If both the arc length and the intrinsic curvature of a curve or surface are preserved, then the flow of the curve or surface is said to be inextensible. The absence of motion-induced strain energy is the physical characteristic of inextensible curve and surface flows. In this paper, we study inextensible tangential, normal and binormal ruled surfaces generated by a curve with constant torsion, which is also called a Salkowski curve. We investigate whether or not these surfaces are minimal or can be developed. In addition, we prove some theorems which are related to inextensible ruled surfaces within three-dimensional Euclidean space.
Citation: Nural Yüksel, Burçin Saltık. On inextensible ruled surfaces generated via a curve derived from a curve with constant torsion[J]. AIMS Mathematics, 2023, 8(5): 11312-11324. doi: 10.3934/math.2023573
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If both the arc length and the intrinsic curvature of a curve or surface are preserved, then the flow of the curve or surface is said to be inextensible. The absence of motion-induced strain energy is the physical characteristic of inextensible curve and surface flows. In this paper, we study inextensible tangential, normal and binormal ruled surfaces generated by a curve with constant torsion, which is also called a Salkowski curve. We investigate whether or not these surfaces are minimal or can be developed. In addition, we prove some theorems which are related to inextensible ruled surfaces within three-dimensional Euclidean space.
This paper considers the Cauchy problem for the coupled non-linear Schrödinger system, shortened to CNLS,
{i∂tuj+Δuj=τ(∑1≤k≤najk|uk|σ)|uj|σ−2uj;u|t=0=u0. | (CNLS) |
Here and hereafter, for j∈[1,n], uj is a complex valued function of the variable (t,y)∈R+×RN, and u:=(u1,…,un). The constant 0≠τ∈C and the coupling parameters satisfy:
ajk=akj≥0 and ajj>0for anyj,k∈[1,n]. |
The coupled Schrödinger system (CNLS) models many physical phenomena, such as the propagation in birefringent optical fibers, Kerr-like photorefractive media in optic and Bose-Einstein condensates [1,4,7], see also [2,3,5,6]. The passage of a light beam with two components along an optical fiber produces the decomposition of the ray into two. Then, the model of a scalar Schrödinger equation can be improved by a system of coupled Shrödinger equations [8]. As a consequence, new kinds of solutions appear, first observed by Manakov [9] in a Kerr medium.
In mathematical point of view, many authors focused their attention on coupled nonlinear equations of Schrödinger type. So, the list of references is necessarily incomplete. The local well-posedness in the energy space was proved in [10,11]. The existence of ground states was investigated in [12,13,14]. The scattering of defocusing global solutions was treated in [15]. The scattering in the repulsive regime with small data was obtained in [16,17]. See also [18,19,20,21] for the inhomogeneous case and [22] for the case of harmonic potential and [23] for the parabolic context.
To the author's knowledge, the above works treat only the case σ≥2 in (CNLS). This is to avoid a singularity of the source term for σ<2. The contribution of this paper is to try to fill in this gap in the literature.
The purpose of this paper is to prove the existence of a local solution to the coupled Schrödinger problem (CNLS) in the case 32<σ<2. Then, one establishes the existence of global solutions which scatter in some Sobolev spaces and the existence of non-global solutions with finite variance.
The paper is organized as follows: Section 2 contains the main results and some standard estimates needed in the sequel. Section 3 proves the local existence of solutions to (CNLS). Section 4 establishes the scattering of global solutions and the existence of non-global solutions to (CNLS).
Throughout this paper, we denote the spaces and norms
Ws,p:=Ws,p(RN),Hs:=Ws,2,Lr:=Lr(RN);‖⋅‖r:=‖⋅‖Lr,‖⋅‖:=‖⋅‖2;‖(u1,…,un)‖p:=(∑1≤j≤n‖uj‖pp)1p,‖(u1,…,un)‖:=‖(u1,…,un)‖2. |
Lastly, T∗>0 denotes the lifespan for an eventual solution to (CNLS).
In this section, we give the main results and some standard estimates.
Solutions of (CNLS) formally satisfy the conservation of the following real quantities, respectively the mass and the energy
M(u):=∑1≤j≤n∫RN|uj(t,y)|2dy;E(u):=∑1≤j≤n(|uj(t,y)|2+|∇uj(t,y)|2dy+τσ∑1≤k≤najk|uj(t,y)uk(t,y)|σ)dy. |
Let the non-linear terms
F:=F(u):=[(∑1≤k≤na1k|uk|σ)|u1|σ−2u1,…,(∑1≤k≤nank|uk|σ)|un|σ−2un];Fj:=Fj(u):=(∑1≤k≤najk|uk|σ)|uj|σ−2uj;Fj,k:=Fj,k(u):=|uk|σ|uj|σ−2uj. |
Take for s∈R,
⟨y⟩2:=1+|y|2andIγ:=(1−Δ)s2. |
Here and hereafter, one defines some real numbers
2−σ<α<min{σ−N2m,p−1}; | (2.1) |
max{N4(σ−1),N2(σ−1−α),N2}<m<N2(2−σ); | (2.2) |
M0>N+m; | (2.3) |
M>4E[N2]+5+m, | (2.4) |
where E[⋅] refers to the integer part. Take the Banach space
Y:={u∈(H−N+M0+M)n,‖u‖Y:=∑1≤j≤n(∑|α|≤E[N2]‖⟨y⟩m∂αuj‖∞+∑E[N2]<|α|≤M‖⟨y⟩m∂αuj‖+∑M<|α|≤−N+M0+M‖∂αuj‖)<∞}. |
Let the centered ball of radius R>0, denoted by BT(R):={u∈CT(Y),‖u‖L∞T(Y)≤R}, and for ν>0, take
Bν,T(R):={u∈BT(R),2inft,x|⟨y⟩muj(t,y)|≥ν,∀j∈[1,n]}. |
Finally, we define the vector Schrödinger group
ei⋅Δ(u1,…,un):=(ei⋅Δu1,…,ei⋅Δun). |
In what follows, we list the results proved in this paper. The first purpose of this note is to prove the existence of local solutions to (CNLS).
Theorem 2.1. Let N≥1, 32<σ<2, and u0∈Y, satisfying
infx∈RN|⟨y⟩mu0(x)|>0. | (2.5) |
Then, there are T>0 and a unique solution to (CNLS) denoted by u∈CT(Y). Moreover, the flow is locally continuous.
Remarks 2.1.
1)The condition σ>32, which seems to be technical, is necessary in order to have (2.1);
2) the continuity of the flow follows with standard arguments;
3) assumption (2.5) avoids in some meaning the singularity of the source term for σ<2;
4) the proof follows ideas in [24], where the scalar case is treated;
5) some tools needed in the proof are taken from [25];
6) the proof is based on a Picard fixed point argument.
The second result is about the scattering of global solutions to (CNLS) in some Sobolev spaces.
Theorem 2.2. Let N≥2 and max{1+1N,32}<σ<2. Take v0∈Y satisfying (2.5), and u0:=eiκt|x|24v0 for some real number κ≫1. Take 0≤s<m−N2. Then, there is a unique global solution to (CNLS) in C(R+,Hs)∩⟨y⟩−N2L∞(R+×RN), which scatters in Hs.
Remarks 2.2.
1)The proof is based on the pseudo-conformal transformation [26] and Theorem 2.1;
2) a similar result was first proved [24] in the scalar case;
3) a part of the proof is omitted because it follows [25];
4) the real number κ depends on ν,‖v0‖Y,N,m.
Finally, one proves the existence of non-global solutions to (CNLS).
Proposition 2.1. Take the assumptions of Theorem 2.1, m>1+1N and τ<0. Then, the solution to (CNLS) is non-global if one of the following statements holds:
1)p≥1+2N and E(u0)<0;
2) E(u0)<d and I(u)<0.
Remarks 2.3.
1)The blow-up in the first case follows with classical variance method;
2) the second case follows like [18,Theorem 2.8];
3) d denotes the ground state energy d:=inf0≠u∈(H1)m{E(u),I(u)=0}, where I(u):=BE(u)−(B−2)‖∇u‖2 and B:=N(σ−1).
Some intermediate results are listed in what follows.
In order to investigate the blow-up of solutions, one needs the next variance identity [27].
Proposition 2.2. Assume that u∈CT([H1]n) is a solution to (CNLS) for τ=−1, and satisfies xuj(t)∈L2, for any 1≤j≤n. Then,
18(∑1≤j≤n‖xuj‖2)″=∑1≤j≤n‖∇uj‖2−B2σ∑1≤j,k≤najk∫RN(|ukuj|)σdy. |
The following estimate will be useful [25,Lemma 2.9].
Lemma 2.1. Let α≥0. Then, for all real number: t, we have
‖⟨y⟩αeitΔu‖≤cα⟨t⟩α(‖Iαu‖+‖⟨y⟩αu‖). |
Recall also [25,Proposition 2.10].
Lemma 2.2. Let α≥0, k,K∈N, and s≥2k+K+3+E[N2]+α. Then, for any t∈R and any |μ|≤K,
‖⟨y⟩α∂μeitΔu‖∞≲⟨t⟩k∑|γ|≤k‖⟨y⟩α∂γu‖∞+⟨t⟩1+k+α(‖Isu‖+∑k<|γ|≤2k+K+3+E[N2]‖⟨y⟩α∂γu‖). |
One recalls the nonlinear estimates [25,Proposition 3.1].
Lemma 2.3. Let α≥0, u∈Bν,T(R), and r∈[1,∞]. Then, for all |μ|≤−N+M0+M, we have
1)
‖⟨y⟩α∂μ(|u|σ)‖L∞((0,T),Lr)≲Rσ‖⟨y⟩α−mσ‖r+∑1≤k≤|μ|ν−(2k−σ)R2k−1(R‖⟨y⟩α−mσ‖r+∑E[N2]<|γ|≤|μ|‖⟨y⟩α+m(1−σ)∂γu‖L∞((0,T),Lr)). |
2)
‖⟨y⟩α∂μ(|u|σ−2u)‖L∞((0,T),Lr)≲∑0≤k≤|μ|ν−(2k+2−σ)R2k(R‖⟨y⟩α−m(σ−1)‖r+∑E[N2]<|γ|≤|μ|‖⟨y⟩α+m(2−σ)∂γu‖L∞((0,T),Lr)). |
Finally, let us give some nonlinear estimates [25,Proposition 3.2].
Lemma 2.4. Let α≥0, u,v∈Bν,T(R), and r∈[1,∞]. Then, for all |μ|≤−N+M0+M, we have
1)
‖⟨y⟩α∂μ(|u|σ−|v|σ)‖L∞((0,T),Lr)≲(Rσ−1+ν−2(3−σ)R5−σ)‖⟨y⟩α−mσ‖r‖u−v‖L∞T(Y)+∑1≤k≤|μ|ν−2(2k−σ)R4k−σ−2(R‖⟨y⟩α−mσ‖r+∑E[N2]<|γ|≤|μ|‖⟨y⟩α+m(1−σ)∂γ(u−v)‖L∞((0,T),Lr))+∑1≤k≤|μ|ν−(2k−σ)(R2k−1[‖u−v‖L∞T(Y)‖⟨y⟩α−mσ‖r+∑E[N2]<|γ|≤|μ|‖⟨y⟩α+m(1−σ)∂γ(u−v)‖L∞((0,T),Lr)]+R2(k−1)‖u−v‖L∞T(Y)[∑E[N2]<|γ|≤|μ|‖⟨y⟩α+m(1−σ)∂γu‖L∞((0,T),Lr)+‖⟨y⟩α+m(1−σ)∂γv‖L∞((0,T),Lr)]); |
2)
‖⟨y⟩α∂μ(|u|σ−2u−|v|σ−2v)‖L∞T(L2)≲ν−2(3−σ)R4−σ‖⟨y⟩α−mσ‖r‖u−v‖L∞T(Y)+∑1≤k≤|μ|ν−2(2k+2−σ)R4k−σ+1(R‖⟨y⟩α−m(σ−1)‖r+∑E[N2]<|γ|≤|μ|‖⟨y⟩α+m(2−σ)∂γ(u−v)‖L∞((0,T),Lr))+∑1≤k≤|μ|ν−(2(1+k)−σ)(R2k[‖u−v‖L∞T(Y)‖⟨y⟩α−m(σ−1)‖r+∑E[N2]<|γ|≤|μ|‖⟨y⟩α+m(2−σ)∂γ(u−v)‖L∞((0,T),Lr)]+R2k−1‖u−v‖L∞T(Y)[∑E[N2]<|γ|≤|μ|‖⟨y⟩α+m(2−σ)∂γu‖L∞((0,T),Lr)+‖⟨y⟩α+m(2−σ)∂γv‖L∞((0,T),Lr)]). |
This section is devoted to Theorem 2.1, which deals with the existence of a unique solution of (CNLS) in CT(Y). Take the function
f(u):=ei⋅Δu0−iτ∫⋅0ei(⋅−s)ΔFds:=(f1(u),…,fm(u)), |
where u0:=(u0,1,…,u0,n), and
fj(u):=ei⋅Δu0,j−iτ∫⋅0ei(⋅−s)ΔFjds. |
Let us prove that f(Bν,T(R))⊂Bν,T(R). By Lemma 2.2, for the choice
k:=E[N2],s=−N+M0+M:=K+2k+3+E[N2]+M0−N, |
and for |α|≤E[N2], we have
‖⟨y⟩m∂αfj(u(t,y))‖L∞T(L∞)≲⟨T⟩E[N2]+1+m(∑|γ|≤E[N2]‖⟨y⟩m∂γu0‖∞+‖I−N+M0+Mu0‖+∑E[N2]<|γ|≤M‖⟨y⟩m∂γu0‖)+T⟨T⟩E[N2]+1+m(∑|γ|≤E[N2]‖⟨y⟩m∂γF‖L∞T(L∞)+‖I−N+M0+MF‖L∞T(L2)+∑E[N2]<|γ|≤M‖⟨y⟩m∂γF‖L∞T(L2)). |
Because p≥1, by Lemma 2.3, for |γ1|≤E[N2], we have
‖⟨y⟩m(2−σ)∂γ1(|uk|σ)‖L∞T(L∞)≲Rσ‖⟨y⟩−2m(σ−1)‖∞+∑1≤l≤E[N2]ν−(2l−σ)R2l−1(R‖⟨y⟩−2m(σ−1)‖∞+∑E[N2]<|γ|≤|γ1|‖⟨y⟩m(3−2σ)∂γuk‖L∞T(L∞))≲Rσ+∑1≤l≤E[N2]ν−(2l−σ)R2l. |
Also, by Lemma 2.3, we get
‖⟨y⟩m(σ−1)∂γ2(|uj|σ−2uj)‖L∞T(L∞)≲∑0≤l≤E[N2]ν−(2l+2−σ)R1+2l. |
Thus, with the Leibniz rule, if |γ|≤E[N2], we get
‖⟨y⟩m∂γF‖L∞T(L∞)≲n∑j,k=1∑γ=γ1+γ2‖⟨y⟩m(2−σ)∂γ1(|uk|p)‖L∞T(L∞)‖⟨y⟩m(σ−1)∂γ2(|uj|σ−2uj)‖L∞T(L∞)≲(Rσ+∑1≤l≤E[N2]ν−(2l−σ)R2l)(∑0≤l≤E[N2]ν−(2l+2−σ)R1+2l). | (3.1) |
Now, with the Leibniz rule via Lemma 2.3, for E[N2]≤|γ|≤M, 12=1r1+1r2, and α∈R,
‖⟨y⟩m∂γF‖L∞T(L2)≲n∑j,k=1∑γ=γ1+γ2,|γ2|≤N‖⟨y⟩mα∂γ1(|uk|p)‖L∞((0,T),Lr1)‖⟨y⟩m(1−α)∂γ2(|uj|σ−2uj)‖L∞((0,T),Lr2)+n∑j,k=1∑γ=γ1+γ2,|γ2|≥N‖⟨y⟩mα∂γ1(|uk|p)‖L∞((0,T),Lr1)‖⟨y⟩m(1−α)∂γ2(|uj|σ−2uj)‖L∞((0,T),Lr2):=(I|γ2|≤N)+(I|γ2|≥N). |
Now, (2.2) enables to take 2−σ<α<σ−N2m, r1=2, and r2=∞. This implies that ⟨y⟩m(α−σ)∈L2. Thus, we have
(I|γ2|≤N)≲n∑j,k=1∑γ=γ1+γ2[Rσ‖⟨y⟩m(α−σ)‖+∑1≤l≤|γ1|ν−(2l−σ)R2l−1(R‖⟨y⟩m(α−σ)‖+∑E[N2]<|μ|≤|γ1|‖⟨y⟩m(α+1−σ)∂μuk‖L∞T(L2))][∑0≤l≤|γ2|ν−(2l+2−σ)R2l(R‖⟨y⟩m(2−α−σ)‖∞+∑E[N2]<|μ|≤|γ2|‖⟨y⟩m(3−α−σ)∂μuj‖L∞T(L∞)]≲n∑j,k=1∑γ=γ1+γ2[Rσ+∑1≤l≤|γ1|ν−(2l−σ)R2l−1(R+∑E[N2]<|μ|≤|γ1|‖⟨y⟩m∂μuk‖L∞T(L2))][∑0≤l≤|γ2|ν−(2l+2−σ)R2l(R+∑E[N2]<|μ|≤|γ2|‖⟨y⟩m∂μuj‖L∞T(L∞))]≲n∑j,k=1∑γ=γ1+γ2[Rσ+∑1≤l≤|γ1|ν−(2l−σ)R2l][∑0≤l≤|γ2|ν−(2l+2−σ)R2l(R+∑E[N2]<|μ|≤|γ2|+E[N2]+1‖⟨y⟩m∂μuj‖L∞T(L2))]. |
Assuming that |γ2|≤N, M≥N+E[N2]+1 gives
(I|γ2|≤N)≲n∑j,k=1∑γ=γ1+γ2[Rσ+∑1≤l≤|γ1|ν−(2l−σ)R2l][∑0≤l≤|γ2|ν−(2l+2−σ)R2l(R+∑E[N2]<|μ|≤|γ2|+E[N2]+1‖⟨y⟩m∂μuj‖L∞T(L2))]≲n∑j,k=1∑γ=γ1+γ2[Rσ+∑1≤l≤|γ1|ν−(2l−σ)R2l][∑1≤l≤|γ2|ν−(2l+1−σ)R2l(R+∑E[N2]<|μ|≤M‖⟨y⟩m∂μuj‖L∞T(L2))]≲n∑j,k=1∑γ=γ1+γ2[Rσ+∑1≤l≤Mν−(2l−σ)R2l][∑0≤l≤Nν−(2l+2−σ)R1+2l]. | (3.2) |
Now, we assume that |γ2|≥N, thus |γ1|≤M−N≤M−E[N2]−1. Since by (2.2) we have m>N2(σ−1−α) and ‖⟨y⟩m(1+α−σ)‖<∞. So, with Lemma 2.3, we get
(I|γ2|≥N)≲n∑j,k=1∑γ=γ1+γ2[Rσ‖⟨y⟩m(α−σ)‖∞+∑1≤l≤|γ1|ν−(2l−σ)R2l−1(R‖⟨y⟩m(α−σ)‖∞+∑E[N2]<|μ|≤|γ1|‖⟨y⟩m(α+1−σ)∂μuk‖L∞T(L∞))][∑0≤l≤|γ2|ν−(2l+2−σ)R2l(R‖⟨y⟩m(1+α−σ)‖+∑E[N2]<|μ|≤|γ2|‖⟨y⟩m(3−α−σ)∂μuj‖L∞T(L2))]≲n∑j,k=1∑γ=γ1+γ2[Rσ+∑1≤l≤|γ1|ν−(2l−σ)R2l−1(R+∑E[N2]<|μ|≤|γ1|‖⟨y⟩m(α+1−σ)∂μuk‖L∞T(L∞))][∑0≤l≤|γ2|ν−(2l+2−σ)R2l(R+∑E[N2]<|μ|≤|γ2|‖⟨y⟩m(3−α−σ)∂μuj‖L∞T(L2))]. |
So,
(I|γ2|≥N)≲n∑j,k=1∑γ=γ1+γ2[Rσ+∑1≤l≤|γ1|ν−(2l−σ)R2l(R+∑1+2E[N2]<|μ|≤M‖∂μuk‖L∞T(L2))][∑0≤l≤|γ2|ν−(2l+2−σ)R1+2l]≲n∑j,k=1∑γ=γ1+γ2[Rσ+∑1≤l≤Mν−(2l−σ)R1+2l][∑0≤l≤Mν−(2l+2−σ)R1+2l]. |
It follows that
‖⟨y⟩m∂γF‖L∞T(L2)≲n∑j,k=1∑γ=γ1+γ2[Rσ+∑1≤l≤Mν−(2l−σ)R1+2l][∑0≤l≤Mν−(2l+2−σ)R1+2l]. | (3.3) |
We break down the next term as follows:
‖∂γF‖L∞T(L2)≲n∑j,k=1∑γ=γ1+γ2,|γ1|<N‖⟨y⟩m(2−σ)∂γ1(|uk|p)‖L∞T(L∞)‖⟨y⟩−m(2−σ)∂γ2(|uj|σ−2uj)‖L∞T(L2)+n∑j,k=1∑γ=γ1+γ2,N≤|γ1|≤M‖⟨y⟩m(2−σ)∂γ1(|uk|p)‖L∞T(L2)‖⟨y⟩−m(2−σ)∂γ2(|uj|σ−2uj)‖L∞T(L∞)+n∑j,k=1∑γ=γ1+γ2,M<|γ1|≤−N+M0+M‖∂γ1(|uk|p)‖L∞T(L2)‖∂γ2(|uj|σ−2uj)‖L∞T(L∞):=(A|γ1|<N)+(AN≤|γ1|≤M)+(AM<|γ1|≤−N+M0+M). | (3.4) |
Now, with the Leibniz rule and Lemma 2.3, for |γ|≤−N+M0+M, we have
(A|γ1|<N)≲n∑j,k=1∑γ=γ1+γ2‖⟨y⟩m(2−σ)∂γ1(|uk|p)‖L∞T(L∞)‖⟨y⟩−m(2−σ)∂γ2(|uj|σ−2uj)‖L∞T(L2)≲n∑j,k=1∑γ=γ1+γ2[Rσ‖⟨y⟩2m(1−σ)‖∞+∑1≤l≤|γ1|ν−(2l−σ)R2l−1(R‖⟨y⟩2m(1−σ)‖∞+∑E[N2]<|μ|≤|γ1|‖⟨y⟩m(3−2σ)∂μuk‖L∞T(L∞))][∑0≤l≤|γ2|ν−(2l+2−σ)R2l(R‖⟨y⟩−m‖+∑E[N2]<|μ|≤|γ2|‖∂μuj‖L∞T(L2)]. | (3.5) |
(2.2) gives ⟨y⟩−m∈L2, and
(A|γ1|<N)≲n∑j,k=1∑γ=γ1+γ2[Rσ+∑1≤l≤|γ1|ν−(2l−σ)R2l−1(R+∑E[N2]<|μ|≤|γ1|‖⟨y⟩m(3−2σ)∂μuk‖L∞T(L∞))][∑0≤l≤|γ2|ν−(2l+2−σ)R2l(R+∑E[N2]<|μ|≤M‖⟨y⟩m∂μuj‖L∞T(L2)+∑M<|μ|≤−N+M0+M‖∂μuj‖L∞T(L2))]≲n∑j,k=1∑γ=γ1+γ2[Rσ+∑1≤l≤|γ1|ν−(2l−σ)R2l−1(R+∑E[N2]<|μ|≤|γ1|‖⟨y⟩m(3−2σ)∂μuk‖L∞T(L∞))][∑0≤l≤|γ2|ν−(2l+2−σ)R1+2l]. | (3.6) |
By Sobolev injection, we have
‖⟨y⟩m(3−2σ)∂μuk‖L∞T(L∞)≲‖∂μuk‖L∞T(L∞)≲‖∂μuk‖L∞((0,T),˙H1+E[N2]). |
Thus, by (2.2), we get
(A|γ1|<N)≲n∑j,k=1∑γ=γ1+γ2[Rσ+∑1≤l≤|γ1|ν−(2l−σ)R2l−1(R+∑1+2E[N2]<|μ|≤1+N+E[N2]‖∂μuk‖L∞T(L2))][∑0≤l≤|γ2|ν−(2l+2−σ)R1+2l]≲n∑j,k=1∑γ=γ1+γ2[Rσ+∑1≤l≤Nν−(2l−σ)R2l][∑0≤l≤−N+M0+Mν−(2l+2−σ)R1+2l]. | (3.7) |
Now, with Lemma 2.3, we have
(AN≤|γ1|≤M)≲n∑j,k=1∑γ=γ1+γ2‖⟨y⟩m(2−σ)∂γ1(|uk|p)‖L∞T(L2)‖⟨y⟩−m(2−σ)∂γ2(|uj|σ−2uj)‖L∞T(L∞)≲n∑j,k=1∑γ=γ1+γ2[Rσ‖⟨y⟩2m(1−σ)‖+∑1≤l≤|γ1|ν−(2l−σ)R2l−1(R‖⟨y⟩2m(1−σ)‖+∑E[N2]<|μ|≤|γ1|‖⟨y⟩m(3−2σ)∂μuk‖L∞T(L2))][∑0≤l≤|γ2|ν−(2l+2−σ)R2l(R‖⟨y⟩−m‖∞+∑E[N2]<|μ|≤|γ2|‖∂μuj‖L∞T(L∞)]. |
Moreover, (2.2) gives ⟨y⟩2m(1−σ)∈L2, and so
(AN≤|γ1|≤M)≲n∑j,k=1∑γ=γ1+γ2[Rσ+∑1≤l≤|γ1|ν−(2l−σ)R2l−1(R+∑E[N2]<|μ|≤M‖⟨y⟩m(3−2σ)∂μuk‖L∞T(L2))][∑0≤l≤|γ2|ν−(2l+2−σ)R2l(R+∑E[N2]<|μ|≤|γ2|‖∂μuj‖L∞T(L∞))]≲n∑j,k=1∑γ=γ1+γ2[Rσ+∑1≤l≤|γ1|ν−(2l−σ)R2l][∑0≤l≤|γ2|ν−(2l+2−σ)R2l(R+∑E[N2]<|μ|≤1+E[N2]+|γ2|‖∂μuj‖L∞T(L2))]. | (3.8) |
Since N+|γ2|≤|γ1|+|γ2|≤−N+M0+M, we have |γ2|≤M+M0−2N, and
(AN≤|γ1|≤M)≲n∑j,k=1∑γ=γ1+γ2[Rσ+∑1≤l≤|γ1|ν−(2l−σ)R2l][∑0≤l≤|γ2|ν−(2l+2−σ)R2l(R+∑E[N2]<|μ|≤1+E[N2]+|γ2|‖∂μuj‖L∞T(L2))]≲n∑j,k=1∑γ=γ1+γ2[Rσ+∑1≤l≤|γ1|ν−(2l−σ)R2l][∑0≤l≤|γ2|ν−(2l+2−σ)R2l(R+∑E[N2]<|μ|≤−N+M0+M‖∂μuj‖L∞T(L2))]≲n∑j,k=1∑γ=γ1+γ2[Rσ+∑1≤l≤Mν−(2l−σ)R2l][∑0≤l≤−N+M0+Mν−(2l+2−σ)R1+2l]. | (3.9) |
Finally, assume that M<|γ1|≤−N+M0+M. Thus, |γ2|≤M0−N≤M−E[N2]−1. So, with Sobolev injections via Lemma 2.3, we have
(AM<|γ1|≤−N+M0+M)≲n∑j,k=1∑γ=γ1+γ2‖∂γ1(|uk|p)‖L∞T(L2)‖∂γ2(|uj|σ−2uj)‖L∞T(L∞)≲[Rσ‖⟨y⟩−mσ‖+∑1≤k≤|γ1|ν−(2k−σ)R2k−1(R‖⟨y⟩−mσ‖+∑E[N2]<|γ|≤|γ1|‖⟨y⟩m(1−σ)∂γu‖L∞T(L2))][∑0≤k≤|γ2|ν−(2k+2−σ)R2k(R‖⟨y⟩−m(σ−1)‖∞+∑E[N2]<|γ|≤|γ2|‖⟨y⟩m(2−σ)∂γu‖L∞T(L∞))]≲[Rσ+∑1≤k≤|γ1|ν−(2k−σ)R2k−1(R+∑E[N2]<|γ|≤|γ1|‖⟨y⟩m(1−σ)∂γu‖L∞T(L2))][∑0≤k≤|γ2|ν−(2k+2−σ)R2k(R+∑E[N2]<|γ|≤|γ2|+1+E[N2]‖⟨y⟩m∂γu‖L∞T(L2))]. |
Thus,
(AM<|γ1|≤−N+M0+M)≲[Rσ+∑1≤k≤|γ1|ν−(2k−σ)R2k−1(R+∑E[N2]<|γ|≤M‖⟨y⟩m∂γu‖L∞T(L2)+∑M<|γ|≤−N+M0+M∂γu‖L∞T(L2))][∑0≤k≤M−E[N2]−1ν−(2k+2−σ)R1+2k]≲[Rσ+∑1≤k≤−N+M0+Mν−(2k−σ)R2k][∑0≤k≤M−E[N2]−1ν−(2k+2−σ)R1+2k]. | (3.10) |
Collecting (3.7), (3.9), and (3.10), we get
‖IM+M0−NF‖L∞T(L2)≲[Rσ+∑1≤k≤−N+M0+Mν−(2k−σ)R2k][∑0≤k≤−N+M0+Mν−(2k+2−σ)R1+2k]. | (3.11) |
Thus, with (3.1)–(3.3) and (3.11), we get
‖⟨y⟩m∂αfj(u)‖L∞T(L∞)≲⟨T⟩E[N2]+1+m(∑|γ|≤E[N2]‖⟨y⟩m∂γu0‖∞+‖I−N+M0+Mu0‖+∑E[N2]<|γ|≤M‖⟨y⟩m∂γu0‖)+T⟨T⟩E[N2]+1+m[Rσ+∑1≤l≤−N+M0+Mν−(2l−σ)R2l][∑0≤l≤−N+M0+Mν−(2l+2−σ)R1+2l]. |
Now, by Lemma 2.1, (3.3), and (3.11), we write
‖⟨y⟩m∂αfj(u)‖L∞T(L2)≲⟨T⟩m(‖I−N+M0+Mu0‖+‖⟨y⟩m∂αu0‖)+T⟨T⟩m(‖I−N+M0+MF‖L∞T(L2)+‖⟨y⟩m∂αF‖L∞T(L2))≲⟨T⟩m(‖I−N+M0+Mu0‖+‖⟨y⟩m∂αu0‖)+T⟨T⟩m[Rσ+∑1≤l≤−N+M0+Mν−(2l−σ)R2l][∑0≤l≤−N+M0+Mν−(2l+2−σ)R1+2l]. | (3.12) |
Moreover, with Lemma 2.1 and (3.11), for |α|≤−N+M0+M, yields
‖∂αfj(u)‖L∞T(L2)≲‖I−N+M0+Mu0‖+T‖∂αF‖L∞T(L2)≲‖u0‖H−N+M0+M+T[Rσ+∑1≤l≤−N+M0+Mν−(2l−σ)R2l][∑0≤l≤−N+M0+Mν−(2l+2−σ)R1+2l]. |
So, taking R:=2c‖u0‖Y, we get
‖f(u)‖L∞T(Y)≤c⟨T⟩E[N2]+1+m‖u0‖Y+cT⟨T⟩E[N2]+1+m[Rσ+∑1≤l≤−N+M0+Mν−(2l−σ)R2l][∑0≤l≤−N+M0+Mν−(2l+2−σ)R1+2l]≤⟨T⟩E[N2]+1+mR2+cT⟨T⟩E[N2]+1+m[Rσ+∑1≤l≤−N+M0+Mν−(2l−σ)R2l][∑0≤l≤−N+M0+Mν−(2l+2−σ)R1+2l]:=⟨T⟩E[N2]+1+mR2+cT⟨T⟩E[N2]+1+mF1(ν,R)F2(ν,R). | (3.13) |
Then, choosing 0<T<<1, it follows that f(BT(R))⊂BT(R). Now, we prove that
2inf(t,y)∈[0,T]×RN|⟨y⟩mf(u(t,y))|≥ν. |
Using the time derivative identity (eitΔ)(k)=(iΔ)keitΔ, we get
eitΔ=∑0≤k≤E[N2](it)kk!Δk+i1+E[N2](E[N2])!∫t0(t−s)E[N2]Δ1+E[N2](eisΔ)ds. |
Thus, with Lemma 2.1 and Sobolev injections via (2.2), we write
‖⟨y⟩m(eitΔu0−u0)‖∞≲∑1≤j≤n∑1≤k≤E[N2]tk‖⟨y⟩mΔku0,j‖∞+∫t0(t−s)E[N2]‖⟨y⟩mΔ1+E[N2](eisΔu0,j)‖∞ds≲t⟨t⟩E[N2](∑1≤|α|≤E[N2]‖⟨y⟩m∂αu0‖∞+∑E[N2]<|α|≤N‖⟨y⟩m∂αu0‖∞)+∫t0(t−s)E[N2]‖⟨y⟩mΔ1+E[N2](eisΔu0)‖∞ds≲t⟨t⟩E[N2](∑1≤|α|≤E[N2]‖⟨y⟩m∂αu0‖∞+∑E[N2]<|α|≤M‖⟨y⟩m∂αu0‖)+∫t0(t−s)E[N2]‖⟨y⟩mΔ1+E[N2](eisΔu0)‖∞ds. |
Now, by Lemma 2.2, for s=5+m+5E[N2] and K=2(1+E[N2]),
‖⟨y⟩mΔ1+E[N2](eisΔu0)‖∞≲⟨t⟩E[N2]∑|γ|≤E[N2]‖⟨y⟩m∂γu‖∞+⟨t⟩1+E[N2]+m(‖Isu‖+∑E[N2]<|γ|≤2(1+E[N2])+3+3E[N2]‖⟨y⟩m∂γu‖). |
Now, since M≥4E[N2]+5+m and M0−N>m>N2, we get
‖⟨y⟩mΔ1+E[N2](eisΔu0)‖∞≲⟨t⟩E[N2]∑|γ|≤E[N2]‖⟨y⟩m∂γu‖∞+⟨t⟩1+E[N2]+m(‖I−N+M0+Mu‖+∑E[N2]<|γ|≤5+5E[N2]‖⟨y⟩m∂γu‖)≲⟨t⟩E[N2]∑|γ|≤E[N2]‖⟨y⟩m∂γu‖∞+⟨t⟩1+E[N2]+m(‖I−N+M0+Mu‖+∑E[N2]<|γ|≤M‖⟨y⟩m∂γu‖). |
Thus,
‖⟨y⟩m(eitΔu0−u0)‖∞≲t⟨t⟩E[N2](∑1≤|α|≤E[N2]‖⟨y⟩m∂αu0‖∞+∑E[N2]<|α|≤M‖⟨y⟩m∂αu0‖)+t⟨t⟩E[N2][⟨t⟩E[N2]∑|γ|≤E[N2]‖⟨y⟩m∂γu‖∞+⟨t⟩1+E[N2]+m(‖I−N+M0+Mu‖+∑E[N2]<|γ|≤M‖⟨y⟩m∂γu‖)]. |
Now, with Lemma 2.2 with (3.13),
‖⟨y⟩m(f(u)−eitΔu0)‖L∞T(L∞)≤cT⟨T⟩E[N2]+1+mF1(ν,R)F2(ν,R). |
Thus, there is C(T)→0 as T→0 such that
|⟨y⟩mf(u)|≥|⟨y⟩meitΔu0|−|⟨y⟩m(eitΔu0−u0)|−|⟨y⟩m(f(u)−eitΔu0)|≥ν−C(T). | (3.14) |
So, taking 0<T<<1, we get
inf{(t,y)∈[0,T]×RN}|⟨y⟩mf(u)|≥ν2. |
Thus, f(Bν,T(R))⊂Bν,T(R). Now, we prove that f is a contraction. For u,v∈CT(Y) and w:=u−v, we have
‖f(u)−f(v)‖L∞T(Y)=‖∫⋅0ei(⋅−s)Δ[F(u)−F(v)]ds‖L∞T(Y)≲‖∫⋅0ei(⋅−s)Δ[(|uk|σ|u1|σ−2u1−|vk|σ|v1|σ−2v1,…,|uk|σ|un|σ−2un−|vk|σ|vn|σ−2vn)]ds‖L∞T(Y)≲∑1≤j≤n(∑|α|≤E[N2]‖∫⋅0⟨y⟩m∂αei(⋅−s)Δ[(|uk|σ|uj|σ−2uj−|vk|σ|vj|σ−2vj)]‖L∞T(L∞)+∑E[N2]<|α|≤M‖∫⋅0⟨y⟩m∂αei(⋅−s)Δ[(|uk|σ|uj|σ−2uj−|vk|σ|vj|σ−2vj)]‖L∞T(L2)+∑M<|α|≤−N+M0+M‖∫⋅0∂αei(⋅−s)Δ[(|uk|σ|uj|σ−2uj−|vk|σ|vj|σ−2vj)]‖L∞T(L2)). |
Let us control the three above terms. By Lemma 2.2 via (2.4), we have
(I):=∑|α|≤E[N2]‖∫⋅0⟨y⟩m∂αei(⋅−s)Δ[(|uk|σ|uj|σ−2uj−|vk|σ|vj|σ−2vj)]‖L∞T(L∞)≲T⟨T⟩E[N2]+1+m(∑|γ|≤E[N2]‖⟨y⟩m∂γ(Fj(u)−Fj(v))‖L∞T(L∞)+‖I−N+M0+M(Fj(u)−Fj(v))‖L∞T(L2)+∑E[N2]<|γ|≤M‖⟨y⟩m∂γ(Fj(u)−Fj(v))‖L∞T(L2)). |
Let |γ|≤E[N2], and write
(I1):=‖⟨y⟩m∂γ(Fj(u)−Fj(v))‖L∞T(L∞)=‖⟨y⟩m∂γ((|uk|σ−|vk|σ)|uj|σ−2uj+|vk|σ(|uj|σ−2uj−|vj|σ−2vj))‖L∞T(L∞)≲∑γ=γ1+γ2(‖⟨y⟩m∂γ1(|uk|σ−|vk|σ)∂γ2(|uj|σ−2uj)‖L∞T(L∞)+‖⟨y⟩m∂γ1|vk|σ∂γ2(|uj|σ−2uj−|vj|σ−2vj)‖L∞T(L∞))≲∑γ=γ1+γ2(‖⟨y⟩m(2−σ)∂γ1(|uk|σ−|vk|σ)‖L∞T(L∞)‖⟨y⟩m(σ−1)∂γ2(|uj|σ−2uj)‖L∞T(L∞)+‖⟨y⟩m(2−σ)∂γ1|vk|σ‖L∞T(L∞)‖⟨y⟩m(σ−1)∂γ2(|uj|σ−2uj−|vj|σ−2vj)‖L∞T(L∞)). |
Taking account of Lemma 2.4, we have
‖⟨y⟩m(2−σ)∂γ1(|uk|σ−|vk|σ)‖L∞T(L∞)≲(Rσ−1+ν−2(3−σ)R5−σ)‖⟨y⟩2m(1−σ)‖∞‖w‖L∞T(Y)+∑1≤k≤E[N2]ν−2(2k−σ)R4k−σ−1‖⟨y⟩2m(1−σ)‖∞‖w‖L∞T(Y)+∑1≤k≤E[N2]ν−(2k−σ)R2k−1‖⟨y⟩2m(1−σ)‖∞‖w‖L∞T(Y)+∑1≤k≤E[N2]ν−(2k−σ)R2(k−1)‖w‖L∞T(Y)≲[Rσ−1+ν−2(3−σ)R5−σ+∑1≤k≤E[N2](ν−2(2k−σ)R4k−σ−1+ν−(2k−σ)R2k−1)]‖w‖L∞T(Y). |
Moreover, also with Lemma 2.4, we have
‖⟨y⟩m(σ−1)∂γ2(|uk|σ−2uk−|vk|σ−2vk)‖L∞T(L∞)≲[ν−2(3−σ)R4−σ+∑1≤k≤E[N2](ν−2(2(1+k)−σ)R4k−σ+2+ν−(2(1+k)−σ)R2k)]‖w‖L∞T(Y). |
Now, by (3.1), it follows that
(I1)≲([Rσ−1+ν−2(3−σ)R5−σ+∑1≤k≤E[N2](ν−2(2k−σ)R4k−σ−1+ν−(2k−σ)R2k−1)]F2(ν,R)+[ν−2(3−σ)R4−σ+∑1≤k≤E[N2](ν−2(2(1+k)−σ)R4k−σ+2+ν−(2(1+k)−σ)R2k)]F1(ν,R))‖w‖L∞T(Y). | (3.15) |
Let E[N2]<γ≤M. Then,
(I2):=‖⟨y⟩m∂γ(Fj(u)−Fj(v))‖L∞T(L2)=‖⟨y⟩m∂γ((|uk|σ−|vk|σ)|uj|σ−2uj+|vk|σ(|uj|σ−2uj−|vj|σ−2vj))‖L∞T(L2)≲∑γ=γ1+γ2(‖⟨y⟩m∂γ1(|uk|σ−|vk|σ)∂γ2(|uj|σ−2uj)‖L∞T(L2)+‖⟨y⟩m∂γ1|vk|σ∂γ2(|uj|σ−2uj−|vj|σ−2vj)‖L∞T(L2))≲∑γ=γ1+γ2(‖⟨y⟩mα∂γ1(|uk|σ−|vk|σ)‖L∞T(L2)‖⟨y⟩m(1−α)∂γ2(|uj|σ−2uj)‖L∞T(L∞)+‖⟨y⟩mα∂γ1|vk|σ‖L∞T(L2)‖⟨y⟩m(1−α)∂γ2(|uj|σ−2uj−|vj|σ−2vj)‖L∞T(L∞)). |
Now, by Lemma 2.4 and using (2.1), we get
‖⟨y⟩mα∂γ1(|uk|σ−|vk|σ)‖L∞T(L2)≲(Rσ−1+ν−2(3−σ)R5−σ)‖⟨y⟩m(α−σ)‖‖w‖L∞T(Y)+∑1≤k≤Mν−2(2k−σ)R4k−σ−2(R‖⟨y⟩m(α−σ)‖+∑E[N2]<|μ|≤|γ1|‖⟨y⟩m(1+α−σ)∂μuk‖L∞T(L2))‖w‖L∞T(Y)+∑1≤k≤Mν−(2k−σ)(R2k−1(‖⟨y⟩m(α−σ)‖‖w‖L∞T(Y)+∑E[N2]<|μ|≤|γ1|‖⟨y⟩m(1+α−σ)∂μ(uk−vk)‖L∞T(L2))+R2(k−1)(∑E[N2]<|μ|≤|γ1|‖⟨y⟩m(1+α−σ)∂μuk‖L∞T(L2)+∑E[N2]<|μ|≤|γ1|‖⟨y⟩m(1+α−σ)∂μvk‖L∞T(L2))‖w‖L∞T(Y))≲(Rσ−1+ν−2(3−σ)R5−σ+∑1≤k≤Mν−2(2k−σ)R4k−σ−1+∑1≤k≤Mν−(2k−σ)R2k−1)‖w‖L∞T(Y). | (3.16) |
Assume that |γ2|≤N, thus, M≥N+E[N2]+1 gives by Sobolev embedding and Lemma 2.3 via (2.1), we get
‖⟨y⟩m(1−α)∂γ2(|uj|σ−2uj)‖L∞T(L∞)≲∑0≤l≤|γ2|ν−(2l+2−σ)R2l(R‖⟨y⟩m(2−α−σ)‖∞+∑E[N2]<|μ|≤|γ2|‖⟨y⟩m(3−α−σ)∂μuj‖L∞T(L∞))≲∑0≤l≤|γ2|ν−(2l+2−σ)R2l(R+∑E[N2]<|μ|≤|γ2|+1+E[N2]‖⟨y⟩m∂μuj‖L∞T(L2))≲∑0≤l≤Mν−(2l+2−σ)R1+2l≲F2(ν,R). |
Moreover, with Lemma 2.4, we write
‖⟨y⟩m(1−α)∂γ2(|uj|σ−2uj−|vj|σ−2vj)‖L∞T(L∞)≲ν−3(2−σ)R4−σ‖⟨y⟩m(2−α−σ)‖∞‖w‖L∞T(Y)+∑1≤k≤Mν−2(2(1+k)−σ)R4k−σ+1(R‖⟨y⟩m(2−α−σ)‖∞+∑E[N2]<|μ|≤|γ2|‖⟨y⟩m(3−α−σ)∂μuk‖L∞T(L∞))‖w‖L∞T(Y)+∑1≤k≤Mν−(2(1+k)−σ)(R2k(‖⟨y⟩m(2−α−σ)‖∞‖w‖L∞T(Y)+∑E[N2]<|μ|≤|γ2|‖⟨y⟩m(3−α−σ)∂μ(uk−vk)‖L∞T(L∞)+R2k−1(∑E[N2]<|μ|≤|γ2|‖⟨y⟩m(3−α−σ)∂μuk‖L∞T(L∞)+∑E[N2]<|μ|≤|γ2|‖⟨y⟩m(3−α−σ)∂μvk‖L∞T(L∞))‖w‖L∞T(Y)). |
So, with Sobolev embeddings,
‖⟨y⟩m(1−α)∂γ2(|uj|σ−2uj−|vj|σ−2vj)‖L∞T(L∞)≲ν−3(2−σ)R4−σ‖w‖L∞T(Y)+∑1≤k≤Mν−2(2(1+k)−σ)R4k−σ+1(R+∑E[N2]<|μ|≤|γ2|+1+E[N2]‖⟨y⟩m∂μuk‖L∞T(L2))‖w‖L∞T(Y)+∑1≤k≤Mν−(2(1+k)−σ)(R2k(‖w‖L∞T(Y)+∑E[N2]<|μ|≤|γ2|+1+E[N2]‖⟨y⟩m∂μ(uk−vk)‖L∞T(L2)+R2k−1(∑E[N2]<|μ|≤|γ2|+1+E[N2]‖⟨y⟩m∂μuk‖L∞T(L2)+∑E[N2]<|μ|≤|γ2|+1+E[N2]‖⟨y⟩m∂μvk‖L∞T(L2))‖w‖L∞T(Y))≲(ν−3(2−σ)R4−σ+∑1≤k≤Mν−2(2(1+k)−σ)R4k−σ+2+∑1≤k≤Mν−(2(1+k)−σ)R2k)‖w‖L∞T(Y). | (3.17) |
Finally, with Lemma 2.3, we have
‖⟨y⟩mα∂γ1|vk|σ‖L∞T(L2)≲Rσ‖⟨y⟩m(α−σ)‖+∑1≤k≤|γ1|ν−(2k−σ)R2k−1(R‖⟨y⟩m(α−σ)‖+∑E[N2]<|γ|≤|γ1|‖⟨y⟩m(1+α−σ)∂γu‖L∞T(L2))≲Rσ+∑1≤k≤|γ1|ν−(2k−σ)R2k−1(R+∑E[N2]<|γ|≤|γ1|‖⟨y⟩m(1+α−σ)∂γu‖L∞T(L2))≲Rσ+∑1≤k≤|γ1|ν−(2k−σ)R2k≲F1(ν,R). |
Collecting the above estimates, we get
∑γ=γ1+γ2,|γ2|≤N(‖⟨y⟩mα∂γ1(|uk|σ−|vk|σ)‖L∞T(L2)‖⟨y⟩m(1−α)∂γ2(|uj|σ−2uj)‖L∞T(L∞)+‖⟨y⟩mα∂γ1|vk|σ‖L∞T(L2)‖⟨y⟩m(1−α)∂γ2(|uj|σ−2uj−|vj|σ−2vj)‖L∞T(L∞))≲[(Rσ−1+ν−2(3−σ)R5−σ+∑1≤k≤Mν−2(2k−σ)R4k−σ−1+∑1≤k≤Mν−(2k−σ)R2k−1)F2(ν,R)+F1(ν,R)(ν−3(2−σ)R4−σ+∑1≤k≤Mν−2(2(1+k)−σ)R4k−σ+2+∑1≤k≤Mν−(2(1+k)−σ)R2k)]‖w‖L∞T(Y). | (3.18) |
Now, assume that M≥|γ2|≥N. Thus, arguing as in (3.3), we have
∑γ=γ1+γ2,M≥|γ2|≥N(‖⟨y⟩mα∂γ2(|uk|σ−|vk|σ)‖L∞T(L2)‖⟨y⟩m(1−α)∂γ1(|uj|σ−2uj)‖L∞T(L∞)+‖⟨y⟩mα∂γ2|vk|σ‖L∞T(L2)‖⟨y⟩m(1−α)∂γ1(|uj|σ−2uj−|vj|σ−2vj)‖L∞T(L∞))≲∑γ=γ1+γ2,M≥|γ2|≥N(‖⟨y⟩mα∂γ2(|uk|σ−|vk|σ)‖L∞T(L2)F2(ν,R)+‖⟨y⟩m(1−α)∂γ1(|uj|σ−2uj−|vj|σ−2vj)‖L∞T(L∞)F1(ν,R)). |
Moreover, since |γ1|≤M−N≤M−E[N2]−1, by (3.16) and (3.17), we write
∑γ=γ1+γ2,M≥|γ2|≥N(‖⟨y⟩mα∂γ2(|uk|σ−|vk|σ)‖L∞T(L2)‖⟨y⟩m(1−α)∂γ1(|uj|σ−2uj)‖L∞T(L∞)+‖⟨y⟩mα∂γ2|vk|σ‖L∞T(L2)‖⟨y⟩m(1−α)∂γ1(|uj|σ−2uj−|vj|σ−2vj)‖L∞T(L∞))≲∑γ=γ1+γ2((Rσ−1+ν−2(3−σ)R5−σ+∑1≤k≤Mν−2(2k−σ)R4k−σ−1+∑1≤k≤Mν−(2k−σ)R2k−1)F2(ν,R)‖w‖L∞T(Y)+(ν−3(2−σ)R4−σ+∑1≤k≤Mν−2(2(1+k)−σ)R4k−σ+2+∑1≤k≤Mν−(2(1+k)−σ)R2k)F1(ν,R))‖w‖L∞T(Y). | (3.19) |
Now, with the Leibniz rule via Lemma 2.3, for |γ|≤−N+M0+M, we have
(I3):=‖∂γ(Fj(u)−Fj(v))‖L∞T(L2)≲∑γ=γ1+γ2(‖⟨y⟩m(2−σ)∂γ1(|uk|σ−|vk|σ)‖L∞T(L∞)‖⟨y⟩−m(2−σ)∂γ2(|uj|σ−2uj)‖L∞T(L2)+‖⟨y⟩m(2−σ)∂γ1|vk|σ‖L∞T(L2)‖⟨y⟩−m(2−σ)∂γ2(|uj|σ−2uj−|vj|σ−2vj)‖L∞T(L∞)). |
Let us start with estimating the first term. By (3.6),
(A1):=‖⟨y⟩m(2−σ)∂γ1(|uk|σ−|vk|σ)‖L∞T(L∞)‖⟨y⟩−m(2−σ)∂γ2(|uj|σ−2uj)‖L∞T(L2)≲‖⟨y⟩m(2−σ)∂γ1(|uk|σ−|vk|σ)‖L∞T(L∞)F2(ν,R). |
With Lemma 2.4, we get
‖⟨y⟩m(2−σ)∂γ1(|uk|σ−|vk|σ)‖L∞T(L∞)≲(Rσ−1+ν−2(3−σ)R5−σ)‖⟨y⟩2m(1−σ)‖∞‖w‖L∞T(Y)+∑1≤k≤Mν−2(2k−σ)R4k−σ−2(R‖⟨y⟩2m(1−σ)‖∞+∑E[N2]<|μ|≤|γ1|‖⟨y⟩m(3−2σ)∂μuk‖L∞T(L∞))‖w‖L∞T(Y)+∑1≤k≤Mν−(2k−σ)(R2k−1(‖⟨y⟩2m(1−σ)‖∞‖w‖L∞T(Y)+∑E[N2]<|μ|≤|γ1|‖⟨y⟩m(3−2σ)∂μ(uk−vk)‖L∞T(L∞)+R2(k−1)(∑E[N2]<|μ|≤|γ1|‖⟨y⟩m(3−2σ)∂μuk‖L∞T(L∞)+∑E[N2]<|μ|≤|γ1|‖⟨y⟩m(3−2σ)∂μvk‖L∞T(L∞))‖w‖L∞T(Y)). |
Since σ>32, we get
‖⟨y⟩m(2−σ)∂γ1(|uk|σ−|vk|σ)‖L∞T(L∞)≲(Rσ−1+ν−2(3−σ)R5−σ)‖w‖L∞T(Y)+∑1≤k≤Mν−2(2k−σ)R4k−σ−2(R+∑E[N2]<|μ|≤|γ1|‖⟨y⟩m(3−2σ)∂μuk‖L∞T(L∞))‖w‖L∞T(Y)+∑1≤k≤Mν−(2k−σ)(R2k−1(‖w‖L∞T(Y)+∑E[N2]<|μ|≤|γ1|‖⟨y⟩m(3−2σ)∂μ(uk−vk)‖L∞T(L∞)+R2(k−1)(∑E[N2]<|μ|≤|γ1|‖⟨y⟩m(3−2σ)∂μuk‖L∞T(L∞)+∑E[N2]<|μ|≤|γ1|‖⟨y⟩m(3−2σ)∂μvk‖L∞T(L∞))‖w‖L∞T(Y)). |
If |γ1|<N, we write by Sobolev injections,
∑E[N2]<|μ|≤|γ1|‖⟨y⟩m(3−2σ)∂μ(uk−vk)‖L∞T(L∞)≲∑E[N2]<|μ|≤N+1+E[N2]‖∂μ(uk−vk)‖L∞T(L2)≲∑E[N2]<|μ|≤−N+M0+M‖⟨y⟩m∂μ(uk−vk)‖L∞T(L2)≲‖w‖L∞T(Y). | (3.20) |
Thus,
(A1)≲(Rσ−1+ν−2(3−σ)R5−σ+∑1≤k≤Mν−2(2k−σ)R4k−σ−1+∑1≤k≤Mν−(2k−σ)R2k−1)F2(ν,R)‖u−v‖L∞T(Y). | (3.21) |
If |γ1|≥N, it is sufficient to estimate the term
(A′1):=‖⟨y⟩m(2−σ)∂γ1(|uk|σ−|vk|σ)‖L∞T(L2)‖⟨y⟩−m(2−σ)∂γ2(|uj|σ−2uj)‖L∞T(L∞)≲‖⟨y⟩m(2−σ)∂γ1(|uk|σ−|vk|σ)‖L∞T(L2)F2(ν,R), |
where we used (3.9). Moreover, Lemma 2.4 via (2.2) gives
‖⟨y⟩m(2−σ)∂γ1(|uk|σ−|vk|σ)‖L∞T(L2)≲(Rσ−1+ν−2(3−σ)R5−σ)‖u−v‖L∞T(Y)+∑1≤k≤Mν−2(2k−σ)R4k−σ−2(R+∑E[N2]<|μ|≤|γ1|‖⟨y⟩m(3−2σ)∂μuk‖L∞T(L2))‖u−v‖L∞T(Y)+∑1≤k≤Mν−(2k−σ)(R2k−1(‖u−v‖L∞T(Y)+∑E[N2]<|μ|≤|γ1|‖⟨y⟩m(3−2σ)∂μ(uk−vk)‖L∞T(L2))+R2(k−1)(∑E[N2]<|μ|≤|γ1|‖⟨y⟩m(3−2σ)∂μuk‖L∞T(L2)+∑E[N2]<|μ|≤|γ1|‖⟨y⟩m(3−2σ)∂μvk‖L∞T(L2))‖u−v‖L∞T(Y)). |
Using the fact that
∑E[N2]<|μ|≤|γ1|‖⟨y⟩m(3−2σ)∂μuk‖L∞T(L2)≲∑E[N2]<|μ|≤M‖⟨y⟩m∂μuk‖L∞T(L2)+∑M<|μ|≤−N+M0+M‖∂μuk‖L∞T(L2), |
we have
‖⟨y⟩m(2−σ)∂γ1(|uk|σ−|vk|σ)‖L∞T(L2)≲(Rσ−1+ν−2(3−σ)R5−σ+∑1≤k≤Mν−2(2k−σ)R4k−σ−1+∑1≤k≤Mν−(2k−σ)R2k−1)‖u−v‖L∞T(Y). |
Thus,
(A′1)≲(Rσ−1+ν−2(3−σ)R5−σ+∑1≤k≤Mν−2(2k−σ)R4k−σ−1+∑1≤k≤Mν−(2k−σ)R2k−1)F2(ν,R)‖u−v‖L∞T(Y). | (3.22) |
By (3.9), let us estimate the term
(A2):=‖⟨y⟩m(2−σ)∂γ1|vk|σ‖L∞T(L2)‖⟨y⟩−m(2−σ)∂γ2(|uj|σ−2uj−|vj|σ−2vj)‖L∞T(L∞)≲F1(ν,R)‖⟨y⟩−m(2−σ)∂γ2(|uj|σ−2uj−|vj|σ−2vj)‖L∞T(L∞). |
If |γ2|≤N, by a similar reasoning to (3.17), (for α=3−σ), we have
‖⟨y⟩m(1−α)∂γ2(|uj|σ−2uj−|vj|σ−2vj)‖L∞T(L∞)≲(ν−3(2−σ)R4−σ+∑1≤k≤Mν−2(2(1+k)−σ)R4k−σ+2+∑1≤k≤Mν−(2(1+k)−σ)R2k)‖u−v‖L∞T(Y). |
It follows that
(A2)≲(ν−3(2−σ)R4−σ+∑1≤k≤Mν−2(2(1+k)−σ)R4k−σ+2+∑1≤k≤Mν−(2(1+k)−σ)R2k)(Rσ+∑1≤l≤|γ1|ν−(2l−σ)R2l)‖u−v‖L∞T(Y). |
If |γ2|≥N, it is sufficient to estimate the term
(A′2):=‖⟨y⟩m(2−σ)∂γ1|vk|σ‖L∞T(L∞)‖⟨y⟩−m(2−σ)∂γ2(|uj|σ−2uj−|vj|σ−2vj)‖L∞T(L2). | (3.23) |
Moreover, with Lemma 2.4 via (2.2), we write
‖⟨y⟩−m(σ−2)∂γ2(|uj|σ−2uj−|vj|σ−2vj)‖L∞T(L2)≲ν−3(2−σ)R4−σ‖u−v‖L∞T(Y)+∑1≤k≤Mν−2(2(1+k)−σ)R4k−σ+1(R+∑E[N2]<|μ|≤|γ2|‖⟨y⟩m∂μuk‖L∞T(L2))‖u−v‖L∞T(Y)+∑1≤k≤Mν−(2(1+k)−σ)(R2k(‖u−v‖L∞T(Y)+∑E[N2]<|μ|≤|γ2|‖⟨y⟩m∂μ(uk−vk)‖L∞T(L2)+R2k−1(∑E[N2]<|μ|≤|γ2|‖⟨y⟩m∂μuk‖L∞T(L2)+∑E[N2]<|μ|≤|γ2|‖⟨y⟩m∂μvk‖L∞T(L2))‖u−v‖L∞T(Y))≲(ν−3(2−σ)R4−σ+∑1≤k≤Mν−2(2(1+k)−σ)R4k−σ+2+∑1≤k≤Mν−(2(1+k)−σ)R2k)‖u−v‖L∞T(Y). |
Since N+|γ1|≤|γ1|+|γ2|≤−N+M0+M, we have |γ1|≤M+M0−2N. So, arguing as in (3.6), we get
‖⟨y⟩m(2−σ)∂γ1(|vk|p)‖L∞T(L∞)≲Rσ+∑1≤l≤|γ1|ν−(2l−σ)R2l−1(R+∑E[N2]<|μ|≤|γ1|‖⟨y⟩m(3−2σ)∂μvk‖L∞T(L∞))≲F1(ν,R). |
Then,
(A′2)≲(ν−3(2−σ)R4−σ+∑1≤k≤Mν−2(2(1+k)−σ)R4k−σ+2+∑1≤k≤Mν−(2(1+k)−σ)R2k)F1(ν,R)‖u−v‖L∞T(Y). | (3.24) |
Taking (3.15), (3.18), (3.19), (3.21), (3.22), (3.23), and (3.24), we get
(I1)+(I2)+(I3)≲F(ν,R)‖u−v‖L∞T(Y). | (3.25) |
Now, taking Lemma 2.1, (3.2), (3.3), and (3.11) via the estimates of (I), we write
‖⟨y⟩m∂α(fj(u)−fj(v))‖L∞T(L2)≲T⟨T⟩m(‖I−N+M0+M(Fj(u)−Fj(v))‖L∞T(L2)+‖⟨y⟩m∂α(Fj(u)−Fj(v))‖L∞T(L2))≲T⟨T⟩mF(ν,R)‖u−v‖L∞T(Y). |
Moreover, taking Lemma 2.1, and the estimate of (I3), for |α|≤−N+M0+M, we get
‖∂α(fj(u)−fj(v))‖L∞T(L2)≲T‖∂α(Fj(u)−Fj(v))‖L∞T(L2)≲TF(ν,R)‖u−v‖L∞T(Y). |
Finally, f is a contraction of BT(R) for small T>0, and the result follows with a classical Picard argument.
In this section, we prove Theorem 2.2 and Proposition 2.1.
Let us start with the next auxiliary result.
Proposition 4.1. Let max{1+1N,32}<σ≤2, κ≫1, and v0∈Y such that infRN|v0(x)|≥ν. Then, there is a unique v∈C([0,1κ],Y) solution to
ivj+Δvj=τ(1−κt)−(2−N(σ−1))(∑1≤k≤najk|vk|σ)|vj|σ−2vj,∀j∈[1,n]. | (4.1) |
Proof of Proposition 4.1: For simplicity and without loss of generality, let us fix τ=1. One applies a Picard fixed point argument. Let the function
g(v):=ei⋅Δv0+i∫⋅0(1−κs)−(2−N(σ−1))ei(⋅−s)ΔFds:=(g1(v),…,gm(v)), |
on the space Bν,1κ(R). Taking (3.1), (3.3), and (3.11), via σ>1+1N, we write
‖g(v)‖L∞T(Y)≲⟨κ−1⟩E[N2]+1+m‖v0‖Y+κ−1⟨κ−1⟩E[N2]+1+mF1(ν,R)F2(ν,R). |
Moreover, arguing as in (3.14), we write
|⟨y⟩mg(v)|≥ν−cκ−1⟨κ−1⟩E[N2]+1+m‖v0‖Y−cκ−1⟨κ−1⟩E[N2]+1+mF1(ν,R)F2(ν,R). |
Moreover, arguing as in (3.25), and the last lines of the previous section, we get
‖g(u)−g(v)‖L∞T(Y)≤cκ−1⟨κ−1⟩E[N2]+1+mF(ν,R)‖u−v‖L∞T(Y). |
This implies that, for large κ≫1, an application of the Picard Theorem finishes the proof.
Take the pseudo conformal transformation [26],
vj(t,y):=(1−κt)−N2eiκ|x|24(1−κt)uj(t1−κt,x1−κt),∀j∈[1,n]. |
Here, 0<T<1κ, and v, given by Proposition 4.1, is a solution to (4.1). Thus, u resolves (CNLS). Now, let define u+j:=eiκ|x|24−iκΔvj(1κ,⋅). So, following [25,Section 4], we have uj∈C(R+,Hs)∩L∞(R+×RN,⟨y⟩N2dydt), and
limt→+∞‖e−itΔuj(t)−u+j‖Hs=0. |
Here, one proves Proposition 2.1. Let u∈CT(Y) be a solution to (CNLS). Thus, m>1+N2 implies that ⟨y⟩1−m∈L2, and
‖⟨y⟩u‖≤‖⟨y⟩mu‖∞‖⟨y⟩1−m‖≲‖⟨y⟩mu‖∞. |
This implies that the solution has a finite variance ∫RN|u(t,y)|2|x|2dy. Let us check that the energy is well-defined. Indeed,
‖ˉujFj,k‖1=‖|uk|σ|uj|p‖1=‖⟨y⟩−2mσ⟨y⟩2mσ|uk|σ|uj|p‖1≤‖⟨y⟩−mσ‖2‖⟨y⟩mu‖2σ∞≲‖u‖2σL∞T(Y). |
Thus, one can apply [18,Theorem 2.8].
This work examines the singular coupled non-linear Schrödinger system (CNLS) with three main objectives. First, it investigates the local existence of solutions. Second, it establishes the existence of global solutions that scatter in certain Sobolev spaces. Finally, it demonstrates the existence of non-global solutions. The primary difficulty arises from the condition σ<2, which introduces a singularity in the term |uj|σ−2 near zero. This singularity renders the classical contraction method in the energy space ineffective. This paper aims to address this gap in the literature by leveraging ideas from [24]. The approach highlights that the singularity issue is localized near zero, requiring the solution to avoid this region, see assumption (2.5). This is challenging because the Schrödinger equation lacks a maximum principle. The global solutions that scatter are obtained through a pseudo-conformal transformation based on the local solutions. Lastly, the existence of non-global solutions is demonstrated using the classical variance method.
Saleh Almuthybri and Tarek Saanouni: study conception and design, data collection, analysis and interpretation of results, and manuscript preparation. All authors have read and approved the final version of the manuscript for publication.
The researchers would like to thank the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support (QU-APC-2024-9/1).
On behalf of all authors, the corresponding author states that there is no conflict of interest. No data-sets were generated or analyzed during the current study.
[1] |
G. Chirikjian, J. Burdick, A modal approach to hyperredundant manipulator kinematics, IEEE Transactions on Robotics and Automation, 10 (1994), 343–354. http://dx.doi.org/10.1109/70.294209 doi: 10.1109/70.294209
![]() |
[2] |
M. Kass, A. Witkin, D. Terzopoulos, Snakes: active contour models, Int. J. Comput. Vision, 1 (1988), 321–331. http://dx.doi.org/10.1007/BF00133570 doi: 10.1007/BF00133570
![]() |
[3] |
H. Lu, J. Todhunter, T. Sze, Congruence conditions for nonplanar developable surfaces and their application to surface recognition, CVGIP: Image Understanding, 58 (1993), 265–285. http://dx.doi.org/10.1006/ciun.1993.1042 doi: 10.1006/ciun.1993.1042
![]() |
[4] |
M. Gage, R. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom., 23 (1986), 69–96. http://dx.doi.org/10.4310/jdg/1214439902 doi: 10.4310/jdg/1214439902
![]() |
[5] | M. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom., 26 (1987), 285–314. |
[6] | M. Gage, On an area-preserving evolution equation for plane curves, Contemp. Math., 51 (1986), 51–62. |
[7] |
D. Kwon, F. Park, Evolution of inelastic plane curves, Appl. Math. Lett., 12 (1999), 115–119. http://dx.doi.org/10.1016/S0893-9659(99)00088-9 doi: 10.1016/S0893-9659(99)00088-9
![]() |
[8] |
D. Kwon, F. Park, D. Chi, Inextensible flows of curves and developable surfaces, Appl. Math. Lett., 18 (2005), 1156–1162. http://dx.doi.org/10.1016/j.aml.2005.02.004 doi: 10.1016/j.aml.2005.02.004
![]() |
[9] |
N. Yüksel, The ruled surfaces according to Bishop frame in Minkowski 3-space, Abstr. Appl. Anal., 2013 (2013), 810640. http://dx.doi.org/10.1155/2013/810640 doi: 10.1155/2013/810640
![]() |
[10] |
N. Yüksel, On dual surfaces in Galilean 3-space, AIMS Mathematics, 8 (2023), 4830–4842. http://dx.doi.org/10.3934/math.2023240 doi: 10.3934/math.2023240
![]() |
[11] |
R. Abdel-Baky, M. Saad, Some characterizations of dual curves in dual 3-space D3, AIMS Mathematics, 6 (2021), 3339–3351. http://dx.doi.org/10.3934/math.2021200 doi: 10.3934/math.2021200
![]() |
[12] |
Y. Li, A. Abdel-Salam, M. Khalifa Saad, Primitivoids of curves in Minkowski plane, AIMS Mathematics, 8 (2023), 2386–2406. http://dx.doi.org/10.3934/math.2023123 doi: 10.3934/math.2023123
![]() |
[13] |
Y. Li, K. Eren, K. Ayvacı, S. Ersoy, The developable surfaces with pointwise 1-type Gauss map of Frenet type framed base curves in Euclidean 3-space, AIMS Mathematics, 8 (2023), 2226–2239. http://dx.doi.org/10.3934/math.2023115 doi: 10.3934/math.2023115
![]() |
[14] |
Y. Li, M. Aldossary, R. Abdel-Baky, Spacelike circular surfaces in Minkowski 3-space, Symmetry, 15 (2023), 173. http://dx.doi.org/10.3390/sym15010173 doi: 10.3390/sym15010173
![]() |
[15] |
H. Abdel-Aziza, H. Serryb, F. El-Adawyb, A. Khalila, On admissible curves and their evolution equations in pseudo-Galilean space, J. Math. Comput. Sci., 25 (2022), 370–380. http://dx.doi.org/10.22436/jmcs.025.04.07 doi: 10.22436/jmcs.025.04.07
![]() |
[16] |
A. Ahmad, H. Abdel Aziz, A. Sorour, Ruled surfaces generated by some special curves in Euclidean 3-space, Journal of the Egyptian Mathematical Society, 21 (2013), 285–294. http://dx.doi.org/10.1016/j.joems.2013.02.004 doi: 10.1016/j.joems.2013.02.004
![]() |
[17] |
R. Hussien, T. Youssef, Evolution of special ruled surfaces via the evolution of their directrices in Euclidean 3-space E3, Appl. Math. Inf. Sci., 10 (2016), 1949–1956. http://dx.doi.org/10.18576/amis/100536 doi: 10.18576/amis/100536
![]() |
[18] |
K. Eren, H. Kosal, Evolution of space curves and the special ruled surfaces with modified orthogonal frame, AIMS Mathematics, 5 (2020), 2027–2039. http://dx.doi.org/10.3934/math.2020134 doi: 10.3934/math.2020134
![]() |
[19] |
A. Kelleci, K. Eren, On evolution of some associated type ruled surfaces, Mathematical Sciences and Applications E-Notes, 8 (2020), 178–186. http://dx.doi.org/10.36753/mathenot.750639 doi: 10.36753/mathenot.750639
![]() |
[20] | T. Körpınar, S. Baş, E. Turhan, V. Asil, On inextensible flows of dural tangent developable surfaces in the dual space D3, Mathematica Aeterna, 2 (2012), 325–333. |
[21] |
Z. Yüzbası, D. Yoon, Inextensible flows of curves on lightlike surfaces, Mathematics, 6 (2018), 224. http://dx.doi.org/10.3390/math6110224 doi: 10.3390/math6110224
![]() |
[22] | N. Yüksel, B. Saltık, M. Karacan, The Characterizations of the curves generated by a curve with constant torsion, Konuralp Journal of Mathematics, 11 (2023), 1–7. |
[23] | E. Kruppa, Analytische und constructive differential geometrie (German), Wien: Springer, 1957. |
[24] | B. O'Neill, Semi-Riemannian geometry with applications to relativity, New York: Academic Press, 1983. |
[25] |
S. Nurkan, İ. Güven, M. Karacan, Characterizations of adjoint curves in Euclidean 3-space, Proc. Natl. Acad. Sci. India, Sect. A Phys. Sci., 89 (2019), 155–161. http://dx.doi.org/10.1007/s40010-017-0425-y doi: 10.1007/s40010-017-0425-y
![]() |
[26] |
Y. Li, Z. Chen, S. Nazra, R. Abdel-Baky, Singularities for timelike developable surfaces in Minkowski 3-space, Symmetry, 15 (2023), 277. http://dx.doi.org/10.3390/sym15020277 doi: 10.3390/sym15020277
![]() |
[27] |
Y. Li, S. Nazra, R. Abdel-Baky, Singularity properties of timelike sweeping surface in Minkowski 3-space, Symmetry, 14 (2022), 1996. http://dx.doi.org/10.3390/sym14101996 doi: 10.3390/sym14101996
![]() |
[28] |
Y. Li, F. Mofarreh, R. Abdel-Baky, Timelike circular surfaces and singularities in Minkowski 3-space, Symmetry, 14 (2022), 1914. http://dx.doi.org/10.3390/sym14091914 doi: 10.3390/sym14091914
![]() |
[29] |
Y. Li, M. Erdoğdu, A. Yavuz, Differential geometric approach of Betchov-Da Rios soliton equation, Hacet. J. Math. Stat., 52 (2023), 114–125. http://dx.doi.org/10.15672/hujms.1052831 doi: 10.15672/hujms.1052831
![]() |
[30] |
Q. Zhao, L. Yang, Y. Wang, Geometry of developable surfaces of Frenet type framed base curves from the singularity theory viewpoint, Symmetry, 14 (2022), 975. http://dx.doi.org/10.3390/sym14050975 doi: 10.3390/sym14050975
![]() |