Lecture No 2 Degenerate Diffusion Free boundary problems

Transcription

1 Lecture No 2 Degenerate Diffusion Free boundary problems Columbia University IAS summer program June, 2009

2 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.

3 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.

4 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.

5 The Barenblatt Solution 0 < t 1 < t 2 < t 3 z t 1 t 2 t 3

6 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.

7 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 )).

8 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 ( ).

9 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.

10 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.

11 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.

12 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.

13 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.

14 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.

15 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.

16 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.

17 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.

18 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.

19 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.

20 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.