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<p>We establish the global well-posedness for generalized derivative KdV equations with small rough data in specific modulation spaces <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(M_{2,1}^{1/m}\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>M</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mi>m</mi> </mrow> </msubsup> </math></EquationSource> </InlineEquation> by the method of smoothing effect estimates combined with the frequency-uniform decomposition. Furthermore, we demonstrate that the mapping from data to solutions is not <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(C^{m+1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> continuous in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(M_{2,1}^{s}\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>M</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> <mi>s</mi> </msubsup> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(s&lt;1/m\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>&lt;</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mi>m</mi> </mrow> </math></EquationSource> </InlineEquation>, indicating the sharpness of the well-posedness result.</p>

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Global well-posedness for generalized derivative KdV equations with small rough data

  • 期刊论文

  • 发表日期：

    2025年4月17日

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  • Yufeng Lu

摘要

We establish the global well-posedness for generalized derivative KdV equations with small rough data in specific modulation spaces \(M_{2,1}^{1/m}\) M 2 , 1 1 / m by the method of smoothing effect estimates combined with the frequency-uniform decomposition. Furthermore, we demonstrate that the mapping from data to solutions is not \(C^{m+1}\) C m + 1 continuous in \(M_{2,1}^{s}\) M 2 , 1 s for \(s<1/m\) s < 1 / m , indicating the sharpness of the well-posedness result.