<p>This article studies the norm-inflation phenomena of a periodic initial-value Good Boussinesq equation in low-regularity Sobolev spaces. Particularly, this article demonstrates that the initial-value problem is ill-posed in the periodic Sobolev spaces <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$H^{-s}_p(0,2 \pi ) \times H^{-s-2}_p(0,2 \pi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>H</mi> <mi>p</mi> <mrow> <mo>-</mo> <mi>s</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> <mo>×</mo> <msubsup> <mi>H</mi> <mi>p</mi> <mrow> <mo>-</mo> <mi>s</mi> <mo>-</mo> <mn>2</mn> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$s>1/2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>></mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. Our proof is based on a constructive method: we provide smooth initial data that generates solutions with arbitrarily high-norms in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$H^{-s}_p(0,2 \pi ) \times H^{-s-2}_p(0,2 \pi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>H</mi> <mi>p</mi> <mrow> <mo>-</mo> <mi>s</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> <mo>×</mo> <msubsup> <mi>H</mi> <mi>p</mi> <mrow> <mo>-</mo> <mi>s</mi> <mo>-</mo> <mn>2</mn> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and for arbitrarily short times. The result is sharp in the sense that previous work (Cerpa and Rivas in J Evol Equ 18:1501–1519, 2018; Kishimoto in J Differ Equ 254:2393–2433, 2013) has shown well-posedness for periodic Sobolev indexes of the form <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$H^{-s}_p(0,2 \pi ) \times H^{-s-2}_p(0,2 \pi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>H</mi> <mi>p</mi> <mrow> <mo>-</mo> <mi>s</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> <mo>×</mo> <msubsup> <mi>H</mi> <mi>p</mi> <mrow> <mo>-</mo> <mi>s</mi> <mo>-</mo> <mn>2</mn> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$s\le 1/2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>≤</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

Norm-inflation results for the good Boussinesq equation

期刊论文
发表日期：

2025年7月29日

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George J. Bautista,
Leyter Potenciano-Machado

摘要

This article studies the norm-inflation phenomena of a periodic initial-value Good Boussinesq equation in low-regularity Sobolev spaces. Particularly, this article demonstrates that the initial-value problem is ill-posed in the periodic Sobolev spaces $H^{-s}_p(0,2 \pi ) \times H^{-s-2}_p(0,2 \pi )$ $H_{p}^{- s} (0, 2 π) \times H_{p}^{- s - 2} (0, 2 π)$ for all $s>1/2$ $s > 1 / 2$ . Our proof is based on a constructive method: we provide smooth initial data that generates solutions with arbitrarily high-norms in $H^{-s}_p(0,2 \pi ) \times H^{-s-2}_p(0,2 \pi )$ $H_{p}^{- s} (0, 2 π) \times H_{p}^{- s - 2} (0, 2 π)$ and for arbitrarily short times. The result is sharp in the sense that previous work (Cerpa and Rivas in J Evol Equ 18:1501–1519, 2018; Kishimoto in J Differ Equ 254:2393–2433, 2013) has shown well-posedness for periodic Sobolev indexes of the form $H^{-s}_p(0,2 \pi ) \times H^{-s-2}_p(0,2 \pi )$ $H_{p}^{- s} (0, 2 π) \times H_{p}^{- s - 2} (0, 2 π)$ with $s\le 1/2$ $s \leq 1 / 2$ .