Maximum principle for the fractional diusion equations and its applications

Transcription

1 Maximum principle for the fractional diusion equations and its applications Yuri Luchko Department of Mathematics, Physics, and Chemistry Beuth Technical University of Applied Sciences Berlin Berlin, Germany Fractional PDEs: Theory, Algorithms and Applications ICERM at Brown University June 18-22, 218

2 Outline of the talk: Maximum principle for a time-fractional diusion equation with the general fractional derivative Maximum principle for the weak solutions of a time-fractional diusion equation with the Caputo derivative Maximum principle for an abstract space- and time-fractional evolution equation in the Hilbert space Short survey of other results Yuri Luchko (BHT Berlin) Maximum principle June 21, / 33

3 1. Diusion equation with the general fractional derivative What is a maximum principle? Maximum principle: A function satises a dierential inequality or equation in a domain D It achieves its maximum on the boundary of D. A very elementary example: f (x) >, x ]a, b[ and f C([a, b]) f achieves its maximum value at one of the endpoints of the interval. Other examples: Maximum principles for ordinary dierential equations and inequalities Maximum principles for partial dierential equations and inequalities Very recently: maximum principles for fractional PDEs Yuri Luchko (BHT Berlin) Maximum principle June 21, / 33

4 1. Diusion equation with the general fractional derivative General fractional derivatives Let k be a nonnegative locally integrable function. The general fractional derivative of the Caputo type: t (D C (k) f )(t) = k(t τ)f (τ) dτ. The general fractional derivative of the Riemann-Liouville type: (D RL (k) f )(t) = d t k(t τ)f (τ) dτ. dt For an absolutely continuous function f with the inclusion f L loc 1 (IR +), we get (D C (k) f )(t) = d dt t k(t τ)f (τ) dτ k(t)f () = (D RL (k) f )(t) k(t)f () Yuri Luchko (BHT Berlin) Maximum principle June 21, / 33

5 1. Diusion equation with the general fractional derivative Particular cases 1) The conventional Caputo and Riemann-Liouville fractional derivatives: k(τ) = (D α 1 f )(t) = Γ(1 α) (D α f )(t) = d ( 1 dt Γ(1 α) τ α Γ(1 α), < α < 1. t t (t τ) α f (τ) dτ, ) (t τ) α f (τ) dτ. Yuri Luchko (BHT Berlin) Maximum principle June 21, / 33

6 1. Diusion equation with the general fractional derivative Particular cases 2) The multi-term derivatives k(τ) = n k=1 a k τ α k Γ(1 α k ), < α 1 < < α n < 1 (D C (k) f )(t) = n (D RL (k) f )(t) = k=1 a k (D α k f )(t), n a k (D α k f )(t). k=1 Yuri Luchko (BHT Berlin) Maximum principle June 21, / 33

7 1. Diusion equation with the general fractional derivative Particular cases 3) Derivatives of the distributed order: k(τ) = 1 where ρ is a Borel measure on [, 1]: τ α Γ(1 α) dρ(α), 1 (D C (k) f ) = (D α f )(t) dρ(α), 1 (D C (k) f ) = (D α f )(t) dρ(α). Yuri Luchko (BHT Berlin) Maximum principle June 21, / 33

8 1. Diusion equation with the general fractional derivative Conditions on the kernel function K1) The Laplace transform k of k, exists for all p >, k(p) = k(t) e pt dt, K2) k(p) is a Stiltjes function, K3) k(p) and p k(p) as p, K4) k(p) and p k(p) as p. Yuri Luchko (BHT Berlin) Maximum principle June 21, / 33

9 1. Diusion equation with the general fractional derivative Properties of the general derivatives (A) For any λ >, the initial value problem for the fractional relaxation equation (D C (k) f )(t) = λf (t), t >, u() = 1 has a unique solution u λ = u λ (t) that belongs to the class C (IR + ) and is a completely monotone function. (B) There exists a completely monotone function κ = κ(t) with the property t k(t τ)κ(τ) dτ 1, t >. (C) For f L loc 1 (IR +), the relations (D C (k) I (k)f )(t) = f (t), (D RL (k) I (k)f )(t) = f (t) hold true, where the general fractional integral I (k) is dened by the formula (I (k) f )(t) = t κ(t τ)f (τ) dτ. Yuri Luchko (BHT Berlin) Maximum principle June 21, / 33

10 1. Diusion equation with the general fractional derivative General time-fractional diusion equation Let Ω be an open and bounded domain in IR n with a smooth boundary Ω (for example, of C 2 class) and T >. The general time-fractional diusion equation: (D C (k) u(x, ))(t)=d 2(u)+D 1 (u) q(x)u(x, t)+f (x, t), (x, t) Ω (, T ], where q C( Ω), q(x), x Ω, D 1 (u) = n i=1 b i (x) u x i, D 2 (u) = n i,j=1 a i,j (x) 2 u x i x j and D 2 is a uniformly elliptic dierential operator. Yuri Luchko (BHT Berlin) Maximum principle June 21, / 33

11 1. Diusion equation with the general fractional derivative Cauchy problem Kochubei considered the Cauchy problem with the initial condition u(x, ) = u (x), x IR n for the homogeneous general time-fractional diusion equation with D 2 =, D 1 and q. His main results: 1) The Cauchy problem has a unique appropriately dened solution for a bounded globally Hölder continuous initial value u. 2) The fundamental solution to the Cauchy problem can be interpreted as a probability density function and thus the general time-fractional diusion equation describes a kind of (anomalous) diusion. Yuri Luchko (BHT Berlin) Maximum principle June 21, / 33

12 1. Diusion equation with the general fractional derivative Initial-boundary-value problem Luchko and Yamamoto: Analysis of uniqueness and existence of solution to the initial-boundary-value problem for the general time-fractional diusion equation with the initial condition and the Dirichlet boundary condition u(x, t) t= = u (x), x Ω u(x, t) (x,t) Ω (,T ] = v(x, t), (x, t) Ω (, T ]. Their main results: 1) Uniqueness of solution both in the strong and in the weak senses (Estimates of the general fractional derivatives > Maximum principle for the general diusion equation > A priory norm estimates of solutions > Uniqueness of solutions). 2) Existence of solution in the weak sense (Separation of variables > Formal solutions in form of generalized Fourier series > Convergence analysis of the formal solutions). Yuri Luchko (BHT Berlin) Maximum principle June 21, / 33

13 1. Diusion equation with the general fractional derivative Estimates of the general fractional derivatives Let the conditions L1) k C 1 (IR + ) L loc 1 (IR +), L2) k(τ) > and k (τ) < for τ >, L3) k(τ) = o(τ 1 ), τ. be fullled. Let a function f C([, T ]) attain its maximum over the interval [, T ] at the point t, t (, T ], and f C(, T ] L 1 (, T ). Then the following inequalities hold true: (D RL (k) f )(t ) k(t )f (t ), (D C (k) f )(t ) k(t )(f (t ) f ()). Yuri Luchko (BHT Berlin) Maximum principle June 21, / 33

14 1. Diusion equation with the general fractional derivative Maximum principle Let us dene the operator P (k) (u) := (D C (k) f )(t) D 2(u) D 1 (u) + q(x)u(x, t). Let the conditions L1)-L3) be fullled and a function u satisfy the inequality P (k) (u), (x, t) Ω (, T ]. Then the following maximum principle holds true: max (x,t) Ω [,T ] u(x, t) max{max x Ω u(x, ), max (x,t) Ω [,T ] u(x, t), }. Yuri Luchko (BHT Berlin) Maximum principle June 21, / 33

15 1. Diusion equation with the general fractional derivative A priori norm estimates Let the conditions K1)-K4) and L1)-L3) be fullled and u be a solution of the initial-boundary-value problem for the general diusion equation. Then the following estimate of the uniform solution norm holds true: where u C( Ω [,T ]) max{m, M 1 } + M f (T ), M = u C( Ω), M 1 = v C( Ω [,T ]), M = F C(Ω [,T ]), and f (t) = t κ(τ) dτ, t k(t τ)κ(τ) dτ 1, t >. Yuri Luchko (BHT Berlin) Maximum principle June 21, / 33

16 1. Diusion equation with the general fractional derivative Uniqueness of the solution The initial-boundary-value problem for the general diusion equation equation possesses at most one solution. This solution continuously depends on the problem data in the sense that if u and ũ solutions to the problems with the sources functions F and F and the initial and boundary conditions u and ũ and v and ṽ, respectively, and F F C( Ω [,T ]) ɛ, u ũ C( Ω) ɛ, v ṽ C( Ω [,T ]) ɛ 1, then the following norm estimate holds true: u ũ C( Ω [,T ]) max{ɛ, ɛ 1 } + ɛ f (T ) with f (t) = t κ(τ) dτ, t k(t τ)κ(τ) dτ 1, t >. Yuri Luchko (BHT Berlin) Maximum principle June 21, / 33

17 1. Diusion equation with the general fractional derivative Literature A.N. Kochubei, General fractional calculus, evolution equations, and renewal processes. Integr. Equa. Operator Theory 71 (211), Yu. Luchko and M. Yamamoto, General time-fractional diusion equation: Some uniqueness and existence results for the initial-boundary-value problems. Fract. Calc. Appl. Anal. 19 (216), Yuri Luchko (BHT Berlin) Maximum principle June 21, / 33

18 2. Diusion equation with the Caputo derivative Single-term time-fractional diusion equation Initial-boundary-value problem: α t u(x, t) = n i (a ij (x) j u(x, t))+c(x)u(x, t)+f (x, t), x Ω R n, t > i,j=1 u(x, t) =, x Ω, t >, u(x, ) = a(x), x Ω with < α < 1 and in a bounded domain Ω with a smooth boundary Ω. In what follows, we always suppose that the spatial dierential operator is a uniformly elliptic one. Yuri Luchko (BHT Berlin) Maximum principle June 21, / 33

19 2. Diusion equation with the Caputo derivative Weak solution in the fractional Sobolev space For u C 1 [, T ], the Caputo fractional derivative is dened by α t u(x, t) = 1 Γ(1 α) t (t s) α s u(x, s)ds, x Ω. Recently, the Caputo fractional derivative t α was extended to an operator dened on the closure H α (, T ) of C 1 [, T ] := {u C 1 [, T ]; u() = } in the fractional Sobolev space H α (Ω). In what follows, we interpret α t u as this extension with the domain H α (, T ). Thus we interpret the problem under consideration as the fractional diusion equation subject to the inclusions { u(, t) H 1 (Ω), t >, u(x, ) a(x) H α (, T ), x Ω. Yuri Luchko (BHT Berlin) Maximum principle June 21, / 33

20 2. Diusion equation with the Caputo derivative Results regarding the maximum principle Luchko: Maximum principle for the strong solution under the assumption c(x), x Ω. Luchko and Yamamoto: Maximum principle for the weak solution in the case c C(Ω) without the non-negativity condition c(x), x Ω. Yuri Luchko (BHT Berlin) Maximum principle June 21, / 33

21 2. Diusion equation with the Caputo derivative Consequences from the maximum principle Let us now denote the solution to the initial-boundary value problem for the fractional diusion equation by u a,f. 1) Non-negativity property: Let a L 2 (Ω) and F L 2 (Ω (, T )). If F (x, t) a.e. (almost everywhere) in Ω (, T ) and a(x) a.e. in Ω, then u a,f (x, t) a.e. in Ω (, T ). 2) Comparison property: Let a 1, a 2 L 2 (Ω) and F 1, F 2 L 2 (Ω (, T )) satisfy the inequalities a 1 (x) a 2 (x) a.e. in Ω and F 1 (x, t) F 2 (x, t) a.e. in Ω (, T ), respectively. Then u a1,f 1 (x, t) u a2,f 2 (x, t) a.e. in Ω (, T ). Yuri Luchko (BHT Berlin) Maximum principle June 21, / 33

22 2. Diusion equation with the Caputo derivative Comparison property regarding the coecient c = c(x) Let us now x a source function F = F (x, t) and an initial condition a = a(x) and denote by u c = u c (x, t) the weak solution to the time-fractional diusion equation with the coecient c = c(x). Then the following comparison property is valid: Let c 1, c 2 C(Ω) satisfy the inequality c 1 (x) c 2 (x) in Ω. Then u c1 (x, t) u c2 (x, t) in Ω (, T ). Yuri Luchko (BHT Berlin) Maximum principle June 21, / 33

23 2. Diusion equation with the Caputo derivative Positivity property Conditions: 1) the initial condition a L 2 (Ω), a, a a.e. in Ω, 2) the weak solution u belongs to C((, T ]; C(Ω)), 3) the source function is identically equal to zero, i.e., F (x, t), x Ω, t >. Then the weak solution u is strictly positive: u(x, t) >, x Ω, < t T. Yuri Luchko (BHT Berlin) Maximum principle June 21, / 33

24 2. Diusion equation with the Caputo derivative Literature Yu. Luchko, Maximum principle for the generalized time-fractional diusion equation. J. Math. Anal. Appl. 351 (29), R. Goreno, Yu. Luchko and M. Yamamoto, Time-fractional diusion equation in the fractional Sobolev spaces. Fract. Calc. Appl. Anal. 18 (215), Y. Liu, W. Rundell, M. Yamamoto, Strong maximum principle for fractional diusion equations and an application to an inverse source problem. Fract. Calc. Appl. Anal. 19 (216), Yu. Luchko, M. Yamamoto, On the maximum principle for a time-fractional diusion equation. Fract. Calc. Appl. Anal. 2 (217), Yu. Luchko, M. Yamamoto, Maximum principle for the time-fractional PDEs. Chapter in Handbook of Fractional Calculus with Applications, to be published in 218. Yuri Luchko (BHT Berlin) Maximum principle June 21, / 33

25 3. Abstract time- and space-fractional diusion equation Abstract time- and space-fractional diusion equation Let X be a Hilbert space over IR with the scalar product (, ). For < α, β < 1, we consider the following evolution equation in the Hilbert space X : D α t u(t) = ( A) β u in X, t > along with the initial condition u() = a X. Assumptions: the operator A is self-adjoint, has compact resolvent, and (, ] ρ( A), ρ( A) being the resolvent of A. We note that u(, t) := u(t) D(( A) β ) for t > and so a boundary condition is incorporated into the equation. Yuri Luchko (BHT Berlin) Maximum principle June 21, / 33

26 3. Abstract time- and space-fractional diusion equation Non-negativity property For < α, β < 1, let us denote a solution to the abstract time- and space-fractional diusion equation by u α,β (t). If a in Ω, then u α,β (, t) in Ω for t. Yuri Luchko (BHT Berlin) Maximum principle June 21, / 33

27 3. Abstract time- and space-fractional diusion equation Sketch of the proof The main idea is rst to prove non-negativity of u α,β in the case α = 1 (space-fractional equation) and then to extend this result for the general case. We start with the following important result: u 1,β (, t) in Ω for t if a in Ω. Yuri Luchko (BHT Berlin) Maximum principle June 21, / 33

28 3. Abstract time- and space-fractional diusion equation Sketch of the proof Ingredients for the proof: 1) Integral representation (( A) β + 1) 1 a = sin πβ π µ β ( A + µ) 1 a µ 2β + 2µ β cos πβ + 1 dµ, a X 2) Maximum principle for A => ( A + µ) 1 a for µ and a in Ω. 3)µ 2β + 2µ β cos πβ + 1 > for µ and < β < 1. Hence Then (1 + ( A) β ) 1 a if a(x), x Ω. ( u 1,β (, t) = e ( A)βt a = lim 1 + t l l ( A)β) l a. Yuri Luchko (BHT Berlin) Maximum principle June 21, / 33

29 3. Abstract time- and space-fractional diusion equation Sketch of the proof Let < α, β < 1. Then with u α,β (x, t) = Φ α (η) = Φ α (η)u 1,β (x, t α η)dη, x Ω, t > l= ( η) l l!γ( αl + 1 α) being a particular case of the Wright function (also known as the Mainardi function). Because u 1,β (x, t) for x Ω and t for a and we get the inequality Φ α (η), η > u α,β (x, t), x Ω, t. Yuri Luchko (BHT Berlin) Maximum principle June 21, / 33

30 4. Short survey of other results Some of other results A. Alsaedi, B. Ahmad and M. Kirane, Maximum principle for certain generalized time and space fractional diusion equations. Quart. Appl. Math. 73 (215), M. Al-Refai, Yu. Luchko, Maximum principles for the fractional diusion equations with the Riemann-Liouville fractional derivative and their applications. Fract. Calc. Appl. Anal. 17 (214), M. Al-Refai, Yu. Luchko, Maximum principle for the multi-term time-fractional diusion equations with the Riemann-Liouville fractional derivatives. Appl. Math. Comput. 257 (215), 451. M. Al-Refai, Yu. Luchko, Analysis of fractional diusion equations of distributed order: Maximum principles and their applications. Analysis 36 (216), Yuri Luchko (BHT Berlin) Maximum principle June 21, / 33

31 4. Short survey of other results Some of other results A. Bernardis, F.J. Martín-Reyes, P.R. Stinga, J.L. Torrea, Maximum principles, extension problem and inversion for nonlocal one-sided equations. Journal of Dierential Equations 26 (7) (216), J. Jia, K. Li, Maximum principles for a time-space fractional diusion equation. Applied Mathematics Letters 62 (216), Y. Liu, Strong maximum principle for multi-term time-fractional diusion equations and its application to an inverse source problem. Computers and Mathematics with Applications 73 (217), Z. Liu, Sh. Zeng, Y. Bai, Maximum principles for multi-term space-time variable-order fractional diusion equations and their applications. Fract. Calc. Appl. Anal. 19 (216), Yuri Luchko (BHT Berlin) Maximum principle June 21, / 33

32 4. Short survey of other results Some of other results Yu. Luchko, Initial-boundary-value problems for the generalized multi-term time-fractional diusion equation. J. Math. Anal. Appl. 374 (211), Yu. Luchko, Boundary value problems for the generalized time-fractional diusion equation of distributed order. Fract. Calc. Appl. Anal. 12 (29), H. Ye, F. Liu, V. Anh, I. Turner, Maximum principle and numerical method for the multi-term time-space Riesz-Caputo fractional dierential equations. Appl. Math. Comput. 227 (214), R. Zacher, Boundedness of weak solutions to evolutionary partial integro-dierential equations with discontinuous coecients. J. Math. Anal. Appl. 348 (28) Yuri Luchko (BHT Berlin) Maximum principle June 21, / 33

33 Thank you very much for your attention! t (D C (k) f )(t) = k(t τ)f (τ) dτ D α t u(t) = ( A) β u Questions and comments are welcome! Yuri Luchko (BHT Berlin) Maximum principle June 21, / 33