Revisit of the Faddeev Model in Dimension Two*

Shijie DONGZhen LEI

1SUSTech International Center for Mathematics,Department of Mathematics,Southern University of Science and Technology,Shenzhen 518055,China.E-mail:shijiedong1991@hotmail.com

2Corresponding author.School of Mathematical Sciences,Shanghai Center for Mathematical Sciences,Fudan University,Shanghai 200433,China.E-mail:zlei@fudan.edu.cn

Abstract The Faddeev model is a fundamental model in relativistic quantum field theory used to model elementary particles.The Faddeev model can be regarded as a system of non-linear wave equations with both quasi-linear and semi-linear non-linearities,which is particularly challenging in two space dimensions.A key feature of the system is that there exist undifferentiated wave components in the non-linearities,which somehow causes extra difficulties.Nevertheless,the Cauchy problem in two space dimenions was tackled by Lei-Lin-Zhou(2011)with small,regular,and compactly supported initial data,using Klainerman’s vector field method enhanced by a novel angular-radial anisotropic technique.In the present paper,the authors revisit the Faddeev model and remove the compactness assumptions on the initial data by Lei-Lin-Zhou(2011).The proof relies on an improved L2 norm estimate of the wave components in Theorem 3.1 and a decomposition technique for non-linearities of divergence form.

Keywords Faddeev model in R1+2,Global existence,Null condition

1 Introduction

The Faddeev model is an important model in quantum field theory with extensive mathematical studies.The investigations on the static Faddeev model or some related problems can be found in the series of works[18–21]by Lin-Yang.On the other hand,the Cauchy problem of the Faddeev model was first tackled by Lei-Lin-Zhou[15]in two space dimensions.Later on,the sharp global regularity for the two dimensional Faddeev model was shown by Geba-Nakanishi-Zhang[10]under some extra assumptions.Recently,the large data global existence for the two(and three)dimensional Faddeev model was studied by Geba-Grillakis[9]and by Zha-Liu-Zhou[27].

We note that the Faddeev model can be regarded as a generalisation of the harmonic maps R1+n→S2.We recall the remarkable pioneering work[11]by Gu on harmonic maps in one space dimension,which is relevant to our study.He succeeded in treating the harmonic maps R1+1→M,where M is a complete Riemannian manifold of dimension n,including the two dimensional sphere S2as a special case,and proved that the solution to the Cauchy problem exists globally.

We recall the seminal works[13–14]by Klainerman,[3]by Christodoulou,[23]by Lindblad-Rodnianski on three dimensional non-linear wave equations,and[1]by Alinhac on two dimensional case.The Cauchy problem of the Faddeev model in three space dimensions and higher can be solved using these classical theories.This problem is particularly tricky in two space dimensions.Nevertheless,Lei-Lin-Zhou proved the global well-posedness of the Cauchy problem of the Faddeev model in two space dimensions with compactly supported initial data.The prime goal of the present paper is to remove the compactness assumptions on the initial data.We would also like to draw one’s attention to some recent progress on two dimensional wave equations of[2,5,12,16].

Main Theorem We want to show the existence of global solutions to system(1.1)and to derive the pointwise asymptotic behavior of the solutions.Our main result is stated as follows.

Theorem 1.1 Consider the Faddeev model(1.1),and let N ≥5 be an integer.Then there exists a small ε0＞0,such that for all initial data satisfying the smallness condition

In general,the smallness condition onis not assumed when treating wave equations,but we will need it in the proof of Proposition 2.6.We note that similar assumptions on the initial data also exist in the remarkable result[12],where the authors removed the compactness assumptions on the celebrated result[1]by Alinhac.In[12],the authors applied a novel weighted L∞–L∞estimate for the wave equations to achieve the goals.As a comparison,we use the energy method to prove Theorem 1.1.

One key idea is to prove refined estimates on the wave solution itself of[15,Theorem 3.1],which is demonstrated in Proposition 2.6.Since the Faddeev model(1.1)contains also quasilinear non-linearities,the result in Proposition 2.6 cannot be directly applied in the highest-order case,which is the most difficult part of the analysis.Fortunately,utilising a decomposition can help us conquer this difficulty;see the discussion in the begining of Section 4.2.Importantly,this way can also be used to remove the compactness assumptions in[1].

The rest of the paper is organised as follows.In Section 2,we present some preliminaries on wave equations.Then in Section 3,we illustrate the derivation of the equations of the Faddeev model(1.1).Finally,we demonstrate the proof of Theorem 1.1 in Section 4.

2 Preliminary

3 The Faddeev Model

4 Proof of Theorem 1.1

The smallness of C1ε and the smallness of δ lead us to

In a similar manner,we can show the first estimate appearing in(4.7).Thus the proof is complete.

4.1 Refined lower-order energy

4.2 Refined higher-order energy

This part is devoted to show the refined estimates of the highest-order case.

Before we estimate ‖ΓIni‖ for |I|=N,we first introduce the following decomposition.We recall that(the same argument applies to n2)

We reorganise the terms to get

We next introduce the new variables

(m0,m1,m2,m3),

which are solutions to the equations

and

in which

We note that the relation between n1and(mμ,m3)reads as follows

Thus,to estimate the unknown ΓIn1,it suffices to estimate the new variables(mμ,m3).We comment that this strategy can also be applied to remove the compactness assumptions on the model problem studied in[1].

Proposition 4.4Under the assumptions in(4.3),we have

ProofWe only provide the proof for ‖ΓIn1‖ with |I|=N.

Step 1Bounds for ‖?mμ‖.Recall the equations in(4.18),and the energy estimates for waves imply

Step 2Bounds for ‖m3‖.

We rely on Proposition 2.6 to achieve this,so we only need to bound the right-hand side of equation(4.19),i.e.,

We first estimate the cubic term ΓI(ni?μnj?μnk)in ΓIf1,which is the worst term in ΓIf1.We find

In the same way,we obtain

Thus we proceed to get

Very similarly,we can show In this step,we are only left with estimating

We observe that

By recalling the results in Proposition 4.2,we easily get

We note that

The way we show(4.22)leads us to

Thus we obtain

The combination of(4.23)–(4.24)and Proposition 2.6 yields

Step 3Bounds for ‖ΓIni‖.

By the estimates in the first two steps,we arrive at

The same also holds for n2,thus the proof is complete.

Recall the expressions of g1,g2in(4.2),and we rewrite them as

in which

Thus the model equations(1.1)can be written as

Acting ΓIwith |I|=N to the equations,we further get

in which

We note that

and

which guarantee the hyperbolicity of the quasi-linear system.

We first show the estimates for the source terms in(4.30).

Lemma 4.2For |I|=N we have

ProofWe will only provide the detailed estimates for the term -ΓIf1+Q1as the term-ΓIf2+Q2can be bounded in the same way.

We recall that the estimates of(4.22)and(4.23)can be applied to show

so we will only need to consider

We observe that

For the third term in the right-hand side of the above equation,the commutator estimates yield

in which C’s are constants.Gathering the above two identities and recalling(4.31)give us

We find all of the terms to be estimated are null terms,so the analysis in Lemma 4.1 can be used to deduce

The proof is complete.

Proposition 4.5Under the assumptions in(4.3),we have

ProofAccording to the ghost weight energy estimates in Proposition 2.5,we only need to show

in which(with w1=ΓIn1,w2=ΓIn2)

We divide these terms into three classes

· Class I:R11,R12.

· Class II:R13,R14,R21,R22,R23,R24,R31,R33,R41,R43.

· Class III:R32,R34,R42,R44.

In each class,we will only illustrate the details of the estimates for one representative term,and others can be estimates analogously.

By the estimates in Lemma 4.2,we get

and similarly

Next,we treat the term R13.We first ignore the denominator in,and find

in which we note each term in the right-hand side is null.By Proposition 2.3,we have

and we proceed to get

In the same manner,we can treat other terms in R13and show

Thus,similarly we get the same bound for other terms in this class.

Now we estimate the term R32.By the smallness of n1,n2,we have

in which we used the relationSince the last two terms can be bounded in the same way as we did for the term R13,so we will only estimate the first term in the right-hand side of the above inequality.We observe that

which lead to

By the estimates for null forms in Proposition 2.3,we have

By the smallness of δ,we get

Thus,we get the same bound for other terms in this class.

The proof is done.

We are now ready to provide the proof for our main result.

Proof of Theorem 1.1By the refined estimates in Propositions 4.3–4.5,we can choose C1?1 very large,and ε ?1 sufficiently small,such that the estimates in(4.5)hold.This means the solution to the Faddeev model(1.1)exists globally.

The pointwise decay in(1.3)can be seen from(4.15)–(4.16)and the Klainerman-Sobolev inequality in Proposition 2.4.