Local existence of solutions and blow-up criteria for the Boussinesq equations in Lei-Lin-Gevrey Spaces
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| Format: | Preprint |
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2026
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| _version_ | 1866914618153107456 |
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| author | Guidolin, Patrícia L. Melo, Wilberclay G. Santos, Thyago S. R. |
| author_facet | Guidolin, Patrícia L. Melo, Wilberclay G. Santos, Thyago S. R. |
| contents | This paper studies the local well-posedness and the behavior at potential finite blow-up times of mild solutions to the three-dimensional fractional Boussinesq equations in Lei-Lin and Lei-Lin-Gevrey spaces $\mathcal{X}_{a,σ}^s(\mathbb{R}^3)$. By combining fixed point arguments with Fourier estimates adapted to these spaces, we obtain local existence and uniqueness results for data in $\mathcal{X}_{a,σ}^s(\mathbb{R}^3)$, including the usual Lei-Lin case $a=0$. We also establish several blow-up criteria for maximal mild solutions and derive lower bounds for the growth of the corresponding norms as the maximal time is approached. In the strict Lei-Lin-Gevrey regime, and when the dissipative exponents coincide, these estimates yield an exponential type blow-up criterion. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_31461 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Local existence of solutions and blow-up criteria for the Boussinesq equations in Lei-Lin-Gevrey Spaces Guidolin, Patrícia L. Melo, Wilberclay G. Santos, Thyago S. R. Analysis of PDEs 35B44, 35A01, 35A02, 35Q30, 35Q35 This paper studies the local well-posedness and the behavior at potential finite blow-up times of mild solutions to the three-dimensional fractional Boussinesq equations in Lei-Lin and Lei-Lin-Gevrey spaces $\mathcal{X}_{a,σ}^s(\mathbb{R}^3)$. By combining fixed point arguments with Fourier estimates adapted to these spaces, we obtain local existence and uniqueness results for data in $\mathcal{X}_{a,σ}^s(\mathbb{R}^3)$, including the usual Lei-Lin case $a=0$. We also establish several blow-up criteria for maximal mild solutions and derive lower bounds for the growth of the corresponding norms as the maximal time is approached. In the strict Lei-Lin-Gevrey regime, and when the dissipative exponents coincide, these estimates yield an exponential type blow-up criterion. |
| title | Local existence of solutions and blow-up criteria for the Boussinesq equations in Lei-Lin-Gevrey Spaces |
| topic | Analysis of PDEs 35B44, 35A01, 35A02, 35Q30, 35Q35 |
| url | https://arxiv.org/abs/2605.31461 |