Lecture No 2 Degenerate Diffusion Free boundary problems Columbia University IAS summer program June, 2009
Outline We will discuss non-linear parabolic equations of slow diffusion. Our model is the porous medium equation u t = u m = div (m u m 1 u), m > 1. It describes various diffusion processes, for example the flow of gas through a porous medium, where u is the density of the gas and f := u m 1 is the pressure of the gas. Since, the diffusivity D(u) = m u m 1 0, as u 0 the equation becomes degenerate at u = 0, resulting to the phenomenon of finite speed of propagation.
Other examples of degenerate diffusion Other examples of slow (degenerate) diffusion are: Evolution p-laplacian Equation (quasi-linear) u t = ( 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 Σ 2 R 3 which is deformed by a normal speed which is proportional to the Gaussian curvature K of the surface. Then, u satisfies u t = det D 2 u (1 + u 2 x + u 2 y ) 3/2 which becomes degenerate on flat regions where the Gaussian Curvature K det D 2 u vanishes.
Scaling and the Barenblatt solution Scaling: If u solves the p.m.e, then ũ(x, t) = γ 1 u(α x, β t) also ) 1 m 1 solves the p.m.e iff γ =. ( α 2 β Self-Similar solution: The above scaling properties lead in 1950 Zeldovich, Kompaneets and Barenblatt to find a source-type self-similar solution of the p.m.e. given by: with λ = ( ) 1 U(x, t) = t λ C k x 2 m 1 t 2µ + n n (m 1) + 2, µ = λ λ (m 1), k =. n 2mn This plays the role of the fundamental solution.
The Barenblatt Solution 0 < t 1 < t 2 < t 3 z t 1 t 2 t 3
Finite Speed of propagation The Barenblatt solution shows that solutions to the p.m.e have the following properties: Finite speed of propagation: If the initial data u 0 has compact support, then at all times the solution u(, t) will have compact support. Free-boundaries: The interface Γ = (suppu) behaves like a free-boundary propagating with finite speed. Solutions are not smooth: Solutions with compact support are only of class C α near the interface. Weak solutions: Since solutions are not smooth the notion of weak solutions needs to be introduced.
The Cauchy problem with L 1 initial data Definition. We say that u 0 is a weak solution of the p.m.e if it is continuous and satisfies u t = u m in the distributional sense, i.e. u φ t + u m φ dx dt = 0 R n (0, ) for all test functions φ C 0 (Rn (0, ). Existence and uniqueness. Given an initial data u 0 L 1 (R n ), there exists a unique weak solution of the Cauchy problem { u t = u m in R n (0, ) u(, 0) = u 0 on R n such that u C([0, T ]; L 1 (R n )).
Contraction property If u 1, u 2 C([0, T ]; L 1 (R n )) are two weak solutions of the Cauchy problem { u t = u m in R n (0, ) u(, 0) = u 0 on R n with u0 i L1 (R n ), then ( ) u 1 (x, t) u 2 (x, t) dx R n u0(x) 1 u0(x) 2 dx. R n The uniqueness of solutions in this class follows easily from ( ).
The Aronson-Bénilan inequality Aronson-Bénilan Inequality: Every solution u to the p.m.e. satisfies the differential inequality ( ) u t k u t, k = 1 (m 1) + 2. n The pressure v := m m 1 um 1 which evolves by the equation v t = (m 1) v v + v 2 satisfies the sharp differential inequality ( ) v k t. Remark: The Aronson-Bénilan ( ) inequality follows from ( ). The differential inequality ( ) becomes an equality when v is the Barenblatt solution.
The Li-Yau Harnack inequality The Aronson-Bénilan inequality v k t imply the inequality: v t + (m 1) k v t v 2. and the equation for v Li-Yau Harnack Inequality: (Auchmuty-Bao and Hamilton) If 0 < t 1 < t 2, then ( ) µ [ t2 v(x 1, t 1 ) v(x 2, t 2 ) + δ ] x 2 x 1 t 1 4 t2 δ. tδ 1 with µ = (m 1) k < 1 and δ = 2k n. Application: If v(0, T ) <, then for all 0 < t < T ɛ we have: v(x, t) t µ (v(0, T ) + C(n, m, ɛ) 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 u t = u m on R n (0, T ]. The initial trace µ 0 exists; there exists a Borel measure µ such that lim t 0 u(, t) = µ 0 in D (R n ) and satisfies the growth condition 1 ( ) R n+2/(m 1) sup R>1 x <R dµ 0 <. The trace µ 0 determines the solution uniquely. For every measure µ 0 on R n satisfying ( ) there exists a continuous weak solution u of the p.m.e. with trace µ 0. All solutions satisfy the estimate u(x, t) C t (u) x 2/(m 1), as x.
The regularity of solutions Assume that u is a continuous weak solution of equation u t = u m, m > 1 on Q := B ρ (x 0 ) (t 1, t 2 ). Question: What is the optimal regularity of the solution u? Caffarelli and Friedman: The solution u is of class C α, for some α > 0. It follows from parabolic regularity theory that if u > 0 in Q then u C (Q). Proof: If 0 < λ u Λ in Q, then u t = div (m u m 1 u) is strictly parabolic with bounded measurable coefficients. It follows from the Krylov-Safonov estimate that u C γ, for some γ > 0, hence D(u) := m u m 1 C α. 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 u 0 has compact support and let u be the unique solution of u t = u m in R n (0, ), u(, t) = u 0. Question: What is the optimal regularity of the free-boundary Γ := (suppu) and the solution u up to the free-boundary? Caffarelli-Friedman: The free-boundary is Hölder Continuous. Caffarelli-Vazquez-Wolanski: If suppu 0 B R, then the pressure v := m m 1 um 1 is Lipschitz continuous for t t 0, where t 0 is such that B R supp u(, t 0 ). Caffarelli-Wolanski: The free-boundary is of class C 1+α, for t t 0.
Equations and non-degenercy conditions Consider the Cauchy problem for the p.m.e: { u t = u m in R n (0, ) u(, 0) = u 0 on R n with u 0 0 and compactly supported. It is more natural to consider the pressure v = m m 1 um 1 which satisfies { v t = (m 1) v v + v 2 in R n (0, ) ( ) v(, 0) = v 0 in R n. Our goal is to prove the existence of a solution v of ( ) which is C smooth up to the interface Γ = (supp v). In particular, the free-boundary Γ will be smooth.
Short time C regularity Non-degeneracy Condition: We will assume that the initial pressure v 0 satisfies: ( ) v 0 c 0 > 0, at suppv 0 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 v 0 Cs 2+α and satisfies ( ). Then, there exists τ 0 > 0 and a unique solution v of the Cauchy problem ( ) on R n [0, τ 0 ] which is smooth up to the interface Γ. In particular, the interface Γ is smooth. Remark: The space Cs 2+α is Hölder 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 coordinates which fixes the free-boundary: Let P 0 Γ(t) s.t. v x > 0 and v y = 0, at P 0. Solve z = v(x, y, t) near P 0 w.r to x = h(z, y, t) to transform the free-boundary v = 0 into the fixed boundary z = 0. The function h evolves by the quasi-linear, degenerate equation ( 1+h 2 ) (#) h t = (m 1) z y h zz 2hy h z h zy + h yy 1+h2 y h z h 2 z Outline: Construct a sufficiently smooth solution of (#) via the Inverse function Theorem between appropriate Hölder spaces, scaled according to a singular metric.
The Model Equation Our problem is modeled on the equation h t = z (h zz + h yy ) + ν h z, on z > 0 with ν > 0. The diffusion is governed by the cycloidal metric ds 2 = dz2 + dy 2, on z > 0 z We define the distance function according to this metric: s((z 1, y 1 ), (z 2, y 2 )) = z 1 z 2 + y 1 y 2 z1 + z 2 + y 1 y 2. The parabolic distance is defined as: s((q 1, t 1 ), (Q 2, t 2 )) = s(q 1, Q 2 ) + t 1 t 2.
Hölder Spaces: Let Cs α denote the space of Hölder continuous functions h with respect to the parabolic distance function s. C 2+α s : h, h t, h z, h y, z h zz, z h zy, z h yy C α s. Theorem (Schauder Estimate) Assume that h solves h t = z (h zz + h yy ) + ν h z + g, on Q 2 with ν > 0 and Q r = {0 z r, y r, t 0 r t t 0 }. Then, h C 2+α s (Q 1 ) C { } h C 0 s (Q 2 ) + g C α s (Q 2 ). Proof: We prove the Schauder estimate using the method of approximation by polynomials introduced by L. Caffarelli and l. Wang.
Short time regularity - summary Using the Schauder estimate, we construct a sufficiently smooth solution of ( ) via the Inverse function Theorem between the Hölder spaces Cs α and Cs 2+α, which are scaled according to the singular metric s. Once we have a Cs 2+α solution we can show that the solution v is C smooth. Hence, the free-boundary Γ C. Observation: To obtain the optimal regularity, degenerate equations need to be scaled according to the right singular metric. Remark: You actually need a global change of coordinates which transforms the free-boundary problem to a fixed boundary problem for a non-linear degenerate equation.
Long time regularity It is well known that the free-boundary will not remain smooth (in general) for all time. Advancing free-boundaries may hit each other creating singularities. Koch: (Long time regularity) Under certain natural initial conditions, the pressure v will be become smooth up to the interface for t T 0, with T 0 sufficiently large. Question: Under what geometric conditions the interface will become smooth and remain so at all time? Theorem (All time Regularity) (D., Hamilton and Lee) If the initial pressure v 0 is root concave, then the pressure v will be smooth and root-concave at all times t > 0. In particular, the interface will remain convex and smooth.