Mark Ainsworth ·Zhiping Mao

Abstract

Keywords Fractional diff erential equation·Non-local energy·Well-posedness·Fourier spectral method

1 Introduction

The fractional diff erential operatorarises in relativistic quantum mechanics where the kinetic energy for a free relativistic particle with momentumpand massmis given byand units are chosen so that the speed of light is unity.Insertinggives the operatorMore generally,one can consider thefractional non-local operatordefined byand consider the related energy functional

whereFcorresponds to the classical double-well potential [7,8]

which favors the statess=±1 .Alternatively,we may write

where

In the current work,we consider the gradient flow associated with the free energy functional (1) in the case∈(1∕2,1] andm∈(0,1) with an eye identifying the impact of using the non-zero mass.Specifically,considering the gradient flow associated with (1) inleads to the fractional equation

The fractional non-local operatoris defined by using the spectral decomposition.Specif cially,ifuis-periodic in each variable,then

where i2=−1 ,and the Fourier coefficients are given by

In the first part of the work,we establish the well-posedness of (2) along with some basic properties of the solution.As remarked earlier,in the casem=0,=1,the equation reduces to the Allen–Cahn equation (0 ) or the Cahn–Hilliard equation (=1 ).The chief diff erence between the Allen–Cahn equation and the Cahn–Hilliard equation is that the latter conserves the quantityu(x,t)dxover time whilst the former does not.The steady states of the Cahn–Hilliard equation consist of regions in whichu=±1 ,corresponding to minimizers ofF(u),separated by diff use interfaces of widthO() [13].Properties of the fractional Cahn–Hilliard equation for=1,∈(1∕2,1] in the massless case were studied in [2] where it was shown that the steady states again consist of regions in whichu=±1 ,separated by diff use interfaces of sizeO() .In the second part of this work,we study what the eff ect of using a non-zero mean has on the structure of the solution.In particular,we show that,analogous to the Cahn–Hilliard case,the solutions consist of regions in which the solution isagain separated by dif ufse interfaces of widthO() .We also observe that the average value of the initial conditionu0has a significant eff ect on the nature of the steady states: ifthen the asymptotic state will beconstantover the whole of Ω.

2 Fourier–Galerkin Approximation and Well-Posedness

2.1 Fourier–Galerkin Approximation

We consider the Fourier–Galerkin discretisation of (2),establish its existence and uniqueness,and derive various properties of the discrete approximation.

We start with the weak formulation of the problem (2).Multiplying by a test functionv∈(Ω) on both sides of (2),integrating over Ω and using the fractional integration by parts (using the same argument as in Lemma A.1 of [1]),we arrive at the following weak formulation of (2): findsuch that

with the initial conditionuN(0)=ΠNu0.Here ΠNis the usualL2-projection operator in,defined by

The next theorem shows that the Fourier–Galerkin approximation preserves the average value ofuand reduces the energy monotonically in time.

Theorem 2.1Letthen the Fourier spectral–Galerkin method pre-serves the average value of ufor all t≥0,i.e.,

Furthermore,the free energy functional satisfies:for0 ≤s

and hence E(uN(t))is monotonically decreasing.Finally,the Fourier spectral Galerkin method(6)is stable and for all >0,there holds

where C()is a positive constant independent oftand N.

The proof of this result closely follows that of Lemmas 2.1 and 2.2 in [1].

We next show the existence and uniqueness of the Fourier–Galerkin solution.

Theorem 2.2ForThen,(6)admits a unique solution uN(t)satisfying the following energy inequalities:

for all t∈ℝ+,whereandare two constants depending onand u0but not on Nand t.

The proof of this result follows the argument used in the proof of Theorem 3 in [2].

2.2 Well-Posedness

The following result concerns the well-posedness of the weak formulation (5):

Theorem 2.3For1∕2≤1,Then,the FCHE(2)admits a unique weak solutionwithfor all T≥0,which additionally satisfies

and

where C()is a positive constant depends onbut not on t.

The proof of this result follows that of Theorem 1 in [2].In particular,the uniform inNestimates for the Fourier–Galerkin approximation means we can use compactness arguments to deduce results for the original equation.

The FCHE (2) enjoys similar properties to those of the Fourier–Galerkin approximation.In particular,we have the following corollary to Theorem 2.1.

Corollary 2.1LetThen,the solution of FCHE satis-fies∫Ωu(t)is constant for allt>0 ,i.e.,

and the energy is non-increasing,i.e.,there holds for t >s>0,

The proof of this result follows that of Corollary 1 in [2].

2.3 Error Estimate for the Fourier Galerkin Semi-Discritization

Finally,we turn to the rate of convergence of the Fourier–Galerkin approximation to the solution of FCHE (2),which shows that we have spectral accuracy provided the solution is smooth enough.

Theorem 2.4Suppose that the solutions of problem(5)and(6)are uniformly bounded and that u(t)∈Hr(Ω),r >1+,t∈[0,T],then the error estimate holds for1∕2≤1,0≤:

whereCandare two constants independent ofN.

The proof of this result follows that of Theorem 4 in [2].

2.4 Time Discretization

For the time integration,we use a stabilized first-order semi-implicit scheme in time [12].For a given partitionwhereis the time step,andK=T∕,we use the Fourier–Galerkin scheme (6) in space,and the stabilized firstorder time semi-implicit scheme given by

whereunNdenotes the numerical approximation touN(t) at timetnandSis a positive stabilization constant.

The implementation is as follows: writing

and takingv=eipx+iqy,p,q=−N∕2,…,N∕2 we get,thanks to the definition of the fractional Laplacian,we obtain

3 Numerical Examples Illustrating Behaviour of Solutions

In this section we shall present some numerical examples which illustrate the basic nature of solutions to (2) with regard to the eff ect of using a non-zero meanmand the fractional derivative.In the following section,we will provide formal analysis of the general properties of solutions and use the results to explain the observation in the current section.We consider the canonical test case of two bubbles.

Example 1We simulate the coalescence of two bubbles centered at (x1,0) and (x2,0) with radiiand,respectively,i.e.,the initial condition is

Example 1 has been well-studied in the casem=0 [1,2].Here we are especially concerned with the eff ect of changing the values of the size of bubblemassmand fractional derivative,on the character of the solution of FCHE.We also discuss how the values of,mandaff ect the width of the interface layer.The solutions were obtained using the Fourier Galerkin method withN=128 and a time step size=0.01 .Theorem 2.4 shows that the scheme is convergent and we assume that the numerical solution provides an accurate approximation of the true solution.In this section we shall content ourselves with observing the eff ects of varying the parameters of the nature of the steady states,and postpone explanations until the next section.

3.1 Influence of Size of the Smaller Bubble

We begin by examining the influence of the size of the bubble.Specifically,we fix the size of one bubble,i.e.,and vary the size of the other bubble,i.e.,the value ofwith the other parameters are set to beandS=50 .The snapshots of the numerical solutions with three diff erent values ofare shown in Fig.1.

Observe that in all cases the two bubbles coalesce into a single bubble.However,in the case ofthe bubble eventually disappears,corresponding to the solutionutending to a constant on the whole domain.Conversely in the other two casesthe steady-state solution corresponds to a single isolated bubble.

Fig.1 Evolution of solutions of FCHE with the initial condition (17) for dif ferent sizes of the small bubble.From top to bottom

3.2 Influence of the Mass

We now examine the eff ect of varying the massmon the steady state solutions.LetandS=50 .In this case,we again fix the two bubbles,i.e.,andand increase the value of the massm.The snapshots of the numerical solutions with three diff erent values ofm2=0,0.2,0.36 are shown in Fig.2.

We observe again that in all cases the two bubbles coalesce into a single bubble,but when the value ofmis small (e.g.,m2=0,0.2 ) we obtain a single bubble,whilst increasing the value ofmresults in the solution tending to a constant steady state (m2=0.36).

3.3 Influence of the Value of Fractional Order

In this subsection,we examine influence of the value of fractional order.LetIn this case,we use the same initial condition as in the previous subsection and present the results obtained using the numerical simulation for various values of the fractional order.Figure 3 shows the snapshots of the numerical solutions with three diff erent values of=1.0,0.8,0.6 and diff erent values ofm2=0,0.36.

Fig.2 Evolution of solutions of FCHE with the initial condition (17) for diff erent values of m.From top to bottom m2 =0,0.2,0.36

Fig.3 Evolution of solutions of FCHE with the initial condition (17) for diff erent values of the fractional order m2 =0 for left two columns whist m2 =0.36 for right two columns,From top to bottom =1.0,0.8,0.6

Figure 3 shows that we again obtain two basic types of stationary solutions,a single bubble or the constant solution,depending on the particular combinations of the parametersandm.

3.4 Influence of the Value of Fractional Order ˛

We now study the influence of the fractional order.As stated in Theorem 2.1 and Corollary 2.1,the average of the solution is conserved for>0 .For=0 ,it has been shown in [1] that the average of the solution is not conserved form=0 .The results obtained whenm>0 are shown in Fig.4,where the other parameters are set tom2=0.2,Observe that the average of the solution is not conserved for=0 ,and the solution converges to a constant whose value is one of the minima of the double-well potential,i.e.,Moreover,we can see that,for>0 ,the steady state is independent of the value of the fractional order.

Fig.4 Evolution of solutions of FCHE with the initial condition (17) for dif ferent values of fractional order From top to bottom =0,0.1,1.0

3.5 Analytical Explanation

In order to better understand the phenomena observed in the previous section,we now turn to analytical arguments.

The first eff ect we wish to examine is the basic nature of the steady states in terms of (a) the values of the piecewise constant nature of the solution and (b) whether a single bubble is obtained or whether the solution is constant over the whole domain.In order to examine these issues,we can (formally) choose=0 since,as observed above,the answer to (a) and (b) are not particularly sensitive to the choice of(providedis small).Thus,we have −Δu=0 in each region.Consequently,the stationary solutionsu∞of the fractional problem (2) will be dictated by the minima of the double-well potential

Ifm<1 ,the above potential attains minima at the values ±,where

Whenm=0 ,we recover the two states ±1 which are familiar from the standard Cahn–Hilliard equation.However,in general,the above values depend onmand the fractional order

We now try to deduce the steady stateuof the solution of (5).First,since=0 ,we assume the interface width is negligible so that we can ignore the eff ect of the interface.Without loss of generality,we suppose thatand that the stationary solution is given by

where Ω0is the union of regions on whichu=and Ω1is the union of regions on whichu=−.Sinceremains constant in time,we have

whilst,on the other hand,Ω0and Ω1satisfy

Consequently,the above two equations yield

Clearly,we have |Ω0|,|Ω1|>0 ,so this solution is only possible providedIft hen the steady-state solution will be constant over all of Ω ,taking the valuewe obtain the solution (20).By the symmetry,for<0 ,we arrive at that if<−,we have the constant solution while otherwise we have the solution (20).The above analysis shows that if the absolute value of the average of the initial conditionu0exceeds,then a single constant state is obtained over all of Ω .Otherwise,a single bubble solution is obtained in which the values are ±.This conclusion is consistent with the observations in Sects.3.1– 3.3 where the value ofpositive for all tests.Actually,in the test of the influence of the size of bubble,the value ofincreases if the size of the bubble decreases,and in the test of the influence of the massmand the fractional orderthe initial condition for all cases is the same,i.e.,the value ofis the same,then the value of the threshold,i.e.,the value ofdecreases by increasing the value ofmor decreasing the value of the fractional order.Furthermore,we compute the ratio |Ω0|∕|Ω| for diff erent cases of phase-separated solutions in Table 1 from which we observe that the computed ratios are close to the predicted ones based on the above analysis; this verifies our analysis.

Table1 The predicted and computed ratios |Ω0|∕|Ω|

3.6 The Scaling of the Width of the Layer

The numerical simulations show that the steady state is either constant over all of Ω ,or takes the values ±in regions separated by a thin layer.Now we turn to discuss that how the values of,mandaff ect the width of the interface layer.To gain insight,we perform a numerical test with a much bigger value of,i.e.,=0.1 ,and an initial condition consisting of two equal size kissing bubbles for diff erent values fractional order.The remaining parameters are chosen asandS=10 .The two bubbles are set to be (x1,0)=(−1,0),=1 and

We present plots of the solution for diff erent values of the fractional orderat various times in Fig.5.Furthermore,the profiles of the steady state with respect toare shown in Fig.6.

Fig.5 Evolution of solutions of FCHE with the initial condition (17) for dif ferent values of the fractional order From top to bottom =1.0,0.8,0.6

Fig.6 The solution of the steady state to FCHE (2) at time T =300 with m2 =0.2 and diff erent values

Observe that in all cases,as time evolves,the two bubbles coalesce into a single bubble.However,the sizes of the single bubble are diff erent from each other,as also the values of the steady state,namely the maximum and the minimum of the solution are diff erent which is consistent with our previous analysis.

We know how the values ofmandwould aff ect the constant states,and we now turn to the question of how these values aff ect the width of the layer.

We use the similar argument to the one we used in [2].In particular,in the region away from the interface,the free energy depends on the value of(u) ,where(u) is given by (18).In particular,since(±)=0 ,we conclude that the contribution to the free energy from region away from the interface vanishes.Therefore,we need only consider the energy around the interface area.To this end,we let Ωldenote the regions occupied by the transition layer and assume that the width of the interface layer between two pure phases is.Using the mean value theorem we have

whilst

It has been shown that the above is of the orderConsequently,since each of the above terms should be of the same order [2],we obtain

This means that the scaling of the width of the interface layerwith respect to the value ofisindependentof the values ofmand the fractional order.

In order to verify these conclusion,we consider a one-dimensional example.

Fig.7 Layer width against for diff erent values of m=0.1, N =1 024,=0.000 1

Fig.8 Layer width against for diff erent values of m,=0.8, N =1 024, 0.000 1

Example 2Consider the 1D version of problem (2) with the initial condition

and we approximate using Fourier–Galerkin with=0.5,N=1 024,=0.000 1.

We obtain Fig.7 (resp.8) by producing plots of solutions for various choices ofand(resp.m),and measuring the width of the layerFigure 7 shows the layer widthagainstwith the value ofm=0.1 and diff erent values of=0.6,0.8,1.0 whilst the Fig.8 shows the layer widthagainstwith the value of the fractional order=0.8 and diff erent values ofm=0.1,0.2,0.36 .Observe that,in all cases,the plots show that the layer widthis proportional to the value ofwhich agrees with our analysis.

4 Conclusions

In the current work,we have considered a modified free energy functional that arises from relativistic quantum mechanics and which reduces to the usual Cahn–Hilliard equation in the casem=0 .We also included fractional derivatives in the free energy which corresponds to the energy of boson stars in the case=1∕2 .It was shown that the resulting equation is well-posed and can be approximated using the Fourier–Galerkin method.Formal arguments were presented to explain the eff ect of the values of the initial condition on the steady states along with the choices ofandm.Finally,we showed that the width of the interface depends only on the value of

References

1.Ainsworth,M.,Mao,Z.: Analysis and approximation of a fractional Cahn–Hilliard equation.SIAM J.Numer.Anal.55,1689–1718 (2017)

2.Ainsworth,M.,Mao,Z.: Well-posedness of the Cahn–Hilliard equation with fractional free energy and its Fourier Galerkin approximation.Chaos Solitons Fractals.102,264–273 (2017)

3.Allen,S.M.,Cahn,J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening.Acta Metall.27,1085–1095 (1979)

4.Ambrosio,V.: On the existence of periodic solutions for a fractional Schrödinger equation.In: Proceedings of the American Mathematical Society (2018)

5.Antoine,X.,Tang,Q.,Zhang,Y.: On the ground states and dynamics of space fractional nonlinear Schrödinger/Gross–Pitaevskii equations with rotation term and nonlocal nonlinear interactions.J.Comput.Phys.325,74–97 (2016)

6.Bogdan,K.,Byczkowski,T.,Kulczycki,T.,Ryznar,M.,Song,R.,Vondracek,Z.: Potential analysis of stable processes and its extensions.In: Graczyk,P.,Stos,A.(eds.) Lecture Notes in Mathematics,vol.1980.Springer,Berlin (2009)

7.Cahn,J.W.: Free energy of a nonuniform system.II.Thermodynamic basis.J.Chem.Phys.30,1121–1124 (1959)

8.Cahn,J.W.,Hilliard,J.E.: Free energy of a nonuniform system.I.Interfacial free energy.J.Chem.Phys.28,258–267 (1958)

9.Kopriva,D.A.: Implementing spectral methods for partial diff erential equations: algorithms for scientists and engineers.Springer,Berlin (2009)

10.Lieb,E.H.,Yau,H.-T.: The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics.Comm.Math.Phys.112,147–174 (1987)

11.Lieb,E.H.,Yau,H.-T.: The stability and instability of relativistic matter.Comm.Math.Phys.118,177–213 (1988)

12.Shen,J.,Yang,X.: Numerical approximations of Allen–Cahn and Cahn–Hilliard equations.Discrete Cont.Dyn.Syst.28,1669–1691 (2010)

13.Yue,P.,Feng,J.J.,Liu,C.,Shen,J.: A diff use-interface method for simulating two-phase flows of complex fluids.J.Fluid Mech.515,293–317 (2004)