Integral representation bound of the true solution to the BVP of double-sided fractional diffusion advection reaction equation

Abstract

This paper is devoted to developing the regularity results of the true (classic) solution to the homogeneous BVP of double-sided fractional diffusion advection reaction equation with variable coefficients on the bounded interval. This topic has been controversial in modelling in recent years, especially on the regularity issue. We use a different strategy of raising regularity of the weak solution and prove that, under suitable conditions, the true solution exists and can be represented in the form of fractional integration; furthermore, we show that usually this integral representation cannot be further improved even with smooth coefficients and right-hand side function in the equation. And we find the precise bound for this integral representation to hold, which measures the “best” regularity guaranteed and is in sharp contrast to the case of integer-order elliptic PDEs.

Appendices

Appendix

1.1 Riemann–Liouville integrals and their properties

Definition 2

Let $w:(c,d) \rightarrow {\mathbb {R}}, (c,d) \subset {\mathbb {R}}$ and $\sigma >0$. The left and right Riemann–Liouville fractional integrals of order $\sigma$ are, formally respectively, defined as

$$\begin{aligned} ({_a}D_{x}^{-\sigma }w)(x)&:= \dfrac{1}{\Gamma (\sigma )}\int _{a}^{x}(x-s)^{\sigma -1}w(s) \, \mathrm{d}s, \end{aligned}$$

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$$\begin{aligned} ({_x}D_{b}^{-\sigma } w)(x)&:= \dfrac{1}{\Gamma (\sigma )}\int _{x}^{b}(s-x)^{\sigma -1}w(s) \, \mathrm{d}s, \end{aligned}$$

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where $\Gamma (\sigma )$ is Gamma function. For convenience, when $c=-\infty$ or $d=\infty$ we set

$$\begin{aligned} ({\varvec{D}}^{-\sigma }w)(x) :={_{-\infty }}D_{x}^{-\sigma } w \text { and } ({\varvec{D}}^{-\sigma * }w)(x) :={_{x}}D_{\infty }^{-\sigma } w. \end{aligned}$$

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In particular, if $\sigma =0$, ${_a}D_{x}^{-\sigma }$, ${_x}D_{b}^{-\sigma }$, ${\varvec{D}}^{-\sigma }$ and ${\varvec{D}}^{-\sigma * }$ are regarded as identity operators.

Property 1

([31], Eq. (2.72), (2.73), p. 48) Given $\sigma > 0$, fractional operators ${_aD_x^{-\sigma }}$ and ${_xD^{-\sigma }_b}$ are bounded in $L^p(\Omega ) (p\ge 1)$:

$$\begin{aligned} \Vert {_aD_x^{-\sigma }}\psi \Vert _{L^p(\Omega )}\le K\Vert \psi \Vert _{L^p(\Omega )}, \, \Vert {_xD^{-\sigma }_b}\psi \Vert _{L^p(\Omega )}\le K\Vert \psi \Vert _{L^p(\Omega )}, \, K=\dfrac{(b-a)^\sigma }{\sigma \Gamma (\sigma )}. \end{aligned}$$

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Property 2

([31], Eq. (2.19), p. 34) Let $\sigma >0$ and $(Qf)(x)=f(a+b-x)$, then the following operators are reflective:

$$\begin{aligned} Q{_aD_x^{-\sigma }}Q={_xD_b^{-\sigma }}. \end{aligned}$$

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Property 3

If $0<\sigma <1$, $1<p<1/\sigma$, then the fractional operators ${_aD_x^{-\sigma }}$, ${_xD_b^{-\sigma }}$ are bounded from $L^p(\Omega )$ into $L^q(\Omega )$ with $q=\frac{p}{1-\sigma p}$.

Property 4

If $0<\frac{1}{p}<\sigma <1+\frac{1}{p}$, fractional operators ${_aD_x^{-\sigma }}$ and ${_xD^{-\sigma }_b}$ map the space $L^p(\Omega )$ into the H$\ddot{\text {o}}$lderian space $H^{\sigma -1/p}(\overline{\Omega })$.

Remark 1

Property 3 is a combination of Theorem 3.5 ([31], p. 66) and Property 2, and Property 4 is a combination of Corollary of Theorem 3.6 ( [31], p. 69) and Property 2.

1.2 Riemann–Liouville derivatives and their properties

Definition 3

Let $w:(c,d) \rightarrow {\mathbb {R}}, (c,d) \subset {\mathbb {R}}$ and $\sigma >0$. Assume n is the smallest integer greater than $\sigma$ (i.e., $n-1 \le \sigma < n$). The left and right Riemann–Liouville fractional derivatives of order $\sigma$ are, formally respectively, defined as

$$\begin{aligned} ({_a}D_x^{\sigma }w)(x) := \frac{\mathrm{d}^n}{\mathrm{d}x^n} {_a}D_x^{\sigma -n} w \text { and }~ ({_x}D_b^{\sigma }w)(x) := (-1)^n \frac{\mathrm{d}^n}{\mathrm{d} x^n} {_x}D_b^{\sigma -n} w. \end{aligned}$$

For ease of notation, when $c=-\infty$ or $d=\infty$ we set

$$\begin{aligned} ({\varvec{D}}^{\sigma } w )(x)= {_{-\infty }}D_{x}^{\mu }w \text { and } ({\varvec{D}}^{\sigma *}w)(x) = {_{x}}D^{\mu }_{\infty }w . \end{aligned}$$

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Property 5

([1], Theorem 4.1) For $v, w \in C_0^\infty (\Omega )$ and $\sigma \ge 0$, it is true that

$$\begin{aligned} \begin{aligned}&({\varvec{D}}^{\sigma } v , {\varvec{D}}^{\sigma } w )_{{\mathbb {R}}} = ({\varvec{D}}^{\sigma *} v , {\varvec{D}}^{\sigma *} w )_{{\mathbb {R}}} = (2\pi )^{2\sigma }\int _{\mathbb {R}}|\xi |^{2\sigma } \widehat{v}(\xi ) \overline{\widehat{w}(\xi )} \,\mathrm{d}\xi ,\\&({\varvec{D}}^{\sigma } v, {\varvec{D}}^{\sigma *} w )_{{\mathbb {R}}} +({\varvec{D}}^{\sigma } w, {\varvec{D}}^{\sigma *} v )_{{\mathbb {R}}} =2\cos (\sigma \pi )({\varvec{D}}^{\sigma } v , {\varvec{D}}^{\sigma } w )_{{\mathbb {R}}}. \end{aligned} \end{aligned}$$

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Property 6

([1], Property 2.4) Let $0<\sigma$ and $w\in C_0^\infty ({\mathbb {R}})$, then ${\varvec{D}}^\sigma w, {\varvec{D}}^{\sigma *} w \in L^p({\mathbb {R}})$ for any $1\le p<\infty$.

1.3 Fractional Sobolev spaces and properties

Fractional Sobolev spaces $\widehat{H}^s({\mathbb {R}})$ can be equivalently described in different ways, serving convenient tools for deriving various properties under different contexts.

Definition 4

(Via Fourier transform [32, 33]) Given $0\le s$, let

$$\begin{aligned} \widehat{H}^s({\mathbb {R}}) := \left\{ w \in L^2({\mathbb {R}}) : \int _{{\mathbb {R}}} (1 + |2\pi \xi |^{2s}) |\widehat{w}(\xi ) |^2 \, \mathrm{d} \xi < \infty \right\} . \end{aligned}$$

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It is endowed with semi-norm and norm

$$\begin{aligned} |w|_{\widehat{H}^s({\mathbb {R}})}:=\Vert (2\pi \xi )^s \widehat{w}\Vert _{L^2({\mathbb {R}})}, \Vert w\Vert _{\widehat{H}^s ({\mathbb {R}})}:=\left( \Vert w\Vert ^2_{L^2({\mathbb {R}})} +|w|^2_{\widehat{H}^s({\mathbb {R}})}\right) ^{1/2}. \end{aligned}$$

Another equivalent definition is achieved with the aid of left or right fractional-order weak derivative, which is a generalization of integer-order weak derivative:

Definition 5

( [1], Sect. 3) Given $0\le s$ and assume $u(x)\in L^2({\mathbb {R}})$, then $u(x) \in \widehat{H}^s({\mathbb {R}})$ if and only if there exists a unique $\psi _1(x) \in L^2({\mathbb {R}})$ such that

$$\begin{aligned} \int _{\mathbb {R}} u \cdot {\varvec{D}}^s \psi =\int _{\mathbb {R}} \psi _1 \cdot \psi \quad \end{aligned}$$

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for any $\psi \in C^\infty _0({\mathbb {R}})$.

Similarly,

$u(x) \in \widehat{H}^s({\mathbb {R}})$ if and only if there exists a unique $\psi _2(x) \in L^2({\mathbb {R}})$ such that

$$\begin{aligned} \int _{\mathbb {R}} u \cdot {\varvec{D}}^{s*} \psi =\int _{\mathbb {R}} \psi _2 \cdot \psi \end{aligned}$$

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for any $\psi \in C^\infty _0({\mathbb {R}})$.

With above definitions, the following property can be deduced, which guarantees the existence of fractional derivatives and provides equivalent semi-norm and norm.

Property 7

([1], Sect. 3) Assume $u\in \widehat{H}^s({\mathbb {R}})$, $s\ge 0$, then ${\varvec{D}}^s u, {\varvec{D}}^{s*}u$ exist a.e. and

$$\begin{aligned} |u|_{\widehat{H}^s({\mathbb {R}})} = \Vert {\varvec{D}}^s u\Vert _{L^2({\mathbb {R}})}=\Vert {\varvec{D}}^{s*}u\Vert _{L^2({\mathbb {R}})}. \end{aligned}$$

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By restricting to the bounded interval we can define the following analogue.

Definition 6

Given $0\le s$.

$$\begin{aligned} \widehat{H}^s_0(\Omega ):=\{\text {Closure of}\, \, u\in C_0^\infty (\Omega ) \, \text {with respect to norm}\, \Vert \tilde{u}\Vert _{\widehat{H}^s({\mathbb {R}})} \}, \end{aligned}$$

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where notation $\tilde{u}$ denotes the extension of u(x) by 0 outside $\Omega$. It is endowed with semi-norm and norm

$$\begin{aligned} |u|_{\widehat{H}^s_0(\Omega )}:=|\tilde{u}|_{\widehat{H}^s({\mathbb {R}})}, \Vert u\Vert _{\widehat{H}^s_0(\Omega )}:=\Vert \tilde{u}\Vert _{\widehat{H}^s({\mathbb {R}})}. \end{aligned}$$

And we have the following properties.

Property 8

Given $\frac{1}{2}<s<1$, then $u\in \widehat{H}^s_0(\Omega )$ can be represented as

$$\begin{aligned} u(x)={_aD_x^{-s}\psi _1}={_xD_b^{-s}}\psi _2, \end{aligned}$$

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for certain $\psi _1$, $\psi _2 \in L^2(\Omega )$. As a consequence, ${_aD_x^s}u$ and ${_xD_b^s}u$ exist a.e. and coincide with $\psi _1$, $\psi _2$, respectively.

Property 9

Given$1/2<s<1$, $g(x)\in C^1(\overline{\Omega })$, then there exists a positive constant C such that

$$\begin{aligned} \tilde{g}\tilde{u}\in \widehat{H}^s({\mathbb {R}})\quad \text {and}\quad \Vert \tilde{g}\tilde{u}\Vert _{\widehat{H}^s({\mathbb {R}})}\le C \Vert \tilde{u}\Vert _{\widehat{H}^s({\mathbb {R}})} \end{aligned}$$

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for any $u(x)\in \widehat{H}^s_0(\Omega )$. (Notation $\tilde{\cdot }$ denotes the extension by zero outside $\Omega$.)

Remark 2

The proof of Property 8 is readily verified by using Definition 5 and the fact that $u(a)=u(b)=0$ implied by $\frac{1}{2}<s<1$, $u\in \widehat{H}^s_0(\Omega )$. And the proof of Property 9 can be referred to Lemma 3.2, [2] or simply Exercise 21, p. 309, [28].

1.4 Spaces $H^*(\Omega )$, ${H^*}_{\sigma (\Omega )}$ and properties

Property 10

([31], Theorem 13.14, p. 248) Let $0<\sigma <1$, the fractional integration operators ${_aD_x^{-\sigma }}$ and ${_xD_b^{-\sigma }}$ map the space $H^*(\Omega )$ one-to-one and onto the space ${H^*}_\sigma (\Omega )$, respectively. Consequently, ${_aD_x^{-\sigma }}(H^*(\Omega ))={_xD_b^{-\sigma }}(H^*(\Omega ))$.

Property 11

Let $0<\sigma <1$, $\gamma _1, \gamma _2>0$, $\gamma _1{_aD_x^{-\sigma }}+\gamma _2{_xD_b^{-\sigma }}$ maps $H^*(\Omega )$ one-to-one and onto the space ${H^*}_\sigma (\Omega )$.

Remark 3

Property 11 is a combination of Theorem 30.7 ([31], p. 626) and Property 10.

Calrification

This article is a refinement of partial results of the original draft [34] (unsubmitted, unpublished, serves as a public reference only) that is available at https://arxiv.org/abs/2005.04405. This refined work is aimed for publication in the journal.