# Global Energy Solution to the Schrödinger Equation Coupled with the Chern-Simons Gauge and Neutral Field

Authors: Hyungjin Huh; Bora Moon
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

We study the Cauchy problem of the Chern-Simons-Schrödinger equations with a neutral field, under the Coulomb gauge condition, in energy spaceH1(R2). We prove the uniqueness of a solution by using the Gagliardo-Nirenberg inequality with the specific constant. To obtain a global solution, we show the conservation of total energy and find a bound for the nondefinite term.


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## Body

## 1. Introduction

In this paper, we are interested in the Cauchy problem of the Chern-Simons-Schrödinger equations coupled with a neutral field (CSSn) inR1+2:(1)iD0ψ+DjDjψ=ψ2ψ+2Nψ,(2)∂00N-ΔN+N=-2ψ2,(3)∂0A1-∂1A0=2Im⁡ψ¯D2ψ,(4)∂0A2-∂2A0=-2Im⁡ψ¯D1ψ,(5)∂1A2-∂2A1=ψ2.Here, ψ(t,x):R1+2→C is the matter field, N(t,x):R1+2→R is the neutral field, and Aμ(t,x):R1+2→R is the gauge field. Dμ=∂μ-iAμ is the covariant derivative, i=-1, ∂0=∂t, ∂j=∂xj, and Δ=∂j∂j. We use notation A=(A0,Aj)=(A0,A1,A2). From now on, Latin indices are used to denote 1,2 and the summation convention will be used for summing over repeated indices.The CSSn system exhibits both conservation of the charge,(6)Qt≔ψt,·L2=Q0,and conservation of the total energy(7)Et≔2∑j=1,2Djψt,·L22+∇Nt,·L22+∂tNt,·L22+Nt,·L22+ψt,·L44+4∫R2Nψ2t,xdx=E0.The CSSn system is invariant under the following gauge transformations:(8)ψ⟶ψeiχ,N⟶N,Aμ⟶Aμ+∂μχ, where χ:R1+2→R is a smooth function. Therefore, a solution to the CSSn system is formed by a class of gauge equivalent pairs (ψ,N,A). In this paper, we fix the gauge by adopting the Coulomb gauge condition ∂jAj=0, which provides elliptic features for gauge fields A. Under the Coulomb gauge condition, the Cauchy problem of the CSSn system is reformulated as follows:(9)i∂tψ+Δψ=-A0ψ+Aj2ψ+2iAj∂jψ+ψ2ψ+2Nψ,(10)∂ttN-ΔN+N=-2ψ2,(11)ΔA0=2Im⁡∂2ψ¯∂1ψ-∂1ψ¯∂2ψ+2∂2A1ψ2-2∂1A2ψ2,(12)ΔA1=-∂2ψ2,(13)ΔA2=∂1ψ2,with the initial data ψ(0,x)=ψ0(x),  N(0,x)=n0(x),  ∂tN(0,x)=n1(x). Note that ψ,  N are dynamical variables and A are determined by ψ through (11)–(13).The CSSn system is derived from the nonrelativistic Maxwell-Chern-Simons model in [1] by regarding Maxwell term in the Lagrangian as zero. Compared with the Chern-Simons-Schrödinger (CSS) system which comes from the nonrelativistic Maxwell-Chern-Simons model by taking the Chern-Simons limit in [1], the CSSn system has the interaction between the matter field ψ and the neutral field N. The CSS system reads as(14)iD0ψ+DjDjψ=-ψ2ψ,∂0A1-∂1A0=2Im⁡ψ¯D2ψ,∂0A2-∂2A0=-2Im⁡ψ¯D1ψ,∂1A2-∂2A1=ψ2and has conservation of the total energy(14)Et≔2∑j=1,2Djψt,·L22-ψt,·L44=E0.We remark that ψt,·L44 has opposite sign in (7) compared with (14). In fact, this difference causes different global behavior of solution. The local well-posedness of the CSS system in H2, H1 was shown in [2, 3], respectively. We can prove the existence of a local solution of the CSSn system by applying similar argument. On the other hands, due to the nondefiniteness of total energy, the CSS system has a finite-time blow-up solution constructed in [2, 4]. The CSSn system also has difficulty with nondefiniteness of Nψ2 in the total energy, but we could obtain a global solution by controlling it with H1-norm.Considering conservation of the energy (7), it is natural to study the Cauchy problem with the initial data ψ0,n0,n1∈H1×H1×L2. Our first result is concerned with a local solution in energy space.Theorem 1.
For the initial data(ψ0,n0,n1)∈H1(R2)×H1(R2)×L2(R2), there are T>0 and a unique local-in-time solution (ψ,N,A) to (9)–(13) such that(15)ψ∈L∞0,T;H1R2∩C0,T;L2R2,N∈L∞0,T;H1R2∩C0,T;L2R2,∂tN∈L∞0,T;L2R2,A0∈L∞0,T;LqR2∩L∞R2∩H˙1R2,Aj∈L∞0,T;LqR2∩H˙1R2, where 2<q<∞. Moreover, the solution has continuous dependence on initial data.Our second result is concerned with a global solution in energy space.Theorem 2.
For the initial data(ψ0,n0,n1)∈H1(R2)×H1(R2)×L2(R2), there exists a unique global solution (ψ,N,A) to (9)–(13) such that(16)ψ∈L∞0,∞;H1R2∩C0,∞;L2R2,N∈L∞0,∞;H1R2∩C0,∞;L2R2,∂tN∈L∞0,∞;L2R2,A0∈L∞0,∞;LqR2∩L∞R2∩H˙1R2,Aj∈L∞0,∞;LqR2∩H˙1R2, where 2<q<∞. Moreover, the solution has continuous dependence on initial data.Note that, considering (11)–(13), Aj can be determined by ψ as(17)Aj=-1j+12πxj′x2∗ψ2,and then A0 can be determined as(18)A0=∑j=12-1j+1πxjx2∗Im⁡ψ¯∂j′ψ+∑j=12-1j+1πxjx2∗Aj′ψ2,where j′=2 if j=1, and j′=1 if j=2. We present estimates for A and refer to [3, 5] for proof.Proposition 3.
Letψ∈H1(R2) and let A be the solution of (11)-(13). Then, we have, for 2<q<∞, (19)AjLq≲ψL21+2/q∇ψL21-2/q,∇AjL2≲ψL2∇ψL2,A0Lq≲1+ψL22ψL22/q∇ψL22-2/q,A0L∞+∇A0L2≲1+ψL22∇ψL22.We will prove Theorems1 and 2 in Sections 2 and 3, respectively. We conclude this section by giving a few notations. We use the standard Sobolev spaces Hs(R2) with the norm fHs=1-Δs/2fL2. We will use c,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

In this section we address the local well-posedness of solution to (9)–(13). We note that if we remove the gauge fields and the term ψ2ψ from the CSSn system, it is the same as the Klein-Gordon-Schödinger system with Yukawa coupling (KGS). There are many studies on the Cauchy problem of the KGS system in the Sobolev spaces Hs [6–9]. Moreover, if we ignore the interaction with the neutral field N which does not cause any difficulty in obtaining a local solution, a local solution for the CSSn system can be obtained in a similar way to the CSS system. We could obtain a local regular solution by referring to [2, 8] and then construct a local energy solution by using the compactness argument introduced in [2, 3, 5, 6]. In other words, a local H1-solution is constructed by the limit of a sequence of more smooth solutions and it satisfies CSSn system in the distribution sense. For the proof, we follow the same argument as in [2]. So we omit the detail of the local existence here. Since the compactness argument does not guarantee the uniqueness and the continuous dependence on initial data of a local solution, we would rather contribute this section to show the uniqueness and the continuous dependence on initial data of a local solution.Theorem 4.
Let(ψ,N,A) and (ψ~,N~,A~) be solutions to (9)–(13) on (0,T)×R2 in the distribution sense with the same initial data (ψ0,n0,n1)∈H1(R2)×H1(R2)×L2(R2) satisfying(20)ψ,ψ~,N,N~∈L∞0,T;H1R2∩C0,T;L2R2,∂tN,∂tN~∈L∞0,T;L2R2,ψLT∞H1,ψ~LT∞H1,NLT∞H1,N~LT∞H1,∂tNLT∞L2,∂tN~LT∞L2≤M,for some M>0. Then, we have (21)ψ-ψ~t,·L2=0,N-N~t,·H1=0, for 0≤t≤T. Moreover, the solution depends on initial data continuously.Before beginning the proof, we gather lemmas used for the proof of Theorem4. We use the following Lp–Lp′ estimate proved in [10] which plays an important role to control the difference of solutions. It was used in [6] for the uniqueness of the KGS system.Lemma 5.
Letf(t,x):R1+2→R be a solution to (22)∂ttf-Δf+f=F,t,x∈R1+2,f0,x=0,(23)∂tf0,x=0, and T(t) be the Klein-Gordon propagator. Then, we have (24)ft,x=∫0tTt-sFsds, and(25)TtFL6R2≲t-1/3FL6/5R2.The Hardy-Littlewood-Sobolev inequality is also used to control the difference of solutions. For the proof, we refer to Theorem6.1.3 in [11].Lemma 6.
LetI1 be the operator defined by (26)I1fx=∫R2fyx-ydy.If 1/q=1/p-1/2, 1<p<2, then we have (27)I1fLqR2≤2πq1/2fLpR2.The following Gagliardo-Nirenberg inequality with the explicit constant depending onq is used to show the uniqueness. It was proved in [12, 13] and used in [3, 5, 12, 13] to show the uniqueness of the nonlinear Schrödinger equations.Lemma 7.
For2≤q<∞, we have (28)fLqR2≤4π1/q-1/2q21/2fL2R22/q∇fL2R21-2/q.We need the following Grönwall type inequality.Lemma 8.
Letf(t) be a continuous nonnegative function defined on I=0,a and has zero only at 0. Suppose that f satisfies(29)ft≤α∫0tfsds1-2/q+β∫0tfsdsfort∈I,where α, β>0 and q>2. Then we have(30)∫0tfsds≤αe2βt/q-αβq/2fort∈I.Proof.
Define(31)ht=q2∫0tfsds2/q+qα2β.Then, the assumption (29) implies (32)h′t=∫0tfsds2/q-1ft≤α+β∫0tfsds2/q=2βqht, and the standard Grönwall’ inequality gives (33)ht≤h0e2βt/q=qα2βe2βt/q. Considering the definition of h(t) in the above inequality, we have (30).We also need the following inequality to show that the solution is continuously dependent on initial data. We refer to [14].Lemma 9.
Letq>1 and a,b>0. Let f:[0,∞)→[0,∞) satisfy (34)ft≤a+b∫0tf1-1/qsdsfor all t≥0. Then, f(t)≤(a1/q+bq-1t)q for all t≥0.Now we are ready to prove Theorem4. The basic rationale is borrowed from [3, 5, 15]. Let (ψ,N,A) and (ψ~,N~,A~) be solutions of (9)–(13) with the same initial data. If we set (35)u=ψ-ψ~andv=N-N~,then the equations for u and v satisfy(36)i∂tu+Δu=A~0-A0ψ-A~0u+2iAj-A~j∂jψ+2iA~j∂ju+Aj2-A~j2ψ+A~j2u+ψ2-ψ~2ψ+ψ~2u+2vψ+2N~u,(37)∂ttv-Δv+v=-2ψ2-ψ~2,where(38)u,v∈L∞0,T;H1R2∩C0,T;L2R2,∂tv∈L∞0,T;L2R2.First of all, we will derive, forq>2,(39)sup0,t⁡uL22≤α∫0tsup0,suL22ds1-2/q+β∫0tsup0,s⁡uL22ds,where (40)α=T2/qqM21+M4/q+M2+4/qandβ=M2. Once we obtain (39), considering u(0,·)L2=0, Lemma 8 gives (41)∫0tsup0,suL22ds≤T1+M4/q+M2+4/qq/2qe2M2T/q-1q/2, for 0≤t≤T. We note that (42)limt→0+⁡ebt-1t=b.Let us take the time interval T′≤T satisfying (2+M2)(2M2T′)<1/2. Letting q→∞ we have that u(t,·)L2=0 for 0≤t≤T′. Using this argument repeatedly, we conclude that u(t,·)L2=0 for 0≤t≤T.To derive the estimate (39), multiplying (36) by u¯ and integrating the imaginary part on [0,t]×R2, we have(43)utL22=∫0t∫R22A~0-A0Im⁡ψu¯︸I+4Aj-A~jRe⁡∂jψu¯︸II+2A~j∂ju2︸III+2Aj2-A~j2Im⁡ψu¯︸IV+2ψ2-ψ~2Im⁡ψu¯︸V+4vIm⁡ψu¯︸VIdxds.Considering ∂jA~j=0, we have (III)=0. Except for the integral (VI), the right-hand side of (43) is bounded, by adopting the same manner described in [3, 5], as follows.(44)I+II+IV+V≲T2/qqM21+M4/q+M2+4/q∫0tsup0,suL22ds1-2/q.We will provide, for instance, the bound for (II) and (IV). The rest can be proved in a similar way. Due to (17), Lemma 6 and Lemma 7 lead to (45)AjL6≲I1ψ2L6≲ψL32and (46)Aj-A~jLq≲I1ψ2-ψ~2Lq≲q1/2ψ+ψ~uLp≲q1/2ψLq+ψ~LquL2≲qMuL2≲qM1+2/quL21-2/q,where p is determined by 1/q=1/p-1/2. For 1/r+1/q=1/2, the Hölder’s inequality and Gagliardo-Nirenberg inequality yield (47)∫R24Aj-A~jRe∂jψu¯dx≲Aj-A~jLq∇ψL2uLr≲M2-2/ruL22/rAj-A~jLq≲qM2+4/quL22-4/q.Thus, the Hölder’s inequality gives (48)∫0t∫R24Aj-A~jRe⁡∂jψu¯dx≲qM2+4/q∫0tusL22-4/qds≲T2/qqM2+4/q∫0tusL22ds1-2/q.For the integral (IV), similar estimate shows (49)∫R22Aj2-A~j2Im⁡ψu¯dx≲Aj-A~jLqAjL6+A~jL6ψL3uLr≲qM1+2/quL21-2/qψL32+ψ~L32ψL3uLr≲qM4+4/quL22-4/q,which implies (50)∫0t∫R22Aj2-A~j2Im⁡ψu¯dx≲T2/qqM4+4/q∫0tusL22ds1-2/q.For the integral (VI), we first apply Lemma 5 to (37) which leads to (51)vsL6≲∫0sTs-τψτ2-ψ~τ2L6dτ≲∫0ss-τ-1/3uτL2ψτL3+ψ~τL3dτ≲Ms2/3sup0,s⁡uL2.Then, we have(52)∫0t∫R24vIm⁡ψu¯dxds≲∫0tvsL6ψsL3usL2ds≲M2∫0tsup0,s⁡uL22ds.Collecting these bounds (44), (52), we obtain (39) which implies (53)ut,·L2=0for0≤t≤T.On the other hand, multiplying (37) by ∂tv and integrating over [0,t]×R2, we have (54)∂tvtL22+∇vtL22+vtL22=-4∫0t∫R2ψ-ψ~ψ+ψ~∂tvdxds,for 0≤t≤T. The Hölder’s inequality and Gagliardo-Nirenberg inequality give us (55)∂tvtL22+∇vtL22+vtL22≲∫0tusL6ψsL3+ψ~sL3∂tvsL2ds≲M2∫0tusL21/3∇usL22/3ds=0.Finally, continuous dependence on initial data follows from the same estimates above and the same argument in [14]. Let (ψ,N,A) and (ψ~,N~,A~) be solutions of (9)–(13) with the initial data (ψ0,n0,n1) and (ψ~0,n~0,n~1), respectively. If we set u=ψ-ψ~ and u0=ψ0-ψ~0, the above estimates show(56)sup0,t⁡uL22≤u0L22+qM21+M4/q+M2+4/q∫0tsup0,s⁡usL221-2/qds.Applying Lemma 9 to (56), we have (57)sup0,t⁡uL22≤u0L24/q+M21+M4/q+M2+4/qTq/2,and this implies that the solution depends on initial data continuously in L2 locally uniformly in time.


## 3. Proof of Theorem2

In this section we study the existence of a global solution to (9)–(13). Firstly, we derive the conservation laws (6) and (7). Multiplying (1) by ψ¯ and taking its conjugate, we have (58)i∂tψψ¯+Δψψ¯=-A0ψ2+Aj2ψ2+2iAj∂jψψ¯+i∂jAjψ2+ψ4+2Nψ2,(59)-i∂tψ¯ψ+Δψ¯ψ=-A0ψ2+Aj2ψ2-2iAj∂jψ¯ψ-i∂jAjψ2+ψ4+2Nψ2.Subtracting (59) from (58), we obtain (60)i∂tψ2+Δψψ¯-Δψ¯ψ=2i∂jAjψ2.Then, integration by parts on R2 gives (61)ddtψt,·L2=0,which implies (6).Multiplying (1) by ∂tψ¯ and taking its conjugate, we have (62)i∂tψ∂tψ¯+Δψ∂tψ¯=-A0ψ∂tψ¯+Aj2ψ∂tψ¯+2iAj∂jψ∂tψ¯+i∂jAjψ∂tψ¯+ψ2ψ∂tψ¯+2Nψ∂tψ¯,-i∂tψ¯∂tψ+Δψ¯∂tψ=-A0ψ¯∂tψ+Aj2ψ¯∂tψ-2iAj∂jψ¯∂tψ-i∂jAjψ¯∂tψ+ψ2ψ¯∂tψ+2Nψ¯∂tψ.Summing the both sides and integrating by parts on R2, we obtain(63)-∫R2∂t∇ψ2dx-∫R2Aj2∂tψ2dx︸i+∫R24AjIm⁡∂jψ∂tψ¯+2∂jAjIm⁡ψ∂tψ¯dx︸ii=-∫R2A0∂tψ2dx+12∫R2∂tψ4dx+2∫R2N∂tψ2dx.Considering (64)i=∫R2∂tAj2ψ2-∂tAj2ψ2dx,ii=∫R22AjIm⁡∂jψ∂tψ¯+2AjIm⁡ψ¯∂j∂tψdx=∫R22∂tAjIm⁡ψ¯∂jψ-2∂tAjIm⁡ψ¯∂jψdx,and (65)∂tDjψ2=∂t∇ψ2-2∂tAjIm⁡ψ¯∂jψ+∂tAj2ψ2,the left side of (63) becomes(66)-ddt∫R2Djψ2dx+∫R2∂tAj2ψ2-2∂tAjIm⁡ψ¯∂jψdx︸iii.On the other hands, multiplying (3), (4) by ∂tA2, ∂tA1, respectively, we have (67)∂tA2∂0A1-∂1A0=2∂tA2Im⁡ψ¯∂2ψ-2A2∂tA2ψ2,-∂tA1∂0A2-∂2A0=2∂tA1Im⁡ψ¯∂1ψ-2A1∂tA1ψ2.Adding the both sides, we have (68)∂tAj2ψ2-2∂tAjIm⁡ψ¯∂jψ=∂tA2∂1A0-∂tA1∂2A0.Replacing (iii) with this, integration by parts gives(69)iii=-∫R2A0∂t∂1A2-∂2A1dx=-∫R2A0∂tψ2dx,where (5) is used. Inserting (66) and (69) into (63), we have(70)ddt∫R22Djψ2+ψ4dx=-4∫R2N∂tψ2dx.Now, multiplying (2) by ∂tN and integrating on R2 provide(71)ddt∫R2∇N2+∂tN2+N2dx=-4∫R2∂tNψ2dx.Adding (70) and (71), we finally obtain (72)ddtEt=0,which leads to (7).Now we are ready to prove the existence of global solution. By the conservation laws (6) and (7), we have(73)Qt=ψtL2=Q0,and(74)Et=2∑j=1,2DjψtL22+∇NtL22+∂tNtL22+NtL22+ψtL44+4∫R2Nψ2t,xdx=E0.Because we do not know the sign of the last term Nψ2, the energy conservation (74) does not imply a global energy solution directly. Therefore we would find a bound for the last term and then a uniform bound for H1-norm of solution which leads to global existence. We refer to [8].Using the Hölder’s inequality, Lemma7, and Young’s inequality, we have(75)-4∫R2Nψ2t,xdx≤4NtL4ψtL4ψtL2≤4NtL21/2∇NtL21/2ψtL4ψtL2≤14∇NtL22+14NtL22+14ψtL44+26ψtL24.From (75) and (74), it follows that (76)2DjψtL22+∇NtL22+∂tNtL22+NtL22+ψtL44≤E0+26Q04+14∇NtL22+14NtL22+14ψtL44 which implies (77)DjψtL22+∇NtL22+∂tNtL22+NtL22+ψtL44≤c. Referring to Proposition 3, the Hölder’s inequality and Young’s inequality give (78)∇ψtL22≤2DjψtL22+2AjtL42ψtL42≤2DjψtL22+2C∇ψtL2ψtL23ψtL42≤2DjψtL22+12∇ψtL22+2C2ψtL26ψtL44, which yields (79)∇ψtL22≤4c1+C2Q06.



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