<p>We demonstrate inflation of Fourier–Lebesgue norms for solutions to the focusing modified Korteweg–de Vries equation posed on the real line. For <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$p\ne 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≠</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and all <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$s\in \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>, we construct a sequence of solutions <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$u_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>u</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> whose initial data <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$u_n(0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> converges to zero in the Fourier–Lebesgue spaces <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {F}L^p_s(\mathbb {R})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">F</mi> <msubsup> <mi>L</mi> <mi>s</mi> <mi>p</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, but whose evolutions at later times <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$t_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>t</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> diverge to infinity.</p>

Growth of Fourier–Lebesgue norms for mKdV

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2026年6月30日

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Saikatul Haque,
Rowan Killip,
Monica Vişan,
Yunfeng Zhang

摘要

We demonstrate inflation of Fourier–Lebesgue norms for solutions to the focusing modified Korteweg–de Vries equation posed on the real line. For $p\ne 2$ $p \neq 2$ and all $s\in \mathbb {R}$ $s \in R$ , we construct a sequence of solutions $u_n$ $u_{n}$ whose initial data $u_n(0)$ $u_{n} (0)$ converges to zero in the Fourier–Lebesgue spaces $\mathcal {F}L^p_s(\mathbb {R})$ $F L_{s}^{p} (R)$ , but whose evolutions at later times $t_n$ $t_{n}$ diverge to infinity.