Free boundaries in fractional filtration equations

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1 Free boundaries in fractional filtration equations Fernando Quirós Universidad Autónoma de Madrid Joint work with Arturo de Pablo, Ana Rodríguez and Juan Luis Vázquez International Conference on Free Boundary Problems Isaac Newton Institute, Cambridge, UK Thursday, June 26th, 2014 Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

2 Fractional filtration equation t u + ( ) σ/2 (ϕ(u)) = 0, x R N, t > 0 u(, 0) = f EXPONENTS: 0 < σ < 2 INITIAL DATA: f L 1 (R N ), f 0 NON-LINEARITIES: ϕ continuous, non-decreasing, ϕ(0) = 0 u m, m > 0 Examples: ϕ(u) = log(1 + u) (u 1) + Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

3 A α A non-negative, self-adjoint, linear operator A α u(x) = 1 Γ( α) 0 ( e ta u(x) u(x) ) dt t 1+α e ta u: solution to { v (t) + Av(t) = 0 v(0) = u λ α = 1 Γ( α) 0 ( ) dt e tλ 1 t 1+α, λ > 0 Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

4 ( ) σ/2 ( ) σ/2 u(x) = 1 Γ( σ 2 ) 0 ( e t u(x) u(x) ) dt t 1+ σ 2 F ( ( ) σ/2 u ) (ξ) = ξ σ F(u)(ξ) ( ) σ/2 u(x) u(y) u(x) = C N,σ P.V. dy R N x y N+σ NONLOCAL OPERATOR Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

5 Motivation Non-linear generalization of the fractional heat equation t u + ( ) σ/2 u = 0 Non-local generalization of the filtration equation t u (ϕ(u)) = 0 (Other possible generalizations [Caffarelli-Vázquez, 2009]) Hydrodynamic limit of zero range processes [Jara, 2009] ϕ(u) = log(1 + u) Nonlocal critical viscous transport equation [dpqrv, 2014], [Córdoba-Córdoba-Fontelos, 2005] Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

6 Linear case: fundamental solution t P + ( ) σ/2 P = 0, x R N, t > 0 P(x, 0) = δ 0 (x), x R N P(ξ, t) = e ξ σ t P(x, t) = t N/σ Φ(xt 1/σ ) self-similar! Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

7 Linear case: fundamental solution t P + ( ) σ/2 P = 0, x R N, t > 0 P(x, 0) = δ 0 (x), x R N P(ξ, t) = e ξ σ t P(x, t) = t N/σ Φ(xt 1/σ ) self-similar! Properties of Φ [Blumenthal-Getoor, 1960]: C, positive, radially decreasing Φ(z) z N σ, z Explicit only if σ = 1, 2 (Poisson, Gauss). Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

8 Integration by parts / Functional space R N ( ) σ/2 ϕ ψ = = ξ σ ˆϕ ˆψ = ξ σ/2 ˆϕ ξ σ/2 ˆψ R N R N ( ) σ/4 ϕ ( ) σ/4 ψ R N Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

9 Integration by parts / Functional space R N ( ) σ/2 ϕ ψ = = ξ σ ˆϕ ˆψ = ξ σ/2 ˆϕ ξ σ/2 ˆψ R N R N ( ) σ/4 ϕ ( ) σ/4 ψ R N ϕ Ḣσ/2 = = ( ) 1/2 ξ σ ˆϕ 2 dξ = ( ) σ/4 ϕ 2 R ( N RN CN,σ ϕ(x) ϕ(y) 2 2 x y N+σ dx dy R N ) 1/2 Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

10 Weak (L 1 -energy) solutions u C([0, ) : L 1 (R N )), ϕ(u) L 2 loc ((0, ) : Ḣσ/2 (R N )) u ψ 0 R N t dxds ( ) σ/4 (ϕ(u))( ) σ/4 ψ dxds = 0, 0 R N ψ C 1 c(r N (0, )) u(, 0) = f a.e. Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

11 Weak (L 1 -energy) solutions u C([0, ) : L 1 (R N )), ϕ(u) L 2 loc ((0, ) : Ḣσ/2 (R N )) u ψ 0 R N t dxds ( ) σ/4 (ϕ(u))( ) σ/4 ψ dxds = 0, 0 R N ψ C 1 c(r N (0, )) u(, 0) = f a.e. THEOREM [dpqrv, 2011], [dpqrv, 2012]: u 0 L 1 (R N ) L (R N ) unique bounded weak solution Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

12 σ-harmonic extension [Caffarelli-Silvestre, 2007] v = E(g) : L σ v div(y 1 σ v) = 0, R N+1 + = {x R N, y > 0} v(x, 0) = g(x), x R N v(x, y) = RN P(x ξ, y)g(ξ) dξ, P(x, y) = d N,σ y σ v lim y1 σ y 0 + = σ lim y 0 + y = σ lim v(x, y) v(x, 0) y 0 + y σ P(x ξ, y) R N y σ ( x 2 + y 2 ) (N+σ)/2 (g(ξ) g(x)) dξ = σd N,σ C N,σ ( ) σ/2 g v y σ µ v σ lim y1 σ y 0 + y = ( )σ/2 g Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

13 An equivalent local problem w = E(ϕ(u)), u = ϕ 1 (Tr(w)) L σ w = 0, (x, y) R N+1 +, t > 0, w y σ = ϕ 1 (w), x R N, y = 0, t > 0, t w = ϕ(f ), x R N, y = 0, t = 0. Some proofs (e.g. existence) may be easier in this formulation! Dynamical boundary conditions (Amann, Escher, Fila, Vitillaro,...) Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

14 L 1 L p L [dpqrv, 2012], [VdPQR, 2014] ϕ (s) C s m 1, s C THEOREM: f L 1 (R N ) L p (R N ), p 1, p > (1 m)n/σ { } sup u(x, t) max C 1, C 2 t N/(N(m 1)+σp) f p σp/(n(m 1)+σp) x R N Moser iteration (Steklov averages) Generalized Stroock-Varopoulos inequality [S, 1984], [V, 1985]: ψ = (Ψ ) 2 R N ( ) σ/4 (ψ(v))( ) σ/4 v R N ( ) σ/4 Ψ(v) 2 Nash-Gagliardo-Nirenberg inequality [dpqrv, 2012]: φ 1+ p 2 N(p+2) 2N σ C φ p/2 p φ Ḣσ/2 Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

15 L C α [Athanasopoulos-Caffarelli, 2009] Non-degeneracy condition: sup ϕ (s) C ϕ(s 2) ϕ(s 1 ), A s 1 < s 2 B. s [s 1,s 2 ] s 2 s 1 THEOREM: Weak + bounded u is C α in {A u(x, t) B} for some α (0, 1) Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

16 L C α [Athanasopoulos-Caffarelli, 2009] Non-degeneracy condition: sup ϕ (s) C ϕ(s 2) ϕ(s 1 ), A s 1 < s 2 B. s [s 1,s 2 ] s 2 s 1 THEOREM: Weak + bounded u is C α in {A u(x, t) B} for some α (0, 1) Particular cases: ϕ (s) c > 0 ϕ(u) = u m, m 1 ϕ(u) = u m, 0 < m < 1 Requires positivity Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

17 L C α [Athanasopoulos-Caffarelli, 2009] Non-degeneracy condition: sup ϕ (s) C ϕ(s 2) ϕ(s 1 ), A s 1 < s 2 B. s [s 1,s 2 ] s 2 s 1 THEOREM: Weak + bounded u is C α in {A u(x, t) B} for some α (0, 1) Particular cases: ϕ (s) c > 0 ϕ(u) = u m, m 1 ϕ(u) = u m, 0 < m < 1 Requires positivity Stefan: only continuity Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

18 C α classical: the main idea [VdPQR, 2014] u 0 = u(x 0, t 0 ), ϕ (u 0 )>0 t u + ϕ (u 0 )( ) σ/2 u = ( ) σ/2 ( ϕ(u) ϕ (u 0 )u ) = ( ) σ/2 ( ϕ(u) ( ϕ(u 0 ) + ϕ (u 0 )(u u 0 ) )) }{{} F(u) Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

19 C α classical: the main idea [VdPQR, 2014] u 0 = u(x 0, t 0 ), ϕ (u 0 )>0 t u + ϕ (u 0 )( ) σ/2 u = ( ) σ/2 ( ϕ(u) ϕ (u 0 )u ) TWO KEY POINTS: 1 u is as regular as F(u) = ( ) σ/2 ( ϕ(u) ( ϕ(u 0 ) + ϕ (u 0 )(u u 0 ) )) }{{} F(u) 2 F(u) is more regular (at (x 0, t 0 )) than u Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

20 C α classical: the main idea [VdPQR, 2014] u 0 = u(x 0, t 0 ), ϕ (u 0 )>0 t u + ϕ (u 0 )( ) σ/2 u = ( ) σ/2 ( ϕ(u) ϕ (u 0 )u ) TWO KEY POINTS: 1 u is as regular as F(u) = ( ) σ/2 ( ϕ(u) ( ϕ(u 0 ) + ϕ (u 0 )(u u 0 ) )) }{{} F(u) 2 F(u) is more regular (at (x 0, t 0 )) than u [Caffarelli-Vasseur, 2010]: QG equation, (σ = 1, RHS local) [dpqrv, 2014]: logarithmic diffusion, (N = σ = 1) Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

21 Linear problem t u + ( ) σ/2 u = ( ) σ/2 g Representation formula, Y = (x, t) Q = R N (0, ) u(y) = P(x x, t)f (x) dx R N t + P(Y Y)( ) σ/2 g(y) dy 0 R N P fractional heat kernel Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

22 Linear problem t u + ( ) σ/2 u = ( ) σ/2 g Representation formula, Y = (x, t) Q = R N (0, ) u(y) = P(x x, t)f (x) dx R N t + ( ) σ/2 P(Y Y)g(Y) dy 0 R N P fractional heat kernel Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

23 Linear problem t u + ( ) σ/2 u = ( ) σ/2 g Representation formula, Y = (x, t) Q = R N (0, ) u(y) = P(x x, t)f (x) dx R N t + ( ) σ/2 P(Y Y)g(Y) dy 0 R N A = ( ) σ/2 P P fractional heat kernel Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

24 Linear problem: a singular kernel u(y) = A(Y Y)χ {t<t} g(y) dg Q Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

25 Linear problem: a singular kernel u(y) = A(Y Y)χ {t<t} g(y) dg Q MAIN DIFFICULTY: A is not explicit, butâ(ξ) = ξ σ e ξ σ t Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

26 Linear problem: a singular kernel u(y) = A(Y Y)χ {t<t} g(y) dg Q MAIN DIFFICULTY: A is not explicit, butâ(ξ) = ξ σ e ξ σ t MAIN CHARACTERISTIC: A is selfsimilar, A(x, t) = t N/σ 1 Ψ(z), z = xt 1/σ R N, t > 0. Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

27 Linear problem: a singular kernel u(y) = A(Y Y)χ {t<t} g(y) dg Q MAIN DIFFICULTY: A is not explicit, butâ(ξ) = ξ σ e ξ σ t MAIN CHARACTERISTIC: A is selfsimilar, A(x, t) = t N/σ 1 Ψ(z), z = xt 1/σ R N, t > 0. MAIN TOOL: Use of σ parabolic topology, ( ) 1/2 Y σ := x 2 + t 2/σ = t 1/σ ( z 2 + 1) 1/2. and σ parabolic Hölder spaces C α σ(q) = {u : u(y 1 ) u(y 2 ) c Y 1 Y 2 α σ, Y 1, Y 2 Q} Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

28 Linear problem: a singular kernel u(y) = A(Y Y)χ {t<t} g(y) dg Q MAIN PROPERTIES: Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

29 Linear problem: a singular kernel u(y) = A(Y Y)χ {t<t} g(y) dg Q MAIN PROPERTIES: 1 Estimates A(Y) c Y N+σ σ, t A(Y) c Y N+2σ σ, x A(Y) c Y N+σ+1 σ Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

30 Linear problem: a singular kernel u(y) = A(Y Y)χ {t<t} g(y) dg Q MAIN PROPERTIES: 1 Estimates A(Y) c Y N+σ σ, t A(Y) c Y N+2σ σ, x A(Y) c Y N+σ+1 σ 2 Cancelation B ± R,ε where B ± R,ε = {ε < Y σ < R, t 0} A(Y) dy = 0 Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

31 Regularity [VdPQR, 2014] THEOREM: u bounded + weak, ϕ C 1,γ (R), 1 + γ > σ, ϕ (s) > 0 u is a classical solution Local version ϕ(u) = u m + positivity Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

32 Regularity [VdPQR, 2014] THEOREM: u bounded + weak, ϕ C 1,γ (R), 1 + γ > σ, ϕ (s) > 0 u is a classical solution Local version ϕ(u) = u m + positivity THEOREM: u bounded + weak, ϕ C k,γ (R), k 2, ϕ (s) > 0 u C k,α (Q) for some 0 < α < 1 Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

33 Positivity f 0 u 0 u(x 0, t 0 ) = 0, u(, t 0 ) 0 t u(x 0, t 0 ) = ( ) σ/2 ϕ(u)(x 0, t 0 ) ϕ(u(x 0, t 0 )) ϕ(u(y, t 0 )) = C N,σ P.V. R N x y N+σ dy ϕ(u(y, t 0 )) = C N,σ P.V. dy R N x y N+σ > 0 Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

34 Free boundaries: Stefan ϕ(u) = (u 1) + u: enthalpy, θ = (u 1) + : temperature Free boundary: {θ > 0}. Localization Open problems: Regularity Regularity of the free boundary Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

35 Free boundaries: mesa problem φ m (u) = u m, m φ (u) = 0, 0 < u < 1, φ (u) [0, ), u = 1 THEOREM [Quirós-Vázquez, WIP]: u m (, t) f := f ( ) σ/2 w in L 1 (R N ) w, ( ) σ/2 w L 1 (R N ), w 0, 1 f + ( ) σ/2 w 0, (1 f + ( ) σ/2 w)w = 0. Baiocchi variable: w m = t u m m 0 Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

36 Free boundaries: mesa problem f 1 f = f w(x) > 0 f (x) = 1 f = R N f R N w compactly supported, f > 0 (unless w 0) f (x) 1 f (x) = 1 f (x) < 1 f (x) f (x) 1 Local case (σ = 2): f = f χ{w=0} + χ {w>0} Not true in general when σ (0, 2) Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

37 References Athanasopoulos, I.; Caffarelli, L. A. Continuity of the temperature in boundary heat control problems. Adv. Math. 224 (2010), no. 1, Blumenthal, R. M.; Getoor, R. K. Some theorems on stable processes. Trans. Amer. Math. Soc. 95 (1960), no. 2, Caffarelli, L.; Silvestre, L. An extension problem related to the fractional Laplacian. Comm. Partial Differential Equations 32 (2007), no. 7-9, Caffarelli, L. A.; Vasseur, A. Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann. of Math. (2) 171 (2010), no. 3, Caffarelli, L. A.; Vázquez, J. L. Nonlinear porous medium flow with fractional potential pressure. ArXiv: [math.ap], to appear in Arch. Rational Mech. Anal. Córdoba, A.; Córdoba, D.; Fontelos, M. A. Formation of singularities for a transport equation with nonlocal velocity. Ann. of Math. (2) 162 (2005), no. 3, Jara, M. Hydrodynamic limit of particle systems with long jumps. arxiv: v2 [math.pr] Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

38 References de Pablo, A.; Quirós, F.; Rodriguez, A.; Vázquez, J. L. A fractional porous medium equation. Adv. Math. 226 (2011), no. 2, de Pablo, A.; Quirós, F.; Rodriguez, A.; Vázquez, J. L. A general fractional porous medium equation. Comm. Pure Appl. Math. 65 (2012), no. 9, de Pablo, A.; Quirós, F.; Rodriguez, A.; Vázquez, J. L. Classical solutions for a logarithmic fractional diffusion equation. J. Math. Pures Appl., 101 (2014), no. 6, Quirós, F.; Vázquez, J. L. The mesa limit for the fractional Porous Media Equation. Work in progress. Vázquez, J. L.; de Pablo, A.; Quirós, F.; Rodriguez, A. Classical solutions and higher regularity for nonlinear fractional diffusion equations. Submitted. Available at arxiv: [math.ap] Stroock, D. W. An introduction to the theory of large deviations. Universitext. Springer-Verlag, New York, ISBN: X Varopoulos, N. Th. Hardy-Littlewood theory for semigroups. J. Funct. Anal. 63 (1985), no. 2, Fernando Quirós International Conference on FPB Isaac Newton Institute, Cambridge Thursday 26th,

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