On some weighted fractional porous media equations

Transcription

1 On some weighted fractional porous media equations Gabriele Grillo Politecnico di Milano September 16 th, 2015 Anacapri Joint works with M. Muratori and F. Punzo Gabriele Grillo Weighted Fractional PME 1 / 38

2 Outline of the talk Introduction to the problem Survey of the literature for related problems Main results for slowly decaying densities Main results for rapidly decaying densities Existence and uniqueness for measure data, a sketch The PME with measure data on manifolds, a sketch The talk is based on the papers: G.G., M. Muratori, F. Punzo, DCDS (2015); G.G., M. Muratori, F. Punzo, Calc. Var. (2015); G.G., M. Muratori, F. Punzo, preprint (2015). Gabriele Grillo Weighted Fractional PME 2 / 38

3 The weighted fractional porous medium equation We consider the following Cauchy problem, that we refer to as weighted fractional porous medium equation (WFPME for short): { ρ(x)u t = ( ) s (u m ) in R d (0, ), where u = u 0 on R d {0}, s (0, 1), d > 2s, m > 1, ρ is a positive weight (or density) satisfying suitable decay conditions that we shall specify later and the initial datum u 0 0 belongs to { } L 1 ρ(r d ) := f : f 1,ρ := f (x) ρ(x)dx <. R d Gabriele Grillo Weighted Fractional PME 3 / 38

4 For all s (0, 1), as usual ( ) s denotes the fractional Laplacian operator, that is, e.g. on test functions, ( ) s φ(x) φ(y) (φ)(x) := p.v. C d,s x y d+2s dy φ C c (R d ), R d where C d,s is a suitable positive constant depending only on d and s. Gabriele Grillo Weighted Fractional PME 4 / 38

5 Related local problems in the literature The local version of the WFPME, that is { ρ(x)u t = (u m ) in R d (0, ), u = u 0 on R d {0}, has largely been studied in the literature: see e.g. [Eidus 90], [Eidus and Kamin 94], [Reyes and Vázquez 08, 09], [Punzo 09], [Kamin, Reyes and Vázquez 10], [Grillo, M. and Porzio 13]. In particular, existence and uniqueness of the so-called weak energy solutions holds true, to some extent regardless of the behaviour of ρ at infinity. Gabriele Grillo Weighted Fractional PME 5 / 38

6 As for the long-time behaviour of solutions, in [Reyes and Vázquez 09] it is proved that if ρ is continuous, positive and satisfies ρ(x) x γ as x with γ [0, 2), then and lim u(t) t u M (t) 1,ρ = 0 lim t tα u(t) um (t) = 0. Gabriele Grillo Weighted Fractional PME 6 / 38

7 Here um is the self-similar Barenblatt solution with mass M um (x, t) x γ dx = M = u 0 (x) ρ(x)dx > 0, R d R d that is with u M (x, t) := t α F ( t κ x ) (x, t) R d (0, ) F(ξ) := (C kξ 2 γ) 1 m 1 + ξ 0 for suitable positive constants C and k depending on d, m, γ, M and α := (d γ)κ, κ := 1 d(m 1) + 2 mγ. Gabriele Grillo Weighted Fractional PME 7 / 38

8 It is worth pointing out that, even if ρ is a locally regular function, um is the solution to the singular problem { x γ u t (u m ) = 0 in R d (0, ), x γ u = Mδ on R d {0}, where δ is the Dirac delta distribution centred at x = 0. Gabriele Grillo Weighted Fractional PME 8 / 38

9 On the other hand, in [Kamin, Reyes and Vázquez 10] the authors prove that if ρ is continuous, positive and satisfies then ρ(x) x γ as x for some γ > 2, 1 t m 1 u(x, t) (m 1) 1 m 1 W 1 m (x) as t, uniformly w.r.t. x R d, which is actually typical of bounded domains. Here W is the minimal positive solution to the sublinear elliptic equation W = ρ W 1 m in R d, which satisfies lim W (x) = 0. x Gabriele Grillo Weighted Fractional PME 9 / 38

10 Related nonlocal problems in the literature Our fractional problem in the non-weighted case ρ 1, namely { u t + ( ) s (u m ) = 0 in R d (0, ), u = u 0 on R d {0}, has been addressed in Vázquez s talk (works of him with de Pablo, Quiros, Rodriguez). Existence, uniqueness and qualitative properties of solutions are discussed in detail. The asymptotic behaviour of solutions as t has then been investigated in [Vázquez 14]. Gabriele Grillo Weighted Fractional PME 10 / 38

11 More precisely, it is first shown that for any M > 0 there exists a unique solution u M to { u t + ( ) s (u m ) = 0 in R d (0, ), u = Mδ on R d {0}. Furthermore, u M still has a self-similar profile: where now u M (x, t) := t α f (t κ x ) (x, t) R d (0, ), α = d d(m 1) + 2s, κ = 1 d(m 1) + 2s and f : [0, ) (0, ) is suitable bounded function, with f (ξ) 0 as ξ. In contrast with the local case, f is in general not explicit. Gabriele Grillo Weighted Fractional PME 11 / 38

12 In view of such properties, it is reasonable to refer again to u M as a Barenblatt-type solution. It then proved that, similarly to the local case, lim t u(t) u M (t) 1 = 0 and lim t tα u(t) um (t) = 0. Gabriele Grillo Weighted Fractional PME 12 / 38

13 Our results Let Ḣs (R d ) be the completion of Cc (R d ) w.r.t. the norm φ Ḣs := ( ) s 2 2 (φ) φ Cc (R d ). Definition 1 A nonnegative function u is a weak solution to the WFPME if: u C([0, ); L 1 ρ(r d )) L (R d (τ, )) for all τ > 0; u m L 2 loc ((0, ); Ḣs (R d )); for any ϕ Cc (R d (0, )) there holds u ϕ t ρ dxdt ( ) s 2 (u m )( ) s 2 (ϕ) dxdt = 0 ; 0 R d 0 R d lim t 0 u(t) = u 0 in L 1 ρ(r d ). Gabriele Grillo Weighted Fractional PME 13 / 38

14 Slowly and rapidly decaying densities ρ We distinguish between two cases: slowly decaying densities: lim x ρ(x) x γ = c for some c > 0 and γ (0, 2s); rapidly decaying densities: for some C 0 > 0 and γ > 2s. ρ(x) C 0 x γ x B c 1, Gabriele Grillo Weighted Fractional PME 14 / 38

15 Slowly decaying densities: an auxiliary problem We first consider the following singular WFPME with measure data: { c x γ u t + ( ) s (u m ) = 0 in R d (0, ), c x γ u = µ on R d {0}, where µ is a positive finite Radon measure and c > 0. In fact, for slowly decaying densities, in the analysis of the long-time behaviour of solutions a crucial role is played by the solution to { c x γ u t + ( ) s (u m ) = 0 in R d (0, ), c x γ u = Mδ on R d {0}. Gabriele Grillo Weighted Fractional PME 15 / 38

16 Definition 2 By a weak solution to the singular WFPME with measure data we mean a nonnegative function u such that: u L ((0, ); L 1 x γ (R d )) L (R d (τ, )) τ > 0, u m L 2 loc((0, ); Ḣs (R d )), 0 c u ϕ t x γ dxdt ( ) s 2 (u m ) ( ) s 2 (ϕ) dxdt = 0 R d 0 R d ϕ Cc (R d (0, )) and lim c x γ u(t) = µ in σ(m(r d ), C b (R d )). t 0 Gabriele Grillo Weighted Fractional PME 16 / 38

17 Theorem 3 Let d > 2s and γ [0, 2s) [0, d 2s]. Then there exists a weak solution u to the singular WFPME with measure data, which moreover satisfies the smoothing estimate u(t) K t α µ(r d ) β t > 0, where K is a suitable positive constant depending only on s, d, m, γ and α := d γ (m 1)(d γ) + 2s γ, β := 2s γ (m 1)(d γ) + 2s γ. In particular, u(t) L 1 x γ (R d ) L (R d ) for all t > 0. In addition, such solution is unique. Gabriele Grillo Weighted Fractional PME 17 / 38

18 We point out that u also satisfies the energy estimates and t2 t 1 R d ( ) s 2 (u m ) (x, t) 2 dxdt+ u m+1 (x, t 2 ) ρ(x)dx = u m+1 (x, t 1 ) ρ(x)dx R d R d t2 t 1 R d (u m+1 2 ) t (x, t) 2 ρ(x)dxdt C for all t 2 > t 1 > 0, where we set ρ(x) = c x γ and C is a positive constant that depends only on m, t 1, t 2 and e.g. on R d u m+1 (x, t 1 /2) ρ(x)dx. Gabriele Grillo Weighted Fractional PME 18 / 38

19 From here on we denote as u c M the unique solution to the singular WFPME with initial datum Mδ. For any λ > 0, it is easy to check that the function u c M,λ (x, t) := λα u c (λκ x, λt) is a solution to the same problem. Hence, as a consequence of our uniqueness result, u c M (x, t) = λα u c M (λκ x, λt) t, λ > 0, for a.e. x R d. We can therefore assert that, to some extent, u c M self-similar Barenblatt-type solution. M is still a Gabriele Grillo Weighted Fractional PME 19 / 38

20 Asymptotic behaviour for slowly decaying density Theorem 4 Let d > 2s and ρ be a slowly decaying density for some γ (0, 2s) (0, d 2s]. Let u be the unique weak solution to the associated WFPME, with R u d 0 (x) ρ(x)dx = M > 0. Then lim u(t) u c M (t) 1, x γ = 0, t Gabriele Grillo Weighted Fractional PME 20 / 38

21 Asymptotic behaviour for rapidily decaying densities In the rapidly decaying case, the long-time behaviour of solutions is deeply linked with the positive minimal solution w to the following fractional sublinear elliptic equation: ( ) s w = ρ w α in R d, where α = 1 m (0, 1). In fact this gives rise to the separable solution (m 1) 1 m 1 t 1 m 1 w 1 m, which replaces the above Barenblatt-type profiles as an attractor for nonnegative solutions. Gabriele Grillo Weighted Fractional PME 21 / 38

22 Definition 5 A bounded, nonnegative function w is a very weak solution to the fractional sublinear elliptic equation if it satisfies w α (x)φ(x) ρ(x)dx = w(x)( ) s (φ)(x) dx R d R d for all φ C c (R d ). Definition 6 A nonnegative function w Ḣs (R d ) is a weak solution to the fractional sublinear elliptic equation if it satisfies w α (x)φ(x) ρ(x)dx = ( ) s s 2 (w)(x)( ) 2 (φ)(x) dx R d R d for all φ C c (R d ). Gabriele Grillo Weighted Fractional PME 22 / 38

23 Existence for the fractional sublinear elliptic equation Theorem 7 Let d > 2s, α (0, 1) and ρ C σ loc (Rd ) for some σ (0, 1), where ρ > 0 is a rapidly decaying density. Then there exists a nontrivial very weak solution w to the fractional sublinear elliptic equation, which moreover satisfies for some C > 0. In particular, w(x) C (I 2s ρ)(x) w(x) 0 as x. Finally, if γ complies with the more restrictive condition then w is also a weak solution. γ > 2s + d 2s α + 2, Gabriele Grillo Weighted Fractional PME 23 / 38

24 The main asymptotic result Theorem 8 Let d > 2s and ρ C σ loc (Rd ) for some σ (0, 1) be a rapidly decaying density, with ρ > 0. Let u be the minimal weak solution to the associated WFPME and w be the minimal very weak and local weak solution to the fractional sublinear elliptic equation with α = 1/m. Then lim t 1 m 1 u(x, t) = (m 1) 1 1 m 1 w m (x) t monotonically and in L p loc for all p [1, ). If in addition d > 4s and γ > 4s, then u and w are unique in the corresponding classes of solutions such that u m L 1 (1+ x ) d+2s (R d (0, T )) T > 0, w L 1 (1+ x ) d+2s (R d ). Gabriele Grillo Weighted Fractional PME 24 / 38

25 Existence with measure data As for existence, the basic idea is to consider the regularized problem { ρ η (x)(u η,ε ) t = ( ) s ( uη,ε m ) in R d (0, ), u η,ε = µ ε on R d {0}, where, for instance, ρ η (x) := ( x + η) γ and µ ε is the mollification of µ. Then one passes to the limit as η 0 + and ε 0 +, exploiting the smoothing effect, energy estimates and monotonicity of potentials. Gabriele Grillo Weighted Fractional PME 25 / 38

26 Uniqueness with measure data Uniqueness is by far the most delicate issue. In order to prove it we adapt a duality method first introduced by [Pierre 82] in the non-weighted (γ = 0) and local (s = 1) case. The problem was open in the fractional case even when ρ 1 More precisely, we take two possibly different solutions u 1 and u 2 with Riesz potentials U 1 := I 2s ( x γ u 1 ) and U 2 := I 2s ( x γ u 2 ), respectively. Given h > 0, we write the differential equation solved by namely with a(x, t) := g(x, t) := U 2 (x, t + h) U 1 (x, t), x γ g t (x, t) = a(x, t)( ) s (g)(x, t) { u m 1 (x,t) u m 2 (x,t+h) u 1 (x,t) u 2 (x,t+h) if u 1 (x, t) u 2 (x, t + h), 0 if u 1 (x, t) = u 2 (x, t + h). Gabriele Grillo Weighted Fractional PME 26 / 38

27 If we could use as a test function a nonnegative solution ϕ to the dual problem { x γ ϕ t = ( ) s (a(x, t) ϕ) in R d (0, T ), ϕ(x, T ) = ψ(x) on R d {T }, for any T > 0 and ψ D + (R d ), then we would end up with g(x, T )ψ(x) x γ dx = g(x, 0)ϕ(x, 0) x γ dx, R d R d whence the conclusion would easily follow in view of the monotonicity of potentials (the r.h.s. is nonpositive). However, existence of such a solution is far from trivial, and requires several approximations. In this regard, a fundamental tool to exploit is the essential self-adjointness of the linear operator x γ ( ) s. Gabriele Grillo Weighted Fractional PME 27 / 38

28 First we consider the solutions {ϕ n,ε } to the approximate problems { x γ (ϕ n,ε ) t = ( ) s [(a n + ε) ϕ n,ε ] in R d (0, T ), ϕ n,ε (x, T ) = ψ(x) on R d {T }, where ε > 0 and {a n } is a suitable approximation of a, such that a n (x, t) is a piecewise constant function of t on the time intervals ( (k + 1)T T, T kt ] k {0,..., n 1}, n n regular in x. Existence, non-negativity and good integrability properties of such solutions are ensured e.g. by standard Markov semigroup theory. Indeed, we can prove the following. Gabriele Grillo Weighted Fractional PME 28 / 38

29 Theorem 9 Let d > 2s and γ [0, 2s). Let D(A) be the space of functions v L 2 γ(r d ) such that x γ ( ) s (v) L 2 γ(r d ). Then the operator A : D(A) L 2 γ(r d ) defined as A(v) := x γ ( ) s (v) v D(A) is densely defined, nonnegative and self-adjoint on L 2 γ(r d ), and the quadratic form associated to it is Q(v, v) := C d,s 2 R d Rd (v(x) v(y)) 2 x y d+2s dxdy, with domain D(Q) = L 2 γ(r d ) Ḣs (R d ). Moreover, Q is a Dirichlet form and A generates a Markov semigroup on L 2 γ(r d ). Gabriele Grillo Weighted Fractional PME 29 / 38

30 Hence, we can actually multiply the equation solved by g by ϕ n,ε and integrate by parts to get (for all 0 < t < T ) g(x, T )ψ(x) x γ dx g(x, t)ϕ n,ε (x, t) x γ dx R d R d T = [a n (x, τ) + ε a(x, τ)] ( ) s (g)(x, τ) ϕ n,ε (x, τ) dxdτ. t R d Recalling that ( ) s (g)(x, τ) = u 2 (x, τ + h) u 1 (x, τ) L (R d (t, T )), taking advantage of the conservation of mass property for ϕ n,ε and of suitable pointwise and Ḣs (R d ) estimates over the Riesz potential I 2s [ x γ ϕ n,ε (x, t) ], we can pass to the limit first as n and then as ε 0. Gabriele Grillo Weighted Fractional PME 30 / 38

31 We find that { x γ ϕ n,ε (x, t)} converges to a family of positive finite Radon measures ν(t), in the sense that ϕ n,ε (x, t)φ(x) x γ dx φ(x) dν(t) R d R d t (0, T ), φ C(R d ) L (R d ), which can be shown to imply g(x, T )ψ(x) x γ dx = g(x, t) dν(t). R d R d Since potentials are monotone decreasing in time, the above identity implies (recalling the definition of g) g(x, T )ψ(x) x γ dx [U 2 (x, h) U 1 (x, t 0 )] dν(t) R d R d for all 0 < t < t 0. Gabriele Grillo Weighted Fractional PME 31 / 38

32 Exploiting the monotonicity of the potentials of ν(t), we can prove that it is possible to pass to the limit in the r.h.s. as t 0, so that g(x, T )ψ(x) x γ dx [U 2 (x, h) U 1 (x, t 0 )] dν R d R d for some positive finite Radon measure ν. Letting t 0 0, by monotone convergence we then obtain g(x, T )ψ(x) x γ dx [U 2 (x, h) U µ (x)] dν 0. R d R d Given the arbitrariness of h, T > 0 and ψ D + (R d ), we deduce that U 2 U 1. By swapping u 1 and u 2 we also get the opposite inequality, whence U 1 = U 2 and so u 1 = u 2. Gabriele Grillo Weighted Fractional PME 32 / 38

33 Existence and uniqueness of an initial trace By means of techniques similar to the ones we used in the proof of existence, we can also establish existence and uniqueness for the initial trace problem. More precisely: Theorem 10 Let d > 2s and γ [0, 2s) [0, d 2s]. Consider a weak solution u to x γ u t + ( ) s (u m ) = 0 without a prescribed initial datum. Then there exists a unique positive finite Radon measure µ which is the initial trace of u in the sense that lim t 0 R d u(x, t)φ(x) x γ dx = R d φ(x) dµ φ C(R d ) L (R d ). In particular, µ(r d ) = R d u(x, t) x γ dx for all t > 0. Gabriele Grillo Weighted Fractional PME 33 / 38

34 The PME on Riemannian manifolds: some statements We shall very briefly discuss some similar results on a non-compact Riemannian manifold M of infinite volume. We consider the non fractional PME, namely { u t = (u m ) in M (0, ), u = µ on M {0}, where is Laplace-Beltrami. Our potential theoretic method in principle work, but several crucial results valid in the Euclidean potential theory do not hold in principle, e.g. no standard mean value inequality for sub or superharmonic functions of any sign hold with the required generality. They are in no way a standard modification of the Euclidean ones. (1) Gabriele Grillo Weighted Fractional PME 34 / 38

35 Our main assumption is that M is nonparabolic, namely that it admits a positive Green function, and a suitable lower bound on Ricci which entails conservation of mass. E.g. we can ask { (i) M is a Cartan-Hadamard manifold (N 3); (ii) Ric o (x) C(1 + dist(x, o) 2 (H) ) for some C 0. Mean value inequalities holds if one considers appropriate mean value functionals which are related to the Green s function (generalization of old results by Fabes-Garofalo and of related ones by Lanconelli-Uguzzoni), this being of independent interest. Gabriele Grillo Weighted Fractional PME 35 / 38

36 Theorem 11 Let assumption (H) be satisfied. Then there exists a (signed) weak solution u to problem (1) corresponding the a finite Radon measure µ, which conserves the quantity µ(m) = u(x, t) dv(x) for all t > 0, and satisfies the smoothing effect M u(t) Kt α µ (M) β for all t > 0, (2) where α := N (m 1)N + 2, β := 2 (m 1)N + 2. (3) Gabriele Grillo Weighted Fractional PME 36 / 38

37 Theorem 12 Let assumption (H) be satisfied. Let u 1 and u 2 be two nonnegative weak solutions to problem (1). Suppose that their initial datum is the same positive Radon measure µ. Then u 1 = u 2. Theorem 13 Let assumption (H) be satisfied. Let u be a weak solution of the differential equation in the problem considered. Suppose in addition that u m L 1 ((0, T ); L 1 loc (M)) for some T > 0. Then there exists a finite Radon measure µ which is an initial trace in the sense of testing with any φ C c (M) and for φ equal to a constant. Under the additional assumption that u 0, then the conclusion holds for any φ C b (M), for some positive µ, without requiring that u m L 1 ((0, T ); L 1 loc (M)). Gabriele Grillo Weighted Fractional PME 37 / 38

38 THANK YOU FOR YOUR ATTENTION! Gabriele Grillo Weighted Fractional PME 38 / 38