Nonlinear and Nonlocal Degenerate Diffusions on Bounded Domains

Nonlinear and Nonlocal Degenerate Diffusions on Bounded Domains Matteo Bonforte Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco 28049 Madrid, Spain matteo.bonforte@uam.es http://www.uam.es/matteo.bonforte BCAM Scientific Seminar BCAM Basque Center for Applied Mathematics Bilbao, Spain, May 30, 207

References References: [BV] M. B., J. L. VÁZQUEZ, A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations on bounded domains. Arch. Rat. Mech. Anal. (205). [BV2] M. B., J. L. VÁZQUEZ, Fractional Nonlinear Degenerate Diffusion Equations on Bounded Domains Part I. Existence, Uniqueness and Upper Bounds Nonlin. Anal. TMA (206). [BSV] M. B., Y. SIRE, J. L. VÁZQUEZ, Existence, Uniqueness and Asymptotic behaviour for fractional porous medium equations on bounded domains. Discr. Cont. Dyn. Sys. (205). [BFR] M. B., A. FIGALLI, X. ROS-OTON, Infinite speed of propagation and regularity of solutions to the fractional porous medium equation in general domains. To appear in Comm. Pure Appl. Math (207). [BFV] M. B., A. FIGALLI, J. L. VÁZQUEZ, Sharp boundary estimates and higher regularity for nonlocal porous medium-type equations in bounded domains. Preprint (206). https://arxiv.org/abs/60.0988 A talk more focussed on the first part is available online: http://www.fields.utoronto.ca/video-archive//event/202/206

Summary Outline of the talk Introduction The abstract setup of the problem Some important examples About Spectral Kernels Basic Theory The Dual problem Existence and uniqueness First set of estimates Sharp Boundary Behaviour Upper Boundary Estimates Infinite Speed of Propagation Lower Boundary Estimates Harnack Inequalities Numerics Regularity Estimates

Homogeneous Dirichlet Problem for Fractional Nonlinear Degenerate Diffusion Equations u t + L F(u) = 0, in (0, + ) Ω (HDP) u(0, x) = u 0 (x), in Ω u(t, x) = 0, on the lateral boundary. where: Ω R N is a bounded domain with smooth boundary and N. The linear operator L will be: sub-markovian operator densely defined in L (Ω). A wide class of linear operators fall in this class: all fractional Laplacians on domains. The most studied nonlinearity is F(u) = u m u, with m >. We deal with Degenerate diffusion of Porous Medium type. More general classes of degenerate nonlinearities F are allowed. The homogeneous boundary condition is posed on the lateral boundary, which may take different forms, depending on the particular choice of the operator L.

About the operator L The linear operator L : dom(a) L (Ω) L (Ω) is assumed to be densely defined and sub-markovian, more precisely satisfying (A) and (A2) below: (A) L is m-accretive on L (Ω), (A2) If 0 f then 0 e tl f, or equivalently, (A2 ) If β is a maximal monotone graph in R R with 0 β(0), u dom(l), Lu L p (Ω), p, v L p/(p ) (Ω), v(x) β(u(x)) a.e, then v(x)lu(x) dx 0 Ω Remark. These assumptions are needed for existence (and uniqueness) of semigroup (mild) solutions for the nonlinear equation u t = LF(u), through a remarkable variant of the celebrated Crandall-Liggett theorem, as done by Benilan, Crandall and Pierre: M. G. Crandall, T.M. Liggett. Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math. 93 (97) 265 298. M. Crandall, M. Pierre, Regularizing Effects for u t = Aϕ(u) in L, J. Funct. Anal. 45, (982), 94 22

Assumption on the nonlinearity F Let F : R R be a continuous and non-decreasing function, with F(0) = 0. Moreover, it satisfies the condition: (N) F C (R \ {0}) and F/F Lip(R) and there exists µ 0, µ > 0 s.t. m = µ ( F F ) µ 0 = where F/F is understood to vanish if F(r) = F (r) = 0 or r = 0. The main example (treated in the rest of the talk) will be F(u) = u m u, with m >, µ 0 = µ = m m <. A simple variant is the combination of two powers: m behaviour when u and m 0 behaviour when u 0 Monotonicity estimates follow by (N): the following maps m 0 t t µ 0 F(u(t, x)) or t t m u(t, x) are nondecreasing in t > 0 for a.e. x Ω. P. Bénilan, M. G. Crandall. Regularizing effects of homogeneous evolution equations, Contributions to Analysis and Geometry, suppl. to Amer. Jour. Math., (98). Pp. 23-39. M. Crandall, M. Pierre, Regularizing Effects for u t = Aϕ(u) in L, J. Funct. Anal. 45, (982), 94 22

Assumption on the inverse operator L Assumptions on the inverse of L We will assume that the operator L has an inverse L : L (Ω) L (Ω) with a kernel K such that L f (x) = K(x, y) f (y) dy, Ω and that satisfies (one of) the following estimates for some γ, s (0, ] and c i,ω > 0 (K) (K2) c 0,Ωδ γ (x) δ γ (y) K(x, y) 0 K(x, y) c,ω x y N 2s ( ) ( ) c,ω δ γ (x) δ γ x y N 2s x y (y) γ x y γ where δ γ (x) := dist(x, Ω) γ. When L has a first eigenfunction, (K) implies 0 Φ L (Ω). Moreover, (K2) implies that Φ dist(, Ω) γ = δ γ and we can rewrite (K2) as ( ) ( ) c,ω Φ(x) Φ(y) (K3) c 0,ΩΦ (x)φ (y) K(x, y) x x 0 N 2s x y γ x y γ

Examples of operators L Reminder about the fractional Laplacian operator on R N We have several equivalent definitions for ( R N ) s : By means of Fourier Transform, (( R N ) s f ) (ξ) = ξ 2sˆf (ξ). This formula can be used for positive and negative values of s. 2 By means of an Hypersingular Kernel: if 0 < s <, we can use the representation ( R N ) s g(x) = c N,s P.V. where c N,s > 0 is a normalization constant. R N g(x) g(z) dz, x z N+2s 3 Spectral definition, in terms of the heat semigroup associated to the standard Laplacian operator: ( R N ) s g(x) = Γ( s) 0 ( e t R N g(x) g(x) ) dt t +s.

Examples of operators L The Spectral Fractional Laplacian operator (SFL) ( Ω) s g(x) = λ s j ĝ j φ j(x) = ( ) e t Ω dt g(x) g(x) Γ( s) t. +s j= 0 Ω is the classical Dirichlet Laplacian on the domain Ω EIGENVALUES: 0 < λ λ 2... λ j λ j+... and λ j j 2/N. EIGENFUNCTIONS: φ j are as smooth as the boundary of Ω allows, namely when Ω is C k, then φ j C (Ω) C k (Ω) for all k N. ĝ j = g(x)φ j(x) dx, with φ j L 2 (Ω) =. Lateral boundary conditions for the SFL Ω u(t, x) = 0, in (0, ) Ω. The Green function of SFL satisfies a stronger assumption than (K2) or (K3), i.e. ( ) ( ) δ γ (x) δ γ (K4) K(x, y) x y N 2s x y (y) γ x y, with γ = γ

Examples of operators L Definition via the hypersingular kernel in R N, restricted to functions that are zero outside Ω. The (Restricted) Fractional Laplacian operator (RFL) ( Ω ) s g(x) g(z) g(x) = c N,s P.V. dz, with supp(g) Ω. R N x z N+2s where s (0, ) and c N,s > 0 is a normalization constant. ( Ω ) s is a self-adjoint operator on L 2 (Ω) with a discrete spectrum: EIGENVALUES: 0 < λ λ 2... λ j λ j+... and λ j j 2s/N. Eigenvalues of the RFL are smaller than the ones of SFL: λ j λ s j for all j N. EIGENFUNCTIONS: φ j are the normalized eigenfunctions, are only Hölder continuous up to the boundary, namely φ j C s (Ω). (J. Serra - X. Ros-Oton) Lateral boundary conditions for the RFL u(t, x) = 0, in (0, ) ( R N \ Ω ). The Green function of RFL satisfies a stronger assumption than (K2) or (K3), i.e. ( ) ( ) δ γ (x) δ γ (K4) K(x, y) x y N 2s x y (y) γ x y, with γ = s γ References. (K4) Bounds proven by Bogdan, Grzywny, Jakubowski, Kulczycki, Ryznar (997-200). Eigenvalues: Blumental-Getoor (959), Chen-Song (2005)

Examples of operators L Introduced in 2003 by Bogdan, Burdzy and Chen. Censored Fractional Laplacians (CFL) a(x, y) Lf (x) = P.V. (f (x) f (y)) x y dy, with N+2s 2 < s <, Ω where a(x, y) is a measurable, symmetric function bounded between two positive constants, satisfying some further assumptions; for instance a C (Ω Ω). The Green function K(x, y) satisfies (K4), proven by Chen, Kim and Song (200) ( ) ( ) δ γ (x) δ γ K(x, y) x y N 2s x y (y) γ x y, with γ = s γ 2. Remarks. This is a third model of Dirichlet fractional Laplacian when [ a(x, y) = const ]. This is not equivalent to SFL nor to RFL. References. Roughly speaking, s (0, /2] corresponds to Neumann boundary conditions. K. Bogdan, K. Burdzy, K., Z.-Q. Chen. Censored stable processes. Probab. Theory Relat. Fields (2003) Z.-Q. Chen, P. Kim, R. Song, Two-sided heat kernel estimates for censored stable-like processes. Probab. Theory Relat. Fields (200)

About the kernels About the kernels of spectral nonlocal operators. Most of the examples of nonlocal operators, but the SFL, admit a representation with a kernel, namely ( ) Lf (x) = P.V. f (x) f (y) K(x, y) dy R N A natural question is: does the SFL admit such a representation? As far as we know, a description of such kernel was not known. Let A be a uniformly elliptic linear operator, we can define the s th power of A, Lg(x) = A s g(x) = Γ( s) The main questions are: 0 ( e ta g(x) g(x) ) dt t +s Does L admit a representation with a Kernel? If yes, What do we know about the Kernel? Positivity, support, (sharp) estimates? The key point is the knowledge of the Heat Kernel of A.

About the kernels Lemma. Spectral kernels. (M.B. A. Figalli and J. L. Vázquez) Let s (0, ), and let L be the s th -power of a linear operator A, and Φ dist(, Ω) the first eigenfunction of A. Let H(t, x, y) be the Heat Kernel of A, and assume that it satisfies the following bounds: there exist constants c 0, c, c 2 > 0 such that H(t, x, y) [ ] [ ] Φ(x) Φ(y) e x y 2 c k t for all 0 < t. t γ/2 t γ/2 t N/2 (c k {c 0, c } maybe different in the upper and lower bounds) and also 0 H(t, x, y) c 2Φ (x)φ (y) for all t. Then the operator L admits a kernel representation, with ( ) ( ) Φ(x) Φ(y) K(x, y) x y N+2s x y γ x y γ In particular, the kernel K is supported in Ω Ω. One line (formal) proof: (recalling that when x y, we have δ x(y) = 0) K(x, y) = Lδ y(x) = Γ( s) (H(t, x, y) δ x(y)) 0 dt t +s = Γ( s) 0 H(t, x, y) t +s dt.

More Examples Spectral powers of uniformly elliptic operators. Consider a linear operator A in divergence form, with uniformly elliptic bounded measurable coefficients: A = N i(a ij j), s-power of A is: Lf (x) := A s f (x) := λ s ˆf k kφ k(x) i,j= L = A s satisfies (K3) estimates with γ = (K3) c 0,Ωφ (x) φ (y) K(x, y) c,ω x y N 2s k= ( ) ( ) φ(x) φ(y) x y x y [General class of intrinsically ultra-contractive operators, Davies and Simon JFA 984]. Fractional operators with rough kernels. Integral operators of Levy-type K(x, y) Lf (x) = P.V. (f (x + y) f (y)) dy. R N x y N+2s where K is measurable, symmetric, bounded between two positive constants, and K(x, y) K(x, x) χ x y < c x y σ, with 0 < s < σ, for some positive c > 0. We can allow even more general kernels. The Green function satisfies a stronger assumption than (K2) or (K3), i.e. ( ) ( ) δ γ (x) δ γ (K4) K(x, y) x y N 2s x y (y) γ x y, with γ = s γ

More Examples Sums of two Restricted Fractional Laplacians. Operators of the form L = ( Ω ) s + ( Ω ) σ, with 0 < σ < s, where ( Ω ) s is the RFL. Satisfy (K4) with γ = s. Sum of the Laplacian and operators with general kernels. In the case L = a + A s, with 0 < s < and a 0, where ( ) A sf (x) = P.V. f (x + y) f (y) f (x) yχ y χ y dν(y), R N the measure ν on R N \ {0} is invariant under rotations around origin and satisfies x 2 dν(y) <, together with other assumptions. R N Relativistic stable processes. In the case ( s L = c c /s ), with c > 0, and 0 < s. The Green function K(x, y) of L satisfies assumption (K 4) with γ = s. Many other interesting examples. Schrödinger equations for non-symmetric diffusions, Gradient perturbation of RFL... References. The above mentioned bounds for the Green functions have been proven by Chen, Kim, Song and Vondracek (2007, 200, 202, 203).

Basic Theory The Dual problem Existence and uniqueness First set of estimates For the rest of the talk we deal with the special case: F(u) = u m := u m u

The dual formulation of the problem The dual formulation of the problem. Recall the homogeneous Cauchy-Dirichlet problem: t u = L u m, in (0, + ) Ω (CDP) u(0, x) = u 0 (x), in Ω u(t, x) = 0, on the lateral boundary. We can formulate a dual problem, using the inverse L as follows where t U = u m, U(t, x) := L [u(t, )](x) = Ω u(t, y)k(x, y) dy. This formulation encodes all the possible lateral boundary conditions in the inverse operator L. Remark. This formulation has been used before by Pierre, Vázquez [...] to prove (in the R N case) uniqueness of the fundamental solution, i.e. the solution corresponding to u 0 = δ x0, known as the Barenblatt solution.

Existence and uniqueness of weak dual solutions Recall that Φ dist(, Ω) γ and w L Φ (Ω) = Ω w(x)φ (x) dx. Weak Dual Solutions for the Cauchy Dirichlet Problem (CDP) A function u is a weak dual solution to the Cauchy-Dirichlet problem (CDP) for the equation tu + Lu m = 0 in Q T = (0, T) Ω if: u C((0, T) : L Φ (Ω)), u m L ( (0, T) : L Φ (Ω) ) ; The following identity holds for every ψ/φ C c((0, T) : L (Ω)) : T 0 Ω L (u) ψ t dx dt T u C([0, T) : L Φ (Ω)) and u(0, x) = u 0 L Φ (Ω). 0 Ω u m ψ dx dt = 0. Theorem. Existence and Uniqueness (M.B. and J. L. Vázquez) For every nonnegative u 0 L Φ (Ω) there exists a unique minimal weak dual solution to the (CDP). Such a solution is obtained as the monotone limit of the semigroup (mild) solutions that exist and are unique. The minimal weak dual solution is continuous in the weighted space u C([0, ) : L Φ (Ω)). In this class of solutions the standard comparison result holds. Remarks. Mild solutions (by Crandall and Pierre) are weak dual solutions. Weak dual solutions are very weak solutions.

Basic Estimates Theorem. First Pointwise Estimates. Let u 0 be a nonnegative weak dual solution to Problem (CDP). Then, for almost every 0 t 0 t and almost every x 0 Ω, we have ( t0 t ) m m u m (t 0, x 0) Theorem. (Absolute upper bounds) Ω u(t 0, x) u(t, x) t t 0 K(x, x 0) dx (M.B. and J. L. Vázquez) ( t t 0 ) m m u m (t, x 0). (M.B. & J. L. Vázquez) Let u be a weak dual solution, then there exists a constant κ 0 > 0 depending only on N, s, m, Ω (but not on u 0!!), such that under the minimal assumption (K): u(t) L (Ω) t κ0 m, for all t > 0. Theorem. (Smoothing effects) (M.B. & J. L. Vázquez) Let ϑ γ = /[2s + (N + γ)(m )] and assume (K2). There exists κ > 0 such that: u(t) L (Ω) κ u(t) 2sϑγ Nϑγ t L Φ (Ω) κ u0 2sϑγ Nϑγ t L Φ (Ω) for all t > 0. Assuming only (K), the above bound holds with L and ϑ 0, instead of L Φ and ϑ γ. More details about this first part online: http://www.fields.utoronto.ca/video-archive//event/202/206

Sharp Boundary Behaviour Upper Boundary Estimates Infinite Speed of Propagation Lower Boundary Estimates

Upper boundary estimates Sharp Upper boundary estimates

Upper boundary estimates Theorem. (Upper boundary behaviour) (M.B., A. Figalli and J. L. Vázquez) Let (A), (A2), and (K2) hold. Let u 0 be a weak dual solution to the (CDP). Let σ (0, ] be σ = 2sm γ(m ) Then, there exists a computable constant κ > 0, depending only on N, s, m, and Ω, (but not on u 0!!) such that for all t 0 and all x Ω : u(t, x) κ Φ (x) σ m t m σγ dist(x, Ω) m t m When σ = we have sharp boundary estimates: we will show lower bounds with matching powers. When σ < the estimates are not sharp in all cases: The solution by separation of variables U(t, x) = S(x)t /(m ) (asymptotic behaviour) behaves like Φ σ/m t /(m ). We will show that for small data, the boundary behaviour is different. In examples, σ < only happens for SFL-type, where γ =, and s can be small, 0 < s < /2 /(2m).

Upper boundary estimates Now consider σ < which means taking s small respect to γ, as follows σ = 2sm γ(m ) < i.e. 0 < s < γ 2 γ 2m. The previous upper bounds are not sharp for small times for small data : Theorem. (Upper boundary behaviour II) (M.B., A. Figalli and J. L. Vázquez) Let (A), (A2), and (K2) hold, and let σ <. Let u 0 be a weak dual solution to (CDP). There exists a constant κ 0 > 0 that depends only on N, s, m, λ, and Ω, such that the following holds: given T > 0, assume that for all x Ω 0 u 0 (x) ε T Φ (x) with ε T = κ 0 e λt. Then for all (t, x) (0, T] Ω we have u(t, x) κ 0 Φ (x). This phenomenon does not happen in the linear case, for u t + Lu = 0. It does not happen either in the nonlinear elliptic case LS m = λs, where S Φ σ/m. This theorem applies for instance to spectral power of an elliptic operators, L = A s. Recall that the domain Ω is smooth, at least C,

Infinite Speed of Propagation Infinite Speed of Propagation and Universal Lower Bounds

Infinite Speed of Propagation Theorem. (Universal lower bounds) (M.B., A. Figalli and J. L. Vázquez) Let L satisfy (A) and (A2), and assume that ( ) Lw(x) = P.V. w(x) w(y) K(x, y) dy, with K(x, y) c0 Φ (x)φ (y) x, y Ω. R N Let u 0 be a weak dual solution to the (CDP) corresponding to u 0 L Φ (Ω). Then there exists a constant κ 0 > 0, so that the following inequality holds: ( u(t, x) κ 0 t ) m m Φ (x) for all t > 0 and all x Ω. t t m Here t = κ u 0 (m ) and κ L Φ (Ω) 0, κ depend only on N, s, γ, m, c0, and Ω. Note that, for t t, the dependence on the initial data disappears u(t) κ 0 Φ t m t t. The assumption on the kernel K of L holds for all examples and represent somehow the worst case scenario for lower estimates. In many cases (RFL, CFL), K satisfies a stronger property: K κ Ω > 0 in Ω Ω. It is remarkable that this minimal behavior is sharp in some situations: for σ <, small data and for small times.