Well-posedness and critical index set of the Cauchy problem for the coupled KdV-KdV systems on $ \mathbb{T} $
نویسندگان
چکیده
Studied in this paper is the well-posedness of Cauchy problem for coupled KdV-KdV systems \[ u_t+a_1u_{xxx} = c_{11}uu_x+c_{12}vv_x+d_{11}u_{x}v+d_{12}uv_{x}, \quad u(x,0)= u_0(x) \] v_t+a_2v_{xxx}= c_{21}uu_x+c_{22}vv_x +d_{21}u_{x}v+d_{22}uv_{x}, v(x,0)=v_0(x)\] posed on torus $\mathbb{T}$ spaces {\cal H}^s_1:=H^s_0 (\mathbb{T})\times H^s_0 (\mathbb{T}), H}^s_2:=H^s_0 H^s(\mathbb{T}), H}^s_3:=H^s H}^s_4:=H^s H^s (\mathbb{T}).\] For $k=1,2,3,4$, it shown that given $a_1$, $a_2$, $(c_{ij})$ and $(d_{ij})$, there exists a unique $s^*_k \in (-\infty, +\infty]$, called critical index, such system analytically well-posed $\cal{H}^s_k$ $s>s^*_k$ while bilinear estimate, key proof analytical well-posedness, fails if $s<s^{*}_k$. Viewing index $s^*_k$ as function coefficients its range $\cal{C}_k$ set space $\cal{H}^s_k$. Invoking some classical results Diophantine approximation number theory, we are able to identify \mbox{$ C}_1= \left \{ -\frac12, \infty \right\} \bigcup \alpha: \frac12\leq \alpha\leq 1 \right \}$ } \quad\text{and}\quad \mbox{${\cal C}_q= -\frac14, $\quad$ $q=2,3,4$.}\] This sharp contrast $R$ case which ${\cal C}$ $H^s (R)\times (R)$ consists exactly four numbers: $ C}=\left -\frac{13}{12}, -\frac34, 0, \frac34 \}.$
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ژورنال
عنوان ژورنال: Discrete and Continuous Dynamical Systems
سال: 2022
ISSN: ['1553-5231', '1078-0947']
DOI: https://doi.org/10.3934/dcds.2022090