10 - SN Partial Differential Equations and Applications Webinars
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Join Jaeyoung Byeon of KAIST as he introduces his recent studies with collaborators on three components systems as basic steps to get general understanding for elliptic systems with large interaction forces. Don’t want to miss another webinar? Register here: https://go.sn.pub/lXgEaxHave an article on nonlinear Schrödinger systems? Submit Your Article: https://go.sn.pub/GT525VLearn more about publishing with SN Partial Differential Equations and Applications: https://go.sn.pub/UFulhi
00:05
pleasure
00:06
to introduce uh jaiyun bien
00:09
from cast spelling out the
00:13
korea advanced the institute
00:16
of science and technology and uh just
00:19
briefly
00:20
and uh jai young studied the mathematics
00:22
in korea
00:23
then on a jsps fellowship and
00:26
he studied in university
00:30
of tokyo and obtained his phd there
00:33
in 1996 and
00:36
after returning to korea and he
00:40
had the post-doc trainings in seoul
00:43
national university
00:44
and the korea institute for advanced
00:48
study then he became a faculty member at
00:51
the post-tech
00:55
pohang university of science and
00:57
technology
00:58
and he was there until 2013
01:02
and as a chair professor then he moved
01:05
to
01:07
cast since then and
01:10
among the uh many scientific
01:13
awards and owners i just mentioned
01:16
uh the uh in in 2004
01:20
he received the very prestigious
01:23
presidential young at work in korea
01:27
and the jail is an expert in non-linear
01:30
electrical problems
01:32
and among the many things
01:36
he's best known for developing new
01:39
techniques
01:40
method for growing solutions
01:43
in degenerate settings and
01:46
so i'm going to just give the stage to
01:50
ion under the talk the title of his talk
01:54
is non-linear shorting a system with a
01:56
large interaction forces
01:58
between different components welcome
02:02
james top thank you very much for your
02:04
introduction
02:06
okay this uh i am very happy to deliver
02:09
my
02:10
some talk in this webinar
02:14
okay so
02:18
okay here is the occasions i would like
02:21
to talk about
02:22
so here is the equation i would like to
02:25
talk about
02:27
so it is a system of elliptic
02:30
equations here lambda i
02:34
and the mu i a positive constant
02:39
so so there are m
02:43
equations so there are interaction term
02:45
here
02:47
so here is an interaction between the ui
02:51
and the uj so that the strength
02:55
the of the interaction is represented by
02:58
the beta i
02:59
j so it is a symmetric the beta i
03:02
j is equal to
03:05
the velocity i the space dimension n
03:10
is 1 2 or 3
03:13
and omega is a domain in rn
03:17
okay so it has a corresponding energy
03:20
function right
03:21
so it is given by this so
03:24
without this interaction term so
03:28
we it is just the addition
03:31
of the m independent linear
03:34
and nonlinear energy function right so
03:37
this interaction is just determined by
03:41
this vector i
03:42
j okay so it comes from the evolution
03:46
some nonlinear schrodinger system it is
03:49
uh
03:50
the occasion for this some standing
03:52
solitary way
03:54
of this form okay
03:57
so after the first
04:00
walk by the tacharin and the chun
04:02
chengwei in
04:03
2005 there are really many works
04:07
for this problem so
04:10
basically there are many many
04:14
studies for the two component problem
04:19
so looking at the energy function
04:23
here so we could
04:26
roughly understand the the
04:29
the effect of the beta ij
04:34
suppose the beta is positive
04:37
then the ui square
04:40
means that some probability density of
04:43
the ice component
04:45
okay u j square is approximate density
04:47
of the j's component
04:50
okay so if they are
04:53
getting closer then
04:57
then it means that this integral
05:00
is getting larger then energy decreases
05:06
so you notice for the
05:09
shooting equation then the low energy
05:14
a solution with the low energy is more
05:16
more stable
05:17
okay so this implies that the
05:22
positive beta i j represents the
05:24
attractive force between two different
05:26
components and u i and u j
05:30
and if beta i j is negative
05:34
then if the it is not correct
05:38
statement as a ui scale and the usc are
05:42
getting further
05:43
then energy decrease okay so
05:47
it implies that the better eyes
05:50
negatively
05:51
is a representative possible force
05:54
between two different
05:55
components you are in the uk
06:00
okay so the my objective
06:04
in the study of this problem or our
06:07
objective here is to understand
06:10
effect of a raji interaction forces
06:14
better ij for
06:18
positive vector solution we want to um
06:22
with a possibly low energy
06:27
okay so here the vector solution means
06:30
that
06:31
it is a solution and each component
06:34
is not zero okay so if the
06:37
one component is zero then it
06:41
reduced it is reduced to two the m
06:44
minus one system
06:48
okay so first
06:52
let's look at the single equation so
06:55
this simple equation so we can solve
07:00
by direct method okay
07:03
so the dissolution is given by this
07:07
so so we want to have some solution with
07:10
a finite energy so we
07:11
require that the u at infinity
07:16
is zero so then solution is given by
07:20
this
07:22
okay looking at this solution so we see
07:26
this phenomena so
07:29
as mu goes to infinity so here we have a
07:33
mu
07:33
here if u goes to infinity this goes to
07:36
zero
07:38
okay so if mu
07:41
is large then the the self-interaction
07:46
is very effective so
07:49
the the energy is
07:52
very small so
07:55
in that case that we have a vanishing
07:58
phenomenon for the some solution
08:02
on the other hand so if you is close to
08:05
g
08:06
zero that means that the intercept
08:08
interaction is very very
08:09
weak in that case so
08:12
this solution blows up as
08:15
some mu goes to zero so in some sense
08:18
it's a counter-intuitive
08:20
but it's a it is a matches
08:23
where uh with some physical
08:26
the principle
08:31
okay so we are interested
08:35
in the system so we first consider the
08:38
system with the two components
08:41
okay so here there are
08:45
many parameters the lambda i
08:48
and the mu i and a beta one two here
08:52
so also depending on the lambda i
08:55
and the ui there's a it is interesting
08:58
to find some solution and study that
09:01
they are asymptotic behavior
09:03
depending on the parameters but
09:06
here my objective is to understand
09:10
that the asymptotic behavior or
09:13
construct of the solution
09:15
when the interspecies the interaction
09:19
term beta 1
09:20
2 is very large okay
09:24
so the in this case
09:28
we normalize in this way so we
09:32
multiply the ui by square root beta 1
09:36
2 then it means that we have
09:40
transformed in this way then we have
09:43
this transform the
09:44
equation okay
09:48
then so the the coefficient
09:52
here one and instead here
09:55
very small constant in the front of it
09:57
you want
09:58
cubic term and you to cubiccon
10:02
so you may take formal limit of this
10:06
system then we have this elastication
10:11
okay so
10:14
it has a pure interaction terms
10:18
here so
10:21
if we could find a solution for this
10:24
limiting problem so it means that we
10:27
could
10:28
find the solution of the original
10:30
problem and we could
10:32
see understand the synthetic behavior
10:35
this is a kind of some inverse procedure
10:39
so we want to solve a reverse problem
10:42
nothing but
10:43
problem so first we understand
10:47
we should understand this equation
10:50
then if we could find a solution for
10:53
this problem
10:54
with a nice structure a kind of
10:57
stability then since the
11:00
the original prop problem is a
11:03
perturbation
11:04
of this limiting problem then keeping
11:07
the structure
11:08
or stability of the solution of the
11:11
limiting problem we could construct
11:13
solution for this part of the problem
11:19
okay so then
11:22
we can find the solution for the limit
11:25
problem
11:26
one here by this minimization this is
11:30
the same it does a multi-paced argument
11:32
so it is just
11:34
an imitation of this quotient
11:38
okay and
11:41
so you know it is a easy to see
11:45
the the existence of the minimize of
11:48
this problem
11:51
okay suppose that the one component is
11:54
zero then this would be very very large
11:58
okay so the we expect
12:01
the minimizer of this variational
12:04
problem has a positive component
12:06
both first and the second component are
12:08
all positive
12:09
or or one side
12:13
okay in fact we it is not difficult it
12:16
is easy
12:16
to check that that there is a minimizer
12:21
that are positive vector solution
12:24
[Music]
12:27
okay since it has a very nice structure
12:30
we follow this minimization argument for
12:33
this
12:34
the particle part of the problem here
12:37
that we consider
12:38
this minimization in the same way
12:41
we could prove this excuse me
12:45
i don't see you turning i don't it seems
12:48
that
12:49
your page is stuck stuck with
12:55
just that equation one the two there and
12:59
no minimization problem i don't see any
13:02
minimization problem
13:04
why i
13:08
use the true computer so
13:11
it is some problem why this happen
13:17
yes i think we can see your screen now
13:20
okay
13:20
uh but we still see equations oh yeah
13:23
yeah that looks good yes
13:24
things are moving okay yeah thanks a lot
13:27
okay so for the limiting problem so for
13:30
the
13:31
limiting problem one so we take
13:34
this minimization so so
13:37
it has a minimizer so it is a very nice
13:40
structure stable
13:42
on the perturbation so for the
13:45
part of the problem so we consider this
13:47
minimization
13:49
so there exists a minimizer which
13:52
is a solution of our the problem
13:56
okay so this energy converts
13:59
to d as beta 1 2 goes to infinity
14:04
okay so
14:08
this implies
14:12
this so if the interaction force
14:16
para 1 2 is very large
14:19
it means that there is a strong
14:21
attraction between two components
14:24
then two component vanishes
14:28
and by a certain denomination we can
14:32
find
14:33
some limiting problem there is the only
14:35
inter
14:36
species interaction terms there
14:40
okay so the the
14:43
lesson we could get from this two system
14:47
is when the paramount is a very large
14:50
this is the most stable
14:52
vector solutions vector solution
14:54
vanishes
14:56
with on aggregation
14:59
as paramount goes to infinity
15:03
so this aggregation is not so strong
15:06
that is
15:07
a weak sense it is a kind of some
15:10
aggregation
15:14
okay okay so
15:18
if paramount is close to minus
15:21
infinity it is a strong repulsion
15:24
force then
15:27
if better want to is negative
15:31
then so by the same same kind of
15:34
minimization it is called as a mountain
15:36
solution
15:37
it is not a better solution anymore so
15:40
we cannot find uh some
15:41
vector solutions through the that kind
15:43
of minimization
15:45
we did just before and
15:48
also if we do the same type of
15:51
some normalization argument then we get
15:54
this limiting problem
15:56
but there is no non-trivial solution for
15:59
this
15:59
so by just a simple integration by up
16:03
the integration by parts we could get
16:05
some
16:06
we want we propose at zero here
16:10
okay so looking at the
16:13
energy functional here so this is the
16:17
interaction energy term
16:20
so if beta 1 2 goes to minus infinity
16:23
since it is minus it is a plus infinity
16:26
together
16:27
then if this integral
16:32
is not 0 it means that it is positive
16:35
then this
16:36
energy diverse to infinity
16:40
okay so we want to construct some
16:44
solution with the low energy so this
16:47
so that means that if there is a low
16:49
energy solution then that this
16:51
top integration would converge to zero
16:55
so that means that the support of u1 and
16:57
the support of u2 are
16:59
showing each other okay
17:04
so the so if we could construct a
17:08
solution
17:09
for large minus bar one two we expect
17:12
this
17:12
the integration interaction the
17:16
integral is very small
17:19
so from this observation
17:22
we may consider this problem so if this
17:26
interaction is small then we we would
17:29
have this two independent problem
17:31
but with a disjointer on disjoint
17:34
domains
17:35
so for some subset subdomain
17:39
of omega prime of omega because this
17:42
problem
17:43
is the least energy level
17:46
okay then it has a this energy solution
17:52
uh okay so there is a type here so omega
17:55
prime
17:55
here also u is zero on omega prime
18:00
okay then we
18:03
may consider this minimization again
18:07
among these the addition of two energy
18:12
for this joint domain omega 1
18:15
omega 2 which are sub sub domain of
18:18
omega here
18:22
so in fact this impliment
18:25
is obtained okay so if omega is
18:28
getting smaller and the smaller this
18:30
energy is very high
18:32
okay so there is they compete each other
18:36
so eventually they take a balance there
18:38
is a
18:39
minimizer among this disjoint
18:42
set okay in fact
18:46
this result was obtained by the
18:49
contracting and the paris journey so
18:51
many years ago
18:54
okay so they consider this minimization
18:58
okay it depends on the beta one two okay
19:02
so so they cons
19:05
that this minimization on the
19:09
the co-dimension two manifold here so
19:12
first
19:13
this one there's according to one
19:15
manipulate
19:16
n1 and also n2 is a good dimension one
19:20
then
19:20
depends the intersection there then take
19:23
the limitation
19:26
then if better one two is negative it is
19:30
obtained
19:31
which is a better solution
19:34
of our problem so we could show
19:39
then furthermore so it is not so
19:42
difficult
19:43
to see this integral
19:47
in total so so better
19:50
without this better one to term so it is
19:53
easy to see that the integration
19:55
comes to zero but together with this
19:57
better one two terms
19:59
here this converts to zero
20:02
okay that means that
20:07
that we could the induce from that
20:11
property that we want with of this
20:13
problem
20:14
the minimizer converts to
20:17
the minimize of this problem so what is
20:20
the difference
20:21
so difference is just the e
20:25
restrict this condition for u1 you you
20:28
want u2 product is zero
20:30
so their support are disjoint each other
20:34
so in fact uh two weeks ago in this
20:36
similar test
20:38
karachini strong attraction gave a very
20:40
nice
20:41
talk about this kind of problem
20:46
okay so in fact there are many many
20:48
further results
20:50
either by terracini and her
20:52
collaborators
20:54
also there are many many issues
20:57
to study further but the regularity of
20:59
free boundary
21:01
or some optimal regularity of the
21:04
minimizer like this
21:05
there are many many some topics to study
21:09
okay so the
21:13
i want to understand some the asymptotic
21:16
behavior
21:17
as interaction term goes to minus
21:20
infinity
21:21
or plus infinity minus infinity here
21:25
when we have a very strong repulsion
21:27
here then
21:28
we observe this so you want you to both
21:32
persist through segregation okay
21:36
you want you to occupy disjunct regions
21:39
okay they survive in the distant regions
21:42
so we could state like this
21:46
for persistence through the segregation
21:53
okay so a lot of understanding for two
21:56
cup
21:56
two component system is this one so if
21:59
better one
21:59
goes to infinity then both
22:03
vanishes while aggregate
22:06
aggregating and if paramount goes to
22:10
minus
22:10
in infinity the both
22:14
survive or persist through segregation
22:18
so that is a simple principle
22:24
we could get through this our study
22:27
under two component system
22:29
now so we moved to three component
22:32
system
22:34
so okay the three component system is
22:36
much more complicated
22:37
so there are basically so
22:41
three parameters here better one two
22:45
there are one three there are two three
22:48
okay so basically i'm interested in the
22:51
delicate problem but we may consider the
22:53
neumann problem but
22:55
i just uh give a talk on the centrifugal
22:58
problem
22:59
here then so energy functional
23:02
is given by this again
23:05
so so if u i
23:09
are in the suburb space
23:14
okay so in this case there are
23:18
four possible cases so all
23:22
the better i j are positive
23:26
let me call the case one so immediate
23:29
interaction forces are all attractive
23:32
second case is that there are two
23:35
positive one negative so two attractive
23:39
one deposit the third case is a one
23:43
attractive so we have a one
23:46
negative term uh one positive
23:50
term two negative ten two degrees the
23:53
last
23:54
case is only possible all
23:57
their ig are negative
24:02
okay so reminder once
24:05
more the the principles we got of
24:08
in the two component case is the strong
24:11
represent course
24:13
we produce some vanishing and
24:15
aggregation
24:17
strong repulsive force okay
24:20
give us some persistence and the
24:22
segregation
24:25
so we hope that this kind of
24:28
principle can be applied to the silicon
24:30
point problem
24:33
so basically the first let me explain
24:36
this
24:37
so fourth case case four
24:40
orbital ij are negative
24:44
then same phenomenon
24:49
so we get so it already it was started
24:52
by the
24:53
quantitative perishing
24:57
so no difference with the two component
25:00
case
25:01
so just the difference is a number of
25:03
some distance
25:04
domain turn into c here since there are
25:08
two
25:09
components involved okay so
25:15
so we have our optimal partitioning of
25:18
domain
25:18
into three distributes of domains
25:21
okay so next
25:25
so we can start this first case so
25:28
all attractive case so
25:32
we follow the approach we did
25:35
for two component case here so first
25:39
i rewrite the equation in this form
25:43
here so i add here alpha
25:46
we multiply alpha beta i 1 2 alpha beta
25:49
1 c so
25:50
now from now on so i fix the beta 1 2
25:54
theta 1
25:54
3 and the beta two three so we take
25:57
alpha
25:58
to infinity okay so in this case
26:02
uh by the same argument
26:06
as in the two component case we
26:08
denominate in this way then
26:10
we get this equation
26:13
so here the instead of this upper
26:16
diagonal term this is this diagonal atom
26:18
goes to zero so we have a formal limit
26:22
so there is no there are no
26:25
self-interactions here there are only
26:27
some some interaction term between the
26:29
different components here
26:32
so by minimization argument
26:35
we could get the solution
26:40
possibly positive vector solution for
26:42
this problem
26:44
so by same type of minimization this
26:47
is same with this minimax argument here
26:51
so it is a mountain pest we could it is
26:53
called as a month
26:54
level so this is a
26:58
huge function i think it's just
27:01
yeah i think you lost your screen again
27:03
uh i think we're not up to date
27:07
okay so sorry since i used it
27:11
again i think we're seeing page 18 at
27:16
the moment which i think is not the one
27:17
you're
27:18
talking about okay here
27:22
so so okay this by this mini
27:25
minimax argument we can find us
27:27
nontrivial solution
27:29
so we should check the whether it is a
27:31
vector solution or not
27:33
okay so in fact
27:36
the this thing solution or multiple
27:39
solution is non-negative
27:41
so we could prove that each component is
27:44
positive
27:46
if this condition does
27:49
hold okay so there is something relation
27:53
between the beta ij
27:54
here so but it is imbibed with this
27:58
lambda i okay
28:01
so if lambda 1
28:05
lambda 2 and lambda 3 are all equals
28:10
then the this condition is quite
28:14
simple so in fact in this case
28:17
better one two these three
28:20
numbers represent the length of the size
28:24
of the for a triangle okay
28:28
so it means that if there is a very nice
28:31
balance between the three interaction
28:34
force
28:35
there's a we could see that that
28:38
that the multiple solution is all
28:40
positive
28:41
vector solution okay
28:45
so that is uh the sha
28:48
that condition is the sharp in some
28:50
sense you know if
28:52
this lambda 1 number 2 lambda 3 are all
28:54
equals then the
28:56
this is a necessary condition okay
28:59
so if this condition is
29:02
violated then there is no vector
29:04
solution anymore
29:06
and no positive vector solution
29:10
and also the
29:14
if because the more general type of
29:17
solution
29:17
sign changing solution we could show
29:20
that
29:21
even though this better one
29:24
i j does not satisfy this condition t
29:28
that so we could find the vector
29:30
solution
29:31
it is not the positive vector solution
29:34
so we could find the vector solution
29:36
so now it is not the most both index one
29:39
it is the higher most index
29:45
okay so in fact if these
29:48
linear coefficients are all equals to
29:51
lambda then we can find
29:54
the solution just comes from this single
29:58
equation
29:58
okay so if u is a solution of this
30:02
problem
30:03
then we have a solution of this problem
30:06
for
30:11
so so so here they
30:14
so you see that the condition is
30:17
necessary
30:18
then the sum of any two
30:21
numbers among threes should be greater
30:23
than the
30:24
the the last one
30:30
okay so let me write this number by the
30:34
a i okay then
30:38
we further know this property
30:42
so this solution is
30:46
non-desired in the linearized sense and
30:49
if
30:50
if and only the u it is a
30:53
solution of a single equation it is
30:56
non-desired
30:57
okay so furthermore these are
31:00
their most indices are same
31:04
okay so if the both index of
31:07
u is one then the the vector solution is
31:10
uh
31:11
also has a mos index one
31:15
and further then any positive solution
31:19
should be of this form but
31:22
we could i couldn't prove that so fully
31:25
so we need
31:26
some condition if the two
31:29
among three interaction parameters are
31:33
same
31:33
then we could show that any positive
31:35
vector solution
31:37
is of this form
31:44
okay so so
31:47
from this result so we get this
31:51
the one okay so
31:54
if we have this condition then for
31:57
substitute large alpha then we take the
32:01
same type of minimization
32:03
for the original problem
32:06
then this energy converts to
32:10
the energy of a limiting problem
32:13
so converts the vector solution since
32:15
this condition is satisfied
32:18
so we see this renewalization
32:22
converge to the solution of the minimax
32:25
solution of a limiting problem
32:27
limit problem u0 okay
32:34
so here even for the the
32:37
city component case we have a same
32:40
phenomenon
32:41
so banishing and aggregation as alpha
32:44
goes to infinity
32:46
but there is a big difference so to
32:49
compound case there was no
32:51
any restriction together some positive
32:54
vector solution but we need uh some
32:57
restriction for better i j so we need a
33:00
certain balance
33:01
between the interaction forces okay if
33:04
there is no
33:06
balance between the interaction forces
33:08
then we can
33:09
we cannot find the positive vector
33:12
solution
33:13
so we should find some higher most index
33:16
solution instead so this balance is
33:20
important
33:22
okay so in fact uh
33:25
i'm much interested in the general case
33:28
if the
33:29
component number of component m is
33:32
strictly greater than
33:33
three i believe there would be a certain
33:36
geometric balance rule between the beta
33:39
i
33:40
j to have a some positive better
33:43
solution
33:44
okay so but i
33:48
in some cases i got some results but i
33:51
couldn't get the sum of
33:53
the complete result
33:56
in general case there would be a nice
33:59
geometric balancer i believe
34:03
okay so many peoples
34:06
also i generalize usually
34:10
for the non-linearity like this okay so
34:13
when p equals 2 it is a correspond to
34:16
our original problem
34:18
so if p is greater than 2 we have this
34:21
one
34:22
there is a big difference between the
34:24
two cases
34:26
okay so if p equals 2
34:29
it is a physical problem but
34:33
when p is strictly greater than 2 it is
34:36
quite different
34:38
so there is no physical that there is no
34:40
any some
34:41
this restriction for the beta i j
34:45
so so in height city case we
34:48
give a restriction p is greater than 200
34:51
less than three
34:52
and one or two dimensional case p is
34:55
strictly greater than two
34:57
then for any better ij
35:01
positive then we can find
35:04
a domain not not general domain
35:08
then above system and the domain has a
35:11
positive vector solution
35:13
okay so the variation
35:17
but looking at the directional structure
35:18
they are very similar
35:21
but the there is a big difference okay
35:24
for the two case when p equals two there
35:28
is a we have a very strong restriction
35:30
for interaction better i j but
35:34
if p is strictly greater than 2 there is
35:37
no any
35:38
we don't need any restriction
35:41
or balance between the interaction
35:43
forces
35:45
okay it is an interesting
35:49
point we observed okay
35:52
then next so this
35:56
third case we have a two repulsive force
35:59
and one attractive force so we may
36:02
write in this form here so
36:06
for the negative beta ij
36:10
i rewrite in this way so better one two
36:13
one three and the beta two three are
36:15
both positive here
36:17
instead i place here minus beta so beta
36:19
is very large
36:21
for attractive force i write here alpha
36:24
beta one two yeah this energy functional
36:27
here
36:28
here this is a norm and this is a kind
36:31
of
36:32
okay so
36:35
look at this case better is zero and
36:38
alpha is large
36:40
okay if beta is 0 then there is no this
36:43
term here
36:44
this term disappear this term disappear
36:47
this
36:48
this term disappear then we have a
36:50
single equation for the last component
36:52
ok type here you see
36:56
so we have a single equation for u3 and
36:59
also you have some equation for u and u2
37:02
without this term so only we have some
37:05
interaction between u and u2
37:07
okay so in this case we could construct
37:10
uh
37:11
some both index one solution for u and u
37:13
two also
37:14
both index one solution for u3
37:18
okay so
37:21
from this observation we could construct
37:23
some solution
37:25
so for large alpha okay so for this
37:29
equation let me denote this energy level
37:31
by capital
37:32
l that the set of the positive resting
37:35
solution by the script
37:36
f here then
37:39
from this observation we could take a
37:42
minimization like this
37:44
so for the single equation for the last
37:46
one that we take a minimization
37:48
similar to this one and the first two
37:51
components we take
37:52
this type of some restriction
37:56
then then we take this minimization
38:00
limit this minimum value by the c alpha
38:03
beta
38:05
then we could prove that
38:08
if alpha is sufficiently large
38:11
independent of a beta
38:12
positive beta then
38:15
it is attained by some better solution
38:19
you alphabetic which is a vector
38:20
solution of
38:22
our problem interesting point
38:25
is uh here energy level is
38:29
close to this l it is energy for single
38:32
equation for the third component
38:34
okay so furthermore there is a vector
38:37
solution converge to this one the first
38:39
and the second component converts to
38:41
zero and the third component converges
38:43
to a distance acceleration of the single
38:45
equation
38:52
okay so in this case we have a one
38:55
dominant
38:55
component third component and the
38:57
vanishing two component as alpha goes to
39:00
infinity but we don't observe any
39:03
aggregation
39:05
here still but in fact there is an
39:08
aggregation
39:09
effect okay so first
39:12
weakens this problem it is defined the
39:14
half space here
39:16
this is a non-autonomous problem we have
39:18
a here
39:19
yn square here also there is some
39:22
dependence under
39:23
the last component
39:27
the solution of a some sort of equation
39:30
here
39:30
single equation let me denote this
39:32
solution by using
39:34
here this normal derivative at p
39:36
boundary point
39:37
p here it doesn't do is uh
39:41
derivative then by the hope maximum
39:44
principle we see that
39:45
this is always positive then it
39:48
may take this minimization okay
39:51
this uh let's see what is that it is uh
39:56
this energy level that we could find
39:59
some distinct solution or multiple
40:00
solution for this problem
40:02
let me denote this that energy level by
40:05
the
40:06
m p usually is dependent of point p
40:09
on the boundary and the solution of this
40:12
the single equation you see
40:16
okay so in fact that we could
40:20
uh take this uh renumerization okay
40:24
so we have this kind of renormalization
40:30
okay so but we keep the third component
40:34
itself
40:34
here we don't transmit within transform
40:39
okay then by this transformation we have
40:42
some equation for v 1 b to precede like
40:44
this
40:46
okay so looking at this one that this
40:48
goes to 0
40:49
this goes to 0 and we hope this
40:52
term goes to 0 and this term
40:57
b3 suppose the b3 converts to the u
41:00
capital u3
41:01
then this skeletal beta
41:04
and this term would converge the normal
41:07
derivative at the
41:08
certain boundary point okay that is the
41:10
idea here
41:13
okay then we need uh some c1 convergence
41:17
of the w3 to the capital u3
41:20
so we need a certain kind of elliptic
41:22
estimate
41:24
so we need some restriction for alphabet
41:28
like this
41:28
so we say the alpha to the a power
41:33
sharp b to the b power goes to input if
41:36
both alpha beta goes to infinity and
41:39
this ratio goes to infinity this means
41:41
that the alpha in some sense
41:44
much larger than beta that goes to
41:47
infinity
41:48
faster than better in certain rate okay
41:52
then if we give us some kind of
41:55
restriction like this
41:56
this condition with some delta
42:00
it depends on space dimension like this
42:03
is it's greater than or equal to quarter
42:06
one decays and uh
42:07
in the high dimensionality case we need
42:09
a disk restriction
42:11
then we have some exact energy expansion
42:14
like this
42:16
okay this this term goes to zero
42:20
this is the least energy level for the
42:22
limiting problem
42:23
for two first two components
42:27
okay then further
42:30
after the normalization we could show
42:32
that
42:33
this normalizes some solution
42:36
converts some solution of limiting
42:38
problem after certain rotation
42:41
or translation okay and
42:44
we further characterize that the
42:48
the concentration point is here x alpha
42:51
beta is a concentration
42:53
point for first two component okay
42:56
then this converts the point p where p
43:01
minimizes this normal derivative for the
43:04
some single equation
43:06
solution of the single equation very
43:08
three so also we should take the
43:10
minimization photo
43:12
even for visually since there will be
43:14
many solution
43:16
in this script error
43:20
okay so the lesson
43:23
we get from in this case is uh
43:26
there is a segregation between the third
43:29
component
43:30
and the banishing two component u1 u2
43:34
okay but even though
43:37
first two component vanishes there is a
43:40
local
43:40
localization and aggregation okay as
43:44
alphabet that goes to infinity okay
43:47
so it is a mixed state between the
43:50
segregation and the banishing
43:53
in this case
43:57
okay so ah i used
44:00
too much time already okay so
44:04
uh if alpha better
44:07
fixed and the better goes to infinity
44:09
then we have a
44:10
first object the segregation between the
44:12
third component the first two component
44:15
and then when we increase the alpha
44:18
then the the region of the use
44:22
of the compound would invade the region
44:25
of the
44:25
first two components okay so it is a
44:28
nice
44:29
the topic to study this what happened
44:31
when the
44:32
beta stands up where the overhead and
44:36
okay
44:39
okay so uh in the radio symmetry case
44:43
it's quite
44:43
interesting in the in the radio symmetry
44:45
case that we
44:47
it's the same type of some problem
44:51
then this same type of
44:55
phenomena for the last component first
44:57
and the
44:58
step further we take a global
45:00
minimization and the latest symmetry
45:02
class
45:03
then first two component compared to
45:05
zero then last
45:06
component it converges to the single
45:08
solution of a single equation
45:10
this thing is solution but
45:14
is the limit asymptotic profile
45:17
depends on the space dimension okay in
45:20
the
45:21
we have some three kind of a limiting
45:24
problem
45:24
for the single equation here and here
45:28
the mb limited note is this energy by
45:31
mb it is a boundary concentration
45:34
here it is a non-autonomous problem
45:37
sometimes
45:38
it may concentrate on center is limit
45:40
this
45:41
the distance by nc okay it is a
45:44
od problem second one is the od problems
45:47
last one is a pd problem
45:49
then for the global minimizer then we
45:52
have this different asymptotic energy
45:55
level depending on the space dimension
45:58
low dimensional case we have a boundary
46:00
concentration
46:01
and a high dimensional case that we have
46:04
as
46:04
the concentration at the center first
46:07
for the first
46:08
two components but we need some
46:10
restriction for alpha beta
46:12
like this okay
46:16
and uh also we can prove that the
46:18
oscillator for the first
46:20
two component okay by renewalization
46:24
okay and further we could find
46:28
some local minimizer then in
46:32
the 3d case we have some concentration
46:35
on the center
46:36
but also we could find the solution
46:40
with the first two component
46:43
concentrating on the boundary near the
46:44
boundary so we have
46:46
this energy level and also
46:49
this energy level in raw dimensional
46:51
case
46:52
it concentrate in the in the center
46:56
okay so also this
46:59
uh asymptotic profile we could find
47:02
so on this narry mini port of code
47:06
dimension 2
47:07
then we can find the two minimizers one
47:10
is a global minimizer
47:12
another one is a local minimizer then
47:15
we expect there will be a new type of
47:18
solution
47:19
in fact that type of solution is
47:21
determined by this
47:23
key function is a determinant of
47:24
function it is
47:27
u is a solution of a single equation
47:29
there is a unique solution in the
47:31
when domain is a ball then we have some
47:34
weight
47:34
here that is a determinant function
47:38
in fact we have corresponding limiting
47:41
problem
47:42
given by this for the first
47:45
two component here then in fact the
47:49
w the determinant function has a unique
47:52
critical point
47:53
between the zero and one in high
47:56
dimensional case in the
47:57
in that one dimensional case it has only
47:59
one maximum point at the zero
48:03
further we could show that uh
48:06
this this has a distinct solution
48:09
so this listing solution is
48:13
linearly non-degenerate okay so if
48:16
better one two and the beta two three
48:18
are same then it is easy to
48:21
see that the non-degeneracy of the
48:23
solution
48:24
but there is no general technique to
48:27
prove that
48:27
that kind of a non-degeneracy if better
48:29
ones three
48:30
and better two three are different but
48:33
in the system case very very difficult
48:35
to check the nondegeneracy
48:36
of the solution but so we could prove
48:39
that the distance solution for this
48:41
this translation for this problem has a
48:45
the non-dignity so it's a non-digestive
48:49
means that
48:50
it kind of comes from that some
48:52
translation invariance
48:53
of this problem okay then
48:56
we could construct a solution with a
49:00
clustering bomb for first
49:03
two component the third component
49:05
converts the does the munich
49:06
acceleration of a single location
49:08
and the first two component has a
49:12
clustering layer that layer
49:15
is close to l0 maximum point of
49:18
determinate
49:19
function and there is a the
49:23
clustering solution this m clustering is
49:25
an m-bomb solution
49:27
but this the amplitude is very very
49:30
small the components to zero
49:32
but the energy level is dependent under
49:35
this one
49:36
this scale and this m is the number of a
49:40
bump
49:40
and m is just this energy level
49:44
okay so there are uh
49:48
some quite the interesting different
49:51
type of
49:51
solution in the radio symmetric case
49:54
the last one last one is the one
49:58
repulsive
49:59
two attractive case it is the most
50:00
difficult case i think
50:03
and i don't explain the detail but here
50:06
we observed all vanishes first
50:10
second third component all converge to
50:13
zero
50:14
as the alpha beta goes to infinity also
50:17
we could
50:17
observe the segregation phenomena
50:20
between
50:21
first and the third component and the
50:23
two and the third component so there is
50:25
some the segregation between the second
50:28
component and
50:29
and second component and this other
50:31
component
50:33
okay okay i i
50:36
would like to test some they explain the
50:39
conclusion
50:40
here so all alternative case
50:43
in the three component cases we observe
50:45
the vanishing aggregation
50:47
also we need a certain balance and the
50:50
only specific case is just the
50:51
segregation
50:53
and the two attractive one repulsive
50:56
case
50:56
and the banishing and the weak
50:58
segregation
50:59
here in the third case the segregation
51:03
means that the third component only
51:04
survived
51:05
and the first two component vanish near
51:08
the
51:09
boundary so okay so partial punishing
51:12
with a segregation
51:13
we observe so
51:17
the for this type of some study
51:20
uh okay so here we are
51:23
so just brief interruption or our chair
51:26
is having technical problems today
51:27
uh so just to let you know that we are
51:29
running short on time
51:31
uh okay so i will finish up thanks
51:34
so it is a conclusion so that there are
51:37
some studies in the general
51:39
component case by the d1
51:42
and the gq and the saw metabolism and
51:45
the way
51:47
but that study is different from this so
51:50
we are interested in this case then this
51:54
the general phenomena would determined
51:57
by this
51:58
network matrix here the n is a
52:01
sign of verizon in this case
52:04
so this the network metrics
52:07
will determine that uh some
52:10
characteristic
52:11
of the whole feature okay so in the
52:15
city component case we have a we
52:17
observed very
52:19
many different behavior okay but in the
52:22
more component case then we would be
52:25
more complicated but
52:27
it would be nice to understand in
52:29
general case
52:31
thank you very much there was some
52:32
therapist i consumed too much
52:36
no problem thanks a lot thank you very
52:38
much for the nice talk i would usually
52:40
hand over to xe jung now but
52:42
unfortunately i think his sound is not
52:43
working and zoom is playing
52:45
tricks on us today so i will
52:50
take over the job maybe of moderating
52:52
questions if there are any so if you
52:53
have any questions feel free to either
52:56
post them in the chat or feel free to
52:58
mute yourself and just to ask directly
53:00
if there are any questions
53:04
see john cannot hear us i can see him in
53:06
the video
53:07
talking but unfortunately i can hear him
53:09
so anybody else does anybody have
53:11
questions uh or remarks
53:23
i see the question in the chat
53:27
so the question is um all the results
53:30
results you mentioned in the talk are in
53:31
dimensions one two and three how about
53:33
dimension four in
53:34
case uh are there any results for a
53:37
system
53:37
with three components yup
53:41
but in the higher dimensions the
53:43
four-dimensional case
53:44
that the non-individual was critical
53:48
so we don't have a compactness
53:51
so we should recover the compactness
53:55
but that so it is another job to do
53:59
so but i have that even though for five
54:02
dimensional cases it does not work in
54:03
the four dimensional case it is a
54:05
critical experiment
54:06
it is quite interesting the case
54:09
okay so i don't have any result
54:16
okay are there any other questions
54:21
feel free to either mute yourself
54:25
or ask them in the chat
54:29
if the sound is working for you
54:30
unfortunately we seem to have technical
54:32
issues today
54:33
for whatever reason
54:38
um
54:41
but if there are no more questions
54:43
unfortunately i don't have any i'm sure
54:45
usually we would have some by our chair
54:47
but i don't have any today so anyway
54:48
thank you
54:49
very much j young for oh sorry there's a
54:52
question actually
54:53
uh have you ever tried to add
54:56
well u subscript i
55:00
superscript k with even k i don't i
55:02
guess you can see the question in the
55:03
chat
55:04
uh i didn't see that
55:09
you don't see it uh so that it's well
55:11
it's a difficult one
55:12
to read out actually so uh
55:17
why i cannot do that uh if you stop
55:20
sharing your screen i think you can you
55:22
can see oh okay
55:23
okay okay
55:26
oh if i steal the screen
55:30
i use a k for even k
55:37
all right uh okay okay so last question
55:43
uh no so i didn't but the
55:47
the for the non-linearity that case
55:50
if even for even k
55:54
that that then make us a problem
55:58
quite differently okay so when p is
56:01
greater than 2 then
56:04
we lose the physical meaning so
56:08
there is no any restriction for better
56:10
ij
56:11
so even for the even number k even
56:14
integer k
56:15
there would be no physical sense
56:19
about this unbalance between the
56:21
interaction force
56:23
i believe
56:30
okay yeah then uh
56:33
thanks a lot for the very nice
56:36
presentation thanks a lot for staying up
56:37
so late i guess it's midnight now
56:39
uh yeah yeah so thank you very much for
56:42
for staying up so late okay
56:43
uh yes it is midnight yes exactly so
56:48
thanks doctor i guess i should ever
56:49
should send everybody in in
56:51
east asia to bed uh and thank you
56:52
everybody for joining today
56:54
and especially thanks to our speaker
56:56
please
56:57
feel free to join us again next week we
56:59
have address tt
57:01
uh i just posted the topic in the in the
57:03
chat feel free to
57:04
join us at the same time same place next
57:07
week in the weeks to come
57:09
uh yes and thanks for joining today