{
  "id": "2312.01906",
  "version": "v1",
  "published": "2023-12-04T14:08:11.000Z",
  "updated": "2023-12-04T14:08:11.000Z",
  "title": "Effect of lower order terms on the well-posedness of Majda-Biello systems",
  "authors": [
    "Xin Yang",
    "Shenghao Li",
    "Bing-Yu Zhang"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-04T14:08:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lower order terms",
      "majda-biello systems",
      "well-posedness",
      "lower-order terms",
      "initial value problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}