Lecture No 2 Degenerate Di usion Free boundary problems · 2009-06-16 · Panagiota Daskalopoulos...
Transcript of 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.
Panagiota Daskalopoulos Lecture No 2 Degenerate Diffusion Free boundary problems
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.
Panagiota Daskalopoulos Lecture No 2 Degenerate Diffusion Free boundary problems
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”.
Panagiota Daskalopoulos Lecture No 2 Degenerate Diffusion Free boundary problems
The Barenblatt Solution
0 < t1 < t2 < t3
-
6 z
t1
t2
t3
x
Panagiota Daskalopoulos Lecture No 2 Degenerate Diffusion Free boundary problems
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.
Panagiota Daskalopoulos Lecture No 2 Degenerate Diffusion Free boundary problems
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)).
Panagiota Daskalopoulos Lecture No 2 Degenerate Diffusion Free boundary problems
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 (∗).
Panagiota Daskalopoulos Lecture No 2 Degenerate Diffusion Free boundary problems
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.
Panagiota Daskalopoulos Lecture No 2 Degenerate Diffusion Free boundary problems
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 | → ∞.
Panagiota Daskalopoulos Lecture No 2 Degenerate Diffusion Free boundary problems
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 | → ∞.
Panagiota Daskalopoulos Lecture No 2 Degenerate Diffusion Free boundary problems
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∞.
Panagiota Daskalopoulos Lecture No 2 Degenerate Diffusion Free boundary problems
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.
Panagiota Daskalopoulos Lecture No 2 Degenerate Diffusion Free boundary problems
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.
Panagiota Daskalopoulos Lecture No 2 Degenerate Diffusion Free boundary problems
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.
Panagiota Daskalopoulos Lecture No 2 Degenerate Diffusion Free boundary problems
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.
Panagiota Daskalopoulos Lecture No 2 Degenerate Diffusion Free boundary problems
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|.
Panagiota Daskalopoulos Lecture No 2 Degenerate Diffusion Free boundary problems
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.
Panagiota Daskalopoulos Lecture No 2 Degenerate Diffusion Free boundary problems
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.
Panagiota Daskalopoulos Lecture No 2 Degenerate Diffusion Free boundary problems
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.
Panagiota Daskalopoulos Lecture No 2 Degenerate Diffusion Free boundary problems
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.
Panagiota Daskalopoulos Lecture No 2 Degenerate Diffusion Free boundary problems
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.
Panagiota Daskalopoulos Lecture No 2 Degenerate Diffusion Free boundary problems