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<p>By constructing a counterexample based on a modified DiPerna–Majda type shear flow, we show that the solution map for the inviscid Boussinesq equation fails to be continuous from <i>C</i><sup>1,<i>α</i></sup> to <i>C</i>(0, <i>T;C</i><sup>1,<i>α</i></sup>) for any 0 &lt; <i>α</i> &lt; 1. In contrast, we establish local well-posedness and continuity of the solution map in the little Hölder space <i>c</i><sup>1,<i>α</i></sup>, where <i>c</i><sup>1,<i>α</i></sup> is the completion of <i>C</i><Stack> <sub><i>c</i></sub> <sup>∞</sup> </Stack> for the norm <i>C</i><sup>1,<i>α</i></sup>. In some sense, this dichotomy highlights that the discontinuity in <i>C</i><sup>1,<i>α</i></sup> is sharp, arising precisely from the non-separability of the classical Hölder space.</p>

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The Ill-posedness and Well-posedness for the Inviscid Boussinesq Equations

  • 期刊论文

  • 发表日期：

    2026年6月5日

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  • Xiaonan Hao,
  • Zhen Li

摘要

By constructing a counterexample based on a modified DiPerna–Majda type shear flow, we show that the solution map for the inviscid Boussinesq equation fails to be continuous from C1,α to C(0, T;C1,α) for any 0 < α < 1. In contrast, we establish local well-posedness and continuity of the solution map in the little Hölder space c1,α, where c1,α is the completion of C c ∞ for the norm C1,α. In some sense, this dichotomy highlights that the discontinuity in C1,α is sharp, arising precisely from the non-separability of the classical Hölder space.