# The Cauchy Problem for Space-Time Monopole Equations in Temporal and Spatial Gauge

Authors: Hyungjin Huh; Jihyun Yim
Journal: Advances in Mathematical Physics
 (2017)

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## Abstract

We prove global existence of solution to space-time monopole equations in one space dimension under the spatial gauge conditionA1=0 and the temporal gauge condition A0=0.


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## Body

## 1. Introduction

In the current article, we study the followingspace-time monopole equations in one space dimension:(1)∂tϕ+A0,ϕ+∂xN+A1,N=0,∂tN+A0,N+∂xϕ+A1,ϕ=0,∂tA1-∂xA0+A0,A1=N,ϕ.Here ϕ,N,A=(A0,A1):R1+1→g, where g is a Lie algebra of a matrix Lie group such as SO(n), SU(n) with Lie bracket [·,·]. We denote space-time derivatives by ∂0=∂t, ∂1=∂x.The space-time monopole equations inR2+1 can be written as follows:(2)∂0ϕ+∂1A2-∂2A1=ϕ,A0+A2,A1,∂0A2+∂1ϕ-∂2A0=A2,A0+ϕ,A1,∂0A1-∂1A0-∂2ϕ=A2,ϕ+A1,A0.Equation (1) is obtained by the dimensional reduction of system (2). More precisely, we consider (2) independent of the y coordinate and renaming A2 as N to get (1). The space-time monopole system (2) is a nonabelian gauge field theory and can be derived by dimensional reduction from anti-self-dual Yang-Mills equations; see [1], for instance. The system is an example of a completely integrable system and has an equivalent formulation as a Lax pair. It was first introduced by Ward in [2] as a hyperbolic analog of Bogomol’nyi equations and discussed from the point of view of twistors.System (1) is invariant under the rescaling(3)ϕλt,x=λϕλt,λx,Aμλt,x=λAμλt,λx,Nλt,x=λNλt,λx, from which we deduce a scale invariant Lebesgue space L1(R) and Sobolev space H˙-1/2(R). Another important property of system (1) is an invariance under the gauge transformation (4)ϕ⟶gϕg-1,A⟶gAg-1+gdg-1,N⟶gNg-1, where g:R1+1→G is smooth and compactly supported map into Lie group G.A broad survey on the space-time monopole equations is given in [1]. In particular, using the inverse scattering transform, they have shown global existence and uniqueness up to a gauge transformation for small initial data in W2,1(R2). The survey [1] also contained a number of other interesting results related to the space-time monopole equations. Czubak showed in [3] that the space-time monopole system in Coulomb gauge is locally well-posed for small initial data in Hs(R2) with s>1/4. The Cauchy problem of the space-time monopole equations in R2+1 under the Lorenz gauge condition was discussed in [4, 5]. In particular almost critical local well-posedness has been proved in [5]. The existence of global solution in R1+1, under Lorenz gauge condition ∂0A0-∂1A1=0, was proved in [6] for Lebesgue space Lp(R) with 1≤p≤∞. In the current article, we will consider two gauge conditions A1=0 and A0=0.First we impose the spatial gauge conditionA1=0. Then system (1) becomes(5)∂tϕ+∂xN=ϕ,A0,∂tN+∂xϕ=N,A0,-∂xA0=N,ϕ.With the notations of ϕ+N=u and ϕ-N=v, system (5) can be rewritten as(6)∂tu+∂xu=u,A0,(7)∂tv-∂xv=v,A0,(8)∂xA0=12v,u.System (6)–(8) has the conservation of charge(9)∫Rut,x2+vt,x2dx=∫Ru0,x2+v0,x2dx.From (8), we have the following representation, with a boundary condition A0(-∞)=0,(10)A0t,x=12∫-∞xv,ut,ydy.We have showed that the initial value problem of (6)–(8) reduces to the study of the following system(11)∂tu+∂xu=u,A0,∂tv-∂xv=v,A0,u0,x=u0x,v0,x=v0x,where A0 is defined by (10).Theorem 1.
For initial datau0,v0∈Hb(R)  (b=0,1), the initial value problem for (11) has a unique, global in time solution which belongs to (12)u,v∈C0,∞;HbR.Moreover we get the following upper bound of H1 norm (13)utH12+vtH12≤cexp⁡ct.We will derive some asymptotic behaviors of solutions to (11) with u0,v0∈Cc(R). We refer to Remark 7 in Section 2.Next we impose the temporal gauge conditionA0=0. Then we have(14)∂tϕ+∂xN=N,A1,∂tN+∂xϕ=ϕ,A1,∂tA1=N,ϕ.With the notations ϕ+N=u and ϕ-N=v, system (14) can be rewritten as(15)∂tu+∂xu=u,A1,(16)∂tv-∂xv=A1,v,(17)∂tA1=12u,v.Theorem 2.
For initial datau0,v0,a0∈Hb(R)(b=0,1), the initial value problem for (15)–(17) has a unique, global in time solution which belongs to (18)u,v,A1∈C0,∞;HbR.Moreover we get the following upper bound of H1 norm: (19)utH12+vtH12+A1tH12≤cexp⁡ct.We obtain the following result for the critical caseL1.Theorem 3.
For anyu0,v0,a0∈L1, the initial value problem for (15)–(17) has a unique, global in time solution which belongs to (20)u,v,A1∈C0,∞;L1.Now we summarize the algebraic definition and properties used for the proof of Theorems1–3. Let g be a Lie algebra of a matrix Lie group such as SO(n), SU(n). We define the following norm for A∈g: (21)A2=A,A=trA∗A.Let 〈·,·〉 and [·,·] denote inner product induced by the trace norm and Lie bracket, respectively. Then we have, for matrices A,B,C∈g,(22)A,B,A=0,A,B,C=A,B,C.Theorem1 is proved in Section 2. We show Theorems 2 and 3 in Sections 3 and 4, respectively. We conclude this section by giving a few notations. We use the standard Sobolev space Ws,p(R) with the norm fWs,p(R)=ΛsfLp(R), where (Λsf^)(ξ)=1+ξ2s/2f^(ξ) and f^ denotes the Fourier transform of f. The space Hs(R) denotes Ws,2(R). We define the space-time norm fLTpLq=∫0Tf(t,·)Lq(R)pdt1/p. We use c to denote various constants. When we are interested in local solutions, we may assume that T≤1. Thus we shall replace smooth function of T,C(T) by C. We use A≲B to denote an estimate of the form A≤cB.


## 2. Proof of Theorem1

To show the existence of local solution to (6)–(8), we introduce a linear estimate (see [7]).Lemma 4.
Letu± be the solution to the inhomogeneous equation(23)∂tu±±∂xu±=F±t,x,u±0,x=u±0x,where u±0∈Lp and F±∈Lloc1[0,T),Lp(R) for 1≤p≤∞. Then for T>0 we have (24)u±t,·Lp≤u±0Lp+∫0TF±s,·Lpds.We will prove the existence of local solution to (11) with u0,v0∈L2(R). The case of u,v∈H1(R) can be treated similarly. Proof follows by standard arguments from a priori estimates of the following Propositions.Proposition 5.
Letu,v be the solution of (11) in a strip [0,T]×R with (25)u,v∈C0,T;L2R.Define (26)JT=sup0≤t≤Tut,·L2R+vt,·L2R.Then there exist constants c>0 and T∗>0, depending only on J(0), such that if T<T∗ then J(T)≤cJ(0).Proposition 6.
Letu,v and u′,v′ be two solutions of (11) verifying the hypothesis of Proposition 5 in a strip R×[0,T] and let J(T) as in a Proposition 5 and J′(T) be the corresponding quantity for the primed solution. Define (27)∆T=sup0≤t≤Tut,·-u′t,·L2R+vt,·-v′t,·L2R.Then there exist constants c>0 and T∗>0, depending only on J(0) and J′(0), such that if T<T∗ then ∆(T)≤c∆(0).Proposition6 follows by the similar argument to Proposition 5. We will only prove Proposition 5. We define (28)XT=∫0Tu,A0s,·L2R+v,A0s,·L2Rds.It is easily shown that J(T)≲J(0)+X(T) by applying Lemma 4. We will derive the inequality X(T)≲T(J(0)+X(T))3. Then the bootstrap argument completes the proof of Proposition 5. From representation (10), we have the following L∞ bound:(29)A0t,·L∞R≤12ut,·L2R2+vt,·L2R2.Applying (29), the integrals in X(T) can be treated as follows: (30)∫0Tu,A0s,·L2Rds≲TJ0+XT3,∫0Tv,A0s,·L2Rds≲TJ0+XT3.Then we get the relation X(T)≲T(J(0)+X(T))3 which completes the proof of Proposition 5 by the bootstrap argument. Global existence for L2 initial data can be proved taking charge conservation (9) into account.We will prove the global existence of solutions to the initial problem (11) for initial data u0,v0∈H1(R). Differentiating both sides of (11), we can obtain (31)∂t+∂xux2=2u,∂xA0,ux≤4uux∂xA0,∂t-∂xvx2=2v,∂xA0,vx≤4vvx∂xA0.Considering (10), we obtain (32)∂tux2+vx2+∂xux2-vx2≤4∂xA0uux+vvx≤4uvuux+vvx≤2u2+v2uux+vvx.Integrating on R and using Gagliardo-Nirenberg inequality (33)uL6R3+vL6R3≤cuL2R2+vL2R2uxL2R+vxL2R,we have (34)ddtuxL2R2+vxL2R2≤cuL2R2+vL2R2uxL2R2+vxL2R2.Then we obtain (35)uxL2R2+vxL2R2≤exp⁡cu0L2R2+v0L2R2tux0,·L2R2+vx0,·L2R2.Therefore we have an upper bound of H1 norm to extend a local solution globally.Remark 7.
Let us consider the initial value problem of (6)–(8) for initial data u0,v0 with compact supp(u0), supp(v0)∈[-R,R]. Considering (22), we obtain from (11) (36)∂t+∂xu2=2u,A0,u=0,∂t-∂xv2=2v,A0,v=0,which implies (37)ut,x=u0x-t,vt,x=v0x+t.Then we have(38)u,vt,x=0fort≥R,x∈R,which implies A0(t,x)=0 for x∈R, t≥R. Therefore we obtain the following system:(39)∂tu+∂xu=0,∂tv-∂xv=0,with initial data u(R,x)=a(x),v(R,x)=b(x). Hence we get the following solution to (39) for t≥R: (40)ut,x=ax-t,vt,x=bx+t.We can check a decay of the local L2 norm in spite of conservation (9) (41)∫lru2+v2t,xdx⟶0ast⟶∞,where [l,r] is any finite interval.


## 3. Proof of Theorem2

We introduce the following bilinear estimates for the proof of existence of local solution to (15)–(17).Lemma 8.
Letu±, A1 be the solution to inhomogeneous equations(42)∂tu±±∂xu±=F±t,x,u±0,x=u±0x,∂tA1=Gt,x,A10,x=a1x,where u±0,a1∈Lp and F±(t,·)∈L1[0,T);Lp(R), G(t,·)∈L1[0,T);Lp(R) for 1≤p≤∞. Then the solutions u±, A1 to (42) satisfy the following estimates:(43)u+u-LTpLp≤121/pu+0Lp+∫0TF+sLpdsu-0Lp+∫0TF-sLpds,(44)u±A1LTpLp≤u±0Lp+∫0TF±sLpdsa1Lp+∫0TGsLpds,for any 0<T<∞.Proof.
We refer to [7] for the proof of (43). We will show estimate (44). We have solution representations of u±, A1(45)u±t,x=u±0x∓t+∫0tF±s,x∓t±sds,A1t,x=a1x+∫0tGs,xds.Applying change of variables and Fubini’s theorem, we can derive the following four estimates: (46)u±0x∓ta1xLTpLpp≤∫R2u±0x∓ta1xpdxdt=∫R2u±0ya1zpdydz=u±0Lppa1Lpp,u±0x∓t∫0tGs,xdsLTpLp≤∫0Tu±0x∓tGs,xLTpLpds≤u±0Lp∫0TGsLpds,a1x∫0tF±s,x±s-tdsLTpLp≤∫0Ta1F±s,x±s-tLTpLpds≤a1Lp∫0TF±sLpds,∫0tF±s,x±s-tds∫0tGs∗,xds∗LTpLp≤∫0T∫0TF±s,x±s-tGs∗,xLTpLpdsds∗≤∫0TF±sLpds∫0TGs∗Lpds∗.Combining the above estimates, we obtain the desired result (44).We will prove the existence of local solution to (15)–(17) with u0, v0,a1∈L2(R). Let us consider the following iteration system corresponding to (15)–(17):(47)∂tun+1+∂xun+1=un,A1n,∂tvn+1-∂xvn+1=A1n,vn,∂tA1n+1=12un,vn,with the initial data u(n+1)(0,x)=u0(x), v(n+1)(0,x)=v0(x), A1(n+1)(0,x)=a1(x), and u(0)=v(0)=A1(0)=0. We will denote R=2u0L2+v0L2+a1L2 and show the following boundedness of iteration:(48)unLT∞L2≤R,vnLT∞L2≤R,A1nLT∞L2≤R,(49)unA1nLT2L2≤R2,A1nvnLT2L2≤R2,unvnLT2L2≤R2,for sufficiently small T>0. It is obvious for n=0.Applying Lemma4, it follows from Hölder inequality that (50)un+1LT∞L2≤u0L2+∫0Tun,A1nL2ds≤u0L2+2TunA1nLT2L2≤12R+2TR2≤R,for sufficiently small T>0. The estimate for v(n+1)LT∞L2≤R is similar. We also have from the third equation of (47) (51)A1n+1LT∞L2≤a1L2+12∫0Tun,vnL2ds≤a1L2+TunvnLT2L2≤12R+TR2≤R,for sufficiently small T>0. Then we obtain (48) for n+1 iteration.We apply Lemma8 to obtain (49) (52)un+1A1n+1LT2L2≤u0L2+2TunA1nLT2L2dsa1L2+TunvnLT2L2ds≤R2,for sufficiently small T>0. Similarly one may obtain the estimate A1(n+1)v(n+1)LT2L2 for small T>0. Applying Lemma 8, we also have for sufficiently small T>0(53)un+1vn+1LT2L2≤12u0L2+2TunA1nLT2L2dsv0L2+2TA1nvnLT2L2ds≤R2.For the difference estimate we denote(54)dnT=un+1-unLT∞L2+vn+1-vnLT∞L2+A1n+1-A1nLT∞L2+un+1A1n+1-unA1nLT2L2+A1n+1vn+1-A1nvnLT2L2+un+1vn+1-unvnLT2L2. We will show that for sufficiently small T>0(55)dnT≤12dn-1T,which implies the existence of a local solution to (15)–(17).Let us show how to estimateu(n+1)-u(n)LT∞L2,A1(n+1)-A1(n)LT∞L2,u(n+1)A1(n+1)-u(n)A1(n)LT2L2 and u(n+1)v(n+1)-u(n)v(n)LT2L2. Other terms can be bounded in a similar way. Applying Lemma 4, we can derive from the first equation (48) that(56)un+1-unLT∞L2≤2TunA1n-un-1A1n-1LT2L2.For u(n+1)A1(n+1)-u(n)A1(n)LT2L2, we estimate from Lemma 8(57)un+1A1n+1-unA1nLT2L2≤un+1A1n+1-A1nLT2L2+un+1-unA1nLT2L2≤Tu0L2+2TunA1nLT2L2unvn-un-1vn-1LT2L2+Ta1L2+TunvnLT2L2unA1n-un-1A1n-1LT2L2≤TR2+2TR2dn-1T.We can derive (58)A1n+1-A1nLT∞L2≤Tunvn-un-1vn-1LT2L2.Applying Lemma 8, we have (59)un+1vn+1-unvnLT2L2≤un+1vn+1-vnLT2L2+un+1-unvnLT2L2≤T2u0L2+2TunA1nL2A1nvn-A1n-1vn-1LT2L2+T2v0L2+2TvnA1nL2unA1n-un-1A1n-1LT2L2≤T2R2+2TR2dn-1T.We can take care of other terms of dn(T) in the same manner as (56) and (57) to obtain (60)dnT≲T1+R2+2TR2dn-1T. We choose T>0 sufficiently small to obtain (55). Then we can prove the local existence, uniqueness, and Lipschitz continuity of solution by applying standard contraction mapping argument. Note that the existence time T depends only on the size of norms of initial data.Now we focus on global existence forH1(R) initial data. Differentiating both sides of (15)–(17), we can obtain(61)∂t+∂xux2=2u,∂xA1,ux≤4uux∂xA1,∂t-∂xvx2=2∂xA1,v,vx≤4vvx∂xA1,(62)∂t∂xA12=12∂xu,v,∂xA1+∂xA1,∂xu,v≤2uxv∂xA1+uvx∂xA1.Integrating (61) on R, we obtain(63)ddtuxL2R2+vxL2R2≤2uL∞R+vL∞R∂xA1L2R2+uxL2R2+vxL2R2.Integrating (62) on R, we obtain(64)ddt∂xA1L2R2≤uL∞R+vL∞RuxL2R2+vxL2R2+∂xA1L2R2.Combining (63) and (64), we arrive at (65)ddtuxL2R2+vxL2R2+∂xA1L2R2≤3uL∞R+vL∞RuxL2R2+vxL2R2+∂xA1L2R2.We estimate L∞ bound from (15), (16) (66)utL∞≤u0L∞≲u0H1,vtL∞≤v0L∞≲v0H1.Then Gronwall’s inequality leads us to (67)uxL2R2+vxL2R2+∂xA1L2R2≤cect.


## 4. Proof of Theorem3

We consider the critical spaceL1. We will show the local existence of solution for small data and construct the local solution for the initial data of arbitrary size applying the finite speed of propagation of solution. Applying the nonconcentration property in the Lebesgue space which was introduced originally in [8], we will extend the local solution globally.First we prove the existence of local solution for a small data. We will show the following boundedness by induction with respect ton.(68)unLT∞L1≤R,vnLT∞L1≤R,A1nLT∞L1≤RunA1nLT1L1≤R2,A1nvnLT1L1≤R2,unvnLT1L1≤R2,for sufficiently small R>0. It is obvious for n=0.Applying Lemma4, we have (69)un+1LT∞L1≤u0L1+2unA1nLT1L1≤R2+2R2≤R,for sufficiently small R>0. The estimate for v(n+1)LT∞L1 is similar. We also have (70)A1n+1LT∞L1≤a1L1+unvnLT1L1≤R2+R2≤R,for sufficiently small R>0.Applying Lemma8, we have (71)un+1A1n+1LT1L1≤u0L1+2unA1nL1dsa1L1+unvnL1ds≤R2,for sufficiently small R>0. The estimate for A1(n+1)v(n+1)LT1L1 is similar. Applying Lemma 8, we also have, for sufficiently small R>0, (72)un+1vn+1LT1L1≤12u0L1+2unA1nL1dsv0L1+A1nvnL1ds≤R2.For the difference estimate we denote(73)dnT=un+1-unLT∞L1+vn+1-vnLT∞L1+A1n+1-A1nLT∞L1+4un+1A1n+1-unA1nLT1L1+4A1n+1vn+1-A1nvnLT1L1+2un+1vn+1-unvnLT1L1. We will show that for sufficiently small R>0(74)dnT≤12dn-1T,which implies the existence of a local solution to (15)–(17).Let us show how to estimateu(n+1)-u(n)LT∞L1, A1(n+1)-A1(n)LT∞L1, u(n+1)A1(n+1)-u(n)A1(n)LT1L1, and u(n+1)v(n+1)-u(n)v(n)LT1L1. Other terms can be bounded in a similar way. Applying Lemma 4, we can derive(75)un+1-unLT∞L1≤2unA1n-un-1A1n-1LT1L1.For u(n+1)A1(n+1)-u(n)A1(n)LT1L1, we estimate from Lemma 8(76)un+1A1n+1-unA1nLT1L1≤un+1A1n+1-A1nLT1L1+un+1-unA1nLT1L1≤u0L1+2unA1nL1unvn-un-1vn-1LT1L1+a1L1+unvnL1unA1n-un-1A1n-1LT1L1≤12R2+2R2dn-1T.We can derive(77)A1n+1-A1nLT∞L1≤unvn-un-1vn-1LT1L1.For u(n+1)v(n+1)-u(n)v(n)LT1L1, we estimate from Lemma 8(78)un+1vn+1-unvnLT1L1≤un+1vn+1-vnLT1L1+un+1-unvnLT1L1≤12u0L1+2unA1nL1A1nvn-A1n-1vn-1LT2L2+12v0L1+2vnA1nL1unA1n-un-1A1n-1LT1L1≤18R2+2R2dn-1T.We can take care of other terms of dn(T) in the same manner as (75)–(78) to obtain (79)dnT≤121+R2+2R2dn-1T.Next we make use of the finite speed of propagation of solution to construct the local solution for the initial data of arbitrary size. Chooser satisfying(80)supx∈R∫x-y<ru0y+v0y+a1ydy<R.With this r, let us split R in two ways (81)I1j=2j-1r,2j+1r,I2j=2jr,2j+1r,for each j∈Z. We also denote that (82)Ω1j≔t,x∈R2:x-2jr≤r-t,t≥0,Ω2j≔t,x∈R2:x-2j+1r≤r-t,t≥0.Applying the finite speed of propagation and considering the smallness condition (80), we can construct the corresponding solution uj,k, vj,k, A1j,k in Ωkj, for each Ikj. Let us consider the following regions: (83)S1j=0,r2×2j-12r,2j+12r,S2j=0,r2×2j+12r,2j+32r.Then we get a solution uj,k, vj,k, A1j,k on Skj. Note that this solution is not influenced by changing the initial data on the complement of Ikj. Using the uniqueness of solution we can gather these solutions to obtain a solution u,v,A1 on ⋃k=1,2⋃j∈ZΩkj. Then we know that the solution u, v, A1 belongs to LT1L1(R).To complete Theorem3, we will show the nonconcentration property of the solution to extend the local solution globally. Let T be the maximal time of the existence such that the solution u,v,A1∈LT1L1(R) to (15)–(17) with initial data u0,v0,a1. Now we assume that T<∞. We will show that there exists r=rT,u0L1,v0L1,a10L1>0 such that(84)sup0<t<Tsupx∈R∫x-y<rut,y+vt,y+A1t,ydy<εfor any ε>0. If condition (84) holds, then there exist 0<t∗<T and r∗=r∗T,ut∗L1,vt∗L1,A1t∗L1>0 satisfying (85)supx∈R∫x-y<r∗ut∗,y+vt∗,y+A1t∗,ydy<R.Taking a new initial data (u(t∗),v(t∗),A1(t∗)), we can extend the solution over [0,T] which gives a contradiction. Therefore we must have T=∞ and then prove the existence of global solution.Let(u,v,A1) be the smooth solution to (15)–(17). From (15) and (16), we obtain (86)∂t+∂xu2=2u,A1,u=0,∂t-∂xv2=2A1,v,v=0,where (22) is taken into account. Then we derive(87)ut,x=u0x-t,vt,x=v0x+t,from which we have(88)supx∈R∫x-y<rut,ydy≤supx∈R∫x-y<ru0y-tdy,supx∈R∫x-y<rvt,ydy≤supx∈R∫x-y<rv0y+tdy.From (17), we derive (89)A1t,x≤a1x+12∫0tu,vs,xds≤a1x+∫0tu0x-sv0x+sds,where (87) is used. Integrating on x-y<r and applying change of variables y-s=z, y+s=w and Fubini’s theorem, we have (90)∫x-rx+r∫0tu0y-sv0y+sdsdy≤∫x-r-tx+r∫2x-z-2r2x-z+2ru0zv0ωdωdz≤∫x-r-tx+ru0z∫2x-z-2r2x-z+2rv0ωdωdz≤supy∈R∫y-ω<2rv0ωdω∫Ru0zdz.Then we obtain(91)supx∈R∫x-y<rA1t,ydy≤supx∈R∫x-y<ra1ydy+supx∈R∫x-y<r∫0tu0y-sv0y+sdsdy≤supx∈R∫x-y<ra1ydy+u0L1supy∈R∫x-y<2rv0ωdω.We can verify the nonconcentration of solution (84) for taking r sufficiently small in (88) and (91).



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