Mathematical Theory of Boundary Layers and
Inviscid Limit Problem
Zhouping Xin
The Institute of Mathematical Sciences
The Chinese University of Hong Kong
International Summer School on Mathematical Fluid
Dynamics
Levico Terme, June 27 - July 2, 2010
Zhouping Xin
CONTENT
§0 Introduction and Overview
• Boundary Conditions;
• Boundary Layer Phenomena;
• Survey of the Progress.
§1 Convergence Theory for no-characteristic boundary condition
§2 Convergence Theory for linearized problem in the case non-slip
B.C.
§3 On Prandtl’s boundary layer equations
§4 Convergence results for Navier-Slip boundary conditions
§5 Criteria for the zero viscosity limit
§6 Asymptotic behavior of boundary layers
§7 A rigorous justification for an anisotropic viscosity case
§8 Questions and Remarks
Zhouping Xin
§0 Introduction and Overview
Problem: Determining the force of resistance experienced by a
solid body moving in a fluid.
Fact: Small forces of viscous friction may perceptibly affect the
motion of a fluid.
L. Prandtl: "Fluid Motion With Very Small Friction", ICM,
Heidelberg, 1904.
⇒ Theory of Boundary Layers: near the physical boundary, the
fluid is governed NOT by the ideal Euler system, but a
"simpler" system, Prandtl system!
Zhouping Xin
References:
• H. Schlichting, Boundary Layer Theory, 7th Edition,
McGraw-Hall, NY, 1987.
• O. A. Oleinik & V. N. Samoklin, Mathematical Models in
Boundary Layer Theory, Chapman Hall, London, 1999.
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Let Ω be a smooth simply-connected domain in Rn with smooth
boundary Γ = ∂Ω. Let the viscous fluid be governed by either
Compressible Navier-Stokes system (CNS)
∂t ρε + div (ρuε ) = 0
∂t (ρε uε ) + div (ρuε ⊗ uε ) + ∇p(ρε ) = div T ε
where
T = µ(∇uε + (∇uε )t ) + µ0 (divuε )I
with
p = p(ρ), p0 (ρ) > 0, ∀ρ > 0,
µ = ε2 , (ε > 0), µ0 = O(1)ε2 ,
or
Zhouping Xin
(CN S)
Incompressible Navier-Stokes equation (NS)
(
∂t u + (u · ∇)u + ∇p = 2 ∆u
∇ · u = 0
(N S)
: viscosity, u : velocity, p : pressure.
We supplement (CNS) (NS) with initial data
(ρε , uε )(x, t = 0) = (ρ0 , u0 )(x)
(or uε (x, t = 0) = u0 (x))
and various boundary conditions:
(1) Nonslip:
u |Γ = 0
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(N LB)
This is the common B.C. used frequently.
(2) Navier boundary condition (1823):
(
u · ~n = 0,
on Γ
(D(uε )~n)τ
= −αuετ
where τ is a tangential direction on Γ, and
D(u) = (∇u + (∇u)t ).
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(N BC)
A very interesting special case is the vorticity free B.C.
uε · ~n = 0,
(∇ × uε ) · τ = 0 onΓ
(SN BC)
which have appeared in many problems, see Beavers-Joseph’s
law, Saffman, Serrin, Marusic-Paloka, etc.
Obviously,
α → 0, (N BC) ⇒ uε |Γ = 0
(non-slip) (NLB)
α → ∞, (N BC) ⇒ uε · ~n|Γ =
∂uετ
∂~
n |τ
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= 0 (complete slip)
Background:
Large eddy simulation (Galdi & Layton, ’00);
Effective BC for a flow over a rough surface (Jager &
Mikelic, ’01); Beavers-Joseph’s law for perforated
boundaries;
Generalized Navier BC to remove the un-physical
singularity for the motion of contact line (Qian, Wang &
Sheng, ’03).
GNBC
slip velocity ∼ shear stress + un-compensated Young stress
Well-posedness theory due to de Veiga (’05), Yodovitch
(1963), J. L. Lions (1969), etc.
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(3) In-flow-out-flow boundary condition (Non-characteristic
B.C.s):
uε = φ,
φ · n 6= 0
on Γ
The corresponding ideal fluid is governed by either the
compressible Euler system:
∂t ρ + div (ρu) = 0
∂t (ρu) + div (ρu ⊗ u) + ∇p = 0
or the incompressible Euler system
∂t u + u · ∇u + ∇p = 0
div u = 0
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(N CBC)
(CE)
(IE)
with the same initial condition and the associated B.C.s
u · n|Γ = 0
(f or N LB or N BC)
or
u · n|Γ = ϕ · n 6= 0
(N CBC)
For compressible flows, more boundary conditions may be
needed depending on the sign of φ · n|Γ in the later case.
Zhouping Xin
Question: → 0?
(1) Whether the viscous flow can be approximated by an ideal
flow? (Inviscid Limited Problem), i.e., the asymptotic
relation between viscous flow and the corresponding ideal
flow.
(2) What is the asymptotic behavior of the viscous solutions to
IBVP uniformly up to the boundary? Are there boundary
layers?
These questions are important both theoretically and physically
due to the appearance of the Boundary-Layer Phenomena.
Indeed, due to the discrepancies of the B.C.s for the viscous and
ideal system, in general, there exists a thin layer near the
boundary Γ = ∂Ω × (0, T ) where the velocity changes
dramatically in the limit ε → 0+ .
Zhouping Xin
By multi-scale asymptotic formal analysis, one expects the
following asymptotic pictures:
I.
O(ε ) = O( µ )
Euler flow
(PDE)
Prandtl’s system (PDE)
Type I. NSB
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II.
N.S. ≈ Euler
Type II. Navier type slip B.C.s
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III.
O(ε 2 ) = O( µ )
Euler System
(PDE)
ODE
Type III. Non-characteristic B.C.s
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Question:
Can one justify rigorously such pictures?
=⇒ Mathematical theory of boundary layers!!
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Main Known results
Case 1: Nonslip BC
This has been the most studied case. Huge literature exists for
studies of large Reynolds number flow with no-slip B.C’s from
various point of views. Yet very little rigorous theory exists for
the unsteady boundary layer behavior of solutions to the N.S.
for both incompressible and compressible fluids, except the case
of analytic data (Asano, Caflisch-Sarmartino) and linearized
problems (Teman-Wang, Xin-Yanagisawa).
Zhouping Xin
(1) T. Kato (’84), R. Temam & X. M. Wang (’01) et al.:
ku − ukL∞ (0,T ;L2 ) → 0
(ZV L)
m
Z
0
T
k∇u (t)k2L2 (Γ ) dt → 0
(Γδ = {x ∈ Ω| dist(x, Γ) ≤ δ})
BUT,
y
( → 0)
kw(t, x, β )kL2x,y → 0
for any β > 0 (with Ω = {x ∈ R, y > 0}), so (ZVL) has no any
information on the behavior of boundary layers.
These criteria implies that the creation of the vorticity near
the boundary ins the key issue to study the inviscit limit
problem.
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(2) Prandtl’s boundary layer system.
Ω = R2+ = {x1 , x2 ), x2 ≥ 0}
x2
x1
~u = 0 (N S),
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u2 = 0 (IE)
By applying L. Prandtl’s ansatz: (ε =
√
(~u, p)(x1 , x2 , t) = (u1 , ε u2 , p)(x, y, t),
µ),
x = x1 ,
y=
x2
,
ε
one obtains that the leading order approximations of the
velocity in the boundary layer are given by

1

 ∂t u1 + ∂x ( u21 ) + u2 ∂y u1 + ∂x p(x, t) = ν ∂y2 u1

2
(P BS)
∂x u1 + ∂y u2 = 0


 (u1 , u2 )|y=0 = 0,
lim u1 (x, y, t) = U (x, t),
y→+∞
where U (x, t) and p(x, t) are determined by the corresponding
Euler flow through the Bernoulli’s law
∂t U + U ∂x U + ∂x p = 0.
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(B)
Main Difficulty I: Well-posedness theory in Hölder Sobolev’s
spaces for the IBVP of (PBS). Since (PBS) is a severely
degenerate Parabolic-elliptic system, even local in time theory
in unknown; (However, some recent progress by Gérard-Varet
and Dormy) except:
Oleinik (’60) et al.: local classical solutions to (Prandtl)
under certain monotonic assumption on flow;
Z. P. Xin & L. Q. Zhang (’04): global classical solution to
(Prandtl) under Oleinik’s monotonic assumption;
Sammartino & Caflisch (’98): well-posedness of the
(Prandtl), and a rigorous justification of Prandtl’s ansatz in
the Analytic Case.
Zhouping Xin
Main Difficulty II: Even in the case that the Prandtl’s system
(PBS) can be solved, the convergence of the viscous solutions as
ε → 0+ still poses an extremely difficult problem due to the
width of the boundary layer region! Even for linear problems,
most of the previous works deals with L2 -convergence theory for
very special flows. This forms a sharp contrast to the
non-characteristic boundary case!
Here, we will present a general result on the linearized N-S
system.
Zhouping Xin
Case 2: Slip Case
Convergence
Yodovich (60’s), Lions (60’s): 2-d, α = +∞;
Xin & Xiao (’07), da Veiga & Crispo (’09): 3-d, α = +∞;
Mikelic, Iftimie, Lopes et al.: α = 1
ku − ukL∞ → 0
Boundary layer: Iftimie & Sueur (’08), α = 1
ku − u −
√
y
w(t, x, √ )kL∞ (0,T ;L2 ) = O()
We will present some recent results on this later.
Major problem: strong convergence for general domains!
Zhouping Xin
Case 3: In-flow-Out-flow Boundary Conditions
This is a relatively easy case at least for short time. In this case,
the boundary is uniformly non-characteristic. The known results
can be summarized as: any "weak" boundary layers are stable
and the viscous solutions converge uniformly to the
corresponding Euler solutions away from the boundaries as
µ → 0+ before singularity formations.
Zhouping Xin
In particular, detailed asymptotic structure of the viscous
solutions to N.S. can be found by matched multi-scale
expansions and justified rigorously. This theory holds for both
incompressible and compressible fluids, see X. M. Wang,
Wang-Temen, Grenier, Xin, etc., Metivier-Geys, Rauch,
Gilslon-Serre, Gue’s.
Some open problems in this case include:
instability of strong boundary layers;
interaction of singularity and boundary layers.
Zhouping Xin
Goals of this talk:
(1) To study the well-posedness theory for Prandt’l boundary
layer equation.
(2) Some convergence results.
(3) Establish some criteria for
(ZVL)
ku − ukL∞ (0,T ;L2 ) −→ 0
for all kinds of dependence of α on .
(4) Asymptotic behavior of boundary layers by using
multi-scale analysis.
(5) Discussion of the rigorous theory on the boundary layer
behavior.
Zhouping Xin
§1 Convergence for Non-Characteristic B.C.s
This is the relatively easy case. Roughly speaking, the most
general results can be stated as:
"Theorem 1.1" Assume that (NCBC) holds. Then any "weak"
boundary layers are nonlinearly stable and the viscous solutions
converge uniformly to the corresponding inviscid solutions away
from the boundaries as ε → 0+ before shock singularity.
To be precise, we consider the special case:
Ω = R1+ × R1 = {(x1 , x2 ) ∈ R2 , x1 ≥ 0}
(1.1)
Then the NCBC becomes
u1 (x1 = 0, x2 , t) = f1 (x2 , t) > 0
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(1.2)
x2
x1
It can be shown that under suitable constraints on the
amplitude of f1 , (1.2) for the (CE) on the region (1.1) satisfies
the "Uniform Lopatinski Condition", so the mixed problem for
(CE), I.C. and (1.2) on Ω ((1.1)) has a unique smooth solution
(ρ0 , u~0 ) with
(ρ0 − ρ, u~0 ) ∈ C l ([0, T∗ ) : H l+3 (R1+ × R1 ))
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(1.3)
where l ≥ 1, and T∗ > 0 is the maximal interval of such a
solution. Set
δ(T ) ≡ max ||u02 (0, ·, t) − f2 (·, t)||L∞ (R1 ) , ∀T < T∗
(1.4)
[0,T ]
In this case, "Theorem 1.1" can be stated more precisely as
follows:
Theorem 1.1 Let (ρ0 , u~0 ) be the solution to the IBVP for (CE)
as in (1.3) with l ≥ 2. Let T ∈ (0, T∗ ) be fixed. Then ∃ positive
constants δ0 and ε0 independent of ε such that if
δ(T ) ≤ δ0 .
Then ∃| solution (ρε , u~ε ) to be IBVP for (CNS) such that
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(1.5)
(ρε − ρ, u~ε ) ∈ C 1 ([0, T ]; H 4 (R1+ × R1 ))
(1.6)
and
sup ||(ρε , u~ε )(x1 , 0, t) − (ρ0 , u~0 )(x1 , ·, t)||L∞ (R1 ) ≤ ch µ ≡ ch ε2 (1.7)
0≤t≤T
x1 ≥h>0
provided that 0 < ε ≤ ε0 . Furthermore, for any given n > 0,
there exists a smooth bounded function
˜)(x, t, n, ε)
(ρ̃, ~u
such that
˜)(·, t)|| ∞ 1 1 ≤ cn ε2n
sup ||(ρε , u~ε )(·, t) − (ρ̃, ~u
L (R+ ×R )
0≤t≤T
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(1.8)
where
˜)(x, t) = (ρ0 , u~0 )(x, t)
(ρ̃, ~u
n
X
x1
i−1 ~
+
µ Bi
, x2 , t
µ
i=1
+
n
X
µi I~i (x1 , x2 , t)
i−1
Boundary Layer Expansion,
faster dynamics
Inner Expansion,
slower dynamics
Remark 1.1 Due to the compatibility of the initial data and
boundary date, (1.5) holds for T small. Thus, the short time
convergence is a simple consequence of Theorem 1.1.
Zhouping Xin
(1.9)
Remark 1.2 (1.8)-(1.9) gives detailed asymptotic behavior of
the solutions to the IBVP for the CNS up to any given order.
˜) in (1.9) can be constructed
Since the asymptotic solution (ρ̃, ~u
explicitly by the method of matched asymptotic expansions.
Similar ideas have been used by Xin, Gues, Gues-Grenier, for
uniform parabolic systems before.
Remark 1.3 How about the stability of strong boundary
layers? The example given by Grenier-Metwier-Rauch for an
uniformly parabolic system suggests that a strong boundary
layer might be UNSTABLE!!! However, this is still OPEN for
the N-S systems. It also should be noted that this instability
never happens for scalar equations (Xin (1998)).
Zhouping Xin
§2 Boundary Layer Theory for Linearized System with
NSB
In the case of non-slip boundary condition, NSB, the rigorous
theory of small dissipation limit problem for either CNS or INS
is extremely difficult in general. Some light can be shed for the
linearized problems. For example, asymptotic analysis of Oseen
type equations in a channel has been successful analyzed
(Teman & Wang, IUMJ, 1996). Here we present a similar
theory for the linearization of the CNS around a given profile,
and justify the Prandtl’s boundary layer theory rigorously in
this case. Roughly speaking, we have
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"Theorem 2.1" Consider a linearized CNS system with
non-slip boundary conditions. Then the boundary layer is
stable, and the viscous solutions converge uniformly to the
inviscid solution away from the boundary.
To be precise, let’s still consider the half-space problem:
Ω = R1+ × R1 . For a given smooth flow
(ρ1 , u01 , u02 )(x1 , x2 , t) ∈ cl ([0, T ], H l+2 (Ω))
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(2.1)
with the property:
1
ρ ≥ c0 > 0
on Ω × [0, T ]
u01 (x1 , x2 , t) ≡ on ∂Ω × [0, T ]
(2.2)
Set


P (ρ)
U =  u1 
u2
(2.3)
Then the linearized N-S problem which we would like to study
is:





 (LCNS)
A0 (U )∂t U


(B.C.)



(I.C.)
M +Uε = 0
on
∂Ω × (0, T )
U ε (x, 0) = (p0 , u10 , u20 )t ≡ U0 (x)
0
ε
+
2
X
0
Aj (U )∂j U
2
2
2
= B(ε , cε )U
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ε
on
Ω × (0, T )
(2.4)
j=1
x∈Ω
where
0 1 0
=
 0 0 1
0
0
t
αu
e
j
j
0 
Aj =
ej βu0j I
β

0 0 0
0
0
0 1 0  ∆U + µ
(0, ∇t )U
∇
0 0 1
M+

α
A0 =  0
0
0
β
0
B(µ, µ0 )U ≡ µ 
α=
ρ0p
,
ρ0
e1 = (1, 0)t ,
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e2 = (0, 1)t ,
β = ρ0
The corresponding linearized inviscid problem is

2
X


0
0

A
(U
)∂
U
+
Aj (U 0 )∂j U 0 = 0 Ω × (0, T ) (LCE)
t
 0
j=1


M0 U0 = 0

 0
U (x, t = 0) = U0 (x)
(2.5)
where
M 0 = (0, 1, 0).
Zhouping Xin
(2.6)
Assume that the I.C. is compatible with the boundary
conditions in (2.4)-(2.5) up to higher order. Then the
well-posedness of both problems (2.4) and (2.5) in the Sobolev
space
C l ([0, T ], H m (Ω))
can be established quite easily. The goal is to give a Pointwise
estimate between U ε and U 0 . In this case, our "Theorem 2.1"
can be stated more precisely as
Theorem 2.1 Assume U0 is compatible with the B.C.,
M + U = 0. Let U 0 ∈ cl ([0, T ]; H m (Ω)) be the unique solution
to the IBVP (2.5) for any 0 < T < ∞. Then the unique smooth
solution U ε to the IBVP (2.4) admits the following estimates:
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sup
0≤t≤T
x1 ≥hε1−δ
x2 ∈R1


2


X
|ρ2 − ρ0 |(x1 , x2 , t) +
|u2j − u0j |(x1 , x2 , t) ≤ cδh ε (2.7)


j=1
where h > 0 is any finite number, 0 < δ < 1, 0 < ε < 1, and cεh
is a positive constant independent of ε. Furthermore, the
following asymptotic ansatz:
X
X
x1
U 2 (x, t) ∼ U 0 +
εi I i (x1 , x2 , t) +
εj B j ( , x2 , t) (AA)
ε
i≥1
j≥0
can be constructed and justified up to any order!
Remark 2.1 Similar conclusions hold for linearized
incompressible N-S system.
Zhouping Xin
Remark 2.2 The main tools are:
Matched Asymptotic Analysis + Weighted Energy Estimates
The key is to estimate a linearized Prandtl’s boundary layer
system. We can show that the linearized Prandtl’s system is
well-posed in a weighted Sobolev space. However, it should be
noted that our estimate here is not enough to handle the
nonlinear problem!
Remark 2.3 The nonlinear problem is still completely open!
Remark 2.4 The asymptotic ansatz (AA) can be constructed
by multi-scale matched asymptotic analysis. The inner functions
(boundary layer) and the outer functions can be constructed
simultaneously by solving IBVP’s for linearized Prandtl’s
systems and CE system.
Zhouping Xin
§3 On Prandtl’s boundary layer equations
For NSB, it is expected that the leading order approximation of
the flow velocity in the boundary layer is governed by the
Prandtl’s system. Thus the solvability of this system in the
standard Hölder or Sobolev’s spaces is of fundamental
importance. Along this line, the major previous rigorous results
are due to Oleinik, who gives some “local” well-posedness theory
in Holder space for a class of data (monotonic data).
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∂t u + u∂x u + v∂y u + ∂x p = ν∂y2 u
∂x u + ∂y v = 0
R = {(x, y, t), 0 ≤ x ≤ L, 0 ≤ y < +∞, 0 ≤ t < T }
(3.1)
(3.2)
The initial condition (I.C.)
u|t=0 = u0 (x, y)
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(3.3)
To be precise, consider a plane unsteady flow of viscous
incompressible fluid in the presence of an arbitrary injection and
removal of the fluid across the boundaries.
y=
Y
ε
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The boundary conditions:
u|x=0 = u1 (y, t), v|y=0 = v0 (x, t)
u|y=0 = 0,
limy→+∞ u(x, y, t) = U (x, t)
(3.4)
where the pressure p = p(x, t) is given by the Bernoulli’s law:
∂t U + U ∂x U + ∂x p = 0
(3.5)
with U = U (x, t) determined by the corresponding Euler flow.
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The class of data studied by Oleinik is described as
(H1 ) : U > 0, u0 > 0, u1 > 0, and v0 ≤ 0; and the data are
compatible.
(H2 ) (Monotonicity): ∂y u0 > 0, ∂y u1 > 0
Under the assumptions (H1 ) and (H2 ), O. A. Oleinik and her
collaborators have obtained a series of results, which can be
summarized as:
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Oleinik’s Theorem:
Under the assumptions (H1 ) and (H2 ) on the data, ∃1 classical
solution (u, v) to the IBVP (3.1)-(3.4) in the domain R provided
that T L is suitably small. The derivatives ∂t u, ∂x u, ∂y u, ∂y2 u,
and ∂y v are all continuous in R̄; moreover, the ration ∂y2 u/∂y u
and (∂y3 u ∂y u − ∂y2 u)/(∂y u3 ) are bounded in R.
One of the open problems listed at the end of the book written
by Oleinik and Sarmokin is:
What are the conditions ensuring the existence and uniqueness
of a solution of the nonstationary Prandtl’s system in the region
R with arbitrary T and L?
Zhouping Xin
The following are some recent results to answer this question:
Theorem 3.1 (Xin-Zhang, 2004)
Assume that
1. (H1 ) and (H2 ) hold;
2. ∂x p(x, t) ≤ 0, (Pressure favorable)
Then the IBVP, (3.1)-(3.4), has a global bounded weak solution
which is Lipsehitz continuous in both space and time.
In fact, such a solution is unique. Indeed,
Theorem 3.2 (Xin-Zhang-Zhao)
Under the same assumptions as in the Theorem 2.1, the weak
solution depends on the data continuously. In particular, such a
weak solution is unique!
More recently, we have shown such a weak solution is in fact a
classical solution.
Zhouping Xin
Theorem 3.3 (Xin-Zhang-Zhao)
Under the same assumptions as in Theorem 3.1, the unique
weak solution to the IBVP, (3.1)-(3.4), is smooth in the interior
of R. Furthermore, if the data are smooth then the weak
solution is a classical solution!
Remark 3.1 The boundary condition, ∂x p ≤ 0, is called
pressure favorable, is the stability condition for stationary
laminar boundary layers.
Remark 3.2 The monotonicity assumption (H2 ) is crucial in
the Theorem 3.3. Indeed, in the case ∂x p ≡ 0, Weinan E and
Bjorn Engquist have shown that a smooth solution to the
Prandtl’s system may blow-up in finite time without (H2 ).
Indeed, in the case that (H2 ) does not hold, even the local
existence of solution has not been established.
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Sketch of The Analysis
Using the generalized Crocco transformation
τ = t,
ξ = x,
η=
u(x, y, t)
,
U (x, t)
w=
∂y u(x, y, t)
,
U (x, t)
one can transform the original IBVP (3.1)-(3.4), into

2 2

 ∂τ w + ηU ∂ξ w + A∂η w + Bw = νw ∂η w, on Ω
∂y u0
w|τ =0 = U ≡ w0 , w|η=1 = 0

∂y u1 (y,t)
 w|
(νw∂η w − u0 w)|η=1 = ∂Ux p
ξ=0 ≡ w1 ≡ U (0,t) ,
(3.6)
(3.7)
where Ω = {(ξ, η, τ ); 0 < τ < ∞, 0 < η < L, 0 < η < 1}
A = (1 − η 2 )∂x U + (1 − η)
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∂t U
Ut
, B = η∂x U +
U
U
(3.8)
Set
ΩT = {(ξ, η, τ ) : 0 < τ < T, 0 < ξ < L, 0 < η < 1}
Definition (Weak solution):
A function w ∈ BV (ΩT ) ∩ L∞ (ΩT ) is said to be a weak solution
to the problem (3.7) if
(i) ∃C = C(T ) > 0 such that
C −1 (1 − η) < w(ξ, η, τ ) < C(1 − η),
on ΩT
1
(1 − η) 2 ∂η w ∈ L2 (ΩT ),
(ii) ∂η2 w is a locally bounded measure in ΩT , and
ZZ
(1 − η)2 d|wηη | < ∞
ΩT
Zhouping Xin
(3.9)
(3.10)
(3.11)
(iii) The boundary conditions in (3.7) are satisfied in the sense
of trace;
(iv) For any ϕ ∈ C 1 (ΩT ) with ϕ|τ =0 = ϕ|ξ=0 = ϕ|ξ=L = 0, the
following identify holds:
Z
ZZ
− w ϕ(1 − η) dξ dη|τ =τ +
w−1 (1 − η)2 ∂τ ϕ dξ dη dτ
ZZ
Z ZΩT
+
((1 − η)2 ϕ)η wη dξ dη dτ +
(ηU ϕ)ξ (1 − η)2 w−1 dξ dη dτ
Ω
Z ZΩT
ZTZ
+
((1 − η)2 Aϕ)η w−1 dξ dη dτ +
(1 − η)2 ϕBw−1 dξ dη dτ
ΩT
ΩT
Z τZ
L
+
v0 (ξ, τ )ϕ(ξ, η = 0, τ )dξ dτ = 0
0
−1
2
0
Zhouping Xin
Then, the main conclusions, Theorems 3.1-3.3, follow from the
following Proposition:
Proposition 3.4 Assume that the data satisfy (H1 ), (H2 ), and
pressure being favorable. Then
1. There exists a weak solution w ∈ BV (ΩT ) ∩ L∞ (ΩT ) to
(3.7) in the sense of the definition. Furthermore, ∀α > 0,
α−1
(1 − η) 2 ∂η w ∈ L2 (ΩT ),
ZZ
(1 − η)α d|wηη | < +∞.
ΩT
Zhouping Xin
2. Such a weak solution is unique and depends on the initial
and boundary data continuously in L1 -norms, i.e.,
Z LZ
1
|w(ξ, η, τ ) − w̃(ξ, η, τ )|dξ dη
0Z 0Z
τ 1
|w(L, η, τ ) − w̃(L, η, τ )|dη dτ
+
Z0 τZ0 L
|w(ξ, 0, τ ) − w̃(ξ, 0, τ )|dξ dτ
0 0
(
Z LZ 1
≤ C0
|w0 (ξ, η) − w̃0 (ξ, η)|dξ dη
0 0
Z τZ 1
|w1 (η, s) − w̃1 (η, s)|dη ds
+
)
Z0 τZ0 L
+
|v0 (ξ, s) − ṽ0 (ξ, s)|dξ ds
+
0
0
for any two weak solutions w(ξ, η, τ ) and w̃(ξ, η, τ ) with
corresponding initial and boundary data.
Zhouping Xin
3. Such a weak solution w(ξ, η, τ ) is smooth in the interior of
ΩT , and if the data are smooth, then it is a classical
smooth solution!
Remark 3.3 The existence of BV (Ωτ ) ∩ L∞ (Ωτ ) solution can
be proved by at least two ways, splitting method of Xin-Zhang
(2004), or by TV uniform estimate as follows:
We approximate (3.7) by (taking ν = 1)

2 2
 ∂τ u − (u + ε) ∂η u + (ηu + ε)∂ξ u + A∂η u + Bu = 0
u|τ =0 = uε0 , u|ξ=0 = uε1 , uε |η=1 = 0,
(3.12)

(u + ε)∂η u|η=0 = (v0ε (u + ε) + ∂Uε p )|η=0
with uε0 , uε1 , and v0ε being suitable regularization of u0 , u1 , and
v0 respectively.
Zhouping Xin
Then one can show that
ε
c+
1 (1 − η) ≤ u (ξ, η, τ ) ≤ c1 (1 − η),
Z
∀(ξ, η, τ ) ∈ QT . (3.13)
|∇uε (ξ, η, τ )dξ dη dτ ≤ c2 (QT )
(3.14)
QT
with c1 and c2 independent of ε. Here (3.13) follows from
comparison arguments while (3.14) involves weighted averging
estimate on the gradient! Then the existence follows from
taking the limit ε → 0+ . However, the arguments for the
uniqueness and continuous dependence are a bit more
complicated. The main strategy is show a L1 -contraction for
weak solutions to (3.7) motivated by the pioneering work of
Krushkov by using Krushkov entropy and BV calculus!
Zhouping Xin
Remark 3.4 The most difficult part of the analysis for the
proof of Proposition 4 is the regularity theory. Indeed, consider
the simple case U (x, t) ≡ 1. Then the equation in (3.7) becomes:
∂τ w + η∂ξ w = νw2 ∂η2 w
in ΩT
(3.15)
one needs to show that a weak solution w ∈ BV (ΩT ) ∩ L∞ (ΩT )
to (3.15) is smooth in the interior of ΩT .
Zhouping Xin
Note that
L0 = ∂η2 − ∂τ + η∂ξ
is a hypoelliptic operator and δλ -homogeneous of degree 2 with
some dialation group {δλ }λ>0
δλ = diag (λ3 , λ, λ2 )
(∵ rank Lie (∂η , ∂τ + η∂x ) = 3)
Thus, the regularity theory can be reduced to study the
C α -regularity of weak solution to
∂τ w + η∂ξ w = A∂η2 w,
Zhouping Xin
A ∈ L∞
(3.16)
(3.16) is a typical ultraparabolic equation while has been studied
extensively, see L. P. Kupcov (1977), Lanconelli-Polidoro (1994),
Polidoro (1998), Mantredini-Polidoro (1998), etc..
Some of the main difficulties:
1. Caccioppoli estimates ; H 1 -estimate;
2. ⇒ one cannot use Sobolev inequalities to do the
Nash-Moser iteration;
3. Even the one-side Nash-Moser iteration can be obtained, it
seems difficult to apply John-Nirenbergy inequality?
Zhouping Xin
Sketch of the regularity analysis (Krushkov’s entropy
estimates)
Consider the C α -regularity of the solution of
∂t u + y∂x u − (Auy )y = 0, y > 0,
(3.17)
at the point (x, y, t) = (0, 1, 0). Let r > 0, and
Br = {(x, y, t)||x−t| ≤ r3 , |y−1| ≤ r, |t| ≤ r2 }, Br+ = Br ∩{t ≤ 0}
Assume that A ∈ L∞ (B1 ) so that for a positive constant λ,
λ−1 < A < λ in B1 .
Define for 0 < α, β, r < 1,
Krt = {(x, y)||y − 1| ≤ r, |x − t| ≤ r3 }
t
Kβr
= {(x, y)||y − 1| ≤ βr, |x − t| ≤ (βr)3 }
Zhouping Xin
Proposition 3.5 Let u(x, y, t) ≥ 0 be a solution to (3.17) in
B1+ , and
mes{(x, y, t) ∈ Br+ , u ≥ 1} ≥ 1/2 mesBr+
(3.18)
Then ∃ positive constants
α, β, h, 0 < α 1, 1 > β ≈ 1, 0 < h < 1/2, depending on λ, such
that for all t ∈ (−αr2 , 0),
+
mes{(x, y) ∈ Kβr
, u ≥ 1} ≥
1
+
mesKβr
.
11
(3.19)
Proof. For u ≥ 0, 0 < h < 1/2, set
V = ln+
1
= (−ln(u + h9/8 ))+ .
u + h9/8
Zhouping Xin
(3.20)
Then V solves
1
∂t̄ V − (AVy )y + Avy2 = (1 − y)(∂t̄ V − ∂x̄ V )
2
(3.21)
where t̄ = (x + t)/2, x̄ = (t − x)/2.
Then a careful modification of the local average estimate of
Krushkov ⇒ (3.19).
The Next key estimate in our analysis is a “weak form" of the
Poincare’s inequality based on the fundamental solution of
(3.17).
For A = 1, the fundamental solution for (3.17) with pole at
(ξ, η, 0).
Zhouping Xin
Γ0 (z, (ξ, η, 0))
n
(√
2
3
exp
− (y−η)
−
2
4t
2πt
=
0,
3
(x
t3
o
− ξ − 2t (y + η)2 ) , t > 0
t ≤ 0.
(3.22)
Here z = (x, y, t). Then Γ0 (z, ζ) is the fundamental solution with
pole at ζ = (ξ, η, τ ) given as in (3.22) with t replaced by t − τ.
Define a ball B̃r centered at (0, 1, 0) as
B̃r = {(x, y, t)||x−ty| < r3 , |y−1| < r, |t| ≤ r2 }, B̃r+ = B̃r ∩{t ≤ 0},
K̃rt = {(x, y)||x − ty| ≤ r3 , |y − 1| ≤ r}.
Zhouping Xin
Then B̃r , K̃rt are equivalent to Br , Krt respectively.
Then Proposition 3.5 implies
Corollary 3.1: Under the same assumption as in Proposition
3.5, it holds that ∀t ∈ (−αr2 , 0),
t
mes{(x, y) ∈ K̃βr
, u ≥ h} ≥
Zhouping Xin
1
t
mesK̃βr
12
(3.23)
Proposition 3.6 Let w be a nonnegative weak subsolution to
(3.17) in B1 . Then ∃ constant C = C(λ) such that
Z
Z
r2
+ 2
|∂y w|2 dy
(3.24)
[(w(z) − I1 ) ] ≤ C
+
(1 − θ)2 B̃r+
B̃θr
where I1 = maxz∈B̃ + I1 (z) with
θr
Z
I1 (z) =
B̃r+
∂η ψ(ζ)∂η Γ0 (z, ζ)w(ζ)dζ
(3.25)
Z
+
B̃r+
(∂τ ψ(ζ) + η∂η ψ(ζ))Γ0 (z, ζ)w(ζ)dζ
ψ(z) = ψ(x, y, t)
= χ([
y+1 2
|t|1/2
1 (y − 1)2 2
(
t + 3(x −
t) − 3tr4 )]1/6 )χ( 3 )
10
4
2
θ
(3.26)
Zhouping Xin
χ ∈ C ∞ ([0, ∞)) such that


χ(s) = 1,
χ(s) = 0,


0 ≤ χ0 (s) ≤
2
(1−2θ)r ,
if s < 2θr,
if s > r,
if 2θr < s < r.
Remark 1) If u is a solution to (3.17), then w = (ln u+hh9/8 )+ is
a subsolution, so one can apply Proposition 3.6 to w.
2) (3.24) yields a Poincare type inequality for a subsolution to
(3.17).
Zhouping Xin
Lemma 3.7 Under the assumption of Proposition 3.5, ∃ an
absolute constant λ0 , 0 < λ0 < 1 such that
|I0 | ≤ λ0 ln(
1
h1/8
)
(3.27)
This follows from the structure of Γ0 and ψ and the following
facts:
Z
Z
+
1=
∂η ψ∂η Γ0 (z, ·)dζ +
(∂τ ψ + η∂ξ ψ)Γ0 (z, ·)dζ, ∀z ∈ B̃θr
+
B̃βr
+
B̃βr
+
+
mes{(B̃βr
\ B̃θr
)∩{z|u
(3.28)
1
+
mesB̃βr , for suitably small θ
≥ h}} ≥
15
(3.29)
Zhouping Xin
Based on Propositions 3.5, 3.6, and Lemma 3.7, it can be shown
that
Proposition 3.8 Suppose that u ≥ 0 be a weak solution to
(3.17) in B1+ and the condition (3.18) is satisfied. Then there
exist positive constants θ and h0 , 0 < h0 , θ < 1, depending only
on λ, such that
+
u(x, y, t) ≥ h0 in Bθr
.
Zhouping Xin
This Proposition can be proved by applying Propositions 3.5,
3.6, and Lemma 3.7 and the Morser’s iterative method
(established already by A. Pascucci & S. Polidoro) to
w = ln+ (
h
9/8
) ⇒ u ≥ h0 .
9/8
u+h
Now if one applies Proposition 3.8 to either
ũ± = 1 ±
u
, M = kukL∞ (B1 )
M
yields the desired oscillation estimate! ⇒ C α -regularity of u.
Zhouping Xin
§4 Convergence results for Navier-Slip
boundary conditions
The 3-dimensional incompressible N-S system can be written as
∂t u − µ∆u + w × u + ∇p = 0 in Ω
(4.1)
∇ · u = 0,
w =∇×u
with initial data
u|t=0 = u0 (x)
(I.C.)
(4.2)
and a slip boundary condition:
u · n = 0,
w×n=0
Zhouping Xin
on ∂Ω
(4.3)
It can be shown that
⇐⇒ u · n = 0,
⇐⇒ u · n = 0,
∂n uτ = 0 ⇐⇒
(4.4)
(D(u)n)τ = −kτ uτ
(4.5)
where kτ is the corresponding principal curvature of the
boundary ∂Ω.
We will assume that Ω ⊂ R3 is a simply connected, bounded
smooth domain.
Zhouping Xin
Set
D(Ω)
Dτ (Ω)
Dn (Ω)
D0 (Ω)
Hτs (Ω)
Hns (Ω)
W
=
=
=
=
=
=
=
C ∞ (Ω̄)
{u ∈ D(Ω); ∇ · u = 0, u · n = 0}
{u ∈ D(Ω); ∇ · u = 0, u × n = 0}
{u ∈ C0∞ (Ω); ∇ · u = 0}
{u ∈ H s (Ω); ∇ · u = 0, u · n = 0}, s ≥ 0
{u ∈ H s (Ω) ∩ H; ∇ · u = 0, u × n = 0}, s ≥ 1
{u ∈ Hτ2 (Ω); (∇×)u ∈ Hn1 (Ω)}.
Zhouping Xin
Then our first convergence result is as follows:
Theorem 4.1 (Xiao-Xin, 2007) Assume that
1. Let Ω = [0, 1]2per × (0, 1).
2. u0 ∈ W ∩ H 3 (Ω)
⇒ ∃ T0 independent of µ such that
(i) The problem (4.1)-(4.3) has a unique strong solution u with
√
the properties (ε = µ)
uε ∈ L2 (0, T0 ; H 4 (Ω))∩C([0, T0 ]; H 3 (Ω)), uε ∈ L2 (0, T0 : W )
Zhouping Xin
(ii) As ε =
√
µ → 0+ ,
in L∞ (0, T0 ; H 3 (Ω))
5
in C([0, T0 ]; H 2 (Ω))
uε −→ u0
uε −→ u0
5
and u0 ∈ C([0, T0 ]; H 2 (Ω)) is a unique solution to
∂t u + w × u + ∇p = 0
in Ω
∇ · u = 0, w = ∇ × u
with I.C. u|t=0 = u0
and B.C. u · n = 0 on ∂Ω.
(iii) The following convergence rate holds:
sup ||uε − u0 ||22 ≤ Cε on [0, T0 ]
0≤t≤T0
Zhouping Xin
However, for general simply-connected bounded domains, we
can only obtain the following result:
Theorem 4.2 Assume that
1. u0 ∈ W ∩ H 3 (Ω), and ∇ × u0 |∂Ω = 0.
2. u(x, t) is a smooth solution to the Euler system on [0, T ],
T > 0.
Then ∃ ε0 > 0, such that for all ε ≤ ε0 , the problem (4.1)-(4.3)
has a unique solution uε (x, t), t ∈ [0, T ] satisfying
ε
sup ||u (·, t) −
0≤t≤T
u(·, t)||21
Z
+ε
0
Zhouping Xin
T
||uε − u||22 dt ≤ cε2
Remark 4.1 In fact, it is not too difficult to show that if the
ideal Euler system has a smooth solution
u0 ∈ c1 ([0, T ], H 5 (Ω)) ∩ L2 ([0, T ], H 6 (Ω)) satisfying the
boundary condition u0 · n|∂Ω = 0 for some T > 0, then the
viscous problem, (4.1)-(4.3), has a unique solution
uε ∈ C([0, T ], H 1 (Ω)) ∩ L2 ([0, T ], H 2 (Ω)) for each fixed ε > 0
such that
1
||uε − u0 ||0 ≤ cε 4
(4.6)
and
uε · n = 0,
(D(uε ) · n)τ = αuετ
Zhouping Xin
(4.7)
for any fixed constant α. However, in general, the convergence
in (3.6) can not hold true for stronger norms, there will be
boundary layers.
Remark 4.2 In the general Navier-slip boundary condition, α
may depend on Reynolds number, α = α(Re ). See Section 6.
Example α = (Re )γ = 0(1)µ−γ . Then similar convergence
theory can be obtained for γ < 12 with varies order of boundary
layer conditions! (Wang-Wang-Xin)
Zhouping Xin
Ideas of proof:
Step 1: Hodge Decomposition and Div-Curl estimate
Set
R
Gc = {u = ∇ϕ; ∇ · u = 0, ∂Ωi u · n = 0 ∀i}
Gh = {u = ∇ϕ; ∇ · u = 0, ϕ = c(i) on ∂Ωi }
Gg = {u = ∇ϕ; ϕ = 0 on ∂Ω}
(4.8)
Then
D(Ω) = Dτ (Ω) ⊕ Gc ⊕ Gh ⊕ Gg
Dτ (Ω) = ∇ × Dn (Ω) = ∇ × (Dn (Ω) ∩ G⊥
h)
(4.9)
Set also
s
Hnc
= Hns (Ω) ∩ G⊥
h
Zhouping Xin
(4.10)
Lemma 1 Let u ∈ D(Ω), ∀ ∈ N . Then
||u||s ≤ c(||∇ × u||s−1 + ||∇ · u||s−1 + |uτ |s− 21 + ||u||s−1 ) (4.11)
=⇒ ||u||s ≤ c(||∇ × u||s−1 + ||u||s−1 )
if u ∈ Dτ (Ω) or u ∈ Dn (Ω), s ∈ N.
(4.12)
1 (Ω). Then
Lemma 2 Let u ∈ Hnc
||u|| ≤ c(||∇ × u||)
||u||s ≤ c||∇ × u||s−1
=⇒
if
u∈
s (Ω)
Hnc
Zhouping Xin
and s ∈ N.
(4.13)
(4.14)
As a consequence
(∇ × u, h) = (u, ∇ × h) = 0
Thus ∇ × ∇× :
∀u ∈ W,
h ∈ Gh
1 (Ω) −→ H 0 (Ω).
W 7−→ Hnc
τ
Proposition 4.3 The Stokes operator −∆ : W 7−→ Hτ0 (Ω) is
bijective and bounded, and its inverse (−∆)−1 is symmetric
positive and compact is Hτ0 (Ω) with
(−∆)−1 (Hτs (Ω)) = Hτs+2 (Ω) for s ≥ 0
In fact, −∆ is self-adjoint on Hτ0 (Ω) with D(−∆) = W .
Zhouping Xin
Step 2: Compatibility of the Nonlinearity with the B.C.
Set
w = ∇ × u,
B(u) = w × u + ∇p.
Proposition 4.4 B(u) ∈ D(Ω) ∩ W if u ∈ D(Ω) ∩ W .
Step 3: Galenkin Approximation and Weak Solution.
Step 4: Strong Solution and vanishing viscosity limit
Zhouping Xin
§5 Criteria for the zero viscosity limit
In this part, we shall give necessary and sufficient conditions to
have
lim ku − ukL∞ (0,T ;L2 ) = 0
(ZV L)
→0
u
where
and u are solutions to the Navier Stokes equations
(N S) and to the Euler equations (IE) respectively.
Only consider the case
α = µγ
for γ ∈ R.
Zhouping Xin
Weak solutions to (N S)-(NBC)
Let V = {u ∈ (H 1 (Ω))2 | ∇ · u = 0 in Ω, u · ~n = 0 on Γ}.
u ∈ Cw ([0, T ], L2 (Ω)) ∩ L2 (0, T ; V ) is a weak solution if:
(i) Energy inequality holds for 0 ≤ t ≤ T :
1 ku (t)k2 + µ
2
Z tZ
0
α−1 |u1 |2 dxds + µ
Γ
Z
0
t
1
k∇u (s)k2 ds ≤ ku0 k2 ;
2
(ii) u satisfies (N S) -(NBC) in weak form, i.e. for any
ϕ ∈ (H 1 (Ω))2 with ∇ · ϕ = 0 in Ω and ϕ · ~n = 0 on Γ, we have
u , ϕ (t) =
Rt
( u , ϕt + u , (u · ∇)ϕ −µ ∇u , ∇ϕ RtR
−µ 0 Γ α−1 u1 ϕ1 dxds+ u , ϕ (0)
0
where k · k ( ·, · resp) is the norm (inner product resp.) in
L2 (Ω).
Zhouping Xin
Initial data: u |t=0 = u0 ∈ L2 (Ω), u|t=0 = u0 ∈ H s (Ω)
(s > 25 ) satisfy
∇ · u0 = ∇ · u0 = 0
ku0 − u0 kL2 → 0 ( → 0)
Zhouping Xin
Theorem 5.1 The slip length α = µγ with γ ∈ R.
(1) When γ <
(2) When
1
2
1
2
(containing α = +∞, 1), (ZVL) holds always.
≤ γ < 1, (ZVL) ⇔
T
Z
k∇u (t)k2L2 (Γδ ) dt → 0
µ
0
for any fixed 0 < δ ≤ .
(3) When γ ≥ 1 (containing α = 0), (ZVL) ⇔
Z
µ
0
T
k∇u (t)k2L2 (Γ ) dt → 0
Zhouping Xin
(ED)
Proof of (2): (i) (ZVL) ⇒ (ED) is straightforward:
Energy inequality satisfied weak solution u , and (ZVL) ⇒
Z
lim sup µ
→0
T
k∇u (t)k2 dt +
0
Z TZ
0
µα−1 |u1 |2 dxdt
=0
Γ
⇒ (ED).
(ii) (ED) ⇒ (ZVL):
(a) Construction of a boundary layer profile
Similar to Kato, construct a boundary layer profile satisfying,
∇ · v = 0 in Ω,
v = u on Γ,
supp v ⊆ Γδ ,
where δ = δ() satisfies lim δ() = 0, to be determined later.
→0
Zhouping Xin
(b) It is easy to have
k(u − u)(t)k2 = ku (t)k2 + ku(t)k2 − 2 u , u (t)
≤ ku0 k2 + ku0 k2 − 2 u , u (t)
= o(1) + 2( u , u − v (0)− u , u − v (t))
Rt
= o(1) − 2 0 ( u , u · ∇(u − v) + u , ∂t (u − v) − ∇u , ∇(u − v) )(s)ds
Rt
= o(1) − 2 0 ( u − u, (u − u) · ∇u − u , u · ∇v + ∇u , ∇v )(s)ds
Rt
≤ o(1) + K 0 k(u − u)(s)k2 ds
Rt
+2 0 ( u , u · ∇v − ∇u , ∇v )(s)ds
Zhouping Xin
(c) Thus, to prove (ZVL), it suffices to verify
T
Z
lim
→0 0
( u , u · ∇v −µ ∇u , ∇v )(s)ds = 0.
Noting that only the tangential component v1 of v has rapidly
change in the normal direction, one has
Z
T
|µ
( ∇u2 , ∇v2 + ∂x u1 , ∂x v1 )(s)ds| = o(1)
0
and
RT
( u2 , (u · ∇)v2 + u1 , u1 ∂x v1 )(s)ds|
RT
≤ K 0 (δ 2 k∇u (s)k2L2 (Γδ ) + δku1 (s)k2L2 (Γ) )ds = o(1)
|
0
when δ ≤ C.
Zhouping Xin
It remains to verify
Z
T
Z
lim
→0 0
(u1 u2 ∂y v1 − µ∂y u1 ∂y v1 )dxdyds = 0
Γδ
Due to ∇ · v = 0 in Ω and v2 |Γ = 0, we can set ϕ = v(t, x, y) in
the weak form of (NS) to get
RT R
− µ∂y u1 ∂y v1 )dxdyds
RT
= u , v (T )− u , v (0) − 0 u , ∂t v (s)ds
RtR
+ 0 Γδ (µ∇u2 · ∇v2 + µ∂x u1 ∂x v1 − u2 (u · ∇)v2 − (u1 )2 ∂x v1 )dxdyds
RT R
+µ1−γ 0 Γ u1 u1 dxds
0
Γδ (u1 u2 ∂y v1
which goes to zero as → 0 when 0 < δ ≤ C under the
assumption (ED).
Zhouping Xin
As Temam-Wang, Kelliher did for the non-slip case, we also
have certain improvement for the previous results:
Theorem 5.2 When the slip length α = o(), (ZVL) if and
only if
Z
T
lim →0
0
k∂τ un (t)k2L2 (Γδ ) dt = 0
µ
= 0.
for a fixed δ = δ() = o(1) satisfying lim→0 δ()
Example: Consider a cylindrically symmetric flow in
Ω = {x ∈ R3 | x1 ∈ R, x22 + x23 ≤ 1}. Denote by (x1 , r, θ) the
cylindric coordinate in Ω. The velocity vector is
u (t, x) = u1 (t, r)~e1 + uθ (t, x1 , r)~eθ
where ~e1 = (1, 0, 0)T and ~eθ = 1r (0, −x3 , x2 )T . Obviously, we
have u · ~n ≡ 0. So, (ZVL) holds always when the slip length is
o().
Zhouping Xin
§6 Asymptotic behavior of boundary layers
As above, we shall only study the case α = µγ . By using
multi-scale analysis, we conclude:
(1) When γ > 12 , near Γ = {y = 0},
(
y
u1 (t, x, y) = uB
1 (t, x, ) + · · ·
y
u2 (t, x, y) = uB
2 (t, x, ) + · · ·
B
with (uB
1 , u2 )(t, x, η) satisfying

B
B
B
B
2 B

∂t uB

1 + u1 ∂x u1 + u2 ∂η u1 + ∂x P (t, x) = ∂η u1


∂ uB + ∂ uB = 0
x 1
η 2
B
B
u1 = u2 = 0
on η = 0



lim
B
η→+∞ u1 (t, x, η) = U (t, x)
(P randtl)
Zhouping Xin
(2) When γ = 12 , near Γ = {y = 0}, the solution u to (N S)
B
has the same expansion as above, with (uB
1 , u2 )(t, x, η)
satisfying

(P randtl)
uB = 0,
2
uB
1 −
Zhouping Xin
∂uB
1
∂η
=0
on η = 0
(3) When 0 < γ < 12 , near Γ = {y = 0},

1
y
−γ
u1 (t, x, y) = µ 21 −γ uB
2
)
1 (t, x, ) + o(µ
u (t, x, y) = µ1−γ uB (t, x, y ) + o(µ1−γ )
2
2
B
with (uB
1 , u2 )(t, x, η) satisfying linearized Prandtl
equations.
(4) When γ ≤ 0, Boundary layer appears at most at the order
O(), and boundary layer profiles satisfy linearized Prandtl
equations.
Zhouping Xin
In summary, we observed:
When the slip length α = µγ , γ =
the boundary layer behavior:
1
2
is critical in determining
I
When γ > 12 , the boundary layer profiles satisfy the same
boundary value problem for the nonlinear Prandtl
equations as in the non-slip case;
I
when γ = 12 , the boundary layer profiles satisfy the
nonlinear Prandtl equations but with a Robin boundary
condition for the tangential velocity;
Zhouping Xin
I
1
when γ < 12 , the boundary layer = O(µ 2 −γ ), and satisfies a
boundary value problem for linearized Prandtl equations,
but with the vorticity
rot u = O(µ−γ )
being unbounded when 0 < γ < 21 .
I
when γ = 0, i.e. the slip length is independent of the
viscosity , the boundary layer appear in the order O(),
which yields both of the convection term (u · ∇)u and the
vorticity rot u being bounded.
Zhouping Xin
§7 A rigorous justification for an anisotropic
viscosity case
Consider the following problem in {t, y > 0, x ∈ R} : (µ = 2 )

∂t u + (u · ∇)u + ∇p = ∂x2 u + µ∂y2 u





 ∇ · u = 0

(7.1)

− µ 41 ∂u1 = 0,
= 0 on y = 0

u
u

1
2
∂y



 u |t=0 = u0 (x, y)
(In this case, the boundary layer profiles satisfy the linearized
1
Prandtl equations, but the vorticity rot u = O(µ− 4 ) is
unbounded.)
Zhouping Xin
Suppose that u0 ∈ H s (Ω) with ∇ · u0 = 0, for a fixed s > 8.
(1) Approximate solutions: Let (u,a , p,a ):

P3
P3
k I,k
k B,k
y

4
4
u,a

1 (t, x, y) =
k=0 µ u1 (t, x, y) +
k=1 µ u1 (t, x, )



P3
3 B,3
k I,k
y
4
4
u,a
2 (t, x, y) =
k=0 µ u2 (t, x, y) + µ u2 (t, x, )




 p,a (t, x, y) = P3 µ k4 pI,k (t, x, y)
k=0
(7.2)
be an approximate solutions satisfying (N S) with errors O(µ).
Specially, (uB,j , pB,j )(x, z, t) → 0 exponentially in z → +∞. uI,0
solves the IE with uI,0 |t=0 = u0 (x, y), pI,1 being a constant. For
all j ≥ 2, (uI,j , pI,j ) solves the following linearized Euler systems:
Zhouping Xin
 I,j
I,0
I,j
I,j ∇)uI,0 + ∇pI,j

∂t u + (u · ∇)u + (u ·P



= ∂x2 uI,j−2 + ∂y2 uI,j−4 −
(uI,k · ∇)uI,j−k



1≤k≤j−1
∇ · uI,j = 0


R +∞

I,j
B,j−2


(t, x, ξ)dξ
u2 |y=0 = − 0 ∂x u1

 I,j
u |t=0 = 0,
(7.3)
and for all j ≥ 1, uB,j
satisfies the boundary value problem for a
1
linear degenerate parabolic equation:

I,0
B,j
I,0
B,j
I,0 B,j
2 B,j = f j

∂t uB,j

1
1 + u1 ∂x u1 + z∂y u2 ∂z u1 + ∂x u1 u1 − ∂z u1



∂ uB,j = β(uI,j−1 + uB,j−1 ) − ∂ uI,j−2 , on {z = 0}
z 1
1
1
y 1

lim uB,j

1 (t, x, z) = 0 exponentially

z→+∞


uB,j |
t=0 = 0,
1
(7.4)
Zhouping Xin
where
f1j =∂x2 uB,j−2
−
1
j−1
X

k
[2]
X
zn

k=1
n=0
n!

B,j−k 
∂x ∂yn uI,k−2n
uB,j−k
+ uB,k
1
1
1 ∂x u1
k
− ∂x p
B,j
−
[2]
j−3 X
X
zn
k=0 n=1
−
k
]+1
j−1 [ X
2
X
k=1 n=1
n!
∂yn+1 uI,k−2n
uB,j−k
1
2
z n n I,k+2−2n
∂ u
∂z u1B,j−k
n! y 2
uB,j+2
is given explicitly by
2
uB,j+2
(t, x, z) =
2
Z
Zhouping Xin
z
+∞
∂x uB,j
1 (t, x, ξ)dξ
(7.5)
and for all j ≥ 5, pB,j (t, x, z) is given by
Z +∞
B,j
p
=−
f2j (t, x, ξ)dξ,
(7.6)
z
where
f2j = ∂z2 u2B,j−2 + ∂x2 uB,j−4
− ∂t u2B,j−2
2

 k
[2]
j−5
n
X
X
z
 ∂x uB,j−k−2

∂ n uI,k−2n + uB,k
−
1
2
n! y 1
k=0
n=0
k
k
−
[2]
j−3 X
X
zn
k=2 n=0
−
j−3
X

n!
X zn
n=0
n!
−
[2]
j−5 X
X
zn
k=0 n=0
[ k2 ]

k=2
∂x ∂yn uI,k−2n
u1B,j−k−2
2

 ∂z u2B,j−k
∂yn uI,k−2n
+ uB,k
2
2
Zhouping Xin
n!
∂yn+1 u2I,k−2n uB,j−k−2
2
(2) Error terms:
Let the exact solutions to (N S) have expansions:
(
u (t, x, y) = u,a (t, x, y) + µR (t, x, y)
p (t, x, y) = p,a (t, x, y) + µπ (t, x, y).
then (R , π ) satisfy:

2
2
,a

 ∂t R + (u · ∇)R + ∇π − (∂x + µ∂y )R + (R · ∇)u = F




 ∇ · R = 0
1 ∂R


R2 = 0, R1 − 4 ∂y1 = r (t, x) on {y = 0}




 R |t=0 = 0
where F , r are bounded in certain weighted spaces.
Zhouping Xin
(3) Estimate:
R
d
2
dt kR (t)kL2
ΩR
· (eq)dxdy = 0 ⇒:
+ k∂x R (t)k2L2 + µk∂y R (t)k2L2
≤ C(kR (t)k2L2 + kF (t)k2L2 + kr (t)k2L2 ) −
R
Ω
R · (R · ∇)u,a dxdy
On the other hand, from ∇ · R and R2 |y=0 = 0, we get
|
R
Ω
R · (R · ∇)u,a dxdy|
1
≤ C1 kR (t)k2L2 + |µ− 4
R
Ω
R2 R1 ∂η uB,1
1 dxdy|
1
≤ C1 kR (t)k2L2 + C2 µ 4 kR1 (t)kL2 k∂y R2 kL2
≤ C2 kR (t)k2L2 + 21 k∂x R1 k2L2
Thus, we have
kR kL∞ (0,T ;L2 ) = O(1).
Zhouping Xin
Theorem 7.1 Suppose that the initial data u0 of (IE) belongs
to H s (Ω) for a fixed s > 8. Then, in L∞ (0, T ; L2 (R2+ )) we have

P3
j
I,j
B,j
y

4
u1 (t, x, y) = uI,0

1 +
j=1 µ (u1 (t, x, y) + u1 (t, x, )) + O(µ)




P
j
3 B,3
 y
4
u2 (t, x, y) = 3j=0 µ 4 uI,j
2 (t, x, y) + µ u2 (t, x, ) + O(µ)


3

P
j



µ 4 pI,j (t, x, y) + O(µ)
 p (t, x, y) =
j=0
(7.7)
Zhouping Xin
Working harder with higher order expansions and high Sobolev
norm estimates, we can get
Theorem 7.2. Assume that the initial data u0 for (IE) belongs
to H s (Ω) for a fixed s > 18, u02 (y = 0, x) = 0, and ∇ · u0 = 0.
Then in L∞ ((0, T ) × R2+ ), the solution to (7.1) has the
asymptotic behavior as in (7.7).
Zhouping Xin
Remark 7.1 (1) The above discussion works for the general
case µγ with 0 < γ < 12 , with the horizontal viscosity being
µ1−2γ while being the vertical viscosity. (2) In the same way
as above, we can study the boundary layer behavior for an
anisotropic Navier-Stokes equations in
{t > 0, −∞ < x1 , x2 < +∞, x3 > 0} as follows:

∂t u + (u · ∇)u + ∇p = ν(∂x21 + ∂x22 )u + µ∂x23 u





 ∇·u =0

u3 |x3 =0 = 0

γ ∂uk


u
−
µ
= 0, on x3 = 0 (k = 1, 2)
k


∂x3

 u |t=0 = u0 (x)
with the horizontal viscosity ν being fixed, for any fixed γ ∈ R.
(The special case, γ = 0 was studied by Iftimie and Planas in
2006.)
Zhouping Xin
§8 Questions and Remarks
(1) Rigorous justification of the Prandtl boundary layer
expansion for the non-slip case, even under Oleinik’s
monotonic assumption? Or even pressure favorable?
(2) Well-posedness of the Prandtl equations with the Robin
boundary condition? Stability? Rigorous justification of
the boundary layer expansion for the case that the slip
length is equal to the square root of the viscosity?
(3) Can we find a "boundary layer profile" w(t, x, √y ) such that
y
u (t, x, y) − u(t, x, y) − w(t, x, √ ) = o(1)
holds in L∞ (0, T ; H 1/2 ) or in L∞ ((0, T ) × Ω), even with
sub-layers and more scales?
(4) Short time well-posedness of Prandtl’s system and
nonlinear instability?
Zhouping Xin
THANK YOU!
Zhouping Xin