Sharp well-posedness for a coupled system of mKdV type equations
Xavier Carvajal Liliana Esquivel Raphael Santos
math.AP 35Q35, 35Q53

	
	Sharp well-posedness]Sharp well-posedness for a coupled system of  mKdV  type equations
	
	Instituto de Matemática, Universidade Federal do Rio de Janeiro-UFRJ. Ilha do Fundão,
		21945-970. Rio de Janeiro-RJ, Brazil
	carvajal@im.ufrj.br
	
	L. Esquivel]Liliana Esquivel
	Gran Sasso Science Institute, CP 67100,  L' Aquila, Italia.
	liliana.esquivel@gssi.it
	
	
	
	Universidade Federal do Rio de Janeiro, Campus Macaé, Brazil
	raphaelsantos@macae.ufrj.br
	
	
	
	[2000]35Q35, 35Q53
	
	
		We consider the initial value problem associated to a  system consisting     modified Korteweg-de Vries  type equations 
		
    ∂_tv + ∂_x^3v + ∂_x(vw^2) =0,   v(x,0)=ϕ(x),
    ∂_tw + α∂_x^3w + ∂_x(v^2w) =0,    w(x,0)=ψ(x),

		and prove  the local well-posedness results for   given data in low regularity Sobolev spaces
		H^s()× H^k(), s,k> -1/2 and |s-k|≤ 1/2, for α≠ 0,1. Also, we prove that: (I) the solution mapping that takes initial data to the solution fails to be C^3 at the origin, when s<-1/2 or k<-1/2 or |s-k|>2; (II) the  trilinear estimates used in the proof of the local well-posedness theorem fail to hold when (a) s-2k>1 or k<-1/2 (b) k-2s>1 or s<-1/2; (c) s=k=-1/2; (III) the local well-posedness result is sharp in a sense that  we can not reduce the proof of the trilinear estimates, proving some related bilinear estimates (as in <cit.>).
	
	
	[
    Raphael Santos
    December 30, 2023
=====================


	Introduction
	
	This paper is devoted to the initial value problem (IVP) for the  system of the  modified Korteweg-de Vries (mKdV)-type equations
	
    ∂_tv + ∂_x^3v + ∂_x(vw^2) =0,   v(x,0)=ϕ(x),
    ∂_tw + α∂_x^3w + ∂_x(v^2w) =0,    w(x,0)=ψ(x),

	where   (x, t) ∈×; v=v(x,t) and w= w(x, t) are real-valued functions,  and α∈ is a constant. 
	
	For α=1, among a vast class of nonlinear evolution equations, the related system was studied by <cit.>, in the context of inverse scattering, showing that this method provides a means of solution of the associated IVP. For existence and estability of solitary waves to the system  (<ref>) we refer the works   <cit.> and <cit.>.
	
	
	The well-posedness for the IVP (<ref>) with initial data in the classical Sobolev spaces H^s()× H^k() was studied by many authors. In 1995, following Kenig, Ponce and Vega <cit.>,  using smoothing properties of the group, Maximal functions ans Strichartz estimates,  Montenegro  <cit.> proved that the IVP (<ref>) with α =1 is locally
	well-posed for given data (ϕ, ψ) in H^s()× H^s(), s≥1/4. He also proved global well-posedness for given data in H^s()× H^s(), s≥ 1, using the conservation laws
	
    I_1(v, w) := ∫_ (v^2 +w^2) dx
    		
    		  and  
    		I_2(v, w) := ∫_ (v_x^2 + w_x^2 - v^2w^2)  dx.

	
	We note that the approach in <cit.> implies the local well-posedness for s≥ 1/4, when 0<α< 1. 
	
	In 1999, Alarcon, Angulo and Montenegro  <cit.> studied some properties of the solutions for the system of nonlinear evolution equation 
	
    {[  ∂_tv+∂_x^3 v+∂_x(v^pu^p+1)=0,   u(x,0)=ϕ(x);                                             ; ∂_tu+ ∂_x^3 u+∂_x(v^p+1u^p)=0,   v(x,0)=φ(x) ].

	for p≥ 1. In this work they proved that (<ref>) has a family of solitary wave solutions, similar to those found for Korteweg de Vries(KdV)-type equations and that it can be stable or unstable depending on the range of p.  We observe the system (<ref>), with α=1, is a special case of (<ref>) with p=1.  In this approach they also uses the smoothing property of the linear group combined with the L_x^pL_t^q Strichartz estimates and maximal function estimates.
	
	In 2001, Tao <cit.> shows that the trilinear estimate is valid for s≥ 1/4
	
	
    ∂_x(uvw)_X_s,b'≲u_X_s,bv_X_s,bw_X_s,b

	when X_s,b is the Bourgain space (see <cit.>). This leads us to get also the local well-posedness for the system (<ref>) when α=1, for s≥ 1/4 in the context of Fourier restriction norm method. It is worth noting that the local well-posedness result for the system (<ref>) with α =1 is sharp and it can be justified in two different way; first the trilinear estimates fail if s<1/4 (see Theorem 1.7 in <cit.>). Second, the solution map is not uniformly continuous if s<1/4 (see Theorem 1.3 in <cit.>). This notion of ill-posedness is a bit strong. 
	For further works in this direction, we refer  <cit.>.  
	
	In 2012, Corcho and Panthee in <cit.> improves the global result in <cit.>,  getting global well-posedness in H^s× H^s, for s>1/4, for α=1, see also <cit.>. 
	
	Recently, in 2019 Carvajal and Panthee  <cit.> proved local well-posedness in H^s× H^s for s>-1/2, when α∈ (0, 1)∪(1,∞). Also, they proved that the key trilinear estimates fails to  hold  and also the solution map is not C^3 at the origin, both when s<-1/2. Observe that this result also is sharp, considering the scaling argument s=-1/2 to the modified KdV equation. 
	
	Many authors studies local well-posedness  for a system with dispersive equations, when the initial data belongs to diferents Sobolev spaces, i.e., in H^s× H^k, k≠ s  (see, e.g., Ginibre, Tsutsumi and Velo <cit.>).
	In this context,  we prove the following local well-posedness result:
	
	
	
	
	
	
	
	
	
	
	
	
		Let α≠ 0,1, b>1/2 and s, k such that s,k  >-1/2 and |s-k|≤ 1/2. 
		Then for any (ϕ,ψ)∈ H^s()× H^k(),  there exist δ = δ((ϕ,ψ)_H^s× H^k) (with δ(ρ)→∞ as ρ→ 0) and a unique solution (v,w)∈ X^δ_s, b× X^α, δ_k, b to the IVP (<ref>) in the time interval [0, δ]. Moreover, the solution satisfies the estimate
		
    (v, w)_X^δ_s, b× X^α, δ_k, b≲(ϕ,ψ)_H^s× H^k,

		where the norms ·_X^δ_s, b and  ·_X^α, δ_s, b are defined in (<ref>).
	
	
	
	The case α∈ (0,1) and k=s, with s>-1/2, of this theorem, was proved in <cit.>. Also, they observed that the LWP in the case 0<α<1 is equivalent to the LWP in the case α>1 by using the transformation v(x,t):=ṽ(α^-1/3x, t) and u(x,t):=ũ(α^-1/3x, t) where 
		
    ∂_tṽ + 1/α∂_x^3ṽ + ∂_x(ṽw̃^2) =0,   
    ∂_tw̃ +∂_x^3w̃ + ∂_x(ṽ^2w̃) =0.

		So we restrict ourselves to prove Theorem <ref> in the case α∈(-∞, 0)∪(1, +∞).
	
	
	
	The main ingredients in the proof of Theorem <ref> are the new trilinear estimates:
	
	
		Let α≠ 0,1,  b=1/2+ϵ, and b'=-1/2+2ϵ, with 0<ϵ<min{2s+1/15,1/6}. Then  the following trilinear estimates 
		
    (vw_1 w_2)_x_X_s,b'≲v_X_s,bw_1_X^α_k,bw_2_X^α_k,b

		and
		
    (v_1 v_2 w)_x_X^α_k,b'≲v_1_X_s,bv_2_X_s,bw_X^α_k,b,

		holds for any  s, k   in the following region: s,k  >-1/2 and |s-k|≤ 1/2. Moreover (<ref>) also hold if 
		s=-1/2 and -1/2<k and  (<ref>) also hold if k=-1/2 and -1/2<s.
		
	
	
	
	Also, we establish some ill-posedness results. The first one is about the smoothness of the solution mapping  associated to the system (<ref>).
	
	
	
		Let α≠  0,1. For any s< -1/2 or k<-1/2 or |s-k|>2 and for given  (ϕ, ψ)∈ H^s()× H^k(), there exist no time T =T((ϕ, ψ)_H^s× H^k) such that the solution mapping that takes initial data (ϕ, ψ) to the solution (v, w) ∈ C([0,T];H^s) × C([0,T];H^k) to the IVP (<ref>) is C^3 at the origin.
	
	
	 We point out that in Section <ref> we prove a little bit more stronger result than  the Theorem <ref>. 
		
	
	
	

In the next figure we represent the regions where we have L.W.P. and C^3-ill-posedness:













	



	The second one is about the failure of the trilinear estimates (<ref>) and (<ref>). We prove the following results:
	
	
	
		Let  α≠ 0,1. 

		(a) The trilinear estimate (<ref>)  fail to hold for any b ∈ whenever s-2k>1  or k<-1/2.

		(b) The trilinear estimate  (<ref>) fail to hold for any b ∈ whenever k-2s>1  or s<-1/2.
	
	
	and 
	
	
		Let  α≠ 0,1. 

		(a) The trilinear estimate (<ref>)  fail to hold  whenever s-k>2, for any ϵ such that 0<ϵ <2/3(s-k-2).

		(b) The trilinear estimate (<ref>)  fail to hold  whenever k-s>2, for any ϵ such that 0<ϵ <2/3(s-k-2).
	
	
	
	Also, at the endpoint we have
	
	
		Let α≠ 0,1, then the estimate
		
    (vw_1 w_1)_x_X_-1/2,-1/2 +4ϵ≲v_X_-1/2,1/2 +ϵw_1_X^ α_-1/2,1/2 +ϵw_2_X^ α_-1/2,1/2 +ϵ

		and
		
    (v_1 v_2 w)_x_X^α_-1/2,-1/2 +4ϵ≲v_1_X_-1/2,1/2 +ϵv_2_X_-1/2,1/2 +ϵw_X^ α_-1/2,1/2 +ϵ

		fails to hold whenever ϵ>0.
	
	
	The plan of this paper is as follows. In Section 2 we fix some notations, define the spaces when we perform the
	iteration process, recall some useful inequalities. In Section 3 we prove the crucial result: Proposition <ref>. In Section 4 we prove the Theorem <ref>. 
	In Section 5 we prove the ill-posedness results and finally, in the Section 6 we give some negative result related to the approach adopted in Section 3.
	
	
	
	
	
	






Function spaces and preliminary estimates

In this section we fix some notations, define the function spaces and remember some preliminary results. First, we introduce the integral equations associated to the system (<ref>),


    v(t)   =U(t)ϕ-∫_0^tU(t-t')∂_x(vw^2)(t')dt',
    
    	w(t)   =U^α(t)ψ -∫_0^tU^α(t-t')∂_x(v^2w)(t')dt',


where U^α(t):=e^-tα∂_x^3 is the unitary group associated to the linear problem ∂_t u+α∂_x^3u=0 and defined via Fourier transform by U^α(t)ϕ={ e^itα(·)^3ϕ̂(·) }. Here U(t) denotes U^1(t). In order to use the Fourier restriction norm method and prove the local result,  we introduce the Bourgain space  X^α_s,b, for s,b∈, to be the completion of the Schwartz class 𝒮(^2) under the norm


    f_X^α_s,b:=U^α(t)f_H^b_t(;H^s_x)=⟨ξ⟩^s⟨τ-αξ^3⟩^bf̃(τ,ξ) _L^2_τ,ξ,


where ⟨·⟩:=1+|·| and f̃ is the Fourier transform in (t,x) variable


    f̃(τ,ξ):=c∫_^2e^-i(xξ+tτ)f(t,x)dtdx.


Hereafter,  for α=1 we will use X_s,b instead of X^1_s,b.

If b>1/2, we have that X_s,b^α↪ C(: H^s_x()) and thus for an interval I=[-δ, δ], we can define the restricted bourgain spaces X^α,δ_s,b endowed with the norm 
    f_X^α,δ_s,b=inf{g_X^α_s,b;  g|_[-δ, δ]=f   }.


Of course, we write X^δ_s,b instead of X^1,δ_s,b.

Now we remember some linear estimates, important parts when we use Fourier Restriction norm method. Let η a smooth function supported on the interval [-2, 2] such that η(t)=1 for all t∈ [-1, 1]. We denote, for each δ >0, η_δ(t)=η(t/δ). The following estimates holds (see e.g. <cit.> or <cit.>)

 Let δ>0, s∈ and -1/2<b'≤ 0≤ b≤ b' +1. Then we have
	
	
		
  (i)    η(t) U^α(t)ϕ_X_s,b^α≲ϕ_H^s.
		 

		
		
  (ii)    η_δ(t)∫_0^tU^α(t-t')f(t')dt'  _X^α_s,b≲δ^1-b+b'f_X^α_s,b'.
	
	







The following lemma will be useful in the proof of the trilinear estimates



	
		
  (i) If a,b>0 and a+b>1, we have
		
    ∫_dx⟨ x -α⟩^a⟨ x -β⟩^b≲1⟨α -β⟩^c,     c=min{a,b, a+b-1}.

		
		
  (ii)
		Let a, η∈, a, η≠ 0, b>1, then
		
    ∫_dx⟨ a(x^2-η^2) ⟩^b≲1|aη|.

		
		
  (iii)
		Let a, η∈, a, η≠ 0, b>1, then
		
    ∫_|x±η| dx⟨ a(x^2±η^2) ⟩^b≲1|a|.

		
		
		
  (iv) For l>1/3,
		
    ∫_dx⟨ x^3+a_2x^2 +a_1 x+a_0 ⟩^l≲ 1.

		
	


	The proof of (<ref>)  can be found in <cit.>,  (<ref>) and (<ref>) in <cit.> and (<ref>)  in  <cit.>.














Trilinear estimates: proof of Proposition <ref>

In this section the ideas in <cit.> plays a central role in the proof of Proposition <ref>. We remember some notations, results and follows the arguments contained therein.  Let k≥ 2 be an integer, a [k;^d+1]-multiplier is any function m:Γ_k(^d+1)→ℂ, where Γ_k(^d+1) denotes  the hyperplane Γ_k(^d+1)={(ξ_1,…,ξ_k)∈ (^d+1)^k; ξ_1+⋯+ξ_n=0} endowed with the measure


    ∫__Γ_k(^d+1)f:=∫_(^d)^k-1f(ξ_1,…,ξ_k-1,-ξ_1-⋯-ξ_k-1)dξ_1dξ_2⋯ dξ_k-1.


The norm of a [k;^d+1]-multiplier m, denoted by m_[k;^d+1], is the best constant such that 


    |  ∫_Γ_k(^d+1)m(ξ)∏_j=1^nf_j(ξ_j) |≤m_[k;^d+1]∏_j=1^kf_j_L^2(^d+1),


holds for all test functions f_j on ^d+1.


Now we start to work on our trilinear estimates. By duality and Plancherel (see e.g. <cit.> or <cit.>),  one can see that  the estimate (<ref>) is equivalent to

    |∫_ξ_1+⋯+ξ_4=0τ_1+⋯+τ_4=0
    	m(ξ_1,τ_1,⋯,ξ_4,τ_4)Π_j=1^4f_j(ξ_j, τ_j)| ≲Π_j=1^4f_j_L^2_ξτ,

where

    m(ξ_1,τ_1,⋯,ξ_4,τ_4):=ξ_4 ⟨ξ_4⟩^s/⟨ξ_1⟩^s⟨ξ_2⟩^k⟨ξ_3⟩^k⟨τ_1-ξ_1^3⟩^1/2+ϵ⟨τ_2-αξ_2^3⟩^1/2+ϵ⟨τ_3-αξ_3^3⟩^1/2+ϵ⟨τ_4-ξ_4^3⟩^1/2-2ϵ.


In this way, recalling the definition of the norm m_[4;^2] of the multiplier m,  the whole matter reduces to showing that

    m_[4;^2]≲ 1.

Observe that

    ξ_4 ⟨ξ_4⟩^s≤⟨ξ_4⟩^s+1≤⟨ξ_4⟩^1/2⟨ξ_4⟩^s+1/2≤⟨ξ_4⟩^1/2( ⟨ξ_1⟩^s+1/2+⟨ξ_2⟩^s+1/2+⟨ξ_3⟩^s+1/2).

We define

    m_1(ξ_1, τ_1, ξ_2,τ_2)=⟨ξ_1⟩^1/2/⟨τ_1-ξ_1^3 ⟩^1/2-2ϵ⟨ξ_2⟩^k ⟨τ_2-αξ_2^3⟩^1/2+ϵ,


    m_2(ξ_1, τ_1, ξ_2,τ_2)=⟨ξ_1⟩^s-k+1/2/⟨τ_1-αξ_1^3 ⟩^1/2+ϵ⟨ξ_2⟩^s ⟨τ_2-ξ_2^3⟩^1/2+ϵ.

From (<ref>) and (<ref>), we get

    m≤   ⟨ξ_4⟩^1/2/⟨τ_4-ξ_4^3⟩^1/2-2ϵ⟨ξ_3⟩^k⟨τ_3-αξ_3^3⟩^1/2+ϵ⟨ξ_1⟩^1/2/⟨τ_1-ξ_1^3 ⟩^1/2+ϵ⟨ξ_2⟩^k ⟨τ_2-αξ_2^3⟩^1/2+ϵ
       + ⟨ξ_4⟩^1/2/⟨τ_4-ξ_4^3⟩^1/2-2ϵ⟨ξ_3⟩^k⟨τ_3-αξ_3^3⟩^1/2+ϵ⟨ξ_2⟩^s-k+1/2/⟨τ_2-αξ_2^3⟩^1/2+ϵ⟨ξ_1⟩^s⟨τ_1-ξ_1^3⟩^1/2+ϵ
       + ⟨ξ_4⟩^1/2/⟨τ_4-ξ_4^3⟩^1/2-2ϵ⟨ξ_2⟩^k⟨τ_2-αξ_2^3⟩^1/2+ϵ⟨ξ_3⟩^s-k+1/2/⟨τ_3-αξ_3^3⟩^1/2+ϵ⟨ξ_1⟩^s⟨τ_1-ξ_1^3⟩^1/2+ϵ
    
    		=:   J_1+J_2+J_3.

Therefore, we have

    J_1
    		   ≤ m_1(ξ_4,τ_4,ξ_3,τ_3)  m_1(ξ_1, τ_1, ξ_2,τ_2)


    J_2
    		   ≤ m_1(ξ_4,τ_4,ξ_3,τ_3) m_2(ξ_2,τ_2,ξ_1,τ_1),


    J_3   ≤ m_1(ξ_4,τ_4,ξ_2,τ_2) m_2(ξ_3,τ_3,ξ_1,τ_1).


Now, using comparison principle, permutation and composition properties (see respectively Lemmas 3.1, 3.3 and 3.7 in <cit.>), it is enough to bound m_j_[3;^2], j=1,2, or equivalently,  
to show the following bilinear estimates

    uv_L^2(^2)≲u_X_-1/2, 1/2-2ϵv_X^α_k, 1/2+ϵ,

and

    uv_L^2(^2)≲u_X^α_k-s-1/2, 1/2+ϵv_X_s, 1/2+ϵ.

This equivalence can be proved using again duality and a similar calculations as the ones used to obtain (<ref>).




Similarly,  the estimate (<ref>) is equivalent to

    |∫_ξ_1+⋯+ξ_4=0τ_1+⋯+τ_4=0
    	M(ξ_1,τ_1,⋯,ξ_4,τ_4)Π_j=1^4f_j(ξ_j, τ_j)| ≲Π_j=1^4f_j_L^2_ξτ,

where

    M(ξ_1,τ_1,⋯,ξ_4,τ_4):=ξ_4 ⟨ξ_4⟩^k/⟨ξ_1⟩^s⟨ξ_2⟩^s⟨ξ_3⟩^k⟨τ_1-ξ_1^3⟩^1/2+ϵ⟨τ_2-ξ_2^3⟩^1/2+ϵ⟨τ_3-αξ_3^3⟩^1/2+ϵ⟨τ_4-αξ_4^3⟩^1/2-2ϵ.


In this way, recalling the definition of the norm M_[4;^2] of the multiplier M,  the whole matter reduces to showing that

    M_[4;^2]≲ 1.



Observe that



    ξ_4 ⟨ξ_4⟩^k≤⟨ξ_4⟩^k+1≤⟨ξ_4⟩^1/2⟨ξ_4⟩^k+1/2≤⟨ξ_4⟩^1/2( ⟨ξ_1⟩^k+1/2+⟨ξ_2⟩^k+1/2+⟨ξ_3⟩^k+1/2).



We define



    M_1(ξ_1, τ_1, ξ_2,τ_2)=⟨ξ_1⟩^1/2/⟨τ_1-αξ_1^3 ⟩^1/2-2ϵ⟨ξ_2⟩^s ⟨τ_2-ξ_2^3⟩^1/2+ϵ,




    M_2(ξ_1, τ_1, ξ_2,τ_2)=⟨ξ_1⟩^k-s+1/2/⟨τ_1-ξ_1^3 ⟩^1/2+ϵ⟨ξ_2⟩^k ⟨τ_2-αξ_2^3⟩^1/2+ϵ.




From (<ref>) and (<ref>), we get



    M≤   ⟨ξ_4⟩^1/2/⟨τ_4-αξ_4^3⟩^1/2-2ϵ⟨ξ_2⟩^s⟨τ_2-ξ_2^3⟩^1/2+ϵ⟨ξ_1⟩^k-s+1/2/⟨τ_1-ξ_1^3 ⟩^1/2+ϵ⟨ξ_3⟩^k ⟨τ_3-αξ_3^3⟩^1/2+ϵ
       + ⟨ξ_4⟩^1/2/⟨τ_4-αξ_4^3⟩^1/2-2ϵ⟨ξ_1⟩^s⟨τ_1-ξ_1^3⟩^1/2+ϵ⟨ξ_2⟩^k-s+1/2/⟨τ_2-ξ_2^3⟩^1/2+ϵ⟨ξ_3⟩^k⟨τ_3-αξ_3^3⟩^1/2+ϵ
       + ⟨ξ_4⟩^1/2/⟨τ_4-αξ_4^3⟩^1/2-2ϵ⟨ξ_1⟩^s⟨τ_1-ξ_1^3⟩^1/2+ϵ⟨ξ_3⟩^1/2/⟨τ_3-αξ_3^3⟩^1/2+ϵ⟨ξ_2⟩^s⟨τ_2-ξ_2^3⟩^1/2+ϵ
    
    		=:   I_1+I_2+I_3.



Therefore, we have



    I_1
    		   ≤ M_1(ξ_4, τ_4, ξ_2,τ_2) M_2(ξ_1,τ_1,ξ_3,τ_3),




    I_2
    		   ≤ M_1(ξ_4,τ_4,ξ_1,τ_1) M_2(ξ_2,τ_2,ξ_3,τ_3),




    I_3   ≤ M_1(ξ_4,τ_4,ξ_1,τ_1) M_1(ξ_3,τ_3,ξ_2,τ_2).




Analogously,  to prove that M_j_[3;^2]≲ 1, j=1,2 is equivalent respectively to show the following bilinear estimates

    uv_L^2(^2)≲u_X^α_-1/2, 1/2-2ϵv_X_s, 1/2+ϵ,

and

    uv_L^2(^2)≲u_X_s-k-1/2, 1/2+ϵv_X^α_k, 1/2+ϵ.




Therefore, the proof of Proposition <ref> follows if we proof  the four bilinear estimates (<ref>), (<ref>), (<ref>)  and (<ref>). In fact we prove the following propositions


	Let s>-1/2,  α≠0,1 and 0<ϵ<min{2s+1/15,1/6}. Then we have the bilinear estimates 
	
	
    fg_L^2(^2)≲f_X_-1/2, 1/2-2ϵg_X^α_s, 1/2+ϵ,

	and
	
    fg_L^2(^2)≲f_X^α_-1/2, 1/2-2ϵg_X_s, 1/2+ϵ.
 




 Let α∈ (-∞, 0)∪(1, +∞) and 0<ϵ<min{2s+1/15,1/6}. Then

	(a)  The inequality (<ref>) holds for any (s,k) in the region:
	
    R_1= {(s,k);   k,s>-1/2,    s-k≤ 1/2 }∪{(s,k);    -1/2< k,    s=-1/2 }.

	
	(b)  The inequality (<ref>)	holds for any  (s,k) in the region:   
	
    R_2={(s,k);    k,s>-1/2,    s-k≥ -1/2 }∪{(s,k);    -1/2< s,    k=-1/2 }.

	





Before we prove these results, we establish some preliminary results.




Let α<1 (α≠0), s>-1/2 and 0<ϵ<min{2s+1/15,1/6}. Then we have
	
    sup_ξ,τ∫_⟨ξ_2⟩/⟨ξ_1 ⟩^2s⟨τ-ξ_2^ 3 -αξ_1^ 3⟩^1-4ϵ dξ_1≲ 1,

	where ξ_2:=ξ-ξ_1, for a fixed ξ∈.


	
	For the case 0<α <1, see  in <cit.>, Lemma 3.2, the estimative of the item labeled by them as (3.12). We will prove the case α< 0. We denote by L_1 the integral in (<ref>). For a fixed  ξ and τ, let
	
    H(ξ_1):=τ-ξ_2^3-αξ_1^ 3=τ-ξ_2^3+|α|ξ_1^ 3.

	We have 
	
    H'(ξ_1)=3[ ξ^ 2_2+|α| ξ_1^ 2]> 0.

	Thus, the function ξ_1↦ H(ξ_1) is monotone on ℝ. We divide the proof into the following two cases:
	
	
	Case 1. (|ξ|<2|ξ_1|) In this case, we have ⟨ξ _2⟩≲⟨ξ_1⟩.  Thus
	
    [                                                                                               χ_{|ξ|<2|ξ_1|}L_1                                                                                                               = ∫_|ξ_1|> |ξ|/2⟨ξ_2⟩⟨ξ_1 ⟩/⟨ξ_1 ⟩^2s+1⟨τ-ξ_2^ 3 +|α|ξ_1^ 3⟩^1-4ϵ dξ_1≲∫_⟨ξ_1 ⟩^ 2/⟨ξ_1 ⟩^2s+1⟨ H(ξ_1)⟩^1-4ϵ dξ_1;                                                                                                                                                                                                                               =                                    ∫_ℝ1/⟨ξ_1 ⟩^2s+1⟨ H(ξ_1)⟩^1-4ϵ dξ_1+ ∫_ξ_1^ 2/⟨ξ_1 ⟩^2s+1⟨ H(ξ_1)⟩^1-4ϵ dξ_1;                                                                                                                                                                                                                               =                                                                                                        J_1+J_2. ]

	Using (<ref>) we have J_1 ≲ 1 provided 0<ϵ< 1/6. In what follows, we estimate J_2 integrating over: (a) |ξ_1|<1 and (b) |ξ_1|≥ 1, separetely. 
	In the first situation, using that 2s+1≥ 0, we have 
	
    [                     χ_{|ξ_1|<1}J_2                                  ≲ ∫_|ξ_1|<11/⟨ H(ξ_1) ⟩^1-4ϵdξ_1≲ 1. ]

	
	For the second case, considering the sets 
    A={ξ_1: ⟨ H(ξ_1) ⟩≲ |ξ_1|^3}   and    B={ξ_1: ⟨ H(ξ_1) ⟩≳ |ξ_1|^3},
 we have 
	
	
    [                                                                                                    χ_{|ξ_1| ≥ 1}J_2                                                                                                                   = ∫_|ξ_1| ≥ 1ξ_1^ 2/⟨ξ_1 ⟩^2s+1⟨ H(ξ_1)⟩^1-4ϵχ__A(ξ_1) dξ_1+∫_|ξ_1| ≥ 1ξ_1^ 2/⟨ξ_1 ⟩^2s+1⟨ H(ξ_1)⟩^1-4ϵχ__B(ξ_1) dξ_1;                                                                                                                                                                                                                                       ≲ ∫_|ξ_1|≥ 1H'(ξ_1) ⟨ H(ξ_1)⟩^5ϵ/⟨ξ_1 ⟩^2s+1⟨ H(ξ_1)⟩^1+ϵχ__A(ξ_1) dξ_1+∫_|ξ_1|≥ 1|ξ_1|^ 2/|ξ_1|^2s+1|ξ_1|^3-12ϵ dξ_1;                                                                                                                                                                                                                                       ≲                    ∫_H'(ξ_1) |ξ_1|^15ϵ/⟨ξ_1 ⟩^2s+1⟨ H(ξ_1)⟩^1+ϵ dξ_1+∫_|ξ_1|≥ 1|ξ_1|^ 2/|ξ_1|^2s+1|ξ_1|^3-12ϵ dξ_1, ]

	where in the first integral we use that ξ_1^ 2≲ H'(ξ_1). Obviously, the second integral is ≲ 1, provided 0<ϵ<2s+1/12. For the first integral, performing the change of variables x=H(ξ_1) on , if 0<ϵ<2s+1/15, we obtain  
	
    ∫_H'(ξ_1) |ξ_1|^15ϵ/⟨ξ_1 ⟩^2s+1⟨ H(ξ_1)⟩^1+ϵ dξ_1		≲∫_dx/⟨ x ⟩^1+ϵ≲ 1.

	Case 2. (2|ξ_1|≤ |ξ|) Because ξ_1+ξ_2=ξ, then we have 
	
    max{⟨ξ_1⟩, ⟨ξ_2⟩}   ≲⟨ξ⟩,
    
    		⟨ξ_2⟩⟨ξ⟩≲⟨ξ_2 ⟩^ 2   ≲ 1+H'(ξ_1).

	
	Fix 0<ϵ<1/6, b=(1-4ϵ)^-1 and a such that  a+b=3. Considering the sets  
    A={ξ_1: ⟨ H(ξ_1) ⟩≲⟨ξ_1⟩^a⟨ξ⟩^b}   and   B={ξ_1: ⟨ H(ξ_1) ⟩≳⟨ξ_1⟩^a⟨ξ⟩^b},
 we have 
	
	
	
    χ_{2|ξ_1| ≤ |ξ|} L_1   =∫_|ξ_1| ≤ |ξ| /2⟨ξ_2⟩⟨ξ⟩/⟨ξ_1 ⟩^2s⟨ξ⟩⟨ H(ξ_1)⟩^1-4ϵχ__A(ξ_1) dξ_1+∫_⟨ξ_2⟩/⟨ξ_1 ⟩^2s⟨ H(ξ_1)⟩^1-4ϵχ__B(ξ_1) dξ_1
    
    		   ≲∫_1/⟨ξ_1 ⟩^2s+1⟨ H(ξ_1)⟩^1-4ϵχ__A(ξ_1)dξ_1+ ∫_|ξ_1| ≤ |ξ| /2H'(ξ_1)/⟨ξ_1 ⟩^2s⟨ξ⟩⟨ H(ξ_1)⟩^1-4ϵχ__A(ξ_1)dξ_1
    
    		        + ∫_⟨ξ⟩⟨ξ_1⟩^2s⟨ξ_1⟩^a(1-4 ϵ)⟨ξ⟩^b(1-4 ϵ)χ__B(ξ_1)dξ_1
    
    		   =J_1+J_2+J_3.

	By (<ref>) we have J_1 ≲ 1 provided 0<ϵ< 1/6 and 2s+1≥ 0. For J_3, remembering the definitions of a and b, if 0<ϵ<(2s+1)/12, we have 
	
	
    J_3≲∫_dξ_1/⟨ξ_1⟩^2+2s-12ϵ≲ 1.

	
	For J_2, taking account (<ref>),  if 0<ϵ<5/9, we have 1-5bϵ>0, and so ⟨ξ_1⟩^1-5bϵ≲⟨ξ⟩^1-5bϵ. Thus, considering that 2s+1≥ 0 and the definition of a and b, if 0<ϵ<(2s+1)/15 we have
	
	
    J_2   =	∫_|ξ_1| ≤ |ξ| /2  H'(ξ_1)  χ__A(ξ_1)⟨ξ_1 ⟩^2s⟨ξ⟩⟨ H(ξ_1) ⟩^1-4ϵ dξ_1≲∫_|ξ_1| ≤ |ξ| /2 H'(ξ_1) ⟨ H(ξ_1) ⟩^5ϵχ__A(ξ_1)⟨ξ_1 ⟩^2s⟨ξ⟩⟨ H(ξ_1) ⟩^1+ϵ dξ_1
    ≲   ∫_|ξ_1| ≤ |ξ| /2 H'(ξ_1)  ⟨ξ_1 ⟩^2s-5 aϵ⟨ξ⟩^1-5b ϵ⟨ H(ξ_1) ⟩^1+ϵ dξ_1≲∫_ H'(ξ_1)  ⟨ξ_1 ⟩^2s+1-5 (a+b)ϵ⟨ H(ξ_1) ⟩^1+ϵ dξ_1
    ≲   ∫_H'(ξ_1)/⟨ H(ξ_1)⟩^1+ϵdξ_1.

	
	Again, making the  change  of variables  x=H (ξ_1), we get the desired bound.



Now we are in position to prove Proposition <ref>.


 First of all, we remember that the case 0<α<1 was proved in <cit.>. So, let's assume that α∈(-∞,0)∪(1,∞).
	We start to prove  that the inequality (<ref>)  hold  if α>1. Let  u ∈ X_-1/2, 1/2-2ϵ and v∈ X^α_s, 1/2+ϵ with ϵ>0 and s>- 1/2. Considering f and g such that
	
    u(x,t)=f(α^-1/3x,t),    v(x,t)=g(α^-1/3x,t),

	using that ⟨ a ξ⟩∼_a⟨ξ⟩, for a≠ 0, and scaling properties of the Fourier transform, we have
	
    f_X_-1/2, 1/2-2ϵ^1/α∼_αu_X_-1/2, 1/2-2ϵ  and  g_X_s, 1/2+ϵ∼_αv_X^α_s, 1/2+ϵ,

	thus f ∈ X_-1/2, 1/2-2ϵ^1/α and g∈ X_s, 1/2+ϵ. Because 1/α∈ (0,1) we can apply the estimate (<ref>)  to obtain
	
    fg_L^2(^2)   ≲f_X^1/α_-1/2, 1/2-2ϵg_X_s, 1/2+ϵ
       ≲_αu_X_-1/2, 1/2-2ϵv_X^α_s, 1/2+ϵ.

	We concludes that (<ref>) holds for α>1,  observing that fg_L^2(^2) =α^-2/3u v_L^2(^2). 
	
	
	
	
	
	
	
	
	Now, if α<0, using Plancherel's identity, one can see that the estimate (<ref>) is equivalent to 
	
	
    B_s(f,g)_L^2_ξ L^2_τ≤ Cf_L^2g_L^2,

	
	where
	
    B_s(f,g)=∫_^2⟨ξ_2 ⟩^1/2f̃(ξ_2,τ_2) g̃(ξ_1,τ_1)/⟨ξ_1 ⟩^s⟨τ_1-αξ_1^ 3⟩ ^1/2+ϵ⟨τ_2-ξ_2^ 3 ⟩^1/2-2ϵdξ_1 dτ_1,

	
	with ξ_2=ξ-ξ_1, τ_2=τ-τ_1.
	Using Cauchy-Schwarz inequality we note (<ref>) holds if 
	
	
	
    ℒ_1:=sup_ξ,τ∫_^ 2⟨ξ_2⟩/⟨ξ_1 ⟩^2s⟨τ_1-αξ_1^ 3⟩ ^1+2ϵ⟨τ_2-ξ_2^ 3 ⟩^1-4ϵ dξ_1 dτ_1≲ 1.

	
	
	In order to see that the estimate (<ref>) hold,  applying the estimate (<ref>) (Lemma <ref>) for  the integral in τ_1, we obtain 
	
    ℒ_1≲sup_ξ,τ∫_⟨ξ_2⟩/⟨ξ_1 ⟩^2s⟨τ-ξ_2^ 3 -αξ_1^ 3⟩^1-4ϵ dξ_1,

	if 0<ϵ<1/4. Applying now (<ref>), we get the desired bound and finish this case. In the same way we can prove the inequality (<ref>) for α<0 or α>1.
	




The next results are usefull in the proof of the Proposition <ref>.





  Let l≥-1/2 and b>1/2.  Considering F a monotone function defined on a Lebesgue-measurable set X⊂ such that 
    |F'(ξ_1)|≳max{ξ_1^2, ξ_2^2},   ∀ξ_1∈ X,
 where ξ_2=ξ-ξ_1, for a fixed ξ∈. Then we have
	
	
    ∫_X⟨ξ_i⟩/⟨ξ_j⟩^2l⟨ F(ξ_1)⟩^2bdξ_1≲∫_X1/⟨ F(ξ_1)⟩^2bdξ_1+1,
	 
	
	for all i, j∈{1, 2}.

 Considering F a monotone increasing function, we have that F'(ξ_1)≳ξ_k^2, for k∈{1, 2}. So 
	
	
    ⟨ξ_i⟩⟨ξ_j⟩   ≲⟨ξ_i⟩^2+⟨ξ_j⟩^2≲ 1+F'(ξ_1),

	
	for all i, j∈{1, 2}. Hence
	
	
    ∫_X⟨ξ_i⟩/⟨ξ_j⟩^2l⟨ F(ξ_1)⟩^2bdξ_1   =∫_X⟨ξ_i⟩⟨ξ_j⟩/⟨ξ_j⟩^2l+1⟨ F(ξ_1)⟩^2bdξ_1
       ≲∫1/⟨ξ_j⟩^2l+1⟨ F(ξ_1)⟩^2bdξ_1+∫_XF'(ξ_1)/⟨ξ_j⟩^2l+1⟨ F(ξ_1)⟩^2bdξ_1
       ≲∫_X1/⟨ F(ξ_1)⟩^2bdξ_1+∫_XF'(ξ_1)/⟨ F(ξ_1)⟩^2bdξ_1

	
	and making the change of variable x=F(ξ_1), we finish the proof.


	Let b>1/2, l ≥ -1/2 and α∈ (-∞, 0)∪(1, +∞), then
	
    J:=∫_B⟨ξ_2⟩/⟨ξ_2⟩^2l⟨ H(ξ_1)⟩^2bdξ_1≲ 1,

	where B= {ξ_1;   ⟨ξ_2⟩ > 1+ι/ι⟨ξ_1⟩}, ι as defined in (<ref>), ξ_2=ξ-ξ_1 and 
	
    H(ξ_1):=τ-ξ_1^ 3 -αξ_2^ 3.



	Of course, (<ref>) holds when α<0 (see the proof of Lemma <ref>). Thus we will suppose α>1. First we will consider |ξ|>1 and let
	
	
    X={ξ_1;   |ξ_1|<ι|ξ|},

	
	where ι∈ (0, 1) will be chosen later, we have
	
	
    J≤∫_B⟨ξ_2⟩/⟨ξ_2⟩^2l⟨ H(ξ_1)⟩^2bχ__X(ξ_1)dξ_1_J_1+∫_B⟨ξ_2⟩/⟨ξ_2⟩^2l⟨ H(ξ_1)⟩^2bχ__\ X(ξ_1)dξ_1_J_2.

	
	Starting with J_2, noting that in \ X={ξ_1;  |ξ_1|≥ι|ξ|> ι}, we have ⟨ξ_2⟩≤1+ι/ι⟨ξ_1⟩, then B∩ (\ X)=∅ and J_2≡0.
	Now, for J_1, we have
	
	
    ⟨ξ_1⟩≲⟨ξ_2⟩≲⟨ξ⟩.

	
	We choose λ∈(0, 3) such that 
	
	
    H'(ξ_1)=3(α-1)ξ_1^2-6αξ_1ξ+3αξ^2≥λαξ^2.

	
	So, we have that H is increasing in X and |H'(ξ_1)|≳ξ^2≳_α,λ,c_2max{ξ_1^2, ξ_2^2}, for all ξ_1∈ X. Thus, collecting this facts and using Lemma <ref> and Lemma <ref> item (iv), we get the desired. We determine ι∈(0, 1) and λ∈(0, 3) such that (<ref>) is valid. If fact, the inequality (<ref>) is equivalent to
	
	
    3(1-α)ξ_1^2+6αξξ_1_quadratic funtion≤(3-λ)αξ^2_constant function.

	
	So we choose ι such that the equality  (<ref>) is true in the interval |ξ_1|≤ι|ξ|. Thus, ι is a root of the quadractic equation
	
	
    3(1-α)ι^2± 6αι+(λ-3)α=0.

	
	One can see that 
	
    ι=(6α-√(36α^2-12α(1-α)(λ-3)) )/[6(α-1)]

	belongs to (0, 1) if λ∈(0, 3).
	
	Now if |ξ |≤ 1,  we have
	
    1/2⟨ξ_2 ⟩≤⟨ξ_1 ⟩≤ 2⟨ξ_2 ⟩

	and using  Lemma <ref> we get
	
    J ≤∫_⟨ξ_2⟩/⟨ξ_2⟩^2l⟨τ-ξ_1^ 3 -αξ_2^ 3⟩^2bdξ_1
    ≲∫_⟨ξ_2⟩/⟨ξ_1⟩^2l⟨1/ατ-1/αξ_1^ 3 -ξ_2^ 3⟩^2bdξ_1
    ≲ 1.

	


With these Lemmas in hands, we can prove Proposition <ref> and therefore the trilinear estimates in Proposition <ref>.

	We only provide a detailed proof of (a), because (b) is analogous. Suppose that s, k>-1/2 and s-k≤ 1/2. As before, using Plancherel identity and Cauchy-Schwarz inequality we need to estimate
	
    ∫_^2⟨ξ_2 ⟩^2(s-k)+1⟨τ_2-αξ_2^3⟩^2b⟨ξ_1⟩^2s⟨τ_1-ξ_1^3⟩^2b dξ_1 dτ_1,

	where ξ_2=ξ-ξ_1, τ_2=τ-τ_1, for τ and ξ fixed and b=1/2+ϵ. Integrating in τ_1 and applying (<ref>), we need to estimate the following integral
	
	
	
	
    ℒ_2=
    		∫_⟨ξ_2⟩^1+2(s-k)/⟨ξ_1 ⟩^2s⟨ H(ξ_1)⟩^2b dξ_1,

	
	where H(ξ_1) is given by (<ref>).
	Our goal is to prove ℒ_2≲ 1. First we consider the case s-k≤ 0, then
	
	
    ℒ_2   ≲∫_⟨ξ_2⟩/⟨ξ_1⟩^2s⟨ H(ξ_1)⟩^2bdξ_1.

	
	If α>1, we write 1/αH(ξ_1)=τ̃-ξ_2^3-βξ_1^3, where τ̃=τ/α and β=1/α. Applying (<ref>), with β∈(0, 1) in place of α, we get ℒ_2≲ 1, remembering that 2b>1-4ϵ. On the other hand, if α<0 we can see that H'(ξ_1)=3αξ_2^2-3ξ_1^2<0 and |H'(ξ_1)|≳_αmax{ξ_1^2, ξ_2^2}. So, combining Lemma <ref> and Lemma <ref> item (iv), we get the same bound.
	
	Considering now 0< s-k ≤ 1/2, let
	
	
    A={ξ_1; ⟨ξ_2⟩≲⟨ξ_1⟩}     and     B={ξ_1; ⟨ξ_1⟩≲⟨ξ_2⟩}.

	
	Thus
	
	
    ℒ_2≤∫_⟨ξ_2⟩^1+2(s-k)/⟨ξ_1 ⟩^2s⟨ H(ξ_1)⟩^2bχ__A(ξ_1) dξ_1_ℒ_2^A+∫_⟨ξ_2⟩^1+2(s-k)/⟨ξ_1 ⟩^2s⟨ H(ξ_1)⟩^2bχ__B(ξ_1) dξ_1_ℒ_2^B.

	
	For the first integral, we have 
	
	
    ℒ_2^A≲∫_⟨ξ_2⟩/⟨ξ_1⟩^2k⟨ H(ξ_1)⟩^2bdξ_1,

	
	and this integral  has already been estimated in (<ref>). For the second integral ℒ_2^B, similarly as above one can see that if s≤ 0,  using Lemma <ref> we have
	
	
    ℒ_2^B≲∫_B⟨ξ_2⟩/⟨ξ_2⟩^2k⟨ H(ξ_1)⟩^2bdξ_1 ≲ 1.

	
	On the other hand, if s>0  again using Lemma <ref> we have
	
	
    ℒ_2^B≲∫_B⟨ξ_2⟩/⟨ξ_2⟩^2(k-s)⟨ H(ξ_1)⟩^2bdξ_1≲ 1,

	
	since k-s ≥ -1/2.   
	Now, we prove the same estimate in the range: s=-1/2 and -1/2< k. Indeed, we need to prove that
	
	
    uv_L^2(^2)≲u_X^α_k, 1/2+ϵv_X_-1/2, 1/2+ϵ.

	This  estimate follows from the estimate (<ref>), noting that 1/2-2ϵ<1/2+ϵ. 
	













Local well-posedness result: proof of Theorem <ref>

In view of the previous sections, we are in position to prove the Local well-posedness result given in Theorem <ref>. Here we understand that a solution of the system (<ref>) is in fact a solution of the associated integral equations (<ref>) and (<ref>). We use standard arguments, so we give the proof for the sake of completeness. 


	Let s,k∈ such that s,k>-1/2 and |s-k|<1/2. Let b=1/2+ϵ and b'=-1/2+2ϵ, with ϵ>0. Let fix α∈(-∞,0)∪(1,∞) (see Remark <ref>). For a>0, to be choosen later,  let
	
	
    B_a={(v,w)∈ X_s,b× X^α_k,b ; (v,w)_X_s,b× X^α_k,b=v_X_s,b+w_X^α_k,b<a  },

	
	
	a complete metric spaces. For δ>0 (choosen later), we define the map 𝔉=𝔉_δ:B_a→ X_s,b× X^α_k,b such that 𝔉(v,w)=(𝔉_1(v,w), 𝔉_2(v,w)) where
	
	
    𝔉_1(v,w)   =η(t)U(t)ϕ-η_δ(t)∫_0^tU(t-t')∂_x(vw^2)(t')dt'
    𝔉_2(v,w)   =η(t)U^α(t)ϕ-η_δ(t)∫_0^tU^α(t-t')∂_x(v^2w)(t')dt'.

	
	
	From the linear estimates in Lemma <ref> and the trilinear estimates in Proposition <ref>, for all (v,w)∈ B_a we get 
	
	
    𝔉(v,w)_X_s,b× X^α_k,b   ≤ C(ϕ,ψ)_H^s× H^k+ Cδ^1-b+b'{v_X_s,bw^2_X^α_k,b + v_X_s,b^2w_X^α_k,b}
       ≤ C(ϕ,ψ)_H^s× H^k+ 2Cδ^ϵa^3.

	
	Now, taking a=2C(ϕ,ψ)_H^s× H^k>0 and δ>0 such that 2Cδ^ϵ<1/2 we conclude that
	
	
    𝔉(v,w)_X_s,b× X^α_k,b≤ a,    for all  (v,w)∈ B_a,

	
	i.e., for a>0 and δ>0 as before, 𝔉_δ(B_a)⊂ B_a. 
	Also, with a similar arguments and taking δ>0 smaller, if necessary, we can conclude that 
	
	
    𝔉(v,w)-𝔉(ṽ,w̃)_X_s,b× X^α_k,b≤(v,w)-(ṽ,w̃)_X_s,b× X^α_k,b,

	
	i.e., 𝔉:B_a→ B_a is a contraction an has a unique fixed point, establishing  a unique solution (v,w) satisfying (<ref>) and (<ref>) for every t∈[-δ, δ]. Because b>1/2,  then we have the persistence property (v,w)∈ C([-δ,δ]:H^s)× C([-δ,δ]:H^k) and also, following a similar arguments, we can conclude that the solution mapping is locally lipchitz from H^s× H^k into C([-δ,δ]:H^s)× C([-δ,δ]:H^k). The uniqueness in the class X_s,b^δ× X^α,δ_k,b can be proved with standard arguments (see e.g. <cit.> or <cit.>).












Ill-posedness results

In this section we prove some ill-posedness results related to the system (<ref>).



 §.§ The solution mapping is not C^3

 




Here we will prove the Theorem <ref>. The proof given here follow the structure of the proofs in <cit.> and <cit.> (see also <cit.>). It is well known that if the LWP results in H^s()× H^k() for (<ref>) is obtained by means of contraction method, then for a fixed r>0 there is a T=T(r,s,k)>0 such that the solution mapping 


    S   :   
    	B_r 
    	     ⟶      C
    	(  [0,T] ; 
    	H^s ) × C
    	(  [0,T] ; 
    	H^k  
    	)
    	
          (ϕ,ψ)   ↦    
    	S_(ϕ, ψ)=(v__(ϕ,ψ) , w__(ϕ,ψ) ) ,


is analitic (see Theorem 3 in <cit.>), where B_r is the r-ball centered at the origin of H^s()× H^k() and v=v__(ϕ,ψ) and w=w__(ϕ,ψ) satisfies, respectively, the integral equations (<ref>) and (<ref>) for initial data ϕ and ψ, in the time interval [0, T]. In this way, if we show that for a certain indices (s, k) the solution mapping is not three times differentiable at the origin (0, 0) for all T>0 fixed,  the contraction method can not be applied to get LWP for these indices (s, k).

Fixing t∈[0, T], 
we define the flow mapping associated to the system (<ref>) the  map 


    S^ t   :   
    	B_r
    	     ⟶      H^s()× H^k()
    	
          (ϕ,ψ)   ↦    
    	S^t_(ϕ, ψ)=S_(ϕ, ψ)(t)=(v__(ϕ,ψ)(t) , w__(ϕ,ψ)(t)).


The Theorem <ref> follows from the next proposition:

 Assume that the system (<ref>) is locally well-posed in the time interval [0, T]. Then we have 
	
		
  (a) The solution mapping (<ref>) is not 3-times Fréchet differentiable at the origin in H^s()× H^k() if |s-k|>2.
		
		
  (b) The flow mapping (<ref>) is not 3-times Fréchet differentiable at the origin in H^s()× H^k() if s<-1/2 or k<-1/2.
	
	


 We point out that the result in (b) implies that the solution mapping is not 3-times differentiable at (0, 0) for the same indices s and k. For a more detailed discussion we refer Remarks 1.4 and 1.5 in <cit.>.
	





Before we start to prove our results, we need to do some calculations.
If S^t is 3-times Fréchet differentiable at the origin in H^s()× H^k(), then its third derivative D^3S^t_(0, 0) belongs to ℬ_1, the normed space of bounded trilinear applications from (H^s× H^k)× (H^s× H^k)× (H^s× H^k) to H^s× H^k and 
we have the following estimate for the third Gâteaux derivative of S^ t 


    ∂^3 S^ t_(0,0)/∂Φ_0∂Φ_1∂Φ_2__H^s× H^k   =
    	 D^3S^ t_(0,0)(Φ_0,Φ_1,Φ_2)__H^k× H^l
       ≤  D^3S^ t_(0,0)__ℬ_1Φ_0__H^k× H^lΦ_1__H^k× H^lΦ_2__H^k× H^l ,
    	 ∀Φ_0,Φ_1,Φ_2∈ H^k× H^l.



Also, if S is 3-times Fréchet differentiable at the origin, we have a similar estimate:




    sup_t∈ [0, T]∂^3 S^ t_(0,0)/∂Φ_0∂Φ_1∂Φ_2__H^s× H^k   =
    	 D^3S_(0,0)(Φ_0,Φ_1,Φ_2)__H^k× H^l
       ≤  D^3S_(0,0)__ℬ_2Φ_0__H^k× H^lΦ_1__H^k× H^lΦ_2__H^k× H^l ,
    	∀Φ_0,Φ_1,Φ_2∈ H^k× H^l,


where ℬ_2 is the normed space of bounded trilinear applications from (H^s× H^k)× (H^s× H^k)× (H^s× H^k) to C(  [0,T] ; 
H^s ) × C(  [0,T] ; H^k ). For Φ_k=(ϕ_k, ψ_k)∈𝒮()×𝒮(), k=0,1,2, we can calculate the third Gâteaux derivative of each component of S^t:


    ∂^3 v_(0,0)/∂Φ_0∂Φ_1∂Φ_2   =-2∫_0^tU(t-t')∂_x{U(t')ϕ_0U^α(t')ψ_1U^α(t')ψ_2 + U(t')ϕ_1U^α(t')ψ_2U^α(t')ψ_0 
           + U(t')ϕ_2U^α(t')ψ_1U^α(t')ψ_0}dt'




and


    ∂^3 w_(0,0)/∂Φ_0∂Φ_1∂Φ_2   =-2∫_0^tU^α(t-t')∂_x{U(t')ϕ_0U(t')ϕ_1U^α(t')ψ_2 + U(t')ϕ_0U(t')ϕ_2U^α(t')ψ_1 
           + U(t')ϕ_1U(t')ϕ_2U^α(t')ψ_0}dt.






So, for directions Φ_0=(ϕ_0, 0), Φ_1=(0, ψ_1) and Φ_2=(0, ψ_2) in 𝒮()×𝒮() we get



    ∂^3 v_(0,0)/∂Φ_0∂Φ_1∂Φ_2   =-2∫_0^tU(t-t')∂_x{U(t')ϕ_0 U^α(t')ψ_1U^α(t')ψ_2 }dt'

and for directions Φ_0=(ϕ_0, 0), Φ_1=(ϕ_1, 0) and Φ_2=(0, ψ_2)




    ∂^3 w_(0,0)/∂Φ_0∂Φ_1∂Φ_2   =-2∫_0^tU^α(t-t')∂_x{U(t')ϕ_0U(t')ϕ_1U^α(t')ψ_2 }dt'.





With these in hands we also need a elementary result, proved  in <cit.>:

 Let A, B, R Lebesgue-measurable subsets of ^n such that[Here X-Y={x-y;  x∈ X  and y∈ Y}.] R-B⊂ A. Then[|X| denotes de Lebesgue measure of the set X]
	
    χ__A∗χ__B_L^2()≳ |B|  |R|^1/2.


The following lemma, which is a version of the elementary lemma above, plays a central role in the proof of the Theorem <ref>.
 Let A, B, C, R Lebesgue-measurable  subsets of ^n such that R-B-C⊂ A. Then
	
    χ__A∗χ__B∗χ__C_L^2()≳ |B||C||R|^1/2.





Now we move to the


	(a)	Suppose that S^t is 3-times differentiable at the origin in H^s× H^k. Because (<ref>) and  (<ref>) are the components of the third Gâteaux derivative at the origin,  we have  the same estimate  with the H^s× H^k norm of (<ref>) or  (<ref>) in the right-hand side of (<ref>). We start with the first component.  Considering  
	A,B,C⊂ bounded subsets and choosing ϕ_0,ψ_1,ψ_2∈𝒮() such that[ For B, bounded subset of , ⟨·⟩^lϕ∼χ__B means
		χ__B≤⟨·⟩^l φ  with  φ_H^l≤2χ__B_L^2 
	]
	
	
	
	
	
	
	⟨·⟩^s ϕ_0∼χ__A, ⟨·⟩^k ψ_1∼χ__B and ⟨·⟩^k ψ_2∼χ__C, 
	we have that[ (*.*)_R(or (*.*)_L) denotes the right(or left)-hand side of an equality or inequality numbered by (*.*)]
	
	
    (<ref>)_L   ≳∫_0^t⟨ξ⟩^s|ξ|/⟨ξ_1 ⟩^s⟨ξ_2  ⟩^k⟨ξ_3⟩^kcos(t'Q_α)χ__A(ξ_1)χ__B(ξ_2)χ__C(ξ_3)  dξ_1dξ_2dt'    _L^2_ξ

	where  ξ_3:=ξ-ξ_1-ξ_2 and 
	
	
	
    Q_α=Q_α(ξ, ξ_1, ξ_2):=ξ^3-ξ_1^3-αξ_2^3-αξ_3^3.
 
	
	
	So combining (<ref>) with (<ref>) we get 
	
	
    sup_t∈[0, T]∫_0^t⟨ξ⟩^s|ξ|/⟨ξ_1 ⟩^s⟨ξ_2  ⟩^k⟨ξ_3⟩^kcos(t'Q_α)χ__A(ξ_1)χ__B(ξ_2)χ__C(ξ_3)  dξ_1dξ_2dt'    _L^2_ξ≲ |A|^1/2|B|^1/2|C|^1/2,

	for all t∈ [0, T]. 
	
	
	In view of Lemma <ref>, now we must choose the sets A, B, C, R and a sequence of times t_N∈ [0, T] in this way: for N∈ 
	
	
    A_N={ξ_1∈ : |ξ_1|<1/2}, 
    		     
    		B_N={ξ_2∈ : |ξ_2|<1/4},      C_N= {ξ_3∈ : |ξ_3-N|<1/8}

	
	
    R_N={ξ∈ : |ξ-N|<1/8}    and     
    		t_N=T/2N^3(1+T).
 
	We have  R_N-B_N-C_N⊂ A_N and t_N∈[0, T]. 
	Because ξ_1∈ A_N, ξ_2∈ B_N and ξ_3∈ C_N, thus
	
	
    ⟨ξ⟩^s|ξ|/⟨ξ_1 ⟩^s⟨ξ_2  ⟩^k⟨ξ_3⟩^k ∼ N^s-k+1    and    cos(t'Q_α)>1/2 , 
    		  ∀ t'∈[0,t_N].
 
	
	
	From Lemma <ref> and (<ref>) yields
	
	
    t_N|R_N|^1/2|B_N||C_N| N^s-k+1 ≲ 
    		|A_N|^1/2|B_N|^1/2|C_N|^1/2, 
    		  ∀ N∈.

	Taking account that |A_N|∼|B_N|∼|C_N|∼ |R_N|∼ 1 and t_N∼ N^-3, then we have s-k<2.
	
	Now, dealing with the second component (<ref>), analogously as we did before, we have
	
	
	
    sup_t∈[0, T]∫_0^t⟨ξ⟩^k|ξ|/⟨ξ_1 ⟩^s⟨ξ_2  ⟩^s⟨ξ_3⟩^kcos(t'P_α)χ__A(ξ_1)χ__B(ξ_2)χ__C(ξ_3)  dξ_1dξ_2dt'    _L^2_ξ≲ |A|^1/2|B|^1/2|C|^1/2,

	for all t∈ [0, T], where 
    P_α=P_α(ξ, ξ_1, ξ_2)=ξ^3-ξ_1^3-ξ_2^3-αξ_3^3.

	
	Interchanging the rules of the sets B_N and C_N given before we can conclude that k-s<2. We finishes the proof of item (a).
	
	(b) In this case we are not able to take a sequence t_N→ 0, because t is fixed number. We need to control de argument in the terms cos(t'Q_α) and cos(t'P_α), and we get this by making both Q_α and P_α suficiently small.  Fixing t∈[0, T], by (<ref>), similarly what we did to get (<ref>), we have that
	
	
	
    ∫_0^t⟨ξ⟩^s|ξ|/⟨ξ_1 ⟩^s⟨ξ_2  ⟩^k⟨ξ_3⟩^kcos(t'Q_α)χ__A(ξ_1)χ__B(ξ_2)χ__C(ξ_3)  dξ_1dξ_2dt'    _L^2_ξ≲ |A|^1/2|B|^1/2|C|^1/2,

	
	For N∈, considering the sets
	
	
	
    A_N={ξ_1∈ : |ξ_1-aN|<ε⟨ t⟩^-1 N^-2}, 
    		     
    		B_N={ξ_2∈ : |ξ_2-bN|<ε14 ⟨ t⟩^-1 N^-2},

	
	
    C_N= {ξ_3∈ : |ξ_3-cN|<ε14 ⟨ t⟩^-1 N^-2}    and     R_N={ξ∈ : |ξ-N|<ε14 ⟨ t⟩^-1 N^-2}

	
	
	
	
	where the constant ε>0 will be small, but fixed, and the positive constants a, b and c satisfies the conditions
	
	
    {[           a+b+c=1,; a^3+α b^3+α c^3=1. ].

	
	First we note that R_N-B_N-C_N⊂ A_N and |A_N|∼|B_N|∼|C_N|∼|R_N|∼ N^2. Also, for ξ_1∈ A_N, ξ_2∈ B_N, ξ_3∈ C_N yields
	
	
    ⟨ξ⟩^s|ξ|/⟨ξ_1 ⟩^s⟨ξ_2  ⟩^k⟨ξ_3⟩^k∼ N^1-2k

	and 
	
	
    |Q_α|   =|ξ^3-N^3+N^3-ξ_1^3-αξ_2^3-αξ_3^3|≤ |ξ^3-N^3|+|(a^3+α b^3+α c^3)N^3-ξ_1^3-αξ_2^3-αξ_3^3
       ≤  |ξ^3-N^3|+|(aN)^3-ξ_1^3|+|α||(bN)^3-ξ_2^3|+|α||(cN)^3-ξ_3^3|
       ≲__a,b,|α| ⟨ t'⟩^-1ε,

	where we use the elementary identity A^3-B^3=(A-B)(A^2+AB+B^2). This implies the desired estimate 
    cos(t'Q_α)>1/2,   for all  t'∈ [0, t].

	
	From the Lemma <ref> and (<ref>) we get
	
	
    t(N^-2)^1/2N^-2N^-2N^1-2k≲ (N^-2)^1/2(N^-2)^1/2(N^-2)^1/2,

	and so k≥ -1/2.
	For the second component (<ref>), in the same way as before, we have for a fixed t in [0, T] the following inequality
	
	
    ∫_0^t⟨ξ⟩^k|ξ|/⟨ξ_1 ⟩^s⟨ξ_2  ⟩^s⟨ξ_3⟩^kcos(t'P_α)χ__A(ξ_1)χ__B(ξ_2)χ__C(ξ_3)  dξ_1dξ_2dt'    _L^2_ξ≲ |A|^1/2|B|^1/2|C|^1/2.

	
	Of course, with the same choice of subsets A_N, B_N, C_N, etc. but with
	
	
    {[         a+b+c=1,; a^3+b^3+α c^3=1, ].

	we can conclude that s≥ -1/2 and we finish the proof of  the item (b) and also the proof of the Theorem.
	













 §.§ Failure of trilinear estimates



 




In this subsection we will prove the failure of the trilinear estimates (<ref>) and (<ref>). 


	We prove only item (a), because the item (b) follows analogously. Using definition of the X_s,b-norm  and Plancherel's identity,  the estimate (<ref>) is equivalent to 
	
    𝒯_s(f,g,h) _L^2_ξ L^2_τ≲f_L^2(^2)g_L^2(^2)h_L^2(^2),

	where 
	
    𝒯_s(f,g,h):= ⟨ξ⟩^s⟨τ-ξ^3⟩^b'ξ∫__^4f(ξ_1, τ_1)g(ξ_2, τ_2) h(ξ_3, τ_3)⟨ξ_1 ⟩^s⟨τ_1-ξ_1^3 ⟩^b⟨ξ_2 ⟩^k⟨τ_2-αξ_2^3 ⟩^b⟨ξ_3 ⟩^k⟨τ_3-αξ_3^3 ⟩^bdξ_1dξ_2dτ_1dτ_2_L^2_ξτ,

	with ξ_3=ξ-ξ_1-ξ_2, τ_3=τ-τ_1-τ_2.
	
	Now, suppose that s-2k>1.	Let c_2 and c_3 be two constants satisfying
	
    c_2+c_3=1,
       α c_2^3+αc_3^3 =1,

	and let  σ_1=τ_1-αξ_1^3, σ_2=τ_2-αξ_2^3, σ_3=τ_3-αξ_3^3  in order to  apply the Lemma <ref>, we define the sets
	
    A_N={(ξ_1, τ_1) ; |ξ_1|< N^-2, |σ_1|<C_α},     B_N={(ξ_2, τ_2) ; |ξ_2-c_2 N|< N^-2/3, |σ_2|<1},

	
	
    C_N={(ξ_3, τ_3) ; |ξ_3-c_3 N|< N^-2/3,  |σ_3|<1}   and   R_N={(ξ, τ) ; |ξ-N|< N^-2/3,  |τ- ξ^3|<1}.

	Then R_N-B_N-C_N ⊂ A_N. In fact , if (ξ_1, τ_1)=(ξ, τ)-(ξ_2, τ_2)-(ξ_3, τ_3) with (ξ, τ) ∈ R_N, (ξ_2, τ_2) ∈ B_N and (ξ_3, τ_3) ∈ C_N, then using (<ref>), we have
	
    |ξ_1|=|ξ-ξ_2-ξ_3| ≤ |ξ-N|+|N-ξ_2-ξ_3| ≤1/3N^-2+|c_2N-ξ_2|+|c_3N-ξ_3| < N^-2.

	On the other hand using again (<ref>)
	
    |σ_1| =   |τ_1 -ξ_1^3|= |τ -τ_2-τ_3 -ξ_1^3|= |τ-ξ^3+ξ^3-σ_2-αξ_2^3- σ_3-αξ_3^3-ξ_1^3|
    ≤    |τ-ξ^3|+|σ_2|+|σ_3|+ |ξ_1|^3+|ξ^3-αξ_2^3-αξ_3^3|
    ≤    3+N^-6+|ξ^3-N^3+α N^3c_2^3 -αξ_2^3+α N^3c_3^3-αξ_3^3|
    ≲    (1+|α|).

	Consider f=χ_A_N, g=χ_B_N, h=χ_C_N. From the definition of A_N, B_N and C_N holds that ⟨ξ_1⟩∼ 1, ⟨ξ_2⟩∼ N,  ⟨ξ_3⟩∼ N, ⟨ξ⟩∼ N and ⟨σ_j ⟩∼ 1, j=1,2,3. By (<ref>) and Lemma <ref> we obtain
	
    N^s-2k+1R_N_L^2(^2)χ_B_N_L^1(^2)χ_C_N_L^1(^2)≤    N^s-2k+1χ_A_N∗χ_B_N∗χ_C_N_L^2(^2)
    ≤   χ_A_N_L^2(^2)χ_B_N_L^2(^2)χ_C_N_L^2(^2)

	which implies
	
    NN^s-2k N^-1 N^-4≲ N^-3

	and the last inequality is false if s-2k>1.
	
	
	Finally, if k<-1/2, let c_1, c_2 and c_3 be three constants satisfying
	
    c_1+c_2+c_3=1,
        c_1^3+α c_2^3+α c_3^3 =1.

	
	Analogously as above, if we consider the sets
	
    A_N={(ξ_1, τ_1) ;   |ξ_1-c_1N|< N^-2, |σ_1|<C_α},      B_N={(ξ_2, τ_2) ;   |ξ_2-c_2N|< N^-2/3, |σ_2|<1},

	
	
    C_N={(ξ_3, τ_3) ;   |ξ_3-c_3N|<  N^-2/3, |σ_3|<1}  and   R_N={(ξ, τ) ;   |ξ-N|< N^-2/3, |τ-ξ^3|<1}.

	Then using the condition (<ref>) we obtain
	
    |ξ_1-c_1N|=|ξ-ξ_2-ξ_3-c_1N| ≤ |ξ-N|+|c_2N-ξ_2|+|c_3N-ξ_3| < N^-2,

	as above and using the condition (<ref>) again
	
    |σ_1| ≤    |τ-ξ^3|+|σ_2|+|σ_3|+ |ξ^3-ξ_1^3-αξ_2^3-αξ_3^3|
    ≲    3+|ξ^3-N^3|+ |c_1^3N^3- ξ_1^3|+|α| |c_2^3N^3- ξ_2^3| +|α| |c_3^3N^3- ξ_3^3|
    ≲    (1+|α|).

	Thus R_N-B_N-C_N ⊂ A_N and ⟨ξ_j⟩∼ N, j=1,2,3, ⟨ξ⟩∼ N and ⟨σ_j ⟩∼ 1, j=1,2,3. Again, by (<ref>) and Lemma <ref> we get
	
    N^1+sN^sN^2kR_N_L^2(^2)χ_B_N_L^1(^2)χ_C_N_L^1(^2)≤   N^1+sN^sN^2kχ_A_N∗χ_B_N∗χ_C_N_L^2(^2)
    ≤   χ_A_N_L^2(^2)χ_B_N_L^2(^2)χ_C_N_L^2(^2)

	therefore
	
    NN^-2k N^-1 N^-4≲ N^-3

	and the last inequality is false if -2k>1.


	We only give the proof of (a), because (b) can be proved in the same way. We will consider the following sets
	
    A_N={(ξ_1, τ_1) ;   |ξ_1|< 1, |σ_1| ∼ N^3},      B_N={(ξ_2, τ_2) ;   |ξ_2|< 1/2, |σ_2|<1},

	
	
    C_N={(ξ_3, τ_3) ;   |ξ_3-N|<1/4, |σ_3|<1}  and   R_N={(ξ, τ) ;   |ξ-N|<1/4, |τ-ξ^3|<1}.

	Then R_N-B_N-C_N ⊂ A_N and ⟨ξ_3⟩∼ N, ⟨ξ_j ⟩∼ 1,  j=1,2, ⟨ξ⟩∼ N and ⟨σ_j ⟩∼ 1, j=2,3. Using (<ref>) we obtain
	
    N^1+sN^3bN^kR_N_L^2(^2)χ_B_N_L^1(^2)χ_C_N_L^1(^2)≤   N^1+sN^3bN^kχ_A_N∗χ_B_N∗χ_C_N_L^2(^2)
    ≤   χ_A_N_L^2(^2)χ_B_N_L^2(^2)χ_C_N_L^2(^2)

	therefore
	
    N^1+sN^3bN^k≲ N^3/2

	and the last inequality is false if s-k>1/2+3b, b=1/2+ϵ.
	


For the next proof we will use another elementary result, proved in <cit.>.

	Let R_j:=[a_j, b_j]×[c_j, d_j]⊂ℝ^2, j=1,⋯,n be rectangles such that b_j-a_j=N and d_j-c_j=M. Then
	
    χ_R_1*χ_R_2*⋯*χ_R_n_L^2(ℝ^2)∼ (NM)^n-1/2=|R_j|^n-1/2.








	We only prove that (<ref>) fails to hold. Using Plancherel's identity, 
	the estimate (<ref>) is equivalent to showing that
	
    ℬ_s   :=⟨ξ⟩^-1/2⟨τ- ξ^3 ⟩^-1/2+4ϵ∫_^4⟨ξ_3 ⟩^1/2⟨ξ_2 ⟩^1/2⟨ξ_1 ⟩^1/2f̃(ξ_1, τ_1)g̃(ξ_2, τ_2)h̃(ξ_3, τ_3) ⟨τ_1- αξ_1^3 ⟩^1/2+ϵ⟨τ_2-αξ_2^3 ⟩^1/2-2ϵ⟨τ_3-ξ_3^3 ⟩^1/2-2ϵ dξ_1 dτ_1dξ_2 dτ_2 _L^2_ξτ
       ≤f_L^2(^2)g_L^2(^2)h_L^2(^2),

	where 
	f(ξ,τ)=⟨ξ⟩^-1/2⟨τ-αξ^3⟩^1/2-2ϵw_1(ξ,τ),  g(ξ,τ)=⟨ξ⟩^-1/2⟨τ-αξ^3⟩^1/2+ϵw_2(ξ,τ), h(ξ,τ)=⟨ξ⟩^-1/2⟨τ-ξ^3⟩^1/2+ϵv(ξ,τ), ξ_1+ξ_2+ξ_3=ξ and τ_1+τ_2+τ_3=τ. 
	
	
	We will construct functions f, g and h for which the estimate (<ref>) fails to hold for all ϵ>0. 	Let c_1, c_2 and c_3 three numbers such that
	
    c_1+c_2+c_3=1,   α c_1^3+α c_2^3+ c_3^3=1,

	and consider two rectangles R_1, R_2 and  R_3  with centres respectively at (c_1N, α(c_1N)^3), (c_2N, α(c_2N)^3) and  (c_3N, (c_3N)^3), and each with dimension N^-(2+r)× N^-r, where -2<r<0. Now, consider f and g defined, via their Fourier transform, by	f = χ_R_1, g= χ_R_2,  h= χ_R_3.
	
	It is easy to see that 
	
    f_L^2(^2) =g_L^2(^2)=h_L^2(^2)=N^-(1+r).

	Also, 
	
    | ξ_1-c_1N|≤1/2N^-(2+r),      |τ_1-α (c_1N)^3|≤1/2N^-r,

	
    |ξ_2-c_2 N|≤1/2N^-(2+r),      |τ_2-α (c_2N)^3|≤1/2N^-r

	and
	
    |ξ_3-c_3 N|≤1/2N^-(2+r),      |τ_3- (c_3N)^3|≤1/2N^-r.

	We have |ξ_j|∼ N and
	
    | ξ-N|   =| ξ_1+ξ_2+ξ_3-c_1N- c_2N-c_3N |≤ | ξ_1-c_1N |+| ξ_2-c_2N |+| ξ_3-c_3N |
    			≤3/2N^-(2+r).

	Also, one can prove  that 
	
    |τ-  ξ^3|=   |τ_1+τ_2+τ_3-  (α c_1^3+α c_2^3+ c_3^3)N^3+  N^3- ξ^3|
    ≤   |τ_1-α (c_1N)^3|+|τ_2-α (c_2N)^3| +|τ_3-  (c_3 N)^3|+ |N^3- ξ^3|
    ≤   3/2N^-r+  1/2N^-(2+r)|N^2+Nξ+ ξ^2|
    			≲ N^-r,

	and for j=1,2:
	
    |τ_j- αξ_j^3|=   |τ_j- α(c_jN)^3+ α(c_jN)^3-αξ_j^3|≤  |τ_j-  α(c_jN)^3|+ |α||(c_jN)^3- ξ_j^3|
    ≤   1/2N^-r+ 1/2N^-(2+r)|c_j^2N^2+c_jNξ_j+ ξ_j^2|
    			≲ N^-r.

	Similarly 
	
    |τ_3-  ξ_3^3|=   |τ_3- (c_3N)^3+  (c_3N)^3- ξ_3^3|≤  |τ_2-  α N^3|+|c_3N- ξ_3| |(c_3N)^2+(c_3N)ξ_3+ ξ_3^2|
    ≲   1/2N^-r+  1/2N^-(2+r) N^2
    			≲ N^-r.

	Thus
	
    ⟨τ_j- αξ_j^3 ⟩≳ N^-r,   j=1,2,   ⟨τ_3-  ξ_3^3 ⟩≳ N^-r,   ⟨τ-  ξ^3 ⟩≳ N^-r

	With these considerations, we get from (<ref>)
	
    ℬ_s(f,g) 
    			   ∼ N^1/2-r (-1/2+4ϵ)∫_^4 N^3/2χ_R_1(ξ_1, τ_1)χ_R_2(ξ_2, τ_2)χ_R_3(ξ_3, τ_3) N^-3(1/2+ϵ)rdξ_1 dτ_1dξ_2 dτ_2_L^2_ξτ(^2)
       ∼ N^2+2r-r ϵχ_R_1*χ_R_2*χ_R_3_L^2(^2)
       = N^2+2r-r ϵ|R_j|^3-1/2 = N^2+2r-r ϵN^-5-5r=N^-3-r(3+ ϵ).

	Now, using  (<ref>) and  (<ref>) in (<ref>), 
	
    N^-3-r(3+ ϵ)≲ N^-3-3r⟺ N^-rϵ≲ 1.

	
	Since r<0, if we choose N large, the estimate (<ref>) fails to hold whenever  ϵ>0 and this completes the proof of the proposition.
	











Failure of Bilinear estimates in Section 3

In this section we will conclude that the Theorem <ref> is sharp in the sense that we cannot use the approach developed in Section <ref>, to improves the Sobolev indices in Proposition <ref>.


	Let α≠ 0,1. 

	(a) If s-k>-1/2 or k<-1/2 then  the following bilinear estimate
	
    uv_L^2(^2)≲u_X^α_k-s-1/2, 1/2+ϵv_X_s, 1/2+ϵ,

	fails to hold.

	(b) If  k-s>-1/2 or s<-1/2 then  the following bilinear estimate
	
    uv_L^2(^2)≲u_X_s-k-1/2, 1/2+ϵv_X^α_k, 1/2+ϵ,

	fails to hold.

 As before, we only prove item (a).	Using Plancherel's identity, 
	the estimate (<ref>) is equivalent to showing that
	
    ℬ_s(f,g):=∫_^2⟨ξ_1 ⟩^s-k+1/2f(ξ_2, τ_2)g(ξ_1, τ_1) ⟨ξ_2⟩^s⟨τ_1-αξ_1^3 ⟩^b⟨τ_2-ξ_2^3 ⟩^b dξ_1 dτ_1 _L^2_ξτ(^2)≲f_L^2(^2)g_L^2(^2),

	where b=1/2+ϵ. 
	
	Let σ_1=τ_1-αξ_1^3, σ_2=τ_2-ξ_2^3, we define the sets
	
    A_N={(ξ_1, τ_1) ;   |ξ_1-N|< N^-2, |σ_1|<C_α}  and   B_N={(ξ_2, τ_2) ;   |ξ_2|< (2N)^-2, |σ_2|<1},

	and 
	
    R_N={(ξ, τ) ;   |ξ-N|< (2N)^-2, |τ-αξ^3+(1-α)ξ^2N|<1}.

	Then R_N-B_N ⊂ A_N. In fact, if (ξ_1, τ_1)=(ξ, τ)-(ξ_2, τ_2) with (ξ, τ) ∈ R_N and (ξ_2, τ_2) ∈ B_N, then
	
    |ξ_1-N|= |ξ -ξ_2 -N|≤ |ξ  -N|+|ξ_2|< N^-2,

	also observe that σ_1+ σ_2= τ -αξ^3-(1-α)ξ_1^3+3 αξξ_1ξ_2, thus
	
    |σ_1| ≤    |σ_2| +|τ -αξ^3-(1-α)ξ_1^3+3 αξξ_1ξ_2|
    ≤   1+|τ-αξ^3+(1-α)ξ^2N|+|1-α|  |ξ^2N -ξ_1^3|+3|αξξ_1ξ_2|
    ≤   2+|1-α| [ N|(ξ-ξ_1)(ξ+ξ_1)|+ ξ_1^2|N-ξ_1| ]+3|αξξ_1ξ_2|
    ≤   C_α.

	Consider f=χ_A_N, g=χ_B_N. We have ⟨ξ_1⟩∼ N, ⟨ξ_2⟩∼ 1, ⟨σ_j ⟩∼ 1, j=1,2. From (<ref>) and Lemma <ref>, we obtain
	
    N^s-k+1/2R_N_L^2(^2)χ_B_N_L^1(^2)≤ N^s-k+1/2χ_A_N∗χ_B_N_L^2(^2)≤χ_A_N_L^2(^2)χ_B_N_L^2(^2)

	
	which implies
	
    N^s-k+1/2 N^-1 N^-2≲ N^-2

	and the last inequality is false if s-k>1/2.
	
	Analogously, if we consider the sets
	
    A_N={(ξ_1, τ_1) ; |ξ_1-N|< N^-2, |τ_1- α N^3|<2},

	
	
    B_N={(ξ_2, τ_2) ; |ξ_2+N|< (2N)^-2, |τ_2+N^3|<1}

	and 
	
    R_N={(ξ, τ) ;   |ξ|< (2N)^-2, |τ-(α-1)N^3|<1},

	then R_N-B_N ⊂ A_N and 
	
    |τ_1-αξ_1^3|≤ |τ_1-α N^3+α (N^3-ξ_1^3)| ≲ 2+|α|,

	
    |τ_2-ξ_2^3|≤ |τ_2+N^3- (N^3+ξ_2^3)| ≲ 1.

	Noting that 
	
    ⟨ξ_1⟩∼ N,   ⟨ξ_2⟩∼ N,   ⟨σ_j ⟩∼ 1,   j=1,2,

	from (<ref>) we obtain
	
    N^-k+1/2R_N_L^2(^2)χ_B_N_L^1(^2)≤ N^-k+1/2χ_A_N∗χ_B_N_L^2(^2)≤χ_A_N_L^2(^2)χ_B_N_L^2(^2),

	
	which implies
	
    N^-k+1/2 N^-1 N^-2≲ N^-2

	and the last inequality is false if k<-1/2. 
	



At the endpoint we have the following result.


	Let α≠ 0, 1, then  the following bilinear estimate
	
    uv_L^2(^2)≲u_X_-1/2, 1/2-2ϵ^αv_X_-1/2, 1/2+ϵ

	fails to hold for all ϵ >0.


	Using Plancherel's identity, 
	the estimate (<ref>) is equivalent to showing that
	
    ℬ_s(f,g):=∫_^2⟨ξ_2 ⟩^1/2⟨ξ_1 ⟩^1/2f̃(ξ_2, τ_2)g̃(ξ_1, τ_1) ⟨τ_1- ξ_1^3 ⟩^1/2+ϵ⟨τ_2-αξ_2^3 ⟩^1/2-2ϵ dξ_1 dτ_1 _L^2_ξτ(^2)≤f_L^2(^2)g_L^2(^2),

	where 
	f(ξ,τ)=⟨ξ⟩^-1/2⟨τ-αξ^3⟩^1/2-2ϵu(ξ,τ),  g(ξ,τ)=⟨ξ⟩^-1/2⟨τ-ξ^3⟩^1/2+ϵv(ξ,τ), ξ_2=ξ-ξ_1 and τ_2=τ-τ_1. 
	
	
	We will construct functions f and g for which the estimate (<ref>) fails to hold for all ϵ>0. 
	
	Consider two rectangles R_1 and  R_2  centered respectively at (N, α N^3), and  (N, N^3), and each with dimension N^-(2+r)× N^-r, where -2<r<0. Now, let f and g defined, via their Fourier transform, by
	f = χ_R_1 and  g= χ_R_2.	It is easy to see that 
	
    f_L^2(^2) =g_L^2(^2)=N^-(1+r).

	Also, 
	
    | ξ_1-N|≤1/2N^-(2+r),      |τ_1- N^3|≤1/2N^-r

	and
	
    |ξ_2-N|≤1/2N^-(2+r),      |τ_2-α N^3|≤1/2N^-r.

	
	We have |ξ_j|∼ N.
	Also, we have   that 
	
    |τ_1- ξ_1^3|=   |τ_1- N^3+ N^3- ξ_1^3|
    			≤  |τ_1-  N^3|+ |N^3- ξ_1^3|
    ≤   1/2N^-r+ 1/2N^-(2+r)|N^2+Nξ_1+ ξ_1^2|
    ≲    N^-r.

	Similarly, 
	
    |τ_2- αξ_2^3|=   |τ_2- α N^3+ α N^3-αξ_2^3|
    			≤  |τ_2-  α N^3|+|α| |N^3- ξ_2^3|
    ≤   1/2N^-r+ |α| 1/2N^-(2+r)|N^2+Nξ_2+ ξ_2^2|
    ≲    N^-r.

	
	With these considerations, we get from (<ref>) and Lemma <ref>
	
    ℬ_s(f,g) 
    			   ≳ N^1+r-r ϵ∫_^2χ_R_1(ξ_1, τ_1)χ_R_2(ξ_2, τ_2) dξ_1 dτ_1 _L^2_ξτ(^2)
       ∼ N^1+r-r ϵχ_R_1*χ_R_2_L^2(^2)
       = N^1+r-r ϵ|R_j|^2-1/2 = N^1+r-r ϵN^-3-3r=N^-2-r(2+ ϵ).

	
	Now, using  (<ref>) and  (<ref>) in (<ref>), 
	
    N^-2-r(2+ ϵ)≲ N^-2-2r⟺ N^-rϵ≲ 1.

	
	Since r<0, if we choose N large, the estimate (<ref>) fails to hold whenever  ϵ>0 and this completes the proof of the proposition.























99
	
	AKN1973
	Mark J. Ablowitz, David J. Kaup, Alan C. Newell, and Harvey Segur,
	Nonlinear-evolution equations of physical significance, Physical
	Review Letters 31 (1973), no. 2, 125–127 (English (US)).
	
	AAM1999
	Eduardo Alarcon, Jaime Angulo, and Jose F. Montenegro, Stability and
		instability of solitary waves for a nonlinear dispersive system, Nonlinear
	Analysis: Theory, Methods and Applications 36 (1999), no.8, 1015 –
	1035.
	
	BT2006
	Ioan Bejenaru and Terence Tao, Sharp well-posedness and ill-posedness
		results for a quadratic non-linear Schrödinger equation, Journal of
	Functional Analysis 233 (2006), no. 1, 228 – 259.
	
	BOP1997
	Daniella Bekiranov, Takayoshi Ogawa, and Gustavo Ponce, Weak
		Solvability and Well-Posedness of a Coupled Schrödinger-Korteweg
		de Vries Equation for Capillary-Gravity Wave Interactions,
	Proceedings of the American Mathematical Society 125 (1997), no. 10,
	2907–2919.
	
	Bourgain1993
	Jean Bourgain, Fourier transform restriction phenomena for certain
		lattice subsets and applications to nonlinear evolution equations, Geometric
	& Functional Analysis GAFA 3 (1993), no. 3, 209–262.
	
	Carvajal2004
	Xavier Carvajal, Local well-posedness for a higher order nonlinear
		Schrödinger equation in sobolev spaces of negative indices, Electronic
	Journal of Differential Equations (EJDE) [electronic only] (2004), 10 p.
	
	Carvajal2006
	Xavier Carvajal, Sharp Global Well-Posedness for a Higher Order
		Schrödinger Equation, Journal of Fourier Analysis and Applications
	12 (2006), no. 1, 53–70.
	
	CP2019
	Xavier Carvajal and Mahendra Panthee, Sharp well-posedness for a coupled
		system of mKdV-type equations, Journal of Evolution Equations
	19 (2019), no. 4, 1167–1197.
	
	CCT2003
	Michael Christ, James Colliander, and Terence Tao, Asymptotics, frequency
		modulation, and low regularity ill-posedness for canonical defocusing
		equations, American Journal of Mathematics 125 (2003), no. 6, 1235
	– 1293.
	
	CP2011
	Adán J. Corcho and Mahendra Panthee, Global well-posedness for a
		coupled modified KdV system, Bulletin of the Brazilian Mathematical
	Society, New Series 43 (2011), 27–57.
	
	Domingues2016
	Leandro Domingues, Sharp well-posedness results for the
		Schrödinger-Benjamin-Ono system, Adv. Differential Equations
	21 (2016), no. 1/2, 31–54.
	
	DS2019
	Leandro Domingues and Raphael Santos, A Note on C^2
		Ill-Posedness Results for the Zakharov System in Arbitrary
		Dimension, 2019 - arXiv:1910.06486.
	
	GTV1997
	J. Ginibre, Y. Tsutsumi, and G. Velo, On the Cauchy Problem for the
		Zakharov System, Journal of Functional Analysis 151 (1997),
	no.2, 384 – 436.
	
	KPV1993
	Carlos Kenig, Gustavo Ponce, and Luis Vega, Well-posedness and
		scattering results for the generalized Korteweg-de Vries equation via the
		contraction principle, Communications on Pure and Applied Mathematics
	46 (1993), no.4, 527–620.
	
	KPV1996
	Carlos Kenig, Gustavo Ponce, and Luis Vega,, A bilinear estimate with applications to the KdV equation,
	Journal of the American Mathematical Society 9 (1996), no.2,
	573–603.
	
	KPV2001
	Carlos Kenig, Gustavo Ponce, and Luis Vega,, On the ill-posedness of some canonical dispersive equations,
	Duke Math. J. 106 (2001), no.3, 617–633.
	
	Montenegro1995
	J.F. Montenegro, Sistemas de equações de evolução não lineares;
		estudo local, global e estabilidade de ondas solitárias, Ph.D. thesis,
	IMPA, Rio de Janeiro, Brazil, 1995.
	
	Takaoka2000
	Hideo Takaoka, Well-posedness for the higher order nonlinear
		Schrödinger equation, Advances in Mathematical Sciences and Applications
	10 (2000), 149–171.
	
	Tao2001
	Terence Tao, Multilinear Weighted Convolution of L^2
		Functions, and Applications to Nonlinear Dispersive Equations,
	American Journal of Mathematics 123 (2001), no.5, 839–908.
	
	Tzvetkov1999
	Nickolay Tzvetkov, Remark on the local ill-posedness for KdV
		equation, Comptes Rendus de l'Académie des Sciences - Series I -
	Mathematics 329 (1999), no. 12, 1043 – 1047.
	






