ESTIMATES OF THE AGGREGATION-DIFFUSION SPLITTING ALGORITHMS FOR THE KELLER-SEGEL EQUATIONS

In this paper, we discuss error estimates associated with three different aggregation-diffusion splitting schemes for the Keller-Segel equations. We start with one algorithm based on the Trotter product formula, and we show that the convergence rate is C∆t, where ∆t is the time-step size. Secondly, we prove the convergence rate C∆t2 for the Strang’s splitting. Lastly, we study a splitting scheme with the linear transport approximation, and prove the convergence rate C∆t.

1. Introduction.In this paper we will consider the following Keller-Segel (KS) equations [8,15] in R d (d ≥ 2): This model is developed to describe the biological phenomenon chemotaxis.Here, ρ(t, x) represents the bacteria density, and c(t, x) represents the chemical substance concentration.
The most important feature of the KS model (1) is the competition between the aggregation term −∇ • (ρ∇c) and the diffusion term ∆ρ.In this paper, we develop three classes of positivity preserving aggregation-diffusion splitting algorithms for the Keller-Segel equations to handle the possible singularity.And we provide a 3464 HUI HUANG AND JIAN-GUO LIU rigorous proof of the fact that the solutions of these algorithms will converge to solutions of the Keller-Segel equations at a certain rate.The precise convergence rate will be given in Theorem 1.1 and Theorem 1.2 stated below after these algorithms have been defined.The convergence analysis for our aggregation-diffusion splitting algorithms are analog to that of the viscous splitting algorithms for the Navier-Stokes equations.
In fluid dynamics, the smooth solutions to the Euler equations are good approximations to the smooth solutions of the Navier-Stokes equations with small viscosity.This idea provides a method to approximate a solution to the Navier-Stokes equations by means of alternatively solving the inviscid Euler equations and a diffusion process over small time steps.Such approximations are called viscous splitting algorithms because they are forms of operator splitting in which the viscous term ν∆v is split from the inviscid part of the equations [12,Chap.3.4],where ν is the viscosity.In 1980, Beale and Majda [1] first proved the convergence rate Cν∆t 2 of the viscous splitting method for the two-dimensional Navier-Stokes equations.
Generally speaking, there are two basic splitting techniques.The first one is based on the Trotter product formula [18,Chap.11,Appendix A] and the convergence rate has been showed to be Cν∆t.The second algorithm is based on the Strang's splitting [17], which has the advantage of converging as Cν∆t 2 with no additionally computational expense.These two basic splitting methods were considered for linear hyperbolic problems by Strang [17] in 1968.He deduced the order of convergence by comparing a Taylor expansion in time of the exact solution with the approximation.Operator splitting is a powerful method for numerical investigation of complex models.Fields of application where splitting is useful to apply include air pollution meteorology [2], fluid dynamic models [9], cloud physics [14] and biomathematics [4].Lastly, we refer to [13] for theoretical and practical use of splitting methods.
For the KS equations (1), the splitting methods can be done as follows.Discretize time as t n = n∆t with time-step size ∆t, and on each time step first solve the aggregation equation, then the heat equation to simulate effects of the diffusion term ∆ρ.We will define this algorithm formally as below.
Denote the solution operator to an aggregation equation by (2) By using Lemma 7.6 in Gilbarg and Trudinger [5], if we define the negative part of the function u as u − := min{u, 0}, then one can easily prove that which leads to that u is nonnegative if u 0 is nonnegative.Also denote the solution operator to the heat equation by H(t), so that ω(t, x) = H(t)ω 0 (x) solves Similarly, we can prove that which also leads to that ω is nonnegative if ω 0 is nonnegative.
Then we can define the first order splitting algorithm by means of the Trotter product formula [18]: where ρ (n) (x) is the approximate value of the exact solution at time t n = n∆t.Furthermore, there is a second order splitting algorithm follows from Strang's method [17]: From the results of ( 3) and ( 5), we know that the splitting schemes ( 6) and ( 7) are positivity preserving.
Since the error estimates are valid when the solution of the KS equations is regular enough, we assume that then the KS system (1) has a unique local solution with the following regularity where T > 0 only depends on ρ 0 L 1 ∩H k (R d ) .The proof of this result is a standard process and it is provided in [7, Appendix A].As a direct result of the Sobolev imbedding theorem, one has The convergence results of our splitting algorithms ( 6) and (7) can be described as follows: Let ρ(t, x) be the regular solution to the KS equations (1) with initial data ρ 0 (x).Then there exist some C * , T * > 0 depending on ρ 0 L 1 ∩H k , such that for ∆t ≤ C * and (n + 1)∆t ≤ T * , the solutions to splitting algorithms are convergent to ρ(t n , x) in L 2 norm.Moreover, the following estimates hold Next, we will set up an aggregation-diffusion splitting scheme with the linear transport approximation as in [6] and provide the error estimate of this method.
First, we recast c(t, x) = Φ * ρ(t, x) with the fundamental solution of the Laplacian equation Φ(x), which can be represented as where , i.e. α d is the volume of the d-dimensional unit ball.Furthermore, the Φ(x) in ( 10) is also called Newtonian potential, and we can take the gradient of Φ(x) as the attractive force F (x). Thus we have where which leads to u (t n + s, X(x, s)) det dX(x, s) dx = u(t n , x).
By using Euler forward method, we have the linear approximation of (12) X(x, s) ≈ x + sv(x, 0) = x + sF * u(t n , x).
Let L(t n + s, X(x, s)) satisfying which leads to Then we can propose the following aggregation-diffusion splitting method with linear transport approximation: ) And here we require that ∆t < 1 The motivation of this scheme comes from the random particle blob method for the KS equations.As a future work, the results obtained in this article will be used to establish the error estimates of the random particle blob method for the KS equations.
One can write ( 13) to (15) in the symbolic form and it is obvious that this scheme also has the positivity preserving property.Moreover, we also prove the convergence theorem of the splitting algorithm ( 16) as below: x) be the regular solution to the KS equations (1) with initial data ρ 0 (x).Then there exist some C * , T * > 0 depending on ρ 0 L 1 ∩H k , such that for ∆t ≤ C * and (n + 1)∆t ≤ T * , the solution to the splitting algorithm In this article, we only present and analyze these semi-discrete splitting schemes and the spatial discretization is not considered.When the solution is regular, the standard spatial discretization such as finite element method, finite difference method and spectral method can be directly applied here and the numerical analysis for these three spatial discretization in the splitting schemes are standard, which is omitted here.However, for the KS equations, solutions can develop singularity.Computing such singular solutions is very challenging, and we refer to [11] for numerical results, where authors prove that the fully discrete scheme is conservative and positivity preserving.Another natural approach in spatial discretization is using the particle method.Actually, the main motivation of current paper is to develop a splitting scheme to analyze the random particle blob method for KS equations.
Notation.For convenience, in this article, we use • p for L p norm of a function.The generic constant will be denoted generically by C, even if it is different from line to line.
To conclude this introduction, we give the outline of this article.In Section 2, we establish the error estimates of the first and second order aggregation-diffusion splitting schemes through three steps: stability, consistency and convergence.Similarly, we provide the error estimate of a splitting scheme with the linear transport approximation in Section 3.

2.
The convergence analysis of the aggregation-diffusion splitting algorithms and the proof of Theorem 1.1.Like always, we follow the Lax's equivalence theorem [16] to prove the convergence of a numerical algorithm, which is that stability and consistency of an algorithm imply its convergence.Therefore, we break the proof of Theorem 1.1 up into three steps.
Step 1.The first step is to prove the stability, which ensures that the solution of the splitting algorithm ( 6) is priori controlled in an appropriate norm.The following proposition shows that our splitting method is There exists some T 1 > 0 depending on ρ 0 L 1 ∩H k , such that for the algorithms (6) and (7), we have Proof.We will only prove (18) in detail and the proof of ( 19) is almost the same.Suppose that 0 ≤ s ≤ ∆t, and we define , 1) .Notice that when s = 0, ρ(t n−1 ) = ρ (n−1) and that when s = ∆t, ρ(t n ) = ρ (n) .The standard regularity of heat equation gives that In order to give the estimate of A(s)ρ (n−1) H k , we need to solve the hyperbolic equation (2).
Multiply (2) by 2u and integrate over R d , then for k > d 2 , we have where −∆c = u and the Soblev imbedding theorem have been used.Now we multiply (2) by 2D 2m u with 1 ≤ |m| ≤ k and integrate over R d , then one has Estimate I 1 first, then we have where we have used the same notation in formula H k , by using Taylor [19,Proposition 3.6], Soblev imbedding theorem and −∆c = u.
Hence we have For I 2 , one has Combining ( 22) and ( 23), it follows that and there exists some Moreover, one has Hence it follows from (20) and (25) by taking s = ∆t Recasting (26), one has Until now, we have finished the proof of ( 18) and we can prove (19) almost the same way.
Step 2. In this step, we will prove our splitting algorithms ( 6) and ( 7) are consistent with the KS equations (1) by using the H k stability in Proposition 1.

Proposition 2. (Consistency) Assume that the initial data
Let ρ(t, x) be the regular solution to the KS equations (1) with local existence time T and T 1 is used in Proposition 1.If we define T * := min{T, T 1 }, then the local errors Proof.We start with proving (28).Recalling the definition of F in (11), we define the bilinear operator B as where we denote ].
(30) For the exact solution ρ(s + t n−1 ) to ( 1), one has Thus the difference between ρ(s + t n−1 ) and ρ(s Take the L 2 inner product of (31) with 2r n (s), then we have We compute that , where we have used the weak Young's inequality [10, P.107 and Young's inequality ab ≤ εa 2 + C(ε)b 2 with ε small enough.
Moreover, we have Next step is to estimate the function f n (s).By the definition of H(s) in ( 4), it satisfies Rewrite f n (s) in ( 30), one has To estimate f n (s), we compute ≤sC Aρ (n−1) F * Aρ (n−1) +3 )s.And similarly, we can compute other terms in (32).Thus for k > d 2 + 3, we have Until now, we have got By using Gronwall's inequality [3, Appendix B, P.624], one concludes that where C 1 , C 2 depends on T * , ρ 0 L 1 ∩H k .Thus, (28) has been proved.Next we are going to prove (29) by using the same procedure in the above arguments, and we can write with ).Thus, using the argument identical to that we have used to estimate f n (s), for k > d 2 + 5, we have fn (s which leads to (29) by using Gronwall's inequality.
Step 3. Finally, we can prove the convergence Theorem 1.1 by using Proposition 2. We estimate r n (∆t) = ρ (n) (x) − ρ(t n , x) as Standard induction implies that for (n + 1)∆t ≤ T * , which concludes the proof of ( 8) in Theorem 1.1.A similar argument holds for (9).Until now, we have completed the proof of Theorem 1.1.

3.
The convergence analysis of the splitting method with linear transport approximation and the proof of Theorem 1.2.In this section, we will prove the convergence estimate of the spitting method with linear transport approximation.Recall this splitting method proposed in Introduction with the initial data ρ(0) (x) = ρ 0 (x): ρ(n+1) (x) = H(∆t)L (n+1) (x).
(37) The proof of Theorem 1.2 can also be divided into three steps like Section 2.
Step 1.As we have done in the last section, firstly, we need to prove that the semi-discrete equations (35) to (37) are stable, i.e.

ρ(n)
In order to do this, we will need the following lemma: Lemma 3.1.Assume that x n+1 ≤ x n + ∆tg(x n ) for some nonnegative and increasing function g(x), then we have where y(t) is a solution to the following ODE Proof.We will prove this lemma by the induction on n.The case n = 0 can be obtained obviously by the initial condition.Since g(x) ≥ 0, we have that y(t) is a nondecreasing function, which leads to y((n + 1)∆t) = y(n∆t) + tn+1 tn g(y(t))dt ≥ y(n∆t) + ∆t g(y(n∆t)).
Hence, we concludes our proof.
To prove (38), if we set x n = ρ(n) H k in Lemma 3.1, then we only need to find the nonnegative and increasing function g(x) satisfying ρ(n+1) Then there exists some C 1 , T 3 > 0 depending on ρ 0 L 1 ∩H k , such that for the algorithm (16) with ∆t ≤ C 1 , we have Proof.
Step 1. (Estimate of the right handside of (36)) We begin with defining Then W 1 (0) = 0 and W 1 (u) is a smooth function with a bound independent of ∆t.
According to [19, Proposition 3.9], we have , where ω 1 (•) and ω 2 (•) are increasing functions.We have to mention here that the functions ω i (•) in the following text are always increasing functions and we denote ω(•) to be a generic function which maybe different from line to line.Moreover, we have where we have used the elliptic regularity of (35) in the second inequality.And (42) implies that Hence, by Moser's inequality [19,Proposition 3.7], from (41) one concludes that where in the second inequality we have used Step 2. (Estimate of the left handside of ( 14)) In this step, we consider the operation η → η to study the left handside of (14), where η x+∆tDG (n) (x) = η(x) for any function η(x).
Like we have done before, we rewrite Continue this process, we know Next, we verify by induction on 1 ≤ s ≤ k such that Recall that we have proved the case s = 1 in (44).For s ≥ 1, one has By the induction hypothesis This completes the proof of (45).Finally since L (n+1) = η, the (45) specializes to the following Step 3. (Estimate of ( 15)) Finally, this step requires H k norm bound for the linear heat equation, and we have ρ(n+1) 46) and ( 47), one has ρ(n+1) where ω is nonnegative and increasing.Now we can apply Lemma 3.1, and the following ODE Step 2. In this step, we will prove the consistency of the algorithm (16) by using Proposition 3, which is described by the following proposition: x) be the regular solution to the KS equations (1) with local existence time T and T 3 is used in Proposition 3. Denote T * := min{T, T 3 }, then the local error (50) Denote V (X(x, s), s) := G (n) (x), then L(t n +s, X) is the solution to the following PDE with initial data L(t n , X) = ρ(n) (X).Thus, it follows from (50) that For the exact solution ρ(t n + s, X) to (1), we have where in the second inequality we have used the regularity of ρ.Moreover, by using the weak Young's inequality, one concludes that Additionally, like we have done in (32) and (33), for k > Step 3. Now we can prove the convergence Theorem 1.2 by using Proposition 4. We estimate rn (∆t) = ρ(n+1) (X) − ρ(t n+1 , X) as Standard induction as we have done in (34) implies that for (n + 1)∆t ≤ T * , which concludes the proof of Theorem 1.2.