The Hele-Shaw Problem as a limit of Stefan Problemsblanki/WCNAtalk.pdf · The Hele-Shaw Problem as a limit of Stefan Problems I. Blank, M. Korten, C. Moore KSU July 3, 2008 I. Blank,

The Hele-Shaw Problem as a limit of Stefan Problems

I. Blank, M. Korten, C. Moore

KSU

July 3, 2008

I. Blank, M. Korten, C. Moore (KSU) Hele-Shaw as a limit of Stefan July 3, 2008 1 / 26

Modeling

The Hele-Shaw Problem or Quasi-Static Stefan Problem is a mathematicalmodel for a number of physical situations...


Modeling


Oil injected between two parallel plates


Modeling



Liquid plastic injected into a molding


Modeling




A one-phase Stefan Problem where the diffusivity is very fast relativeto the speed of the moving interface


Modeling




A one-phase Stefan Problem where the diffusivity is very fast relativeto the speed of the moving interface

Cancerous tumor growth


Classical Formulation

In the most basic formulation there is a pressure function or temperaturefunction u which satisfies:




∆u = 0 in the diffusive region at each moment in time.





u = 0 on the outer boundary of the diffusive region.






There is either Dirichlet or Neumann data given for u on theboundary of the fixed inner slot.







The boundary of the diffusive region flows according to D’arcy’s law,or in other words, the speed of the moving boundary is the one sidednormal derivative of the function u.







The boundary of the diffusive region flows according to D’arcy’s law,or in other words, the speed of the moving boundary is the one sidednormal derivative of the function u.

Fixed BoundaryMoving Boundary

Slot

Fluid or Diffusive Region


Classical Solutions: Background

When beginning with a C 2,α free boundary, short time existence ofclassical solutions was proved by Escher and Simonett in SIAM J.Math. 1997.


Classical Solutions: Background

When beginning with a C 2,α free boundary, short time existence ofclassical solutions was proved by Escher and Simonett in SIAM J.Math. 1997.

Of course it is absolutely impossible to expect all classical solutions toexist globally in time, because of potential collisions of two differentparts of the boundary of the surface. So there is a real need forgeneralized solutions.


Viscosity Solutions

Inwon Kim gave a good definition of a Viscosity solution ofHele-Shaw in her 2003 ARMA paper. Within this paper she showedexistence and uniqueness.


Viscosity Solutions


Idea behind viscosity solutions: Make use of comparison principles,and smooth functions which “touch” the putative solution fromabove or below.


Viscosity Solutions


Idea behind viscosity solutions: Make use of comparison principles,and smooth functions which “touch” the putative solution fromabove or below.

These solutions are further studied by Kim in a sequence of paperswith coauthors including David Jerison, Sunhi Choi, and AntoineMellet.


Weak Solutions of Hele-Shaw after Integrating in Time

In 1981 Elliot and Janovsky introduced a notion of weak solution toHele-Shaw and showed existence and uniqueness within their framework.




To arrive at their notion of weak solution, one would start by integrating aclassical solution in time (the so-called Baiocchi transform), and then lookat the equations satisfied after integrating these transformed functionsagainst a suitable class of test functions.





Of course there is an extra integration in time involved, and it is by

no means clear that the derivative with respect to time of the final

product is a function.





Of course there is an extra integration in time involved, and it is by

no means clear that the derivative with respect to time of the final

product is a function.

In any case, the EJ-solutions turn out to be solutions to the obstacleproblem at each fixed time, and therefore there is a great deal of literaturewhich can be applied to the situation.


Presumably access to all of the obstacle problem theory and theorems forvariational inequalities helped make this EJ-notion of weak solution by farthe most popular in the 1980’s and 1990’s.


Presumably access to all of the obstacle problem theory and theorems forvariational inequalities helped make this EJ-notion of weak solution by farthe most popular in the 1980’s and 1990’s.

There is also a paper by DiBenedetto and Friedman which uses a similarframework to deal with the ill-posed version of the Hele-Shaw problem.


Weak Solutions of Hele-Shaw without Integrating in Time

The first rigorous treatment of a weak solution without the integration intime appears (as best as I can find) in work by Gil and Quiros.




They appear to need constant Dirichlet data to make their theory work.





In Blank, Korten, and Moore ([BKM] TAMS to appear) there is also anotion of weak solution which does not require an additional integration intime, and the existence, uniqueness, and regularity theory works withoutany additional artificial assumptions on the injection Dirichlet data.






In fact, we show a massive shortcut in the case of Dirichlet data which isnondecreasing in time.






In fact, we show a massive shortcut in the case of Dirichlet data which isnondecreasing in time.

Both the work of Gil and Quiros, and the work of B., Korten, and Mooreallow weak solutions of Hele-Shaw with a “Mushy” Region.


Mushy Regions

What is a “Mushy” Region?


Mushy Regions


In the Stefan problem it would be zero degree ice which has some,but not all, of the energy needed to change from zero degree ice tozero degree (liquid) water.


Mushy Regions



In terms of the mathematics, it is any place where the free boundarywill move through faster (given the same normal derivative of thetemperature function), and this notion is easily generalizable toHele-Shaw. In the Hele-Shaw problem we can imagine:


Mushy Regions




1 Left over residue, or


Mushy Regions




1 Left over residue, or2 A narrowing of the plates

as something which would have this effect.


The Equations in Our Formulation

We let

α∞(s) :=

{

0 s < 1[ 0, +∞) s ≥ 1 .



We let

α∞(s) :=

{

0 s < 1[ 0, +∞) s ≥ 1 .

Any pair (u(∞), V (∞)) which satisfies



We let

α∞(s) :=

{

0 s < 1[ 0, +∞) s ≥ 1 .

Any pair (u(∞), V (∞)) which satisfies0 ≤ u(∞) ≤ 1,



We let

α∞(s) :=

{

0 s < 1[ 0, +∞) s ≥ 1 .

Any pair (u(∞), V (∞)) which satisfies0 ≤ u(∞) ≤ 1,V (∞) ∈ α∞(u(∞)) a.e.,



We let

α∞(s) :=

{

0 s < 1[ 0, +∞) s ≥ 1 .

Any pair (u(∞), V (∞)) which satisfies0 ≤ u(∞) ≤ 1,V (∞) ∈ α∞(u(∞)) a.e., and

∫

Dc

∫

∞

0ϕt(x , t)u(∞)(x , t) dt dx +

∫

Dc

∫

∞

0∆xϕ(x , t)V (∞)(x , t) dt dx

=

∫

∂D

∫

∞

0

∂ϕ

∂ν(x , t)p(x , t) dt dHn−1(x) −

∫

Dc

ϕ(x , 0)uI (x) dx .

for all ϕ ∈ C∞ (IRn × [0,∞)) , such that ϕ ≡ 0 on ∂D × [0,∞), andϕ(x , t) → 0 as either t → ∞ or |x | → ∞,



We let

α∞(s) :=

{

0 s < 1[ 0, +∞) s ≥ 1 .

Any pair (u(∞), V (∞)) which satisfies0 ≤ u(∞) ≤ 1,V (∞) ∈ α∞(u(∞)) a.e., and

∫

Dc

∫

∞

0ϕt(x , t)u(∞)(x , t) dt dx +

∫

Dc

∫

∞

0∆xϕ(x , t)V (∞)(x , t) dt dx

=

∫

∂D

∫

∞

0

∂ϕ

∂ν(x , t)p(x , t) dt dHn−1(x) −

∫

Dc

ϕ(x , 0)uI (x) dx .

for all ϕ ∈ C∞ (IRn × [0,∞)) , such that ϕ ≡ 0 on ∂D × [0,∞), andϕ(x , t) → 0 as either t → ∞ or |x | → ∞,

is called a weak solution of the Hele-Shaw problem with boundary datap(x , t) and initial data uI (x).I. Blank, M. Korten, C. Moore (KSU) Hele-Shaw as a limit of Stefan July 3, 2008 10 / 26

Huh?!?!?!

OK, so to try to write the previous slide in shorthand, we have a phase

function u(∞) and a pressure function V (∞) which satisfy: u(∞)t = ∆V (∞)

everywhere outside the injection slot.


Huh?!?!?!




Dealing with the obvious objections...


Huh?!?!?!





u(∞) will be constant in time except at the moment when the freeboundary reaches it, and at that moment will jump from uI (x) to 1instantaneously, and then live happily ever after with that value of 1.(See [BKM].) Thus, ∆V (∞) = 0 where it should.


Huh?!?!?!






It follows from work of Korten that V (∞) will equal p on theboundary of the slot when there is sufficient regularity, and u(∞)(x , 0)will equal uI (x) a.e.


Huh?!?!?!






It follows from work of Korten that V (∞) will equal p on theboundary of the slot when there is sufficient regularity, and u(∞)(x , 0)will equal uI (x) a.e.

The fact that u(∞)t = ∆V (∞) holds in a weak sense across the free

boundary leads to D’arcy’s law. This is shown rigorously by Danielliand Korten (2005).


A Mesa Limit

In [BKM], we produce weak solutions of Hele-Shaw as limits of solutionsto Stefan problems as we let the diffusivity tend to infinity.


A Mesa Limit


Many authors have observed that taking limits of certain PDEs and certainFBPs will lead to the Hele-Shaw problem. In particular,


A Mesa Limit



Caffarelli and Friedman in 1987. They took limits of solutions of theporous medium equation on all of IR

n, and found:


A Mesa Limit




n, and found:

The limiting solutions were independent of time.


A Mesa Limit




n, and found:

The limiting solutions were independent of time.The limiting solution was everywhere equal to the initial value or 1.


A Mesa Limit




n, and found:

The limiting solutions were independent of time.The limiting solution was everywhere equal to the initial value or 1.Under geometric assumptions about the initial conditions, they showedthat the “Mesa” set (i.e. where the solution was 1), had the sameregularity as the noncontact set in an obstacle problem.


A Mesa Limit




n, and found:


Friedman and Huang in 1988: Did the same for ut = limn→∞∆φn(u).


A Mesa Limit




n, and found:


Friedman and Huang in 1988: Did the same for ut = limn→∞∆φn(u).

Others did similar things on subsets of IRn and got convergence to

Hele-Shaw or something similar: Benilan, Boccardo, Bouillet, Elliott,Gil, Herrero, Kim, King, Korten, Marquez, Quiros, Ockendon, etc.


Our m-Approximating Problem

We let αm(s) := m(s − 1)+.



We let αm(s) := m(s − 1)+. If V (m) = αm(u(m)), and




∫

Dc

∫

∞

0ϕt(x , t)u(m)(x , t) dt dx +

∫

Dc

∫

∞

0∆xϕ(x , t)V (m)(x , t) dt dx

=

∫

∂D

∫

∞

0

∂ϕ

∂ν(x , t)p(x , t) dt dHn−1(x) −

∫

Dc

ϕ(x , 0)uI (x) dx

for ϕ as before, then we say




∫

Dc

∫

∞


∫

Dc

∫

∞


=

∫

∂D

∫

∞

0

∂ϕ

∂ν(x , t)p(x , t) dt dHn−1(x) −

∫

Dc

ϕ(x , 0)uI (x) dx

for ϕ as before, then we say (u(m), V (m)) is a weak solution of the Stefanproblem with boundary data p(x , t) and initial data uI (x).




∫

Dc

∫

∞


∫

Dc

∫

∞


=

∫

∂D

∫

∞

0

∂ϕ

∂ν(x , t)p(x , t) dt dHn−1(x) −

∫

Dc

ϕ(x , 0)uI (x) dx

for ϕ as before, then we say (u(m), V (m)) is a weak solution of the Stefanproblem with boundary data p(x , t) and initial data uI (x).

In this formulation, m is the diffusivity, u(m) is the energy, and V (m) is thetemperature.


Letting m → ∞

Since V (m) ≤ max p(x , t) =: M, and since u(m) ≤ 1 + M/m, gettingweak-∗ L∞ convergence is trivial.


Letting m → ∞


This fact in turn leads to the fact that the newly constructed pair(u(∞), V (∞)) will satisfy the integral equation that we require.


Letting m → ∞



Of course, weak-∗ L∞ convergence is far from pointwise convergence, sothere is immediate trouble with some of the pointwise properties that werequire:


Letting m → ∞




We need u(∞) to be a “phase function.” i.e. It should equal uI (x)initially and jump to 1 where it lives forever after the free boundarymoves past it.


Letting m → ∞





We need V (∞) ∈ α∞(u(∞)).


Letting m → ∞





We need V (∞) ∈ α∞(u(∞)).

These properties are trivial if we can get pointwise convergence.


The Easy Case: p Nondecreasing in Time

We suppose first that pt(x , t) ≥ 0.




In this case, we can observe that the functions u(m)(x , t) are allnondecreasing in time by the WMP. (Look at difference quotients in timein the wet region.)





Now we let m < k and look at the barrier function

v (k)(x , t) :=

u(m)(x , t) if u(m)(x , t) < 1

1 + m

k(u(m)(x , t) − 1) if u(m)(x , t) ≥ 1





Now we let m < k and look at the barrier function

v (k)(x , t) :=

u(m)(x , t) if u(m)(x , t) < 1

1 + m

k(u(m)(x , t) − 1) if u(m)(x , t) ≥ 1

This is a reasonable critter to look at because it satisfiesαk(v (k)) ≡ αm(u(m)).


Then we compute

v(k)t (x , t) ≤ u

(m)t (x , t)

= m∆(u(m)(x , t) − 1)+

= k∆(v (k)(x , t) − 1)+


Then we compute

v(k)t (x , t) ≤ u

(m)t (x , t)

= m∆(u(m)(x , t) − 1)+

= k∆(v (k)(x , t) − 1)+

Thus, v (k) is a local subsolution to the k-approximating problem, andtherefore v (k)(x , t) ≤ u(k)(x , t) which makes life very happy...


Then we compute

v(k)t (x , t) ≤ u

(m)t (x , t)

= m∆(u(m)(x , t) − 1)+

= k∆(v (k)(x , t) − 1)+


In particular:

k[u(k) − 1]+ ≥ k[v (k) − 1]+ ≡ m[u(m) − 1]+,


Then we compute

v(k)t (x , t) ≤ u

(m)t (x , t)

= m∆(u(m)(x , t) − 1)+

= k∆(v (k)(x , t) − 1)+


In particular:

k[u(k) − 1]+ ≥ k[v (k) − 1]+ ≡ m[u(m) − 1]+,

So nevermind weak convergence, we have pointwise monotoneconvergence in m.


Then we compute

v(k)t (x , t) ≤ u

(m)t (x , t)

= m∆(u(m)(x , t) − 1)+

= k∆(v (k)(x , t) − 1)+


In particular:

k[u(k) − 1]+ ≥ k[v (k) − 1]+ ≡ m[u(m) − 1]+,

So nevermind weak convergence, we have pointwise monotoneconvergence in m.

If p is allowed to decrease in time we need a totally different strategy ...


The Hard Case: Nonmonotone p

Now and for the duration of the talk p > 0, but there is no monotonicityassumed.




We do know by using only WMP that the range of u(m)(x , ·) is{uI (x)} ∪ [1, 1 + M/m], and so, using very basic analysis one can showthat the range of u(∞)(x , ·) must be a subset of the interval [uI (x), 1].





Of course we want the range to be only {uI (x)} ∪ {1}, but thinking aboutthe Haar basis, for example, reveals that this may not be so easy to do.





Of course we want the range to be only {uI (x)} ∪ {1}, but thinking aboutthe Haar basis, for example, reveals that this may not be so easy to do.

The solution involves the following steps...


Sketch of Proof of a.e. Convergence

1 Baiocchi Transform the solutions of the m-approximating problem.

W (m)(x , t) :=

∫

t

0V (m)(x , s) ds .




W (m)(x , t) :=

∫

t

0V (m)(x , s) ds .

2 Prove some apriori spatial regularity for these functions. This enablesus to say that the W (m)(·, T ) solve obstacle problems. In fact:

∆xW(m)(x , T ) = χ

{W(m)(x,T )>0}

(x)(u(m)(x , T ) − uI (x)) .




W (m)(x , t) :=

∫

t

0V (m)(x , s) ds .


∆xW(m)(x , T ) = χ

{W(m)(x,T )>0}

(x)(u(m)(x , T ) − uI (x)) .

3 We adapt some stability theory found in Blank 2001 to get L1

convergence of the characteristic function of the set whereW (m)(·, t) > 0.




W (m)(x , t) :=

∫

t

0V (m)(x , s) ds .


∆xW(m)(x , T ) = χ

{W(m)(x,T )>0}

(x)(u(m)(x , T ) − uI (x)) .

3 We adapt some stability theory found in Blank 2001 to get L1

convergence of the characteristic function of the set whereW (m)(·, t) > 0.

4 Taking a subsequence gives us pointwise convergence of thesecharacteristic functions, and in that same subsequence we willnecessarily have pointwise convergence of the u(m)’s.


How the Obstacle Problem Arises

∆xW(m)(x , T ) = ∆x

∫

T

0V (m)(x , s) ds

=

∫

T

0∆xV

(m)(x , s) ds

=

∫

T

0u

(m)t (x , s) ds

= u(m)(x , T ) − u(m)(x , 0)

= u(m)(x , T ) − uI (x) .


How the Obstacle Problem Arises

∆xW(m)(x , T ) = ∆x

∫

T

0V (m)(x , s) ds

=

∫

T

0∆xV

(m)(x , s) ds

=

∫

T

0u

(m)t (x , s) ds

= u(m)(x , T ) − u(m)(x , 0)

= u(m)(x , T ) − uI (x) .

By having a bit more fun with stability we show that the equations abovewill hold at every time and not just almost every time. This fact hasimportant consequences for the regularity theory.


Limiting Problem

Of course one can (and should) ask whether the solutions of the limitingproblem will also satisfy the obstacle problem after the Baiocchi Transform.


Limiting Problem


After a bit more work, the answer is yes.


Limiting Problem


After a bit more work, the answer is yes.

So, what are the consequences for the regularity of the free boundary?


Caffarelli Alternative

In 1977 Caffarelli proved the following remarkable theorem:

Theorem (Caffarelli’s Alternative)

Let w ≥ 0 satisfy:

∆w = χ{w>0}

f (x)

where 0 < λ ≤ f ≤ µ, and f ∈ Cα. Then either

a. 0 is a Singular Point of FB(w) : in which case Λ is “cusp-like” near

0, or

b. 0 is a Regular Point of FB(w) : in which case ∂Λ is C 1,α near 0.

In the actual theorem, Caffarelli states things in a much more quantitativeway.


Picture of Caffarelli’s Alternative

Corner

Not a possible contact setby Caffarelli’s Theorem

Cusp

This set could be thecontact set for a solutionto the obstacle problem.


Picture of Caffarelli’s Alternative

Corner

Not a possible contact setby Caffarelli’s Theorem

Cusp

This set could be thecontact set for a solutionto the obstacle problem.

In Blank 2001, I gave all sorts of extensions of this result and sharpversions of this result when f is not assumed to be Holder continuous.The key ideas in proving most of the results involved improvements to theexisting stability theory.


Application to Hele-Shaw

King, Lacey, and Vazquez have a very well known result stating that if aHele-Shaw flow starts with a corner with an angle in a certain range, thenthat corner will persist for some time.




Theorem (BKM)

If a Hele-Shaw flow has a smooth enough slot, Dini-continuous initial

data, and positive Dirichlet or Neumann data on the slot, then the fluid

will never form a corner no matter what the topology of the slot or slots

may be.




Theorem (BKM)

If a Hele-Shaw flow has a smooth enough slot, Dini-continuous initial

data, and positive Dirichlet or Neumann data on the slot, then the fluid

will never form a corner no matter what the topology of the slot or slots

may be.

Basically, the regularity of the wet region is the regularity of thenoncontact set of an obstacle problem, and so it cannot form corners.


Behavior of the FB in Time

Because of focusing effects and because of the increased speed of a cusp,the free boundary will (in general) only be Cα in time.




On the other hand, one can ask whether this nonLipschitz behavior in timecan occur when the free boundary is spatially smooth.





It can’t. [BKM]. In particular, if a free boundary has infinite speed at apoint, then spatially, that point is an irregular point of the CaffarelliAlternative.





It can’t. [BKM]. In particular, if a free boundary has infinite speed at apoint, then spatially, that point is an irregular point of the CaffarelliAlternative.

This result is proved by making use of the uniform linear stability ofsmooth free boundaries to the obstacle problem given in Blank 2001.Basically, the result says that if you have a smooth free boundary, and youperturb the Laplacian of the obstacle by a sufficiently small ǫ, then themost the free boundary will move is Cǫ.


Open Problems


Open Problems

Do cusps persist?


Open Problems

Do cusps persist?

Are spatially smooth “fronts” C 1 in time?


Open Problems

Do cusps persist?


Is V continuous at all spatially smooth FB points?


Open Problems

Do cusps persist?


Is V continuous at all spatially smooth FB points?

Is there a good comparison between our weak solutions and Kim’sviscosity solutions?


The End

Thank you for listening!


Documents

The Hele-Shaw Problem as a limit of Stefan Problemsblanki/WCNAtalk.pdf · The Hele-Shaw Problem as a limit of Stefan Problems I. Blank, M. Korten, C. Moore KSU July 3, 2008 I. Blank,