Effect of lower order terms on the well-posedness of Majda-Biello systems
arxiv(2023)
摘要
This paper investigates a noteworthy phenomenon within the framework of
Majda-Biello systems, wherein the inclusion of lower-order terms can enhance
the well-posedness of the system. Specifically, we investigate the initial
value problem (IVP) of the following system:
\[
\left\{
\begin{array}{l}
u_{t} + u_{xxx} = - v v_x,
v_{t} + \alpha v_{xxx} + \beta v_x = - (uv)_{x},
(u,v)|_{t=0} = (u_0,v_0) \in H^{s}(\mathbb{R}) \times H^{s}(\mathbb{R}),
\end{array}
\right. \quad x \in \mathbb{R}, \, t \in \mathbb{R}, \] where $\alpha \in
\mathbb{R}\setminus \{0\}$ and $\beta \in \mathbb{R}$. Let $s^{*}(\alpha,
\beta)$ be the smallest value for which the IVP is locally analytically
well-posed in $H^{s}(\mathbb{R})\times H^{s}(\mathbb{R}) $ when $s >
s^{}(\alpha, \beta)$.
Two interesting facts have already been known in literature: $s^{*}(\alpha,
0) = 0$ for $\alpha \in (0,4)\setminus\{1\}$ and $s^*(4,0) = \frac34$. Our key
findings include the following:
For $s^{*}(4,\beta)$, a significant reduction is observed, reaching $\frac12$
for $\beta > 0$ and $\frac14$ for $\beta < 0$.
Conversely, when $\alpha \neq 4$, we demonstrate that the value of $\beta$
exerts no influence on $s^*(\alpha, \beta)$.
These results shed light on the intriguing behavior of Majda-Biello systems
when lower-order terms are introduced and provide valuable insights into the
role of $\alpha $ and $\beta$ in the well-posedness of the system.
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