xi+ cu.
Here the coefficients a, b, c may depend on t . We recall that L
is called uniformly elliptic if there exists constant λ > 0 such that
aijξiξj ≥ λ|ξ|2 ∀ ξ ∈ Rn.
∂t + L is called uniformly parabolic if L is uniformly elliptic.
Ching-hsiao Cheng Lecture 4. The parabolic theory and Stokes equations
Boundary conditions
Two types of boundary conditions are considered.
1 Dirichlet boundary condition:
u = 0 on ∂Ω× (0,T ).
2 Neumann boundary condition:
aij ∂u∂xj
Ni = g on ∂Ω× (0,T ).
A Robin type of boundary condition can also be considered, but
the theory behind that is similar to the Neumann problem, so
we ignore the discussion of such boundary condition.
Ching-hsiao Cheng Lecture 4. The parabolic theory and Stokes equations
The parabolic equations
In other words, we consider the Dirichlet problem
ut + Lu = f in Ω× (0,T ),
u = u0 on Ω× t = 0, (D)
u = 0 on ∂Ω× (0,T ),
or the Neumann problem
ut + Lu = f in Ω× (0,T ),
u = u0 on Ω× t = 0, (N)
aij ∂u∂xj
Ni = g on ∂Ω× (0,T ).
Ching-hsiao Cheng Lecture 4. The parabolic theory and Stokes equations
The weak formulation and weak solutions
Assume that aij ,bi , c ∈ L∞(Ω× (0,T )), f ∈ L2(0,T ; L2(Ω)),
and u0 ∈ L2(Ω).
Definition (Weak solutions with Dirichlet boundary conditions)
A function u ∈ L2(0,T ; H10 (Ω)) with ut ∈ L2(0,T ; H−1(Ω)) is
said to be a weak solution of (D) provided that
〈ut , ϕ〉+ B(u, ϕ) =(f , ϕ)
L2(Ω)∀ϕ ∈ H1
0 (Ω), a.e. t ∈ (0,T ),
andu(0) = u0 ,
where B(u, ϕ) is defined by
B(u, ϕ) ≡∫
Ω
[aij ∂u∂xj
∂ϕ
∂xi+ bi ∂u
∂xiϕ+ cuϕ
]dx .
The integral equality is called the variational formulation of (D).
Ching-hsiao Cheng Lecture 4. The parabolic theory and Stokes equations
The weak formulation and weak solutions
Assume further that g ∈ L2(0,T ; L2(∂Ω)).
Definition (Weak solutions with Neumann boundary conditions)
A function u ∈ L2(0,T ; H1(Ω)) with ut ∈ L2(0,T ; H1(Ω)′) is saidto be a weak solution of (D) provided that
〈ut , ϕ〉+ B(u, ϕ) = (f , ϕ)L2(Ω) + (g, ϕ)L2(∂Ω)
∀ϕ ∈ H1(Ω), a.e. t ∈ (0,T ),
andu(0) = u0 ,
where B(u, ϕ) is defined by
B(u, ϕ) ≡∫
Ω
[aij ∂u∂xj
∂ϕ
∂xi+ bi ∂u
∂xiϕ+ cuϕ
]dx .
The integral equality is called the variational formulation of (N).
Ching-hsiao Cheng Lecture 4. The parabolic theory and Stokes equations
The meaning of u(0) = u0
For H = H10 (Ω;Rn) or H = H1(Ω), the initial value of a function
u ∈ L2(0,T ;H) does not make sense. However, since ut is
required to belong to L2(0,T ;H ′) in the definition of the weak
solution, the following “time embedding lemma”
Lemma
Suppose u ∈ L2(0,T ;H), with ut ∈ L2(0,T ;H ′) for some Hilbertspace H so that H →L2(Ω) ⊆ H ′. Then u ∈ C([0,T ]; L2(Ω)),and
maxt∈[0,T ]
‖u(t)‖L2(Ω) ≤(
1 +1T
)[‖u‖L2(0,T ;H) + ‖ut‖L2(0,T ;H ′)
].
suggests that u ∈ C([0,T ]; L2(Ω)); thus u(0) makes sense, and
limt→0+
‖u(t)− u0‖L2(Ω) = 0.
Ching-hsiao Cheng Lecture 4. The parabolic theory and Stokes equations
The existence and uniqueness of the weak solution
Note that the general form of the weak formulation above is
〈ut , ϕ〉+ (∇u,∇ϕ)L2(Ω) = F (ϕ) ∀ϕ ∈ H, a.e. t ∈ (0,T ).
Construction of a weak solution - the Galerkin method: Let
ek∞k=1 be an orthogonal basis in H which is orthogonal in Hand orthonormal in L2(Ω). For each k ∈ N, let
uk (x , t) =k∑
`=1
dk` (t)e`(x)
satisfy
(ukt (t), ϕ)L2(Ω) + B(uk (t), ϕ
)= F (ϕ) ∀ ϕ ∈ span(e1, · · · , ek ).
and
dk` (0) = (u0, e`)L2(Ω) ∀ 1 ≤ ` ≤ k .
Ching-hsiao Cheng Lecture 4. The parabolic theory and Stokes equations
The existence and uniqueness of the weak solution
Since every test function ϕ in the span can be written as a
linear combination of e1, · · · , ek , by the bi-linearity of B we find
that the equality above is equivalent tok∑
`=1
[dk ′` (t)(e`, ej )L2(Ω) + dk
` (t)B(e`, ej
)]= F (ej ) ∀ 1 ≤ j ≤ k .
Then dk (t) = [dk1 (t), · · · ,dk
k (t)]T satisfies the following ODE:
dk ′(t) + M(t)dk (t) = Fk (t),
where Mij = B(ei , ej)T, Fk = [F (e1),F (e2), · · · ,F (ek )]T. The
fundamental theorem of ODE suggests that dk exists in a time
interval [0,Tk ]. The goal is to show that the limit of uk , if exists,
is the weak solution.
Ching-hsiao Cheng Lecture 4. The parabolic theory and Stokes equations
The existence and uniqueness of the weak solution
Question 1: Is there a positive lower bound of Tk? If not, the
limit of uk means nothing.
Question 2: How do we ensure that uk converges? If uk does
converge, in what space and in what sense?
Answer: We need to look at the so-called energy estimates.
The starting point is that uk satisfies(ukt (t), ej
)L2(Ω)
+ B(uk (t), ej
)= F (ej ) ∀ 1 ≤ j ≤ k ;
thus by the bi-linearity of B and linearity of F ,(ukt (t),uk (t)
)L2(Ω)
+ B(uk (t),uk (t)
)= F
(uk (t)
)⇒ 1
2ddt‖uk (t)‖2
L2(Ω) + B(uk (t),uk (t)
)= F
(uk (t)
)
Ching-hsiao Cheng Lecture 4. The parabolic theory and Stokes equations
The existence and uniqueness of the weak solution
By the uniform ellipticity of L (or the parabolicity of ∂t + L),