Regularity of Solutions for the Nonlocal Diusion Equation on Periodic Distributions Ilyas Mustapha Bacim Alali and Nathan Albin

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Regularity of Solutions for the Nonlocal Diﬀusion Equation on

Periodic Distributions

Ilyas Mustapha, Bacim Alali, and Nathan Albin

Department of Mathematics, Kansas State University, Manhattan, KS

October 4, 2022

Abstract

This work addresses the regularity of solutions for a nonlocal diﬀusion equation over the

space of periodic distributions. The spatial operator for the nonlocal diﬀusion equation is given

by a nonlocal Laplace operator with a compactly supported integral kernel. We follow a uniﬁed

approach based on the Fourier multipliers of the nonlocal Laplace operator, which allows the

study of regular as well as distributional solutions of the nonlocal diﬀusion equation, integrable

as well as singular kernels, in any spatial dimension. In addition, the results extend beyond

operators with singular kernels to nonlocal super-diﬀusion operators. We present results on

the spatial and temporal regularity of solutions in terms of regularity of the initial data or

the diﬀusion source term. Moreover, solutions of the nonlocal diﬀusion equation are shown to

converge to the solution of the classical diﬀusion equation for two types of limits: as the spatial

nonlocality vanishes or as the singularity of the integral kernel approaches a certain critical

singularity that depends on the spatial dimension. Furthermore, we show that, for the case of

integrable kernels, discontinuities in the initial data propagate and persist in the solution of the

nonlocal diﬀusion equation. The magnitude of a jump discontinuity is shown to decay overtime.

Keywords: Nonlocal diﬀusion equations, nonlocal Laplace operators, nonlocal superdiﬀusion, Fourier

multipliers, spatial regularity, temporal regularity.

1 Introduction

In this work, we study the regularity of solutions to the nonlocal diﬀusion equation given by

(ut(x, t) = Lδ,βu(x, t) + b(x), x ∈Tn, t > 0,

u(x, 0) = f(x),(1)

over the space of periodic distributions Hs(Tn), with s∈R. Here Tndenotes the periodic torus in

Rnand Lδ,β is a nonlocal Laplace operator deﬁned by

Lδ,βu(x) = cδ,β ZBδ(x)

u(y)−u(x)

ky−xkβdy, x ∈Rn,(2)

where Bδ(x) denotes a ball in Rn,δ > 0 is called the horizon or the nonlocality, and the kernel

exponent βsatisﬁes β < n + 2 [9, 10]. The scaling constant cδ,β is given by

cδ,β =2(n+ 2 −β)Γ(n

2+ 1)

πn

2δn+2−β.

arXiv:2210.00880v1 [math.AP] 23 Sep 2022

Nonlocal integral operators with compact support of the form (2) have their roots in peridynamics

[24, 25] and have been introduced in nonlocal vector Calculus [10]. These nonlocal operators have

been used in diﬀerent applied settings, see for example [4, 6, 7, 17, 19]. The work in [3] proposed

a nonlocal model for transient heat transfer, which is valid when the body undergoes damage or

evolving cracks. There have been many mathematical analysis studies involving nonlocal Laplace

operators and peridynamic operators including the works [2,9,15,21–23]. In general, exact solutions

are not readily available for nonlocal models, however, diﬀerent computational techniques and

numerical analysis methods have been developed for solving nonlocal equations such as [1, 5,8,11–

14, 16, 20, 26].

The work in [2] introduces the Fourier multipliers for nonlocal Laplace operators, studies the

asymptotic behavior of these multipliers, and then applies the asymptotic analysis in the periodic

setting to prove regularity results for the nonlocal Poisson equation. In this work, we apply the

Fourier multipliers approach developed in [2] to study the regularity of solutions to the nonlocal

diﬀusion equation over the space of periodic distributions. The organization of this article and a

brief description of the main contributions of this study are as follows.

•A review of the Fourier multipliers analysis for the nonlocal Laplace operator (2) is provided in

Section 2.

•In Section 3, we present the regularity of solutions analysis for the nonlocal diﬀusion equation

with initial data in Hs(Tn), but without a diﬀusion source.

–Theorem 3 and Proposition 1 provide the spatial and temporal regularity results, respec-

tively, in any spatial dimension. The temporal regularity for a general periodic distribution

in Hs(Tn), with s∈R, is studied in the sense of Gateaux derivative.

–In the case when the Fourier coeﬃcients of the initial data f∈Hs(Tn) are summable

k∈Zn|ˆ

fk|<∞,

then the solution of the nonlocal diﬀusion equation, considered as a function of the spa-

tial variable x, is a regular L2(Tn) function and Proposition 2 of Section 3.3 provides the

temporal regularity of the solution with respect to the classical derivative.

–Theorem 7 and Theorem 8 provide convergence results for the solution of the nonlocal

diﬀusion equation, without a diﬀusion source, to the solution of the corresponding classi-

cal diﬀusion equation with respect to two diﬀerent limits: as δ→0+or as β→n+ 2,

respectively.

•In Section 4, we present the regularity of solutions analysis for the nonlocal diﬀusion equation,

when a diﬀusion source b∈Hs(Tn), for some s∈R, is present.

–Theorem 9 and Proposition 3 provide the spatial and temporal regularity results, respec-

tively, in any spatial dimension. The temporal regularity for a general periodic distribution

in Hs(Tn), with s∈R, is studied in the sense of Gateaux derivative.

–In the case when the Fourier coeﬃcients of the source term b∈Hs(Tn) are summable

k∈Zn|ˆ

bk|<∞,

then the solution of the nonlocal diﬀusion equation, considered as a function of the spa-

tial variable x, is a regular L2(Tn) function and Proposition 4 of Section 4.1 provides the

temporal regularity of the solution with respect to the classical derivative.

–Theorem 12 and Theorem 13 provide convergence results for the solution of nonlocal diﬀu-

sion equation with a non-zero diﬀusion source to the solution of the corresponding classical

diﬀusion equation with respect to two kinds of limits: as δ→0+or as β→n+ 2, respec-

tively.

•In Section 5, we show that, for the case of integrable kernels, that is when β < n, discontinuities

in the initial data propagate and persist in the solution of the nonlocal diﬀusion equation. The

magnitude of a jump discontinuity is shown to decay as time increases.

2 Fourier multipliers

In this section, we give a summary of the Fourier multipliers’ results introduced in [2], which are

relevant to the work presented in Section 3. These multipliers are deﬁned through the Fourier

transform by

Lδ,βu(x) = 1

(2π)nZRn

mδ,β ˆu(ν)eiν·xdν, (3)

where mδ,β(ν) is given by

mδ,β(ν) = cδ,β ZBδ(0)

cos(ν·z)−1

kzkβdz, (4)

for β < n + 2. The following theorem gives the hypergeometric representation of these multipliers.

Theorem 1. Let n≥1, δ > 0and β < n + 2. Then the Fourier multipliers can be written as

mδ,β(ν) = −kνk22F31,n+ 2 −β

2; 2,n+ 2

2,n+ 4 −β

2;−1

4kνk2δ2.(5)

The hypergeometric function 2F3on the right hand side is well-deﬁned for any β6=n+ 4, n +

6,···, hence, using (5), the deﬁnition of the multipliers is extended to the case when β≥n+ 2

with β6=n+ 4, n + 6,···. Consequently, the operator Lδ,β is extended to these larger values of

βusing the Fourier transform. In particular, for the case when β=n+ 2, and since mδ,n+2(ν) is

equal to −kνk2, the extended operator Lδ,β coincides with the classical Laplace operator ∆. For

the case, n+ 2 < β < n + 4, the extended operator Lδ,β corresponds to a nonlocal super-diﬀusion

operator [1].

The representation (5) is used to provide the asymptotic behavior of mδ,β (ν) for large kνk. This

is given by the following result [2].

Theorem 2. Let n≥1,δ > 0and β /∈ {n+ 2, n + 4, n + 6, . . . }. Then, as kνk→∞,

mδ,β(ν)∼









−2n(n+2−β)

δ2(n−β)+ 2 2

δn+2−βΓ(n+4−β

2)Γ(n+2

(n−β)Γ(β

2)kνkβ−n,if β6=n,

−2n

δ22 log kνk+ log δ2

4+γ−ψn

2,if β=n,

(6)

where γis Euler’s constant and ψis the digamma function.

To simplify the notation, throughout this article we will denote mδ,β simply by m. However, in

places in which there is a need to emphasize the dependence of the multipliers on the parameters

δand β, such as when we take limits in those parameters, we will revert to the notation mδ,β .

3 Regularity of solutions for the peridynamic diﬀusion equation

In this section, we focus on the following nonlocal diﬀusion equation with initial data and no

diﬀusion source

(ut(x, t) = Lδ,βu(x, t), x ∈Tn, t > 0,

u(x, 0) = f(x).(7)

In order to study the existence, uniqueness, and regularity of solutions to (7) over the space of

periodic distributions, we consider the identiﬁcation U(t) = u(·, t), with U: [0,∞)→Hs(Tn).

3.1 Eigenvalues on periodic domains

Let Lδ,β be deﬁned on the periodic torus

Tn=

i=1

[0, ri], ri>0, i = 1,2,··· , n.

Deﬁne

νk=2πk1

,2πk2

,...,2πkn

rn,

for any k∈Zn. Let φk(x) = eiνk·x. Then,

Lδ,βφk(x) = mδ,β(νk)φk(x),(8)

which shows that φkis an eigenfunction of Lδ,β with eigenvalues mδ,β (νk). To simplify the notation,

we will often suppress the dependence of the multipliers on δand βand use m(ν) to denote mδ,β(ν).

Consider the nonlocal diﬀusion equation deﬁned in (1). For s∈R, let Hs(Tn) be the space of

periodic distributions hon Tnsuch that

khk2

Hs(Tn):= X

k∈Zn

(1 + kkk2)s|ˆ

hk|2<∞.

3.2 Distributional solutions for nonlocal diﬀusion equation

Let f∈Hs(Tn) and deﬁne U, V : [0,∞)→Hq(Tn) for some q∈R, by

U(t) = X

fkem(νk)teiνk·x,(9)

V(t) = X

fkm(νk)em(νk)teiνk·x.(10)

Observe that for any t≥0, U(t) and V(t) are well-deﬁned periodic distributions, since em(νk)tand

m(νk)em(νk)tare both bounded functions in k.

Theorem 3. Let n≥1, δ > 0and β < n + 4. Let 1and 2be such that 0< 1< 2<1. Assume

that f∈Hs(Tn)for some s∈R. Then for any ﬁxed t > 0,U(t)∈Hp(Tn)and V(t)∈Hr(Tn),

where

p=









s, if β < n,

s+4nt

δ2(1 −1),if β=n,

∞,if β > n,

r=









s, if β < n,

s+4nt

δ2(1 −2),if β=n,

∞,if β > n,

(11)

with H∞(Tn) := T

s∈R

Hs(Tn).

Proof. We observe that

06=k∈Zn

(1 + kkk2)p|ˆ

Uk|2=X

06=k∈Zn

(1 + kkk2)p−s(1 + kkk2)s|ˆ

fkem(νk)t|2.

Since f∈Hs(Tn), then the result that U(t)∈Hp(Tn) follows by showing that

(1 + kkk2)p−se2m(νk)t

is bounded for k6= 0. To see this, we consider three cases. For the case β < n, then p−s= 0 and

e2m(νk)t(1 + kkk2)p−s=e2m(νk)t,

which is bounded since m(νk)≤0.

For the case when n<β<n+ 4, let q∈Rbe arbitrary. Then,

(1 + kkk2)q−se2m(νk)t=(1 + kkk2)q−s

e2t|m(νk)|,

which vanishes as kkk→∞, and hence boundedness follows. Thus, U(t)∈Hq(Tn) for all qand

therefore,

U(t)∈\

q∈R

Hq(Tn) = H∞(Tn).

For the case when β=n, then p−s=4nt

δ2(1 −1). From Theorem 2, we have

m(νk)∼ −4n

δ2log kνkk,

which implies that

lim

kνkk→∞

m(νk)

−4n

δ2log kνkk= 1.(12)

Thus, for any 1>0, there exists N∈Nsuch that

−4n

δ2(1 + 1) log kνkk ≤ m(νk)≤ −4n

δ2(1 −1) log kνkk,(13)

for all kνkk ≥ N. Therefore,

e2m(νk)t≤e−8nt

δ2(1−1) log kνkk=kνkk−8nt

δ2(1−1).(14)

Since there exists A > 0 such that Akkk≤kνkk,then

(1 + kkk2)p−se2m(νk)t≤(1 + kkk2)4nt

δ2(1−1)

kνkk8nt

δ2(1−1)≤(1 + kkk2)4nt

δ2(1−1)

(Akkk)8nt

δ2(1−1),

which is bounded.

Similarly, to show that V(t)∈Hr(Tn),we show that

(1 + kkk2)r−sm(νk)2e2m(νk)t

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RegularityofSolutionsfortheNonlocalDiusionEquationonPeriodicDistributionsIlyasMustapha,BacimAlali,andNathanAlbinDepartmentofMathematics,KansasStateUniversity,Manhattan,KSOctober4,2022AbstractThisworkaddressestheregularityofsolutionsforanonlocaldiusionequationoverthespaceofperiodicdistributions.The...

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Regularity of Solutions for the Nonlocal Diusion Equation on Periodic Distributions Ilyas Mustapha Bacim Alali and Nathan Albin

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