The sharp corner formation in 2d Euler dynamics of patches: infinite double exponential rate of merging

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by  Sergey A. Denisov

THE SHARP CORNER FORMATION IN 2D EULER DYNAMICS
OF PATCHES: INFINITE DOUBLE EXPONENTIAL RATE OF
2 MERGING
1
0
2 SERGEYA.DENISOV

y
a
M
Abstract. For the2dEulerdynamics ofpatches, weinvestigate the conver-
7
gence to the singular stationary solution in the presence of a regular strain.
1
Weprovethattherateofmergingcanbemadedoubleexponential.

]
P
A
.
h 1. Introduction and statement of the results
t
a In this paper, we study the 2d Euler dynamics of patches:
m
θ˙ = θ u, u= ⊥∆−1θ, θ(z,0)=θ , z =(x,y) R2, ⊥ =( ∂ ,∂ ) (1)
[ ∇ · ∇ 0 ∈ ∇ − y x

2 where the θ0 is the characteristic function of some compact set (a patch). In our
case, this compact set will be a centrally symmetric pair of two simply connected
v
0 domains with rather regular boundaries.
1 This problem attracted a lot of attention in both physics and mathematical lit-
2 erature in the last several decades and became a classical one. The existence of
2
weak solutions for Euler equation is due to Yudovich [12]; his result, in particular,
.
1 ensures that the dynamics of patches is well-defined. In [3], Chemin proved that if
0 the boundary of the patch is sufficiently regular then it will retain the same regu-
2
larityforever;anotherproofofthatfactwasgivenlaterbyBertozziandConstantin
1
[1]. We recommend the wonderful books [2, 4] for introduction to the subject and
:
v for simplified proofs. For closely related models (e.g., SQG), there were attempts
i
X recentlytoprovethatasingularitycanoccurinfinitetime(see,e.g.,[5]andrelated
[6, 8, 9]).
r
a Although we have global regularity for the 2d Euler dynamics of contours, very
little is knownaboutthe mechanismleadingto “singularityformation”. The latter
can be described in various ways: growth in time of the curvature, perimeter, etc.
In this paper, we focus on one particular geometric characteristic, the distance
betweentwointeracting patches,andshow thatit candecayas double exponential
infinitely in time. We also reveal the mechanism that drives this process.
Let Ω− = z,z Ω , i.e., the image of Ω under the central symmetry. The
{− ∈ }
boundary of Ω will be denoted by Γ.
In R2 C, we consider the 2d Euler dynamics givenby the initial configuration
∼
Ω(0) Ω−(0). Theareasofpatches,thedistancebetweenthem,thevalueofvorticity
∪
Keywords: Contourdynamics,cornerformation,self-similarbehavior,doubleexponentialrate
of merging
2000 AMS Subject classification: primary76B47,secondary76B03.
1
2 SERGEYA.DENISOV
–allthesequantitiesareoforderoneast=0. Astimeevolves,theEulerevolution
deforms Ω(0) Ω−(0) to a new pair Ω(t) Ω−(t) with Ω(t) = Ω(0) because
∪ ∪ | | | |
the flow preserves the area and the central symmetry. These new patches will be
separated from each other for all times, i.e. Ω(t) Ω−(t) = , but the question,
∩ ∅
however, is how small the distance between them can get? The main result of the
paper is
Theorem 1.1. Let positive δ be sufficiently small. Then, there is a simply con-
1
nected domain Ω(0) satisfying dist(Ω(0),Ω−(0)) 1, and a time-dependent com-
∼
pactly supported incompressible odd strain
S(z,t)=(P(z,t),Q(z,t))
such that
dist(Ω(t),0) dist(Ω(t),Ω−(t)).exp( eδ1t)
∼ −
where Ω(t) Ω−(t) is the Euler dynamics of Ω(0) Ω−(0) in the presence of the
∪ ∪
strain S(z,t). Moreover, for S(z,t) we have
S(z,t)
sup | | < (2)
z ∞
t≥0,z6=0
| |
Remark 1. The corresponding equation for θ(z,t)=χΩ(t)+χΩ−(t) is
θ˙ =∇θ·(u+S), u=∇⊥∆−1θ, θ(z,0)=χΩ(0)+χΩ−(0) (3)
(compare this to (1)).
Remark 2. As it will be clear from the proof, these contours will touch each
other at t = + and the touching point is at the origin. In the local coordinates
∞
around the origin the functions parameterizing the contours converge to x in
±| |
a self-similar way which will be described in detail. In the following elementary
lemma, we show that under the S–strain alone no point can approach the origin
in the rate faster than exponential as long as assumption (2) is made and so it
is the nonlinear term θ u in (3) that produces the “double exponentially” fast
∇ ·
singularity formation.
Lemma 1.1. Let S(z,t)bean odd vector field that satisfies (2). Consider θ(z,t)=
χ (z) which solves
Υ(t)
θ˙ = θ S(z,t), θ(z,0)=χ (z) (4)
Υ(0)
∇ ·
where dist(Υ(0),0)>0 and Υ(0) is a compact set. Then,
dist(Υ(t),0)&e−Ct (5)
Proof. This (4) is a transport equation and we have
z˙ = S(z,t), z(0)=z
0
−
for the characteristics z(z ,t). As S(z,t) is odd, S(0,t)= 0. Therefore, the origin
0
is a stationary point. The estimate (2) yields
S(z,t) <C z , t 0 and z C
| | | | ∀ ≥ ∀ ∈
Therefore,
r˙ Cr, r = z 2
| |≤ | |
and
r(0)e−Ct r(t)
≤
This implies (5) as the patch Υ(t) is convected by the flow. (cid:3)
THE SHARP CORNER FORMATION IN 2D EULER DYNAMICS OF PATCHES 3
OnthePDElevel,wewillconstructanapproximatesolutiontothe2dEulerdy-
namicsofcontourswheretheerrorcanbeinterpretedasthe strain. Itisimportant
to mention here that two centrally symmetric patches can not approach the origin
in the rate faster than double exponential even if one places them in the strain S
from the theorem 1.1. In other words, the following estimate holds true
dist(Ω(t),0)&exp( eqt) (6)
−
with some positive constant q which depends on the initial configuration. This
is an immediate corollary of the lemma 8.1 from [2], page 315. The constant δ
1
from the theorem as well as q in (6) can be changed by a simple rescaling (i.e., by
multiplying the value of vorticity by a constant) thus the size of δ is small only
1
when compared to the parameters of the problem.
In R2, in contrast to T2, the kernel of ∆−1 is easier to write and ∆−1 can be
definedoncompactlysupportedL1 functionssowewilladdresstheproblemonthe
whole plane rather than on T2. On the 2d torus, similar results hold.
The interaction of two vortices was extensively studied in the physics literature
(see,e.g.,[10,11]). Forexample,themergingmechanismwasdiscussedin[10]where
some justifications (both numerical and analytical) were given. In our paper, we
provide rigorous analysis of that process and obtain the sharp bounds.
In [7], the authors study an interesting question of the “sharp front” formation.
Loosely speaking, the sharp front forms if, for example, two level sets of vorticity,
each represented by a smooth time-dependent curve, converge to a fixed smooth
arc as t . Let the “thickness” of the front be denoted by δ (t). In [7], the
front
→∞
following estimate for 2d Euler dynamics is given (see theorem 3, p. 4312)
δ (t)>e−(At+B)
front
with constants A and B depend only on the geometry of the front. The scenario
considered in our paper is different as the singularity forms at a point.
The idea of the proof comes from the following very natural question. Consider
an active-scalar dynamics
θ˙ = θ ⊥(Aθ) (7)
∇ ·∇
whereAistheconvolutionwithakernelK (ξ). Insomeinterestingcases,A=∆−α
A
with α > 0 so K is positive and smooth near the origin. It also obeys some
A
symmetriesinheritedfromitssymbolontheFourierside,e.g.,isradiallysymmetric.
Perhaps, the most interesting cases are α = 1 (2d Euler), which is treated in this
paper, and α = 1/2 (SQG), which is somewhat less established in the physics
literature but still is very interesting mathematically. If one considers the problem
(7)withA=∆−α onthe2dtorusT2 =[ π,π]2,thenthereisastationarysingular
−
weak solution (the author learned about this solution from [9]), a “cross”
θs =χE +χE− χJ χJ− (8)
− −
where E = [0,π]2 and J = [ π,0] [0,π]. One can think about two patches
− ×
touching each other at the origin and each forming the right angle. This picture
is also centrally symmetric. Now, the question is: is this configuration stable? In
other words, can we perturb these patches a little so that they will converge to
the stationary solution at least around the origin? The flow generated by θ is
s
hyperbolic and so is unstable. However, a suitably chosen curve placed into the
stationary hyperbolic flow will converge to a sharp corner as time evolves. The
4 SERGEYA.DENISOV
problemofcourseisthattheactualflowisinducedbythepatchitselfandsoitwill
be changing in time. That suggests that one has to be very careful with the choice
of the initial patch to guarantee that this process is self-sustaining. Nevertheless,
that seems possible and thus the mechanism of singularity formation through the
hyperbolic flow can probably be justified. We do it here by neglecting the smaller
orderterms. Ingeneral,theapplicationofsomesortoffixedpointargumentseems
tobe needed. Eitherway,this scenariois azeroprobabilityevent(atleastthe way
the proof goes) if the “random” initial condition is chosen. However, if one wants
to see merging for a long but fixed time, then this can be achieved for an open set
of initial data so from that perspective our construction is realistic.
We will handle the case α = 1 only without trying to make the strain smooth.
The issue of whether one can choose S(z,t) C∞ or even drop it completely will
∈
be addressed in the separate publication and will, perhaps, require the realization
of the full fixed-point argument idea explained above. We were able to do that so
far only for the model equation where the convolution kernel is a smooth bump.
Forα>1,thesameargumentshowsthatthemerginghappenswithexponential
attraction to the origin. This case is much easier as the convolution kernel is not
singular anymore so, for example, one can take the strain exponentially decaying
in time.
For α < 1, we expect our technique to show that the contours can touch each
other in finite time thus proving the blow-up. This, however, will require serious
refinement of the method.
Thereareotherstationarysingularweaksolutionsknownfor2dEulerdynamics
and for other problems in fluid mechanics. It would be interesting to perform
analogousstabilityanalysisforeachofthemwiththegoalof,e.g.,solvingthefinite
time blow up problem.
The structure of the paper is as follows. In the sections 2,3, and 4, we prove
someauxiliaryresults. Thesection5containstheconstructionofΩ(t),Γ(t)andthe
proofofthe theorem1.1. Inappendix,we findanapproximateself-similarsolution
to the local equation of the curve’s evolution.
2. Velocity field generated by the limiting configuration
In this section, we consider the velocity field generated by one particular pair
of patches which resembles the limiting (t = ) configuration around the origin
of the dynamics described in theorem 1.1. In∞R2, take θs(z) = χE(z)+χE−(z),
where E = [0,1]2. Recall that in the case of T2 analogous configuration (8) is a
steady state. Let x,y > 0, the other cases can be treated using the symmetry of
the problem. We have
(x ξ ,y ξ )
∆−1θ =C − 1 − 2 dξ dξ
∇ s ZE∪E− (x−ξ1)2+(y−ξ2)2 1 2
Integrating, we have for the first component
1
C log(x2+(y+ξ )2) log(x2+(y ξ )2) dξ +r (x,y)
2 2 2 1
− −
Z0 (cid:16) (cid:17)
y
=C log(x2+ξ2)dξ+r (x,y)
2
Z0
THE SHARP CORNER FORMATION IN 2D EULER DYNAMICS OF PATCHES 5
where r (x,y) are odd and smooth around the origin. Integrating by parts, we
1(2)
get the following expression for the integral
ylog(x2+y2) 2y+2xarctan(y/x)
−
By symmetry, for the second component of the gradient we have
xlog(x2+y2) 2x+2yarctan(x/y)
−
Thus, the velocity is
u (z)= ⊥∆−1θ =C( x,y)log(x2+y2)+r(x,y) (9)
s s
∇ −
around the origin, where r(z) . z . By scaling the time in Euler dynamics, we
| | | |
can always assume that C =1/2 in (9) so we get
Lemma 2.1. Around the origin, we have
u (z)= ⊥Λ (z)+r(z) (10)
s s
∇
where
y , y > x
Λs(z)=xylogQ(x,y), Q(x,y)= |x|, |x| >|y| (11)
(cid:26) | | | | | |
and
r(z) . z (12)
| | | |
Proof. For x,y > 0, this is a direct corollary of the calculations above. For the
other cases, this follows from the symmetry of configuration. (cid:3)
Remark 1. The function ⊥Λ is not continuous on the diagonal. That,
s
∇
however, is not so important as we are trying to control not smoothness of the
patches but rather their distance to the origin. This particular choice of Λ was
s
made only to simplify the calculations below.
We will focus later on the first term in (10) as the second one has a smaller size
andwillbeabsorbedintothestrainlater(seetheformulationofthemaintheorem).
The level sets of Λ around the origin are hyperbolas asymptotically. Indeed, take
s
x=ǫx,y =ǫy and consider
Λ (z)=ǫ2logǫ
s
b b
or
logQ(x,y)
xy 1+ =1 (13)
logǫ
(cid:18) (cid:19)
For the velocity, b b
bb
( x(logy+1),ylogy), y >x 0
∇⊥Λs(z)= (−xlogx,y(logx+1)), 0 y <≥x (14)
(cid:26) − ≤
For other x and y, we can use the symmetry of Λ .
s
These formulas show that the flow generated by Λ is hyperbolic around the
s
origin. Moreover,the attraction and repelling is double exponential. For example,
the point (0,y ) will have a trajectory (0,yet). In the next section, we will study
0 0
the Cauchy problem associated to this flow.
6 SERGEYA.DENISOV
3. The Cauchy problem for the dynamics generated by the limiting
configuration
Theconstructionofthepatchesandthestraininthemaintheoremwillbebased
onthe calculationsdoneinthissection. WeconsiderthefollowingCauchyproblem
where t is a parameter:
z˙(τ,t)= ⊥Λ (z(τ,t)), τ 0
s
∇ ≥
the formula for the right hand side is given in (14), and the initial position is
z(0,t)=(ǫ(t),e−1), ǫ(t)=exp( eδt) (15)
−
For now, δ is some fixed positive number. So, at each time t, a point (let us say
it has an index t) with initial position (ǫ(t),e−1) starts moving under the flow so
that at time t+τ (i.e., time τ spent since the start of the motion) we will see the
arc built by points that are indexed by t+τ ,τ [0,τ]. Let us call this arc Γ (t)
1 1 1
∈
(rotatedby π/4 in the anticlockwisedirection, it will be a part of the Γ(t)=∂Ω(t)
from the main theorem).
Our goal in this section is to study the Γ (t) around the origin when t + .
1
→ ∞
This is a straightforwardand rather tedious calculationwhich we have to perform.
For each t, consider the trajectory z(τ,t) within the fixed neighborhood of the
origin: [0,e−1] [0,e−1] . Thistrajectorywillbe aninvariantsetfor potentialΛ
s
{ × }
that corresponds to Λ = e−1ǫ(t). The parameter t, again, can be regarded as
s
−
the tagofthis trajectory. Fort , aroundzerothese trajectorieswilllookmore
→∞
and more like hyperbolas due to (13).
Therewillbetwoimportanteventsforeachtrajectory: thefirstiswhenitcrosses
the line y =x at time τ =T (t) and the second is when it crosses the line x=e−1
1
and thus leaves the domain of interest 0 < x < e−1,0 < y < e−1. We will denote
this time by T (t). Since the trajectory is an invariant set for Λ ,
2 s
z(t,T (t))=(ǫ(t),ǫ(t))
1
and ǫ(t) can be found from the equation
b b
ǫ2(t)logǫ(t)= e−1ǫ(t) (16)
b −
This gives
b b
2ǫ(t)
ǫ(t)= (1+o¯(1)), t (17)
selogǫ−1(t) →∞
One can write an asymptotical expansion up to any order but we are not going to
b
need it.
Within y >x>0, we have y˙ =ylogy, y(0,t)=e−1 and so
y(τ,t)=e−eτ, τ <T (t)
1
Then, the value of T (t) can be found from
1
e−eT1(t) =ǫ(t)
or (due to (16))
2eT1 T =1b+eδt
1
−
That gives us an asymptotics
T (t)=δt log2+e−δt(1 log2+δt)+O(t2)e−2δt (18)
1
− −
THE SHARP CORNER FORMATION IN 2D EULER DYNAMICS OF PATCHES 7
Therefore, at the actual time w,
w =t+T (t)=t(1+δ) log2+e−δt(1 log2+δt)+O(t2)e−2δt
1
− −
the curve we study will intersect the line y = x at the point (ǫ(t),ǫ(t)). Then,
assuming that t is a function of w, we have
t(w)= w + log2 2−δ1(δ1w+1−log2+δ1log2)e−δ1w+b O(be−1.5δ1w)
1+δ 1+δ − 1+δ
δ =δ/(1+δ)<1 (19)
1
and
δ (δ 1)
f(w)=ǫ(t(w))=F(δ)exp eδ1w2−1/(δ+1)+ 1 1− w (1+o(1)) (20)
− 2
(cid:18) (cid:19)
b log2 1 δ δ2log2
F(δ)=exp + 1 δ log2+ 1
1
2 − 2 2 − 2
(cid:18) (cid:19)
3.1. The asymptotical form of the curve within e−1 > y > x > 0. We first
κ
compute the scaling limit of the curve in the z < f(w) neighborhood of
| |
zero where κ is any small positive number. Fixnthe ti(cid:16)me w(cid:17)anod solve the system
backwardin time using auxiliary functions α(τ,w),β(τ,w):
α˙ =α(logβ+1), β˙ = βlogβ
−
and α(0)=f(w),β(0)=f(w). We get then
β(τ,w)=(f(w))e−τ , α(τ,t)=eτ(f(w))2−e−τ (21)
Take T, the moment at which the curve is studied, and let
T =w τ, α=f(T)α, β =f(T)β
−
Given α and β, the parameters w and τ are now the functions of T, α, and β.
b b
We have equations
b beτ(f(w))2−e−τ =f(w τ)α, (f(w))e−τ =f(w τ)β b b
− −
Denoting u=logα,v =logβ, we get
b b
logf(w τ)
b v b e−τ − logf(w−)
=
u τ logf(w τ)
2 e−τ + −
− logf(w) − logf(w)
Now, let τ [0,C] be fixed and take w + which is the same then as taking
∈ → ∞
T . From (20), we immediately get
→∞
v e−τ e−δ1τ
− = s (τ,δ ), ω 0
u → 2 e−τ e−δ1τ − − 1 →
− −
First, notice that
s (τ,δ )>0
− 1
and
1 δ
1
s (τ,δ ) ω, ω = − (0,1), τ 0
− 1
→ 1+δ ∈ →
1
8 SERGEYA.DENISOV
Now, we see that, depending on τ 0, we have different zones for x and different
≥
asymptotical regimes. For example, if τ 0, the limiting curve is
→
y =x−ω (22)
Remark 1. For the distance to the origin we have
b b
min x2+x−2ω =x2 +x−2ω <2
xb>0 m m
(cid:16) (cid:17)
where
b b b b
x =ω1/(2ω+2) <1
m
Notice now that δ 0 as δ 0 and
1
→ →
ω 1 if δ 0
→ →
so in the δ 0 limit the limiting shape for τ 1 is hyperbola, which is approx-
→ ≪
imately the invariant set for Λ . This makes the perfect sense as the initial point
s
(e−eδt,e−1)approachestheseparatrixOY slowerwhenδ 0. Haditbeenconstant
→
in t, the curve would exactly follow the invariant set for Λ .
s
Where precisely on the rescaled space x,y will this uniform convergence take
place? We have
α(τ,t)=eτ f(T +τ) 2−e−τb bf(T) 1+τ1(T) f(T)
∼ ≪
(cid:16) (cid:17) (cid:16) (cid:17)
where τ (T) 0 arbitrarily slowly.
1
→
For each τ > 0, there will be its own part of the curve with the corresponding
scaling limit given by
y =x−s−(τ,δ1)
For this fixed positive τ we can easily identify the domain where this scaling is
going to take place. Indeed, at timbe T,bthe formula (20) yields
δ (δ 1)
f(T)=F(δ)exp eδ1T2−1/(δ+1)+ 1 1− T (1+o(1)) (23)
− 2
(cid:18) (cid:19)
Forthey–coordinateofthepointonthecurveinthedomainrepresentedbyτ,(21)
gives
y exp 2−1/(1+δ)eδ1T d1(τ) exp δ1(δ1−1)T e−τ
∼ − 2
h (cid:16) (cid:17)i (cid:20) (cid:18) (cid:19)(cid:21)
where d1(τ) = e−(1−δ1)τ < 1. Thus, we see that y (f(T))d1(τ)± and for the
∼
x–coordinate we have similarly
x (f(T))l1(τ)±, l (τ)=(2 e−τ)eδ1τ >1
1
∼ −
Foranyfixedτ >0,wecomputedtheasymptoticalshapeofthecurve. However,
we willalso needthe bounds onthe curve for τ(T) + . Givenarbitrarilysmall
→ ∞ µ
fixed µ>0, the the part of the curve in the region y > f(T) will be contained
in the rectangle (cid:16) (cid:17)
µ
0<x<f(T), f(T) <y <e−1 (24)
The first inequality is obvious and can(cid:16)certain(cid:17)ly be improved a lot but this is all
we will need later on.
THE SHARP CORNER FORMATION IN 2D EULER DYNAMICS OF PATCHES 9
3.2. Asymptotical form of the curve within 0 < y < x < e−1. This part of
the curve is more complicated. To study it, consider equations
α˙(τ)= α(τ)logα(τ), β˙(τ)=β(τ)(logα(τ)+1)
−
with initial conditions α(0) = f(w),β(0) = f(w). We are interested in the shape
of the curve at time T =w+τ.
α(τ)=(f(w))e−τ, β(t)=(f(w))2−e−τeτ (25)
We again scale by f(T) as follows
α=f(T)α, β =f(T)β
If u=logα,v =logβ then
b b
τ logf(w+τ)
b bv logf(w) +2−e−τ − logf(w)
=
u logf(w+τ)
e−τ
− logf(w)
v 2 e−τ eδ1τ
− − = s (τ,δ ) (26)
u → e−τ eδ1τ − + 1
−
We again have
s (τ,δ ) ω
+ 1
→
so the rescaled curve converges to the graph of
1 δ
y =x−ω, ω = − 1
1+δ
1
and this convergence is unifobrm obn f(T) < x < f(T) 1−τ1(T) where τ (T) is
1
positive and tends to zero arbitrary slowly. (cid:16) (cid:17)
For any fixed τ (0,τ ), we have
cr
∈
2 e−τ eδ1τ
s (τ,δ )= − − >0
+ 1 − e−τ eδ1τ
−
The critical τ (δ )=logξ (δ ) where ξ (δ ) is the solution of the equation
cr 1 cr 1 cr 1
1
2 ξδ1 =0, ξ >1
− ξ −
Notice that ξ (δ ) 1 as δ 1 and ξ (δ ) + as δ 0. Consequently, for
cr 1 1 cr 1 1
→ → → ∞ →
the critical value
τ (δ ) 0, δ 1
cr 1 1
→ →
and
τ (δ ) + , δ 0
cr 1 1
→ ∞ →
As before, for the subcritical value τ < τ , the curve will have a limiting shape
cr
given by
y =x−s+(τ,δ1)
however s (τ,δ ) decays in τ and s (τ ,δ )=0.
+ 1 + cr 1
The (x,y)–coordinatesofpartobfthebcurvecorrespondingto eachτ <τ iseasy
cr
to find. We again have
δ (δ 1)
f(T)=F(δ)exp eδ1T2−1/(δ+1)+ 1 1− T (1+o(1))
− 2
(cid:18) (cid:19)
10 SERGEYA.DENISOV
and
x (f(T))d2(τ)±, d (τ)=e−τ(1+δ1) <1
2
∼
y (f(T))l2(τ)±, l (τ)=2e−δ1τ e−τ(1+δ1)
2
∼ −
andl (τ)>1forτ <τ . Inthex–coordinate,thedomaincorrespondingtoτ <τ
2 cr cr
will be characterized by f(T)<x<fd2(τcr−)(T). Moreover,
d (τ ) 0, δ 0
2 cr 1
→ →
What can be said about the curve for the region with τ τ ? For the analysis
cr
≥
that follows in the next section we will only need very rough bounds.
Starting at time w, the point with coordinates (f(w),f(w)) will go along the
trajectory which will then cross the line x=e−1 in time
1 log2
τ =loglog δ w +o¯(1)
1
f(w) ∼ − 1+δ
From (25) we get for τ >τ and large w
cr
β 2−2e−τ 2−2e−τcr 2−2e−τcr
=eτ f(w) < f(w) < f(T/(1+δ ))
1
α
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
Therefore, part of the curve corresponding to τ > τ will be contained in the
cr
rectangle
[ f(T) d2(τcr−),e−1] [0, f(T/(1+δ ) ζ(δ1)], ζ(δ )>0 (27)
1 1
×
(cid:16) (cid:17) (cid:16) (cid:17)
3.3. Self-similar behavior around the origin. Summarizing the calculations
above, we conclude that around the origin the rescaled curve will converge to the
graph of the function y = x−ω. How large is the domain of convergence? If the
curve is calculated at time T and if τ (T) is arbitrary positive function such that
1
τ1(T) 0 as T , tbhenbthis convergence will be valid in the region
→ →∞
1+τ1(T) 1−τ1(T)
f(T) <x< f(T)
Outside this window, w(cid:16)e hav(cid:17)e different scal(cid:16)ing lim(cid:17)its for different values of the
parameter τ >0.
Remark 2. What is the nature of the scaling law for τ 0 that we have got?
∼
Well, the Euler dynamics is defined by the convolution with the kernel
K(z) log z
∼ | |
and log(sǫ) = logǫ +O(1) for s [1,M] with any fixed M. Thus, the strength
| | | | ∈
ofthecreatedhyperbolicflowismoreorlessthesamewithinanyannulusǫ< z <
| |
Mǫ. Outside this annulus,say,for z =√ǫ,the sizeofthe kernelis quite different.
| |
Now, assume that we are given the hyperbolic flow in the whole plane defined
by the following equations
x˙ =y
y˙ =x
(cid:26)
Let us find the evolution of the contour C(t) under this flow which would be self-
similarinthesensethatC(t)=C(0)e−δt. AssumethatC(t)isgivenbythegraphof
thefunctiony(x,t)andthenonegetsnonhomogeneousBurgersequationfory(x,t)
y (x,t)= y (x,t)y(x,t)+x
t x
−
By our assumptions,
y(x,t)=e−δtH(eδtx)