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<p>This work establishes conditions for the existence and uniqueness of local mild solutions to the Boussinesq equations with fractional dissipations in Sobolev-Gevrey spaces. We prove that a unique mild solution exists in an appropriate Sobolev-Gevrey class and analyze its behavior up to the maximal time of existence. In particular, we derive quantitative lower bounds describing how the norm of the solution must blow up as it approaches a finite maximal time. As a corollary, we deduce that the solution exhibits exponential growth.</p>

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Exponential blow-up of mild solutions to the fractional Boussinesq equations in the Gevrey class

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月4日

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  • Wilberclay G. Melo,
  • Cilon F. Perusato,
  • Thyago S. R. Santos

摘要

This work establishes conditions for the existence and uniqueness of local mild solutions to the Boussinesq equations with fractional dissipations in Sobolev-Gevrey spaces. We prove that a unique mild solution exists in an appropriate Sobolev-Gevrey class and analyze its behavior up to the maximal time of existence. In particular, we derive quantitative lower bounds describing how the norm of the solution must blow up as it approaches a finite maximal time. As a corollary, we deduce that the solution exhibits exponential growth.