lecture no 2 degenerate di usion free boundary problems · 2009-06-16 · panagiota daskalopoulos...

Lecture No 2Degenerate Diffusion

Free boundary problems

Panagiota Daskalopoulos

Columbia University

IAS summer programJune, 2009

Panagiota Daskalopoulos Lecture No 2 Degenerate Diffusion Free boundary problems

Outline

We will discuss non-linear parabolic equations of slow diffusion.Our model is the porous medium equation

ut = ∆um = div (m um−1∇u), m > 1.

It describes various diffusion processes, for example the flow ofgas through a porous medium, where u is the density of thegas and f := um−1 is the pressure of the gas.

Since, the diffusivity D(u) = m um−1 ↓ 0, as u ↓ 0 theequation becomes degenerate at u = 0, resulting to thephenomenon of finite speed of propagation.


Other examples of degenerate diffusion

Other examples of slow (degenerate) diffusion are:

Evolution p-Laplacian Equation (quasi-linear)

ut = ∇ · (|∇u|p−2∇u), p > 2

which becomes degenerate where ∇u = 0.

Gauss Curvature Flow with flat sides (fully-nonlinear)

Let z = u(x , y , t) be the graph of a surface Σ2 ⊂ R3 which isdeformed by a normal speed which is proportional to theGaussian curvature K of the surface. Then, u satisfies

ut =det D2u

(1 + u2x + u2

y )3/2

which becomes degenerate on flat regions where the GaussianCurvature K ∼ det D2u vanishes.


Scaling and the Barenblatt solution

Scaling: If u solves the p.m.e, then u(x , t) = γ−1 u(α x , β t) also

solves the p.m.e iff γ =(α2

β

) 1m−1

.

Self-Similar solution: The above scaling properties lead in 1950Zeldovich, Kompaneets and Barenblatt to find a source-typeself-similar solution of the p.m.e. given by:

U(x , t) = t−λ(

C − k|x |2

t2µ

) 1m−1

+

with

λ =n

n (m − 1) + 2, µ =

λ

n, k =

λ (m − 1)

2mn.

This plays the role of the ”fundamental solution”.


The Barenblatt Solution

0 < t1 < t2 < t3

-

6 z

t1

t2

t3

x


Finite Speed of propagation

The Barenblatt solution shows that solutions to the p.m.e have thefollowing properties:

Finite speed of propagation: If the initial data u0 hascompact support, then at all times the solution u(·, t) willhave compact support.

Free-boundaries: The interface Γ = ∂(suppu) behaves like afree-boundary propagating with finite speed.

Solutions are not smooth: Solutions with compact support areonly of class Cα near the interface.

Weak solutions: Since solutions are not smooth the notion ofweak solutions needs to be introduced.


The Cauchy problem with L1 initial data

Definition. We say that u ≥ 0 is a weak solution of the p.m.e if it iscontinuous and satisfies ut = ∆um in the distributional sense, i.e.∫∫

Rn×(0,∞)u φt + um ∆φ dx dt = 0

for all test functions φ ∈ C∞0 (Rn × (0,∞).

Existence and uniqueness. Given an initial data u0 ∈ L1(Rn), thereexists a unique weak solution of the Cauchy problem{

ut = ∆um in Rn × (0,∞)

u(·, 0) = u0 on Rn

such that u ∈ C ([0,T ]; L1(Rn)).


Contraction property

If u1, u2 ∈ C ([0,T ]; L1(Rn)) are two weak solutions of the Cauchyproblem {

ut = ∆um in Rn × (0,∞)

u(·, 0) = u0 on Rn

with ui0 ∈ L1(Rn), then

(∗)∫

Rn

|u1(x , t)− u2(x , t)| dx ≤∫

Rn

|u10(x)− u2

0(x)| dx .

The uniqueness of solutions in this class follows easily from (∗).


The Aronson-Benilan inequality

Aronson-Benilan Inequality: Every solution u to the p.m.e. satisfiesthe differential inequality

(∗) ut ≥ −k u

t, λ =

1

(m − 1) + 2n

.

The pressure v := mm−1 um−1 which evolves by the equation

vt = (m − 1) v ∆v + |∇v |2

satisfies the sharp differential inequality

(∗∗) ∆v ≥ −λt.

Remark: The Aronson-Benilan (∗) inequality follows from (∗∗).The differential inequality (∗∗) becomes an equality when v is theBarenblatt solution.


The Li-Yau Harnack inequality

The Aronson-Benilan inequality ∆v ≥ −λt and the equation for v

imply the inequality:

vt + (m − 1)λv

t≥ |∇v |2.

Li-Yau Harnack Inequality:If 0 < t1 < t2, then

v(x1, t1) ≤(

t2

t1

)µ [v(x2, t2) +

δ

4

|x2 − x1|2

tδ2 − tδ1t−µ2

].

with µ = (m − 1)λ < 1 and δ = 2λn .

Application: If v(0,T ) <∞, then for all 0 < t < T − ε we have:

v(x , t) ≤ t−µ (Tµ v(0,T ) + C (n,m, ε) |x |2)

i.e. v grows at most quadratically as |x | → ∞.


The Cauchy problem with general initial data

Let u ≥ 0 be a weak solution of ut = ∆um on Rn × (0,T ].

The initial trace µ0 exists; there exists a Borel measure µ suchthat

limt↓0

u(·, t) = µ0 in D ′(Rn)

and satisfies the growth condition

(∗) supR>1

1

Rn+2/(m−1)

∫|x |<R

dµ0 <∞.

The trace µ0 determines the solution uniquely.

For every measure µ0 on Rn satisfying (∗) there exists acontinuous weak solution u of the p.m.e. with trace µ0.

All solutions satisfy the estimate u(x , t) ≤ Ct(u) |x |2/(m−1), as|x | → ∞.


The regularity of solutions

Assume that u is a continuous weak solution of equation

ut = ∆um, m > 1 on Q := Bρ(x0)× (t1, t2).

Question: What is the optimal regularity of the solution u ?

Caffarelli and Friedman: The solution u is of class Cα, forsome α > 0.

It follows from parabolic regularity theory that if u > 0 in Qthen u ∈ C∞(Q).

Proof: If 0 < λ ≤ u ≤ Λ in Q, then ut = div (m um−1∇u) isstrictly parabolic with bounded measurable coefficients.

It follows from the Krylov-Safonov estimate that u ∈ Cγ , forsome γ > 0, hence D(u) := m um−1 ∈ Cα.

We conclude that from the Schauder estimate that u ∈ C 2+α

and by repeating then same estimate we obtain that u ∈ C∞.


The regularity of the free-boundary

Assume that the initial data u0 has compact support and letu be the unique solution of

ut = ∆um in Rn × (0,∞), u(·, t) = u0.

Question: What is the optimal regularity of the free-boundaryΓ := ∂(suppu) and the solution u up to the free-boundary ?

Caffarelli-Friedman: The free-boundary is Holder Continuous.

Caffarelli-Vazquez-Wolanski: If suppu0 ⊂⊂ BR , then thepressure v := m

m−1um−1 is Lipschitz continuous for t ≥ t0,

where t0 is such that BR ⊂⊂ supp u(·, t0).

Caffarelli-Wolanski: The free-boundary is of class C 1+α, fort ≥ t0.


Equations and non-degenercy conditions

Consider the Cauchy problem for the p.m.e:{ut = ∆um in Rn × (0,∞)

u(·, 0) = u0 on Rn

with u0 ≥ 0 and compactly supported. It is more natural toconsider the pressure v = m

m−1 um−1 which satisfies

(∗)

{vt = (m − 1) v ∆v + |∇v |2 in Rn × (0,∞)

v(·, 0) = v0 in Rn.

Our goal is to prove the existence of a solution v of (∗) which isC∞ smooth up to the interface Γ = ∂(supp v). In particular, thefree-boundary Γ will be smooth.


Short time C∞ regularity

Non-degeneracy Condition: We will assume that the initial pressurev0 satisfies:

(∗∗) |∇v0| ≥ c0 > 0, at suppv0

which implies that the free-boundary will start moving at t > 0.

Theorem (Short time Regularity) (D., Hamilton)Assume that at t = 0, the pressure v0 ∈ C 2+α

s and satisfies (∗∗).Then, there exists τ0 > 0 and a unique solution v of the Cauchyproblem (∗) on Rn × [0, τ0] which is smooth up to the interface Γ.In particular, the interface Γ is smooth.

Remark: The space C 2+αs is Holder space for second derivatives

that it is scaled with respect to an appropriate singular metric s.This is necessary because of the degeneracy of our equation.


Short time Regularity - Sketch of proof

Coordinate change: We perform a change of coordinateswhich fixes the free-boundary: Let P0 ∈ Γ(t) s.t.

vx > 0 and vy = 0, at P0.

Solve z = v(x , y , t) near P0 w.r to x = h(z , y , t) to transformthe free-boundary v = 0 into the fixed boundary z = 0.

The function h evolves by the quasi-linear, degenerateequation

(#) ht = (m − 1) z(

1+h2y

h2z

hzz − 2hy

hzhzy + hyy

)− 1+h2

y

hz

Outline: Construct a sufficiently smooth solution of (#) viathe Inverse function Theorem between appropriate Holderspaces, scaled according to a singular metric.


The Model Equation

Our problem is modeled on the equation

ht = z (hzz + hyy ) + ν hz , on z > 0

with ν > 0.The diffusion is governed by the cycloidal metric

ds2 =dz2 + dy2

z, on z > 0

We define the distance function according to this metric:

s((z1, y1), (z2, y2)) =|z1 − z2|+ |y1 − y2|

√z1 +

√z2 +

√|y1 − y2|

.

The parabolic distance is defined as:

s((Q1, t1), (Q2, t2)) = s(Q1,Q2) +√|t1 − t2|.


Holder Spaces:

Let Cαs denote the space of Holder continuous functions h

with respect to the parabolic distance function s.

C 2+αs : h, ht , hz , hy , z hzz , z hzy , z hyy ∈ Cα

s .

Theorem (Schauder Estimate) Assume that h solves

ht = z (hzz + hyy ) + ν hz + g , on Q2

with ν > 0 and Qr = {0 ≤ z ≤ r , |y | ≤ r , t0 − r ≤ t ≤ t0}.Then,

‖h‖C2+αs (Q1) ≤ C

{‖h‖C0

s (Q2) + ‖g‖Cαs (Q2)

}.

Proof: We prove the Schauder estimate using the method ofapproximation by polynomials introduced by L. Caffarelli andl. Wang.


Short time regularity - summary

Using the Schauder estimate, we construct a sufficientlysmooth solution of (∗) via the Inverse function Theorembetween the Holder spaces Cα

s and C 2+αs , which are scaled

according to the singular metric s.

Once we have a C 2+αs solution we can show that the solution

v is C∞ smooth. Hence, the free-boundary Γ ∈ C∞.

Observation: To obtain the optimal regularity, degenerateequations need to be scaled according to the right singularmetric.

Remark: You actually need a global change of coordinateswhich transforms the free-boundary problem to a fixedboundary problem for a non-linear degenerate equation.


Long time regularity

It is well known that the free-boundary will not remainsmooth (in general) for all time. Advancing free-boundariesmay hit each other creating singularities.

Koch: (Long time regularity) Under certain natural initialconditions, the pressure v will be become smooth up to theinterface for t ≥ T0, with T0 sufficiently large.

Question: Under what geometric conditions the interface willbecome smooth and remain so at all time ?

Theorem (All time Regularity) (D., Hamilton and Lee)If the initial pressure v0 is root concave, then the pressure vwill be smooth and root-concave at all times t > 0. Inparticular, the interface will remain convex and smooth.


Selected References

Aronson, D.G. and Benilan P., Regularite des solutions del’equation de milieux poreux dans Rn, C.R. Acad. Sci. Paris,288, 1979, pp 103-105.

D. G. Aronson, and L. Caffarelli, The initial trace of a solutionof the porous medium equation,’ Trans. Amer. Math. Soc.280 (1983) 351-366.

Auchmuty, G. and Bao, D., Harnack-type inequalities forevolution equations. Proc. Amer. Math. Soc. 122 (1994),no. 1, 117–129.

Caffarelli, Luis A. and Friedman, A., Continuity of the densityof a gas flow in a porous medium. Trans. Amer. Math. Soc.252 (1979), 99–113.

Caffarelli, L. A., Vazquez, J. L. and Wolanski, N. I. ; Lipschitzcontinuity of solutions and interfaces of the N-dimensionalporous medium equation. Indiana Univ. Math. J. 36 (1987),no. 2, 373–401.


Selected References

B.E.J. Dahlberg, and C.E. Kenig, Nonnegative solutions to theporous medium equation, Comm. in P.D.E. 9 (1984) 409-437.

Daskalopoulos, P., Hamilton, R., C∞-Regularity of the FreeBoundary for the porous medium equation, J. of Amer. Math.Soc., Vol. 11, No 4, (1998) pp 899-965.

Daskalopoulos, P., Kenig, C., Degenerate diffusions. Initialvalue problems and local regularity theory, EMS Tracts inMathematics, 1. European Mathematical Society (EMS),Zurich, 2007.

M. Pierre, Uniqueness of the solution of ut −∆ϕ(u) = 0 withinitial datum a measure, Nonlinear Anal. 6 (1982), 175-187.

Vazquez, J.L., An Introduction to the theory of the porousmedium equation; Lecture notes.


lecture no 2 degenerate di usion free boundary problems · 2009-06-16 · panagiota daskalopoulos...

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