Global Strong Solutions to Some Nonlinear Dirac Equations in Super-Critical Space

Abstract

We study the initial value problem of some nonlinear Dirac equations which are 𝐿^𝑚(ℝ) critical. Corresponding to the structure of nonlinear terms, global strong solutions can be obtained in different Lebesgue spaces by using solution representation formula. The uniqueness of weak solutions is proved for the solution 𝑈∈𝐿^∞([0,𝑇];𝑌^𝑚+2(ℝ)).

1. Introduction

In this study, we are interested in global strong solutions of nonlinear Dirac type equations: −𝑖𝜕_𝑡𝑈₁+𝜕_𝑥𝑈₁=𝜕_𝑈1𝐹𝑈₁,𝑈₂,−𝑖𝜕_𝑡𝑈₂−𝜕_𝑥𝑈₂=𝜕_𝑈2𝐹𝑈₁,𝑈₂, (1) where 𝐹(𝑈₁,𝑈₂)∶ℂ²→ℝ is a nonlinear potential function which will be specified later.

The following Thirring model with the potential 𝐹(𝑈₁,𝑈₂)=|𝑈₁|²|𝑈₂|² has been studied in [1–4]: −𝑖𝜕_𝑡𝑈₁+𝜕_𝑥𝑈₁=𝑈₂²𝑈₁,−𝑖𝜕_𝑡𝑈₂−𝜕_𝑥𝑈₂=𝑈₁²𝑈₂,𝑈₁(0,𝑥)=𝑢₁(𝑥),𝑈₂(0,𝑥)=𝑢₂(𝑥). (2) The initial value problem of (2) was studied in [1, 3] in terms of Sobolev space 𝐻^𝑠(𝑠≥1). Low regularity well-posedness was discussed in [4] showing that there exists a time 𝑇>0 and solution 𝑈∈𝐶([0,𝑇],𝐻^𝑠(ℝ)) (𝑠>0) of the Cauchy problem (2). Their results were based on the observation of the null structure of Thirring equations and application of 𝑋^𝑠,𝑏 spaces which is a certain subspace of 𝐶([0,𝑇],𝐻^𝑠(ℝ)). They also proved global well-posedness for 𝑠>1/2 and unconditional uniqueness for 𝑠>1/4. Nonlinear Dirac equations in ℝ¹⁺¹ have been studied by several authors [5–9].

In the context of Bragg grating [10, 11], the nonlinear term 𝐹 takes the form 𝐹=𝑈₁⁴+𝑈₂⁴+4𝑈₁²𝑈₂², (3) which gives 𝜕_𝑈1𝐹=(2|𝑈₁|²+4|𝑈₂|²)𝑈₁ and 𝜕_𝑈2𝐹=(2|𝑈₂|²+4|𝑈₁|²)𝑈₂. We may consider the sixth and higher orders. The following potential term is introduced in the context of the Bose-Einstein condensates [12]: 𝐹=𝑈₁²+𝑈₂²𝑈₁²𝑈₂², (4) which gives 𝜕_𝑈1𝐹=(2|𝑈₁|²|𝑈₂|²+|𝑈₂|⁴)𝑈₁ and 𝜕_𝑈2𝐹=(2|𝑈₂|²|𝑈₁|²+|𝑈₁|⁴)𝑈₂.

Several authors [4, 5, 7] have studied the initial value problem of the following Dirac equations with quadratic nonlinearities: −𝑖𝜕_𝑡𝑈₁+𝜕_𝑥𝑈₁=𝑄₁𝑈₁,𝑈₂,−𝑖𝜕_𝑡𝑈₂−𝜕_𝑥𝑈₂=𝑄₂𝑈₁,𝑈₂. (5) If the nonlinear term is of 𝑄₁=|𝑈₂|², 𝑄₂=|𝑈₁|² type (𝑄₁=𝑈₁𝑈₂, 𝑄₂=𝑈₁𝑈₂, resp.), then one can obtain local well-posedness for the Sobolev space 𝐻^𝑠(𝑠>0) [4] (𝐻^𝑠(𝑠>−1/4) [7], resp.). Note that the scaling properties of quadratic Dirac equations give the critical Sobolev exponent 𝑠_cr=−1/2.

Now it seems natural to consider the following equations: 𝜕_𝑡𝑈₁+𝜕_𝑥𝑈₁=𝑖𝑈₁^𝑘𝑈₂^𝑚−𝑘𝑈₁,𝜕_𝑡𝑈₂−𝜕_𝑥𝑈₂=𝑖𝑈₂^𝑘𝑈₁^𝑚−𝑘𝑈₂, (6) which are generalization of the basic cases of the literature and model problem to investigate regularities of solutions according to the structure of nonlinearities. Here, 𝑚 is positive integer and 𝑘=0,1,…,𝑚.

The system (6) is invariant under the scaling 𝑈^𝜆(𝑡,𝑥)=𝜆𝑈𝜆^𝑚𝑡,𝜆^𝑚𝑥, (7) from which we deduce a scale invariant Lebesgue space 𝐿^𝑚(ℝ). We study the initial value problem of (6) in Lebesgue space. Let us denote 𝑝(𝑚,𝑘)=Max(𝑚−𝑘,2𝑘+1). (8) Note that 𝑝(𝑚,𝑘)≥2 except for 𝑚=1 and 𝑘=0 where 𝑝(1,0)=1. We define 𝑌^𝑝(ℝ)=𝐿²(ℝ)∩𝐿^𝑝(ℝ), (9) where 𝑝≥2. We call 𝑌^𝑝 as subcritical space for 𝑝>𝑚, critical space for 𝑝=𝑚, and supercritical space for 𝑝<𝑚.

For the initial data 𝑢_𝑗∈𝐿²(ℝ)(𝑗=1,2), we have proved in [2] that there exists a global strong solution 𝑈=(𝑈₁,𝑈₂) to the initial value problem (2) in the critical space 𝑈∈𝐶[0,∞);𝐿²(ℝ). (10) Elementary and interesting approach was made. and special structure of nonlinearity was made use of in applying Fubini's theory in integration. The following is concerned with the existence of global strong solutions for the case of 𝑚≥2 or 𝑘≥1.

Theorem 1. For the initial data 𝑣_𝑗∈𝑌^{𝑝(𝑚,𝑘)}(ℝ)(𝑗=1,2), there exists a global strong solution 𝑉=(𝑉₁,𝑉₂) to the initial value problem (6) which satisfies 𝑉∈𝐶[0,∞);𝑌^{𝑝(𝑚,𝑘)}(ℝ). (11)

Remark 2. (1) Related with the title of this paper, we emphasize the case 𝑝(𝑚,𝑘)<𝑚 where global strong solution can be constructed in the super-critical space.
(2) Taking the embedding 𝑌^𝑞(ℝ)↪𝑌^𝑟(ℝ)    (𝑞≥𝑟) into account, it is likely to say that the system (6) with small 𝑝(𝑚,𝑘) has better smoothing property than equations with large 𝑝(𝑚,𝑘). We can check, in the case of (4), the existence of solution 𝑉∈𝐶([0,∞);𝑌⁵(ℝ)) because of 𝑝(4,2)=5>4=𝑝(4,0).
(3) For the case of 𝑚=1 and 𝑘=0, we can construct a global strong solution 𝑉=(𝑉₁,𝑉₂) to the initial value problem (6) which satisfies 𝑉∈𝐶[0,∞);𝐿¹(ℝ) (12) for the initial data 𝑣_𝑗∈𝐿¹(ℝ)(𝑗=1,2).

Our second result is concerned with the uniqueness of weak solutions. The null structure of the nonlinearity of Thirring model which is 𝐿² critical problem was used in [2] to prove the uniqueness in 𝑌⁴. To treat the general nonlinearity in (6) of which we could not find the null structure, a different approach is considered.

Theorem 3. Let 𝑈 and 𝑉 be solutions of (6) in the distribution sense with same initial data. Moreover, one assumes that 𝑈,𝑉∈𝐿^∞[0,𝑇];𝑌^𝑚+2(ℝ). (13) Then, one has ‖(𝑈−𝑉)(𝑡,⋅)‖_𝐿²=0 for 0≤𝑡≤𝑇.

Theorems 1 and 3 imply that if 𝑝(𝑚,𝑘)≥𝑚+2 then a strong solution to the initial value problem (6) is unique and is in fact a well-posed solution.

Theorem 1 is proved in Section 2 and Theorem 3 in Section 2.2. We use 𝐶 to denote various constants and 𝐴≲𝐵 to denote an estimate of the form 𝐴≤𝐶𝐵.

2. Proof of Theorem 1

To construct global strong solution of (6), we basically follow the idea of [2] with modification. We will find a solution representation formula. Then, global strong solutions can be obtained by constructing an explicit approximation and using Fubini's theorem.

2.1. Representation Formula of Classical Solution

In this subsection, we consider 𝐶^∞((0,𝑇)×ℝ) solutions which satisfy (6) in the classical sense. An explicit representation of its solution is given in terms of initial data [2, 13]. It is interesting to express the solution of nonlinear partial differential equations in terms of initial data.

Integrating (6) along the outgoing and ingoing characteristics, we obtain 𝑈₁(𝑡,𝑥)=𝑢₁()𝑖𝑥−𝑡×exp^𝑡₀𝑈₁(𝑠,𝑥−𝑡+𝑠)^𝑘×𝑈₂(𝑠,𝑥−𝑡+𝑠)^𝑚−𝑘,𝑈𝑑𝑠₂(𝑡,𝑥)=𝑢₂()𝑖𝑥+𝑡×exp^𝑡₀𝑈₂(𝑠,𝑥+𝑡−𝑠)^𝑘×𝑈₁(𝑠,𝑥+𝑡−𝑠)^𝑚−𝑘𝑑𝑠. (14) Taking into account 𝑈₁(𝑡,𝑥)=𝑢₁(𝑥−𝑡),𝑈₂(𝑡,𝑥)=𝑢₂(𝑥+𝑡), (15) (14) becomes 𝑈₁(𝑡,𝑥)=𝑢₁()𝑖𝑥−𝑡×exp2𝑢₁(𝑥−𝑡)^{𝑘𝑥+𝑡}_𝑥−𝑡𝑢₂𝑦^𝑚−𝑘,𝑈𝑑𝑦₂(𝑡,𝑥)=𝑢₂()𝑖𝑥+𝑡×exp2𝑢₂(𝑥+𝑡)^{𝑘𝑥+𝑡}_𝑥−𝑡𝑢₁𝑦^𝑚−𝑘𝑑𝑦. (16) In the case of 𝑘=𝑚, we have the following obvious representation formula of solution: 𝑈₁(𝑡,𝑥)=𝑢₁(𝑥−𝑡)exp𝑖𝑡𝑢₁(𝑥−𝑡)^𝑚,𝑈₂(𝑡,𝑥)=𝑢₂(𝑥+𝑡)exp𝑖𝑡𝑢₁(𝑥−𝑡)^𝑚. (17)

2.2. Global Strong Solutions in 𝑌^𝑝(ℝ)

We now present the proof of Theorem 1. Let us introduce a strong solution of the initial value problem of the system (6).

Definition 4. Consider the Cauchy problem (6) with initial data 𝑣=(𝑣₁,𝑣₂)∈𝑌^{𝑝(𝑚,𝑘)}(ℝ)). It is said that 𝑉=(𝑉₁,𝑉₂) is a strong solution to the Cauchy problem on the time interval [0,𝑇] provided that 𝑉₁,𝑉₂∈𝐶[0,𝑇];𝑌^{𝑝(𝑚,𝑘)}(ℝ) (18) satisfy (6) in the sense of distributions. That is, for any 𝜙∈𝐶^∞₀((−𝑇,𝑇)×ℝ), we have ^𝑇_0ℝ𝑉₁𝜕_𝑡𝜙+𝑉₁𝜕_𝑥𝜙+𝑖𝑉₁^𝑘𝑉₂^𝑚−𝑘𝑉₁+𝜙𝑑𝑥𝑑𝑡_ℝ𝑣₁(𝑥)𝜙(0,𝑥)𝑑𝑥=0,_𝑇0ℝ𝑉₂𝜕_𝑡𝜙−𝑉₂𝜕_𝑥𝜙+𝑖𝑉₂^𝑘𝑉₁^𝑚−𝑘𝑉₂+𝜙𝑑𝑥𝑑𝑡_ℝ𝑣₂(𝑥)𝜙(0,𝑥)𝑑𝑥=0. (19)(20)

Remark 5. We say that 𝑉 is a weak solution if 𝑉∈𝐿^∞([0,𝑇];𝑌^{𝑝(𝑚,𝑘)}(ℝ)) satisfies (6) in the sense of distribution.

For the initial data 𝑣_𝑗∈𝑌^{𝑝(𝑚,𝑘)}(ℝ)(𝑗=1,2), we propose that the following functions 𝑉_𝑗 are the global strong solution of (6): 𝑉₁(𝑡,𝑥)=𝑣₁()𝑖𝑥−𝑡×exp2𝑣₁(𝑥−𝑡)^{𝑘𝑥+𝑡}_𝑥−𝑡𝑣₂𝑦^𝑚−𝑘,𝑉𝑑𝑦₂(𝑡,𝑥)=𝑣₂()𝑖𝑥+𝑡×exp2𝑣₂(𝑥+𝑡)^{𝑘𝑥+𝑡}_𝑥−𝑡𝑣₁𝑦^𝑚−𝑘𝑑𝑦. (21) For the smooth function sequence {𝑢_𝑗^(𝑛)}^∞_𝑛=1  (𝑗=1,2) which converges to 𝑣_𝑗 in 𝑌^{𝑝(𝑚,𝑘)}(ℝ) as 𝑛→∞, we consider sequence of classical solutions 𝑈_𝑗^(𝑛) of (6). First of all, we estimate the difference of (𝑉_𝑗,𝑈_𝑗^(𝑛)). Using the representation (16) and (21), we have 𝑉₁−𝑈₁^(𝑛)≤𝑖(𝑡,𝑥)exp2𝑢₁^(𝑛)(𝑥−𝑡)^{𝑘𝑥+𝑡}_𝑥−𝑡𝑢₂^(𝑛)𝑦^𝑚−𝑘×𝑣𝑑𝑦₁−𝑢₁^(𝑛)(+𝑣𝑥−𝑡)₁(𝑥−𝑡)𝑒^{(𝑖/2)|𝑣1(𝑥−𝑡)|𝑘∫𝑥+𝑡𝑥−𝑡|𝑣2(𝑦)|𝑚−𝑘𝑑𝑦}−𝑒^{(𝑖/2)|𝑢1(𝑛)(𝑥−𝑡)|𝑘∫𝑥+𝑡𝑥−𝑡|𝑢2(𝑛)(𝑦)|𝑚−𝑘𝑑𝑦}≤𝑣₁−𝑢₁^(𝑛)(+1𝑥−𝑡)2𝑣₁(𝑥−𝑡)𝑣₁(𝑥−𝑡)^{𝑘𝑥+𝑡}_𝑥−𝑡𝑣₂𝑦^𝑚−𝑘−𝑢𝑑𝑦_1(𝑛)(𝑥−𝑡)^{𝑘𝑥+𝑡}_𝑥−𝑡𝑢₂^(𝑛)𝑦^𝑚−𝑘=(A)+(B),𝑑𝑦 (22) where we used |𝑒^𝑖𝑥−𝑒^𝑖𝑦|≤|𝑥−𝑦| for 𝑥, 𝑦∈ℝ. The 𝐿^𝑞 norm of the first term (A) can be treated as follows: _ℝ𝑣₁−𝑢₁^(𝑛)(𝑥−𝑡)^𝑞𝑑𝑥^1/𝑞=𝑣₁−𝑢₁^(𝑛)_𝐿𝑞. (23) To estimate the 𝐿^𝑝 norm of the second term (B), we consider the following three cases. (i)For the case 𝑘=0, we have with 2≤𝑝≤𝑚_ℝ𝑣₁(𝑥−𝑡)^{𝑝𝑥+𝑡}_𝑥−𝑡𝑣₂𝑦^𝑚𝑑𝑦−^𝑥+𝑡_𝑥−𝑡𝑢₂^(𝑛)𝑦^𝑚𝑑𝑦^𝑝≲𝑣𝑑𝑥₂−𝑢₂^(𝑛)𝑝_𝐿𝑚𝑣_2𝐿^𝑚−1_𝑚+𝑢₂^(𝑛)_𝐿^𝑚−1_𝑚^𝑝𝑣₁^𝑝_𝐿𝑝. (24)(ii)For the case 𝑘=𝑚, we have with 1≤𝑝≤(2𝑚+1)/(𝑚+1)(2𝑡)^𝑝_ℝ𝑣₁(𝑥−𝑡)^𝑝𝑣₁(𝑥−𝑡)^𝑚−𝑢₁^(𝑛)(𝑥−𝑡)^𝑚𝑝≲()𝑑𝑥2𝑡^𝑝𝑣₁^𝑝_{𝐿(𝑚+1)𝑝}𝑣₁−𝑢₁^(𝑛)𝑝_{𝐿(𝑚+1)𝑝}×𝑣_1𝐿^{(𝑚−1)𝑝}_(𝑚+1)𝑝+𝑢₁^(𝑛)_𝐿^{(𝑚−1)𝑝}_(𝑚+1)𝑝. (25)(iii)For other cases 1≤𝑘≤𝑚−1, we have with 1≤𝑝≤𝑝(𝑚,𝑘)/(𝑘+1)_ℝ𝑣₁(𝑥−𝑡)^𝑝𝑣₁(𝑥−𝑡)^{𝑘𝑥+𝑡}_𝑥−𝑡𝑣₂𝑦^𝑚−𝑘−𝑢𝑑𝑦_1(𝑛)(𝑥−𝑡)^{𝑘𝑥+𝑡}_𝑥−𝑡𝑢₂^(𝑛)𝑦^𝑚−𝑘𝑑𝑦^𝑝𝑑𝑥≲sup_𝑥^𝑥+𝑡_𝑥−𝑡|𝑣₂𝑦|^𝑚−𝑘𝑑𝑦^𝑝×_ℝ𝑣₁(𝑥−𝑡)^𝑝𝑣₁(𝑥−𝑡)^𝑘−𝑢₁^(𝑛)(𝑥−𝑡)^𝑘𝑝𝑑𝑥+sup_𝑥^𝑥+𝑡_𝑥−𝑡𝑣₂𝑦^𝑚−𝑘−𝑢₂^(𝑛)𝑦^𝑚−𝑘𝑑𝑦^𝑝×_ℝ𝑣₁(𝑥−𝑡)^𝑝𝑢₁^(𝑛)(𝑥−𝑡)^𝑘𝑝𝑑𝑥≲sup_𝑥^𝑥+𝑡_𝑥−𝑡𝑣₂𝑦^𝑚−𝑘𝑑𝑦^𝑝𝑣₁^𝑝_{𝐿(𝑘+1)𝑝}×𝑣₁−𝑢₁^(𝑛)𝑝_{𝐿(𝑘+1)𝑝}𝑣_1𝐿^{(𝑘−1)𝑝}_(𝑘+1)𝑝+𝑢₁^(𝑛)_𝐿^{(𝑘−1)𝑝}_(𝑘+1)𝑝+sup_𝑥^𝑥+𝑡_𝑥−𝑡𝑣₂𝑦^𝑚−𝑘−𝑢₂^(𝑛)𝑦^𝑚−𝑘𝑑𝑦^𝑝×𝑣₁^𝑝_{𝐿(𝑘+1)𝑝}𝑢₁^(𝑛)_𝐿^𝑘𝑝_(𝑘+1)𝑝, (26)where sup_𝑥^𝑥+𝑡_𝑥−𝑡𝑣₂𝑦^𝑚−𝑘𝑑𝑦^𝑝≲(2𝑡)^𝑝/2𝑣₂^𝑝_𝐿2𝑣for𝑚−𝑘=1,_{2𝐿(𝑚−𝑘)𝑝𝑚−𝑘}for𝑚−𝑘≥2,sup_𝑥^𝑥+𝑡_𝑥−𝑡𝑣₂𝑦^𝑚−𝑘−𝑢₂^(𝑛)𝑦^𝑚−𝑘𝑑𝑦^𝑝≲(2𝑡)^𝑝/2𝑣₂−𝑢₂^(𝑛)𝑝_𝐿2𝑣for𝑚−𝑘=1,₂−𝑢₂^(𝑛)𝑝_{𝐿𝑚−𝑘}𝑣_2𝐿^{𝑚−𝑘−1}_𝑚−𝑘+𝑢₂^(𝑛)_𝐿^{𝑚−𝑘−1}_𝑚−𝑘^𝑝for𝑚−𝑘≥2. (27)

Now, we are ready to prove that 𝑉=(𝑉₁,𝑉₂) is a global strong solution to the Cauchy problem (6). It is easy to check (18) by considering the representation (21). Making use of the classical solution to (6), we have ^𝑇_0ℝ𝑉₁𝜕_𝑡𝜙+𝑉₁𝜕_𝑥𝜙+𝑖𝑉₁^𝑘𝑉₂^𝑚−𝑘𝑉₁+𝜙𝑑𝑥𝑑𝑡_ℝ𝑣₁(𝑥)𝜙()=0,𝑥𝑑𝑥_𝑇0ℝ𝑉₁−𝑈₁^(𝑛)𝜕_𝑡𝜙+𝜕_𝑥𝜙+𝑑𝑥𝑑𝑡_ℝ𝑣₁−𝑢₁^(𝑛)𝜙()+0,𝑥𝑑𝑥_𝑇0ℝ𝑖𝑉₁^𝑘𝑉₂^𝑚−𝑘𝑉₁−𝑈₁^(𝑛)𝑘𝑈₂^{(𝑛)𝑚−𝑘}𝑈₁^(𝑛)=(1)+(2)+(3).𝜙𝑑𝑥𝑑𝑡 (28) Considering (23)–(26) with 𝑞=𝑝=𝑝(𝑚,𝑘)/(𝑘+1), the first integral (1) can be estimated as follows: |(1)|≲𝑇sup_{0≤𝑡≤𝑇}𝜕_𝑡𝜙(𝑡,⋅)_𝐿𝑞′+𝜕_𝑥𝜙(𝑡,⋅)_𝐿𝑞′×sup_{0≤𝑡≤𝑇}𝑉₁−𝑈₁^(𝑛)(𝑡,⋅)_𝐿𝑞, (29) where sup_{0≤𝑡≤𝑇}𝑉₁−𝑈₁^(𝑛)(𝑡,⋅)_𝐿𝑞≲𝑣₁−𝑢₁^(𝑛)_𝐿𝑞+(2𝑇)^𝑝/2𝑣₂^𝑝_𝐿2𝑣₁^𝑝_{𝐿𝑝(𝑚,𝑘)}×𝑣₁−𝑢₁^(𝑛)𝑝_{𝐿𝑝(𝑚,𝑘)}𝑣_1𝐿^{(𝑘−1)𝑝}_{𝑝(𝑚,𝑘)}+𝑢₁^(𝑛)_𝐿^{(𝑘−1)𝑝}_{𝑝(𝑚,𝑘)}+(2𝑇)^𝑝/2𝑣₂−𝑢₂^(𝑛)𝑝_𝐿2𝑣₁^𝑝_{𝐿𝑝(𝑚,𝑘)}𝑢₁^(𝑛)_𝐿^𝑘𝑝_{𝑝(𝑚,𝑘)}^1/𝑝𝑣for𝑚−𝑘=1,₁−𝑢₁^(𝑛)_𝐿𝑞+𝑣_2𝐿^{(𝑚−𝑘)𝑝}_𝑚−𝑘𝑣₁^𝑝_{𝐿𝑝(𝑚,𝑘)}𝑣₁−𝑢₁^(𝑛)𝑝_{𝐿𝑝(𝑚,𝑘)}×𝑣_1𝐿^{(𝑘−1)𝑝}_{𝑝(𝑚,𝑘)}+𝑢₁^(𝑛)_𝐿^{(𝑘−1)𝑝}_{𝑝(𝑚,𝑘)}+𝑣₂−𝑢₂^(𝑛)𝑝_{𝐿𝑚−𝑘}𝑣_2𝐿^{𝑚−𝑘−1}_𝑚−𝑘+𝑢₂^(𝑛)_𝐿^{𝑚−𝑘−1}_𝑚−𝑘^𝑝×𝑣₁^𝑝_{𝐿𝑝(𝑚,𝑘)}𝑢₁^(𝑛)_𝐿^𝑘𝑝_{𝑝(𝑚,𝑘)}^1/𝑝for𝑚−𝑘≥2. (30) The integral (2) can be bounded easily: _ℝ𝑣₁−𝑢₁^(𝑛)𝜙(0,𝑥)𝑑𝑥≤𝑣₁−𝑢₁^(𝑛)_𝐿2𝜙(0,𝑥)_𝐿2. (31) To take care of the integral (3), we decompose the integral by using representation (16) and (21) with the notation 𝐸(𝑣₁,𝑣₂,𝑘)=exp((𝑖/2)|𝑣₁(𝑥−𝑡)|^𝑘∫^𝑥+𝑡_𝑥−𝑡|𝑣₂(𝑦)|^𝑚−𝑘𝑑𝑦): ^𝑇_0ℝ𝑉₁^𝑘𝑉₂^𝑚−𝑘𝑉₁−𝑈₁^(𝑛)𝑘𝑈₂^{(𝑛)𝑚−𝑘}𝑈₁^(𝑛)𝜙()=𝑡,𝑥𝑑𝑥𝑑𝑡_𝑇0ℝ𝑣₁(𝑥−𝑡)^𝑘−𝑢₁^(𝑛)(𝑥−𝑡)^𝑘𝑣₂(𝑥+𝑡)^𝑚−𝑘×𝑣₁(𝑥−𝑡)𝐸𝑣₁,𝑣₂𝜙()+,𝑘𝑡,𝑥𝑑𝑥𝑑𝑡_𝑇0ℝ𝑢₁^(𝑛)(𝑥−𝑡)^𝑘×𝑣₂(𝑥+𝑡)^𝑚−𝑘−𝑢₂^(𝑛)(𝑥+𝑡)^𝑚−𝑘×𝑣₁(𝑥−𝑡)𝐸𝑣₁,𝑣₂𝜙()+,𝑘𝑡,𝑥𝑑𝑥𝑑𝑡_𝑇0ℝ𝑢₁^(𝑛)(𝑥−𝑡)^𝑘𝑢₂^(𝑛)(𝑥+𝑡)^𝑚−𝑘×𝑣₁−𝑢₁^(𝑛)(𝑥−𝑡)𝐸𝑣₁,𝑣₂𝜙()+,𝑘𝑡,𝑥𝑑𝑥𝑑𝑡_𝑇0ℝ𝑢₁^(𝑛)(𝑥−𝑡)^𝑘𝑢₂^(𝑛)(𝑥+𝑡)^𝑚−𝑘𝑢₁^(𝑛)()×𝐸𝑣𝑥−𝑡₁,𝑣₂,𝑘−𝐸𝑢₁^(𝑛),𝑢₂^(𝑛)𝜙()=(I)+()+()+(),,𝑘𝑡,𝑥𝑑𝑥𝑑𝑡IIIIIIV (32) where we understand that the integral (I) does not exist for 𝑘=0 and the integral (II) for 𝑘=𝑚.

Changing variables 𝑥+𝑡=𝑦 and 𝑥−𝑡=𝑠, the first integral (I) can be bounded as follows: |(I)|≤_ℝ𝑠^𝑠+2𝑇𝑣₁(𝑠)^𝑘−𝑢₁^(𝑛)(𝑠)^𝑘𝑣₂𝑦^𝑚−𝑘×𝑣₁(𝑠)𝜙𝑦,𝑠𝑑𝑦𝑑𝑠≲sup_𝑠𝑠^𝑠+2𝑇𝑣₂𝑦^𝑚−𝑘𝜙×𝑦,𝑠𝑑𝑦_ℝ𝑣₁−𝑢₁^(𝑛)(𝑠×𝑣₁)(𝑠)^𝑘−1+𝑢₁^(𝑛)(𝑠)^𝑘−1𝑣₁(𝑠)𝑑𝑠≲sup_𝑠𝑠^𝑠+2𝑇𝑣₂𝑦^𝑚−𝑘𝜙𝑦,𝑠𝑑𝑦𝑣₁−𝑢₁^(𝑛)_𝐿𝑘+1×𝑣_1𝐿^𝑘−1_𝑘+1+𝑢₁^(𝑛)_𝐿^𝑘−1_𝑘+1𝑣_1𝐿𝑘+1, (33) where we note that 𝑘≥1. We also have sup_𝑠𝑠^𝑠+2𝑇𝑣₂𝑦^𝑚−𝑘𝜙≤𝑦,𝑠𝑑𝑦sup_𝑠∫_𝑠^𝑠+2𝑇𝜙𝑣𝑦,𝑠𝑑𝑦if𝑚−𝑘=0,_2𝐿2sup_𝑠∫_𝑠^𝑠+2𝑇𝜙𝑦,𝑠²𝑑𝑦^1/2𝑀𝑣if𝑚−𝑘=1,_{2𝐿𝑚−𝑘𝑚−𝑘}if𝑚−𝑘≥2, (34) where we denote 𝑀=sup_Σ𝑇|𝜙(𝑡,𝑥)| where Σ_𝑇={(𝑥,𝑡)∣𝑥∈ℝ,0≤𝑡≤𝑇}.

For the integral (II), we consider two cases. Note that the integral (II) disappears if 𝑘=𝑚. For the first case 𝑚−𝑘=1, we have |(II)|≤_ℝ𝑠^𝑠+2𝑇𝑢₁^(𝑛)(𝑠)^𝑘𝑣₂𝑦−𝑢₂^(𝑛)𝑦×𝑣₁(𝑠)𝜙𝑦,𝑠𝑑𝑦𝑑𝑠≲sup_𝑠𝑠^𝑠+2𝑇𝑣₂𝑦−𝑢₂^(𝑛)𝑦𝜙×𝑦,𝑠𝑑𝑦_ℝ𝑢₁^(𝑛)(𝑠)^𝑘𝑣₁(𝑠)𝑑𝑠≲sup_𝑠𝑠^𝑠+2𝑇𝜙𝑦,𝑠²𝑑𝑦^1/2×𝑣₂−𝑢₂^(𝑛)_𝐿2𝑢₁^(𝑛)𝑘_𝐿𝑘+1𝑣_1𝐿𝑘+1, (35) where 𝑘≥1 because 𝑚=1 and 𝑘=0 case is excluded. For the second case 𝑚−𝑘≥2, we have |(II)|≤_ℝ^𝑦_𝑦−2𝑇𝑢₁^(𝑛)(𝑠)^𝑘𝑣₂𝑦^𝑚−𝑘−𝑢₂^(𝑛)𝑦^𝑚−𝑘×𝑣₁(𝑠)𝜙𝑦,𝑠𝑑𝑠𝑑𝑦≲sup_𝑦^𝑦_𝑦−2𝑇𝑢₁^(𝑛)(𝑠)^𝑘𝑣₁(𝑠𝜙×𝑣)𝑦,𝑠𝑑𝑠₂−𝑢₂^(𝑛)_{𝐿𝑚−𝑘}𝑣_2𝐿^{𝑚−𝑘−1}_𝑚−𝑘+𝑢₂^(𝑛)_𝐿^{𝑚−𝑘−1}_𝑚−𝑘, (36) where we have sup_𝑦^𝑦_𝑦−2𝑇𝑢₁^(𝑛)(𝑠)^𝑘𝑣₁(𝑠𝜙≤𝑣)𝑦,𝑠𝑑𝑠_1𝐿2sup_𝑦∫^𝑦_𝑦−2𝑇𝜙𝑦,𝑠²𝑑𝑠^1/2𝑀𝑢if𝑘=0,_{1(𝑛)𝑘𝐿𝑘+1}𝑣_1𝐿𝑘+1if𝑘≥1. (37)

The integral (III) can be treated in a similar way to (II). For the case 𝑘=0, we have |(III)|≲sup_𝑦^𝑦_𝑦−2𝑇𝑣₁−𝑢₁^(𝑛)(𝑠)𝜙𝑦,𝑠𝑑𝑠𝑢₂^(𝑛)𝑚_𝐿𝑚≲sup_𝑦^𝑦_𝑦−2𝑇𝜙𝑦,𝑠²𝑑𝑠^1/2×𝑣₁−𝑢₁^(𝑛)_𝐿2𝑢₂^(𝑛)𝑚_𝐿𝑚. (38) For the case 𝑘≥1, we have |(III)|≲sup_𝑠𝑠^𝑠+2𝑇𝑢₂^(𝑛)𝑦^𝑚−𝑘𝜙×𝑣𝑦,𝑠𝑑𝑦₁−𝑢₁^(𝑛)_𝐿𝑘+1𝑢₁^(𝑛)𝑘_𝐿𝑘+1≲𝑀𝑢₂^(𝑛)_{𝐿𝑚−𝑘}𝑣₁−𝑢₁^(𝑛)_𝐿𝑘+1𝑢₁^(𝑛)𝑘_𝐿𝑘+1. (39)

Taking into account |𝑒^𝑖𝑥−𝑒^𝑖𝑦|≤|𝑥−𝑦| for 𝑥, 𝑦∈ℝ, the integral (IV) can be bounded for the case 𝑘=0: _ℝ^𝑦_𝑦−2𝑇𝑢₁^(𝑛)(𝑠)𝑢₂^(𝑛)𝑦^𝑚×^𝑦_𝑠𝑣₂(ℎ)^𝑚𝑑ℎ−^𝑦_𝑠𝑢₂^(𝑛)(ℎ)^𝑚𝑑ℎ𝜙𝑦,𝑠𝑑𝑠𝑑𝑦≲sup_𝑦,𝑠^𝑦_𝑠𝑣₂(ℎ)^𝑚−𝑢₂^(𝑛)(ℎ)^𝑚×𝑑ℎ_ℝ𝑢₂^(𝑛)𝑦^𝑚𝑦_𝑦−2𝑇𝑢₁^(𝑛)(𝑠)𝜙𝑦,𝑠𝑑𝑠𝑑𝑦, (40) where supremum sup_𝑦,𝑠 is taken over 𝑠∈ℝ and 𝑠≤𝑦≤𝑠+2𝑇. Then, we obtain (IV)≲𝑣₂−𝑢₂^(𝑛)_𝐿𝑚𝑣_2𝐿^𝑚−1_𝑚+𝑢₂^(𝑛)_𝐿^𝑚−1_𝑚×𝑢₂^(𝑛)𝑚_𝐿𝑚𝑢₁^(𝑛)_𝐿2sup_𝑦^𝑦_𝑦−2𝑇𝜙𝑦,𝑠²𝑑𝑠^1/2. (41) For the case 𝑘≥1, we have _ℝ𝑠^𝑠+2𝑇𝑢₁^(𝑛)(𝑠)^𝑘+1𝑢₂^(𝑛)𝑦^𝑚−𝑘×𝑣₁(𝑠)^𝑘𝑦_𝑠𝑣₂(ℎ)^𝑚−𝑘−𝑢𝑑ℎ_1(𝑛)(𝑠)^𝑘𝑦_𝑠𝑢₂^(𝑛)(ℎ)^𝑚−𝑘𝜙≤𝑑ℎ𝑑𝑦𝑑𝑠_{ℝ𝑠𝑠+2𝑇}𝑢₁^(𝑛)(𝑠)^𝑘+1𝑢₂^(𝑛)𝑦^𝑚−𝑘𝑣₁(𝑠)^𝑘×^𝑦_𝑠𝑣₂(ℎ)^𝑚−𝑘−𝑢₂^(𝑛)(ℎ)^𝑚−𝑘𝜙+𝑑ℎ𝑦,𝑠𝑑𝑦𝑑𝑠_{ℝ𝑠𝑠+2𝑇}𝑢₁^(𝑛)(𝑠)^𝑘+1𝑢₂^(𝑛)𝑦^𝑚−𝑘×𝑣₁(𝑠)^𝑘−𝑢₁^(𝑛)(𝑠)^𝑘×^𝑦_𝑠𝑢₂^(𝑛)(ℎ)^𝑚−𝑘𝑑ℎ𝜙𝑦,𝑠𝑑𝑦𝑑𝑠≲sup_𝑦,𝑠^𝑦_𝑠𝑣₂(ℎ)^𝑚−𝑘−𝑢₂^(𝑛)(ℎ)^𝑚−𝑘×𝑑ℎ_ℝ𝑢₁^(𝑛)(𝑠)^𝑘+1𝑣₁(𝑠)^𝑘_𝑠^𝑠+2𝑇𝑢₂^(𝑛)𝑦^𝑚−𝑘𝜙𝑑𝑦𝑑𝑠+sup_𝑦,𝑠^𝑦_𝑠𝑢₂^(𝑛)(ℎ)^𝑚−𝑘𝑑ℎ_ℝ𝑢₁^(𝑛)(𝑠)^𝑘+1×𝑣₁(𝑠)^𝑘−𝑢₁^(𝑛)(𝑠)^𝑘_𝑠^𝑠+2𝑇𝑢₂^(𝑛)𝑦^𝑚−𝑘𝜙𝑑𝑦𝑑𝑠. (42) Considering 𝑠≤𝑦≤𝑠+2𝑇, we can bound (IV) as follows. For the case 𝑚−𝑘=0, we have |(IV)|≲2𝑇𝑢₁^(𝑛)_𝐿^𝑚+1_2𝑚+1𝑣₁−𝑢₁^(𝑛)_𝐿2𝑚+1×𝑣_1𝐿^𝑚−1_2𝑚+1+𝑢₁^(𝑛)_𝐿^𝑚−1_2𝑚+1sup_𝑠𝑠^𝑠+2𝑇𝜙𝑦,𝑠𝑑𝑦. (43) For the case 𝑚−𝑘=1, we have |(IV)|≲√2𝑇𝑣₂−𝑢₂^(𝑛)_𝐿2𝑢₁^(𝑛)𝑚_{𝐿2𝑚−1}𝑣_1𝐿^𝑚−1_2𝑚−1×𝑢₂^(𝑛)_𝐿2sup_𝑠𝑠^𝑠+2𝑇𝜙𝑦,𝑠²𝑑𝑦^1/2. (44) For other cases (𝑘≥1 and 𝑚−𝑘≥2), we finally have |(IV)|≲𝑀𝑣₂−𝑢₂^(𝑛)_{𝐿𝑚−𝑘}𝑣_2𝐿^{𝑚−𝑘−1}_𝑚−𝑘+𝑢₂^(𝑛)_𝐿^{𝑚−𝑘−1}_𝑚−𝑘×𝑢₂^(𝑛)_𝐿^𝑚−𝑘_𝑚−𝑘𝑢₁^(𝑛)_𝐿^𝑘+1_2𝑘+1𝑣_1𝐿^𝑘+1_2𝑘+1+𝑀𝑢₂^(𝑛)_𝐿^{2𝑚−2𝑘}_𝑚−𝑘𝑢₁^(𝑛)_𝐿^𝑘+1_2𝑘+1𝑣₁−𝑢₁^(𝑛)_𝐿2𝑘+1×𝑣_1𝐿^𝑘−1_2𝑘+1+𝑢₁^(𝑛)_𝐿^𝑘−1_2𝑘+1. (45) Then, we have |(1)|+|(2)|+|(3)|→0 as 𝑛→∞ which implies that (19) holds.

Proof of Theorem 3

Here, we prove Theorem 3 which shows the uniqueness of weak solutions to (6). Before proving Theorem 3, we introduce a lemma given in [14] for easy reference. Consider the following equations on the time interval [−𝐿,𝐿]: 𝜕_𝑡+𝜕_𝑥𝑣₊=𝑓₊,𝜕_𝑡−𝜕_𝑥𝑣₋=𝑓₋. (46)

Lemma 6. Suppose that 𝑓_±∈𝐿¹([−𝐿,𝐿]×ℝ) and 𝑣_±∈𝐿^∞([−𝐿,𝐿];𝐿¹(ℝ)) satisfy (46) in the sense of distribution. Then, one has ^𝐿−𝑇_−𝐿+𝑇𝑣₊𝑇,𝑦𝑑𝑦≤^𝐿_−𝐿𝑣₊0,𝑦𝑑𝑦+^𝑇₀^𝐿−𝑡_−𝐿+𝑡𝑓₊𝑡,𝑦𝑑𝑦𝑑𝑡,_{𝐿−𝑇−𝐿+𝑇}𝑣₋≤𝑇,𝑦𝑑𝑦_𝐿−𝐿𝑣₋0,𝑦𝑑𝑦+^𝑇₀^𝐿−𝑡_−𝐿+𝑡𝑓₋𝑡,𝑦𝑑𝑦𝑑𝑡. (47)

Now let 𝑈_𝑗 and 𝑉_𝑗 be two weak solutions of (6) with the same initial data. We define 𝜔_𝑗=𝑈_𝑗−𝑉_𝑗. Then, we have equations for 𝜔_𝑗: 𝜕_𝑡𝜔₁+𝜕_𝑥𝜔₁=𝑖𝑈₁−𝑉₁𝑈₁^𝑘−1𝑈₂^𝑚−𝑘𝑈₁+⋯+𝑖𝑉₁^𝑘𝑈₂−𝑉₂𝑈₂^{𝑚−𝑘−1}𝑈₁+⋯+𝑖𝑉₁^𝑘𝑉₂^{𝑚−𝑘−1}𝑈₂−𝑉₂𝑈₁+𝑖𝑉₁^𝑘𝑉₂^𝑚−𝑘𝜔₁,𝜕_𝑡𝜔₂−𝜕_𝑥𝜔₂=𝑖𝑈₂−𝑉₂𝑈₂^𝑘−1𝑈₁^𝑚−𝑘𝑈₂+⋯+𝑖𝑉₂^𝑘𝑈₁−𝑉₁𝑈₁^{𝑚−𝑘−1}𝑈₂+⋯+𝑖𝑉₂^𝑘𝑉₁^{𝑚−𝑘−1}𝑈₁−𝑉₁𝑈₂+𝑖𝑉₂^𝑘𝑉₁^𝑚−𝑘𝜔₂. (48) Multiplying by complex conjugates, respectively, and taking the real parts, we obtain 𝜕_𝑡𝜔₁²+𝜕_𝑥𝜔₁²=2𝑈₁−𝑉₁𝑈₁^𝑘−1𝑈₂^𝑚−𝑘Im𝑈₁𝜔₁+⋯+2𝑉₁^𝑘𝑉₂^{𝑚−𝑘−1}𝑈₂−𝑉₂Im𝑈₁𝜔₁,𝜕_𝑡𝜔₂²−𝜕_𝑥𝜔₂²=2𝑈₂−𝑉₂𝑈₂^𝑘−1𝑈₁^𝑚−𝑘Im𝑈₂𝜔₂+⋯+2𝑉₂^𝑘𝑉₁^{𝑚−𝑘−1}𝑈₁−𝑉₁Im𝑈₂𝜔₂, (49) where we note that Re(𝑖|𝑉₁|^𝑘|𝑉₂|^𝑚−𝑘|𝜔₁|²)=0=Re(𝑖|𝑉₂|^𝑘|𝑉₁|^𝑚−𝑘|𝜔₂|²). Applying Lemma 6 to (49) and considering 𝜔_𝑗(0,𝑥)=0, we have ^𝐿−𝑇_−𝐿+𝑇𝜔₁𝑡,𝑦²𝑑𝑦≲^𝑇₀𝑈₁−𝑉_1𝐿𝑚+2𝑈₁^𝑘_𝐿𝑚+2𝑈_2𝐿𝑚+2𝜔_1𝐿𝑚+2+⋯+𝑉₁^𝑘_𝐿𝑚+2𝑉_2𝐿^{𝑚−𝑘−1}_𝑚+2𝑈₂−𝑉_2𝐿𝑚+2×𝑈_1𝐿𝑚+2𝜔_1𝐿𝑚+2𝑑𝑡, (50) where we understand ‖𝑓‖_𝐿𝑚+2=(∫^𝐿−𝑡_−𝐿+𝑡|𝑓(𝑡,𝑥)|^𝑚+2𝑑𝑥)^1/(𝑚+2).

To estimate ‖|𝑈_𝑗(𝑡,⋅)|−|𝑉_𝑗(𝑡,⋅)|‖_𝐿𝑚+2(𝑗=1,2) in (50) we multiply (6) by |𝑈_𝑗|^𝑚𝑈_𝑗 to obtain 𝜕_𝑡𝑈₁^𝑚+2+𝜕_𝑥𝑈₁^𝑚+2𝜕=0,_𝑡𝑈₂^𝑚+2−𝜕_𝑥𝑈₂^𝑚+2=0. (51) With the same equations for 𝑉_𝑗, we obtain 𝜕_𝑡𝑈₁^𝑚+2−𝑉₁^𝑚+2+𝜕_𝑥𝑈₁^𝑚+2−𝑉₁^𝑚+2𝜕=0,_𝑡𝑈₂^𝑚+2−𝑉₂^𝑚+2−𝜕_𝑥𝑈₂^𝑚+2−𝑉₂^𝑚+2=0. (52) Applying Lemma 6 and considering |𝑈_𝑗(0,𝑥)|^𝑚+2−|𝑉_𝑗(0,𝑥)|^𝑚+2=0, we have ^𝐿−𝑡_−𝐿+𝑡𝑈_𝑗(𝑡,𝑥)−𝑉_𝑗(𝑡,𝑥)^𝑚+2≤𝑑𝑥_{𝐿−𝑡−𝐿+𝑡}𝑈_𝑗(𝑡,𝑥)^𝑚+2−𝑉_𝑗(𝑡,𝑥)^𝑚+2𝑑𝑥≤0. (53) Considering the inequality (50), we conclude that ∫^𝐿−𝑇_−𝐿+𝑇|𝜔₁(𝑡,𝑦)|²𝑑𝑦=0.

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0015866) and also partially supported by the TJ Park Junior Faculty Fellowship.