# The Singularity Formation on the Coupled Burgers–Constantin–Lax–Majda System with the Nonlocal Term

Authors: Linrui Li; Shu Wang
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

In this paper, we study the finite-time singularity formation on the coupled Burgers–Constantin–Lax–Majda system with the nonlocal term, which is one nonlinear nonlocal system of combining Burgers equations with Constantin–Lax–Majda equations. We discuss whether the finite-time blow-up singularity mechanism of the system depends upon the domination between the CLM type’s vortex-stretching term and the Burgers type’s convection term in some sense. We give two kinds of different finite-time blow-up results and prove the local smooth solution of the nonlocal system blows up in finite time for two classes of large initial data.


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## Body

## 1. Introduction

We study the formation of singularities for the following coupled Burgers–Constantin–Lax–Majda system with the nonlocal term:(1)∂tu+aupux=buHu,x∈ℝ,t>0,ut=0=u0,where a>0,b>0,andp>0 are the given constants and H is the Hilbert transform operator defined by(2)Hfx=1πP.V.∫−∞+∞fyx−ydy,or defined by the Fourier transform by(3)Hu^ξ=−isignξu^ξ.In the following, we will use the fractional operatorΛβ with β∈0,1, which is defined by Λβ≡−Δβ/2 and can be given by(4)Λβux=kβ∫ℝux−uyx−y1+βdy,where kβ=Γ1+βcos1−βπ/2/π.System (1) possesses the nonlocal nonlinear term buHu and the local nonlinear term aupux. If a=0, it recovers back to the famous Constantin–Lax–Majda (CLM) system:(5)ωt=ωHω,ωx,0=ω0x,which is one nonlocal nonlinear system proposed by Constantin et al. in [1] as a simplified 1D system for the 3D vorticity model on the incompressible Euler equation. It should be pointed out that the question on the finite-time singularity formation of the CLM model is considered to be closely related to the most outstanding mathematical open problems on the three-dimensional incompressible Euler equation. In (5), the nonlocal nonlinear term ωHω is one kind of one-dimensional approximation of the vortex-stretching term Dωω in the three-dimensional incompressible Euler equation. If b=0 and p=1, system (1) recovers back to the classical Burgers system in the conservation law field, where its general form with the viscosity is given by (see [2, 3])(6)ut+uux=νuxx.It is known that the solutions to two systems, both the CLM system and the Burgers system, have the characteristic of finite-time blow-up singularity formation for the smooth solutions with the smooth initial data. Whena>0 and b>0, system (1) possesses both the CLM type’s nonlocal term uHu and the (local) generalized Burgers type’s convection term upux.We also recall some related problems on the finite-time blow-up singularity regimes about some models with the Hilbert transform and the generalized Burgers equations. Gregorio proposed the following system(7)ωt+vωx−vxω=0,vx=Hω,as another simplified model of the 3D vorticity version of incompressible inviscid Euler flow in [4, 5] and obtained some numeric results which implied that the convection term vωx seems to prevent the appearance of finite-time blow-up singularity. Recently, Lei et al. [6] proved that there exist global smooth solutions to system (7) for a class of large nonnegative initial data. Okamoto et al. [7] suggested the generalized CLM system(8)ωt+avωx−vxω=0,vx=Hω,with a as a parameter. Catro and Córdoba [8] proved the finite-time blow-up singularity for the solution to system (8) when a<0 from a class of smooth initial data. Córdoba et al. [9, 10] studied the system ωt+Hωωx=0 and obtained the finite-time blow-up results for the smooth initial data. Sakajo [11] studied the CLM equation with the generalized viscosity term ωt=ωHω−ν−Δα/2ω, where they gave the explicit solution and obtained the finite-time blow-up result for the smooth initial data. Hou et al. [12] investigated the singularity formation of a nonlinear nonlocal system,(9)ut=2uv,vt=Hω,and proved the finite-time blow-up result for a class of large initial data and the global existence result for another initial data for the Cauchy or periodic problem to system (9). Recently, Choi et al. [13] studied the finite-time blowup of a one-dimensional model from the three-dimensional axisymmetric Euler equations and obtained the finite-time blow-up result for the smooth initial data.Kiselev et al. [14] studied the blow-up and regularity problem for the fractal Burgers equation ∂tu+uux=−−Δβu0≤β≤1. Castro et al. [15] studied the singularity formation of a surface wave model ut+uux=ΛβHu (where 0≤β<1) and obtained the finite-time singularity result when 0≤β<1. Hur [16] discussed the generalized surface wave model ∂tu−2u∂xu+HjΛγu=0 (where j=0 or j=1 and γ∈0,1) and proved the finite-time singularity result. Hunter and Ifrim [17] discussed the lifespan of the smooth solutions of a Burgers–Hilbert equation ut+εuux=Hu over cubically nonlinear time scales on ϵ.Meanwhile, we remark that there are a lot of research studies carried out on nonlocal different equations. Coclite et al. [18] obtained the formation of singularities in finite time in nonlocal Burgers equations with the time-fractional derivative and employed Burgers equations to model a problem arising in the job market. Wang and Zhang [19] established the nonexistence of positive solutions to nonlocal Lane–Emden equations and obtained the “Fujita index” for nonlocal reaction-diffusion equations, which is new in the blow-up theory. In this paper, motivated by the above results on the CLM system and the generalized Burgers system with the nonlocal operator, we study the finite-time blow-up singularity mechanism of system (1). We would like to discuss whether the finite-time blow-up singularity mechanism of the system depends upon the domination between the CLM type’s vortex stretching and the Burgers type’s convection in some sense. We will give two kinds of different finite-time blow-up singularity regimes for two classes of initial data. One mechanism seems to be caused by the domination of the vortex-stretching term of the system, see (17)–(19), which is of CLM’s finite-time blow-up mechanism type, but another seems to be caused by the domination of the convection term of the system, see (42) and (43), which is of Burgers’ finite-time blow-up mechanism type. That is to say that the coupled Burgers–Constantin–Lax–Majda system possesses two kinds of finite-time blow-up singularity regimes of both the Constantin–Lax–Majda system and the Burgers system.The rest of this paper is as follows. The main results of this paper are given in Section2, and Section 3 is devoted to the proofs of the main results of Section 2.


## 2. Main Results

In this section, we give the main results of this paper.Theorem 1.
Letu0∈L1ℝ∩C01+δ0<δ<1 and u0x≥≡0x∈ℝ. Also, let p>0. Assume that there exists α0∈ℝ such that(10)u0α0=0,Hu0α0>0.Then, there is finite timeT:0<T<+∞ such that the solution ux,t to system (1) blows up in finite time.Theorem 2.
Letu0∈L1ℝ∩C01+δ0<δ<1 and u0x≥≡0x∈ℝ. Also, let p>0 and λ1/p+1<λ<1 be given and u0˜>0 be the solution to the algebra equation(11)abC02u0˜p+2u0L1ℝ−b2+2/pCa2/pu0L1ℝ2/p=0,where C0=C0p,λ,πandC=Cp,λ,π which are given by(12)C0p,λ,π=p−p+11−λ1−λp+1π,(13)Cp,λ,π=p⋅22/p−1p+21+2/pp+1πp−p+11−λ1−λ2/p.
Assume that there existsα0∈ℝ such that(14)u0α0>u0˜,Hu0α0>0.Then, there is finite timeT:0<T<+∞ such that the solution ux,t to system (1) blows up in finite time.Remark 1.
For the coupled Burgers–CLM equation, it is obvious that the system isL1-conservative on the L1 norm of the nonnegative solution by using the property of Hilbert transform ∫ℝuxHuxdx=0.Remark 2.
Assumption (10) on initial data in Theorem 1 to guarantee the finite-time blow-up mechanism for system (1) is completely the same as the one for CLM system (5). However, assumption (13) on initial data in Theorem 2 is completely different from assumption (10), which yields to another finite-time blow-up mechanism for system (1). That is to say that there are two kinds of the finite-time blow-up regimes. One blow-up regime, given by Theorem 1, is caused by the nonlocal nonlinear term related to the vortex-stretching term, while another regime given by Theorem 2 comes from Burgers’ type convection term. This reflects that the modes of the finite-time blow-up regimes may depend upon the domination between the CLM type’s vortex-stretching term and the Burgers’ type convection term in some sense.Remark 3.
The assumption conditionHu0α0>0 in (13) can be replaced by Hu0α0≥0, and Theorem 2 still holds.


## 3. The Proof of the Main Results

To finish the proof of the main results, we, firstly, recall some basic properties of the Hilbert transform (see [1]).Lemma 1.
(15)HHf=−f,HfHg+gHf=HfHg−fg,Hfx=Hfx,∫ℝuHuxdx=0.Now, we give the proof of Theorem1 by using Lemma 1.Proof of Theorem1.
For the local existence and uniqueness of the smooth solution in the class ofC0,T,C01+δℝ to system (1), it can be obtained by the iteration technique and by using the existence and uniqueness theory of the linearized hyperbolic equation. This is standard, and we omit it here. Also, it is obvious that the solution ux,t≥0 only if u0x≥0.
Taking the Hilbert operator on (1), we get, with the help of Lemma 1, that(16)Hut+aHupux=12bHu2−u2.
Takingx=α0 in (16) and using assumption (10) on the initial data, we get(17)dHuα0,tdt=12bHu2α0,t−aHupuxα0,t,(18)−aHupuxα0,t=−ap+1π∫ℝup+1α0,t−up+1y,tx−y2dy=ap+1π∫ℝup+1y,tx−y2dy≥0.
Combining (17) and (18), we have(19)dHuα0,tdt≥b2Huα0,t2.
According to the theory on the nonlinear ODE and by usingHu0α0>0 in assumption (10) on the initial data, we can get from (19) that Huα0,t blows up in finite time.
This completes the proof of Theorem1.
Next, to prove Theorem2, we give a local nonnegative lower bound estimate on the nonlocal term upxΛux−1/p+1Λu1/p+1x, which plays the key role on proving the main result of Theorem 2 of this paper. In fact, it is an extension version of Lemma 2 in the paper [15], where the case that p=1 is obtained. For completeness, we give its proof here.Lemma 2.
Letu∈L1ℝ with u≥0,≡0, and let p>0 and λ1/p+1<λ<1 be given. Then, for any fixed point x∈ℝ satisfying ux>0, we have(20)upxΛux−1p+1Λup+1x≥C0up+2xuL1R,where C0 is given by (12) in Theorem 2.Proof.
LetΩ be the set(21)Ω=y∈Ry−x<Δ,where Δ>0 will be given in the following. And let Ω1andΩ2 be the subsets of Ω such that(22)Ω1=y∈Ωux−uy≥λux=y∈Ωuy≤1−λux,Ω2=y∈Ωux−uy<λux=y∈Ωuy>1−λux.
In the following,Δ>0 will be chosen to be such that Ω1>0 and Ω2>0. Obviously, 0<Ω1=Ω−Ω2≤Ω=2Δ.
On the one hand, using the fact(23)Λux=1π∫ℝux−uyx−y2dy,we have(24)upxΛux−1p+1Λup+1x=1π∫ℝupxux−uyx−y2dy−1p+11π∫ℝup+1x−up+1yx−y2dy=1p+11π∫ℝpup+1x−p+1upxuy+up+1yx−y2dy.
By Young’s inequalityab≤ap+bq/q⋅pq/p with 1/p+1/q=1, we get(25)p+1upxuy≤up+1y+pup+1x,which yields to(26)pup+1x−p+1upxuy+up+1yx−y2≥0.
Combining (24) with (26), we have(27)upxΛux−1p+1Λup+1x≥1p+11π∫Ω1pup+1x−p+1upx1−λuxΔ2dy=p−p+11−λp+1πΔ2up+1xΩ1.
Here, we needp−p+11−λ>0, i.e., 1/p+1<λ<1.
On the other hand, we have(28)uL1ℝ=∫ℝuydy=∫ℝuydy≥∫Ω2uydy≥ux1−λΩ2,which yields to(29)Ω2≤uL1Rux1−λ.
SinceΩ1=Ω−Ω2 and Ω=2Δ, we have, by combining (27) and (29), that(30)upxΛux−1p+1Λup+1x≥p−p+11−λp+1πΔ2up+1xΩ1=p−p+11−λp+1πΔ2up+1xΩ−Ω2≥p−p+11−λp+1πΔ2up+1xΩ−uxL1Rux1−λ=p−p+11−λp+1πΔ2up+1x2Δ−uxL1Rux1−λ.
Now, we chooseΔ=uL1ℝ/1−λux for fixed x∈ℝ satisfying ux>0. Thus, we get from (30) that(31)upxΛux−1p+1Λup+1x≥p−p+11−λp+1πΔ2up+1x2Δ−uxL1Rux1−λ=p−p+11−λ1−λp+1πuL1ℝup+2x,which gives (20). This completes the proof of Lemma 2.Remark 4.
Under the assumption of Lemma2, inequality (20) holds still for the point x0∈ℝ such that ux0=0 because upx0Λux0−1/p+1Λup+1x0=−1/p+1Λup+1x0=1/p+1∫ℝup+1y/x0−y2dy≥0.Proof of Theorem2.
First, it is easy to prove that there is a unique smooth solutionux,t∈C0,T,C01+δℝ of system (1) for some time 0<T≤∞.
Now, we want to prove the solution will blow up in finite time for some classes of large initial data in the sense of our assumption. We end this by a contradiction argument. Assume that there exists the global solution for all timet∈0,∞.
Firstly, for the solution of system (1), there exists one L1 conservation law, see Remark 1:(32)utL1ℝ=u0L1ℝ.
Then, we define the particle trajectoriesxα,t by the equation(33)dxα,tdt=aupxα,t,t,xα,tt=0=α.
It follows from systems (1) and (33) that(34)duxα,t,tdt=buHuxα,t,t,which yields to(35)uxα,t,t=u0αeb∫0tHuxα,s,sds.
Thus, it follows from (35) that uxα,t,t≥0 for any t∈0,∞ if u0α≥0 for some fixed α∈ℝ, and uxα,t,t>0 for any finite t∈0,∞ if u0α>0 for some fixed α∈ℝ.
For any fixedα∈ℝ satisfying u0α>0 (especially, this is true for α=α0 in Theorem 2), it follows from (34) that(36)dloguxα,t,tdt=bHuxα,t,t,and, hence, by using (16), (32), and (20) in Lemma 2, we have(37)d2loguxα,tdt2=b∂tHuxα,t,t+∂xHuxα,t,taupxα,t,t=bbHuHu−Haupuxxα,t,t+abupΛuxα,t,t=b2HuHu−abHup+1p+1x+abupΛuxα,t,t=b2HuHuxα,t,t+abupΛu−1p+1Λup+1xα,t,t≥b2HuHuxα,t,t+abC0up+2xα,t,tu0L1R=b22Hu2−u2xα,t,t+abC0up+2xα,t,tu0L1R.
Here, we have usedHupuxx=1/p+1Hup+1x=1/p+1Λup+1x, Λu=Hux, and HuHu=1/2Hu2−u2.
By Young’s inequality, we have(38)b22u2≤abC02up+2u0L1ℝ+b2+2/pa2/pCp,λ,πu0L1ℝ2/p,where Cp,λ,π>0 is given by (13) in Theorem 2.
Putting (38) into (37), we have(39)d2loguxα,t,tdt2≥abC02up+2xα,t,tu0L1ℝ−b2+2/pa2/pCp,λ,πu0L1ℝ2/p.
Now, setting(40)dloguxα,t,tdt=Vxα,t,t,then, by (39) and (40), we have(41)duxα,t,tdt=uxα,t,tVxα,t,t,dVxα,t,tdt≥abC02up+2xα,t,tu0L1ℝ−b2+2/pa2/pCp,λ,πu0L1ℝ2/p,uxα,t,t,Vxα,t,tt=0=u0α,V0α.
By the existence-uniqueness and extension theory of ordinary differential equations, we know that the solutionu,Vxα,t,t of system (40) is nondecreasing with respect to the time t≥0 and for fixed α and satisfies u,Vxα,t,t≥u0˜,0 and u,Vxα,t,t⟶+∞,+∞ as t⟶+∞ if u0,V0α>u0˜,0 for some fixed α∈ℝ, where u=u0˜ is the solution to algebra equation (11). In fact, here, V0α=bHu0α by using (36).
Takingα=α0 especially, where α0 is given in Theorem 2, by the assumption on Theorem 2, we have that u0,V0α0>u0˜,0. Hence, it is obvious that there exists time t0>0 such that uxα0,t,t is nondecreasing for t≥t0>0, and u,Vxα0,t0,t0 satisfies abC0/4up+2xα0,t0,t0/u0L1ℝ≥b2+2/p/a2/pCp,λ,πu0L1ℝ2/p, Vxα0,t0,t0=bHuxα0,t0,t0>0, and loguxα0,t0,t0>0.
Also, takingα=α0 in (39), we have, for all t≥t0>0, that(42)d2loguxα0,t,tdt2≥ab2C0u0L1ℝup+2xα0,t,t−b2+2/pa2/pCp,λ,πu0L1ℝ2/p=ab4C0u0L1ℝup+2xα0,t,t+ab4C0u0L1ℝup+2xα0,t,t−b2+2/pa2/pCp,λ,πu0L1ℝ2/p≥ab4C0u0L1ℝup+2xα0,t,t+ab4C0u0L1ℝup+2xα0,t0,t0−b2+2/pa2/pCp,λ,πu0L1ℝ2/p≥ab4C0u0L1ℝup+2xα0,t,t.
Integrating (42) with respect to t from t0 to t, we have(43)dloguxα0,t,tdt≥Vxα0,t0,t0+abC04u0L1ℝ∫t0tup+2xα0,τ,τdτ,which, by integrating (43) with respect to t from t0 to t, gives(44)loguxα0,t,t≥loguxα0,t0,t0+Vxα0,t0,t0t−t0+abC04u0L1ℝ∫t0t∫t0sup+2xα0,τ,τdτds.
Sinceuxα,t,t≥0, we have(45)loguxα,t,t≤log+uxα,t,t≤uxα,t,t.
Combining (44) with (45), we have(46)uxα0,t,t≥loguxα0,t,t≥Vxα0,t0,t0t−t0+abC04u0L1ℝ∫t0t∫t0sup+2xα0,τ,τdτds.
Let(47)Ft=Vxα0,t0,t0t−t0+abC04u0L1ℝ∫t0t∫t0sup+2xα0,τ,τdτds.
Then, by (46), we have(48)F″t≥abC04u0L1ℝFp+2t,t≥t0,Ft0=0,F′tt=t0=Vxα0,t0,t0.
Multiplying the first equation of (47) by F′t≥0 and then integrating the resulting equation, we have(49)F′t≥abC02p+3u0L1ℝFp+3t+V2xα0,t0,t0=Vxα0,t0,t0abC02p+3u0L1ℝV2xα0,t0,t0Fp+3t+1=Vxα0,t0,t0abC02p+3u0L1ℝV2xα0,t0,t01/p+3Ftp+3+1.
LetG=abC0/2p+3u0L1ℝV2xα0,t0,t01/p+3Ft; we have(50)G′t≥abC02p+3u0L1ℝV2xα0,t0,t01/p+3Vxα0,t0,t0Gp+3t+1.
That is to say(51)G′tGp+3t+1≥abC02p+3u0L1ℝV2xα0,t0,t01/p+3,and, hence,(52)∫t0Gtdξξp+3+1′≥abC02p+3u0L1ℝV2xα0,t0,t01/p+3,which gives(53)∫t0Gtdξξp+3+1≥abC02p+3u0L1ℝV2xα0,t0,t01/p+3t−t0.
TakingT to be(54)+∞>∫t0∞dξξp+3+1=abC02p+3u0L1ℝV2xα0,t0,t01/p+3T−t0,when t⟶T<+∞ in (53), we have(55)+∞>∫t0∞dξξp+3+1≥abC02p+3u0L1ℝV2xα0,t0,t01/p+3T−t0,which implies that the maximal existence time to the solution of (1) cannot be extended more than T<∞. This is a contradiction with the assumption that the solution is global in time. The proof of Theorem 2 is complete.



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