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<p>We consider the Cauchy problem for quadratic derivative fractional nonlinear Schrödinger equations on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">R</mi> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {T}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">T</mi> </math></EquationSource> </InlineEquation>. We determine the sharp exponents of the fractional derivatives for which the Cauchy problem is well-posed in the Sobolev space. Thanks to the global well-posedness result established by Nakanishi and Wang (2025), we can expand the solution as a sum of iterated terms. By deriving estimates for each iterated term, we establish norm inflation with infinite loss of regularity, which in particular implies ill-posedness.</p>

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Norm inflation for quadratic derivative fractional nonlinear Schrödinger equations

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月24日

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  • Toshiki Kondo,
  • Mamoru Okamoto

摘要

We consider the Cauchy problem for quadratic derivative fractional nonlinear Schrödinger equations on \(\mathbb {R}\) R or \(\mathbb {T}\) T . We determine the sharp exponents of the fractional derivatives for which the Cauchy problem is well-posed in the Sobolev space. Thanks to the global well-posedness result established by Nakanishi and Wang (2025), we can expand the solution as a sum of iterated terms. By deriving estimates for each iterated term, we establish norm inflation with infinite loss of regularity, which in particular implies ill-posedness.