Notes on continuity result for conformable diffusion equation on the sphere: The linear case

1 Introduction

Diffusion is a common phenomenon in nature. We can easily encounter the diffusion process in many fields, such as: physics, chemistry, technology, or even economic and finance. For example, diffusion processes are used to model the price movements of financial instruments. Besides, diffusion also occurs in liquids and gases when their particles collide randomly and spread out. In this article, we are interested in studying the diffusion equation on the unit sphere S n ⊂ R n + 1 as follows:

where f is the initial datum, G is the source term function, and D t θ is called the conformable derivative of order θ . Here, Δ ∗ is the Laplace-Beltrami operator defined in Section 2.

Before mentioning some equations involving fractional conformable derivatives, we present a handful of results about classical diffusion equations on the sphere. Le Gia [1] studied the approximation of parabolic equations on unit spheres using spherical basis functions. Some types of elliptic equations on the sphere were also studied by [2,3]. The diffusion equation on the sphere is an inspiration to many interested mathematicians. These equations describe several patterns occurring in geophysics, oceanography, and geology, for example [4,5, 6,7,8, 9,10]. In [11], Wendland provided a novel discretization method for studying (system of) semilinear parabolic equations on Euclidean spheres. In [4], the authors also proved the existence of random attractors for the Navier-Stokes equations on the sphere. The biggest challenge for us when studying the problem on the spheres is that we have to understand well about spherical harmonics. From visual observation, the investigation of diffusion equations on the spheres is more complicated than in the Euclidean domain.

There are many types of definitions for fractional-order derivatives. Among them, the three most common are Riemann-Liouville, Caputo, and Grünwald-Letnikov. Besides, we can encounter many others, such as Atangana-Baleanu, Caputo-Fabrizio, Hilfer, etc. (see [12,13, 14,15,16, 17,18,19, 20,21,22, 23,24] and references therein). Fractional derivatives are usually used for modeling processes of mass transport, diffusion [25], optics, and other phenomena with memory effects such as [26]. With its own characteristics, fractional-order models have demonstrated have advantages when modeling heat equation [27] and Brownian motion [28] or stochastic processes [29].

When investigating fractional diffusion equations, the issues of interest often are the well-posedness of solution by degree of nonlinearity source function [30] and the regularity [31,32,33]. This article also considers those matters for equations equipped with the fractional conformable derivative. It should be emphasized that the research direction of partial differential equations (PDEs) using conformable derivatives is one of the sub-branches of PDEs with fractional derivatives. Here, below is the definition of conformable derivative.

Let A be a Banach space, and the function v : [ 0 , ∞ ) → A . Let D t θ be the conformable derivative operator defined as:

for any v ∈ A (see [34,35]). The conformable derivative can be viewed as one of the extensions of the classical derivative, which was first proposed by Khalil in [34]. With the definition (1.2), we immediately realize that D t θ becomes a classical derivative operator in case θ = 1 . There are thousands of works related to conformable differential equation. However, there are not too many studies for the diffusion equation with conformable derivative on infinite dimensional spaces. Let us refer the reader to the interesting study [35], which investigated the well-posedness of the conformable diffusion equation in both cases, that is, linear and nonlinear source terms. In that work, the authors also show that the conformable derivative model agrees better with the experimental data than the normal diffusion equation.

It is a very interesting issue to observe the solution of the problem (1.1) for both cases θ = 1 and 0 < θ < 1 . If θ = 1 , the mild solution of the problem (1.1) depends on only two variables x and t , while with 0 < θ < 1 , it depends on θ , x , t . The problem of continuity in terms of variables x and t has been carefully investigated in [35]. However, the continuity problem for the mild solution to problem (1.1) with respect to the fractional order 0 < θ < 1 is still open, that is, in this work, we will focus on the question

In (1.3), where u θ and u θ ′ are two solutions to problem (1.1). The motivation and meaning of this question are carefully presented in [36]. The same question for equations with the Caputo fractional derivative has been studied in [37]. In article [38], authors investigated the initial value problem for a system of nonlinear pseudo-parabolic equations with Caputo fractional derivative. Our article expands the problem to the case of a conformable fractional derivative on the sphere.

This article is organized as follows: Section 2 introduces some basic knowledge on spherical harmonics and some necessary spaces on the sphere. The main research results are presented in Section 3. Section 4 gives some conclusion and discusses about further improvements.

2 Preliminaries

In this section, we refer to some knowledge on spherical harmonics from the articles [1,2,3, 4,9].

Let Δ ∗ be the Laplace-Beltrami operator in R n . The Laplace-Beltrami operator can be understood as the restriction of the standard Laplace operator on sphere, which can be written as follows:

where Δ is the Laplace operator, ∇ is the gradient operator, and H is the Hessian matrix. We also know that the Laplace-Beltrami operator is linear, self-adjoint, and negative definite in the spatial variable.

It is so familiar that the eigenvalues of − Δ ∗ are given as follows:

and the respective eigenfunctions are called the spherical harmonics Y l ( x ) of order l . This means, they are spherical polynomials and satisfied

The space of all spherical harmonics of degree l on S n , denoted by V l , has an orthonormal basis { Y l k ( x ) : k = 1 , 2 , 3 , … N ( n , l ) } , where

For any f ∈ L 2 ( S n ) , we can express it in terms of spherical harmonics as follows:

where d S is the surface measure of the unit sphere. The Sobolev space X σ ( S n ) with real parameter σ consists of all distributions f such that

Obviously, the norm induced from an inner product is

3 Continuity results with respect to fractional order

First, we express u ( x , t ) in terms of spherical harmonics as follows:

By simple calculations, we obtain the ordinary differential equation with conformable derivative as follows:

Thanks to the formula in [39], we obtain

The following theorem is shown about the continuity problem with respect to fractional order.

Let us now continue to prove Theorem 3.2. From (3.2), we find that

In view of (3.15), the norm of the first term B 1 is bounded by

By applying (3.19), the term B 2 is bounded by

Let us divide two cases.

Case 1. The function G ∈ L ∞ ( 0 , T ; X μ ( S n ) ) . Under this assumption, we have the following observation

Then, we follow from (3.22) that

Case 2. The function G ∈ L 2 ( 0 , T ; X μ ( S n ) ) . Using the Hölder inequality and noting that θ > 1 2 , we derive that

Then, we follows from (3.22) that

Since (3.4), we have the following observation:

This implies that

Let us divide two cases.

Case 1. The function G ∈ L ∞ ( 0 , T ; X μ ( S n ) ) . If t ≤ 1 , then it is easy to check that ν θ − 1 ≥ ν θ ′ − 1 for 0 ≤ ν ≤ t and 0 < θ < θ ′ < 1 . Hence, we derive that

The integral term on the right of (3.29) is evaluated as follows:

where C 5 depends on a 0 , b 0 , and ε . If t ≥ 1 , then we have that

It is obvious to see that

and

Let us now give the following estimate: