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<p>We prove the unconditional well-posedness for the fourth-order nonlinear Schrödinger-type equations in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(H^s({\mathbb T})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mi>s</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">T</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> when <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(s \ge 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, which includes the non-integrable case. This regularity threshold is optimal because the nonlinear terms cannot be defined in the space-time distribution framework for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(s&lt;1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>&lt;</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. The main idea is to employ the normal form reduction and a kind of cancelation property to deal with derivative losses.</p>

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Unconditional well-posedness for the fourth-order nonlinear Schrödinger-type equations on the torus

  • 期刊论文

  • 发表日期：

    2025年7月28日

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  • Takamori Kato

摘要

We prove the unconditional well-posedness for the fourth-order nonlinear Schrödinger-type equations in \(H^s({\mathbb T})\) H s ( T ) when \(s \ge 1\) s ≥ 1 , which includes the non-integrable case. This regularity threshold is optimal because the nonlinear terms cannot be defined in the space-time distribution framework for \(s<1\) s < 1 . The main idea is to employ the normal form reduction and a kind of cancelation property to deal with derivative losses.