On viscosity solutions of space-fractional diffusion equations of Caputo type

Namba, Tokinaga; Rybka, Piotr

doi:10.48550/arxiv.1905.00168

Cited by 2 publications

(4 citation statements)

References 20 publications

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“…In the next lemma we will show that ∂ ∂x D α f is non positive in the maximum point of f in the interior of the interval. This result, under stronger regularity assumptions, was proved in [12,Lemma 2.2]. Here we present the proof, where we do not demand C 2 regularity of f .…”

Section: Extremum Principlesmentioning

confidence: 67%

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A space-fractional Stefan problem

Ryszewska¹

2019

Preprint

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Section: Extremum Principlesmentioning

confidence: 67%

“…where we used (12). Hence, we may apply the Lebesgue dominated convergence theorem to pass to the limit under the integral sign in I 1 and we get…”

Section: Proof Using Definition (19) We May Rewrite (16) As Followsmentioning

confidence: 99%

A space-fractional Stefan problem

Ryszewska¹

2019

Preprint

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“…u 0 is well defined because, taking ν = α + 1, µ = 1 in Proposition 6 from Appendix, we obtain that E α+1 (λ) > 0 for λ belonging to the sector ϑ. Placing this u 0 in the formula (12) we obtain the solution for (8) which belongs to D α u(x) = (E α+1 (λ)) −1 (g * y α E α+1,α+1 (λy α+1 ))(1)E α+1 (λx α+1 )−g * x α E α+1,α+1 (λx α+1 ).…”

Section: Operator ∂ ∂X D α As a Generator Of An Analytic Semigroupmentioning

confidence: 91%

“…Another paper that presents the probabilistic point of view on space-fractional problems is [4], where the authors consider equations with time-fractional Caputo derivative and non-local space operators. A completely different approach for solving (1) for p ≡ 1 with zero Dirichlet boundary conditions is employed in [12], where the authors obtained the viscosity solutions. Further discussion was made in [3] and [13] where the authors compare the problems with diffusive flux modeled by the Caputo and the Riemann-Liouville derivative and carry a numerical analysis.…”

Section: Introductionmentioning

confidence: 99%

An analytic semigroup generated by a fractional differential operator

Ryszewska

2020

Journal of Mathematical Analysis and Applications

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We study a space-fractional diffusion problem, where the non-local diffusion flux involves the Caputo derivative of the diffusing quantity. We prove the unique existence of regular solutions to this problem by means of the semigroup theory. We show that the operator defined as divergence of product of a positive function with the Caputo derivative is a generator of analytic semigroup. the simplified problem (1). We would like to emphasise that from the modelling point of view, it is important that (1) is a balance low.Our goal is to study the basic solvability problem for equation (1), by the semigroup approach. At this moment we would like to present a broader context. It is worth to mention here, that in the paper [2] the authors studied an array of fractional diffusion equations, for a variety of boundary conditions, but they constructed only the C 0 -semigroup. Besides, we note that the problem (1) was not explicitly studied in [2]. Another paper that presents the probabilistic point of view on space-fractional problems is [4], where the authors consider equations with time-fractional Caputo derivative and non-local space operators. A completely different approach for solving (1) for p ≡ 1 with zero Dirichlet boundary conditions is employed in [12], where the authors obtained the viscosity solutions. Further discussion was made in [3] and [13] where the authors compare the problems with diffusive flux modeled by the Caputo and the Riemann-Liouville derivative and carry a numerical analysis.In the present paper we will present the results concerning solvability of (1) by means of the semigroup theory. At first, we will focus our attention on the case where p ≡ 1. We will describe the domain of ∂ ∂x D α in terms of Sobolev spaces and as a final result we will construct an analytic semigroup. Precisely, we will show that the operator ∂ ∂x D α , considered on the domaingenerates an analytic semigroup. Here, by H α (0, 1) we mean the fractional Sobolev space (see [10, definition 9.1]) and the subspace 0 H α (0, 1) will be introduced in Proposition 3. Subsequently, we will extend our result for the case of any strictly positive lipschitz continuous p. We would like to emphasise that, developing the theory of analytic semigroups for the systems of type (1) seems to be especially important if we notice that the operator ∂ ∂x D α is not self-adjoint, thus we can not expect that its eigenfunctions generate the basis of any natural Hilbert space.The paper is organized as follows. In chapter 2 we give preliminary results concerning fractional operators. Chapter 3 is devoted to the proof of the main result. Namely, we will show that ∂ ∂x D α is a generator of C 0 -semigroup of contractions which can be extended to the analytic semigroup (Theorem 2). In chapter 4, we extend the results of the previous chapter to the case of operator ∂ ∂x (p(x)D α ), where p is a positive lipschitz function (Theorem 3). In chapter 5, we will present the basic applications of this result to solvability of (1). At last, for the sake of comple...

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On viscosity solutions of space-fractional diffusion equations of Caputo type

Cited by 2 publications

References 20 publications

A space-fractional Stefan problem

A space-fractional Stefan problem

An analytic semigroup generated by a fractional differential operator

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