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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.06776 |
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| _version_ | 1866916345726107648 |
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| author | Córdoba, Diego Martínez-Zoroa, Luis Zheng, Fan |
| author_facet | Córdoba, Diego Martínez-Zoroa, Luis Zheng, Fan |
| contents | In this work we establish the formation of singularities of classical solutions with finite energy of the forced fractional Navier Stokes equations where the dissipative term is given by $|\nabla|^α$ for any $α\in [0, α_0)$ ($α_0 = \frac{22-8\sqrt7}{9} > 0$). We construct solutions in $\mathbb{R}^3\times [0,T]$ with a finite $T>0$ and with an external forcing which is in $L^1_t([0, T]) C_x^{1,ε}\cap L^{\infty}_{t}L_{x}^2$, such that on the time interval $0 \le t < T$, the velocity $u$ is in the space $C^\infty\cap L^2$ and such that as the time $t$ approaches the blow-up moment $T$, the integral $\int_0^t |\nabla u| ds$ tends to infinity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_06776 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Finite time blow-up for the hypodissipative Navier Stokes equations with a force in $L^1_t C_x^{1,ε}\cap L^{\infty}_{t}L_{x}^2$ Córdoba, Diego Martínez-Zoroa, Luis Zheng, Fan Analysis of PDEs In this work we establish the formation of singularities of classical solutions with finite energy of the forced fractional Navier Stokes equations where the dissipative term is given by $|\nabla|^α$ for any $α\in [0, α_0)$ ($α_0 = \frac{22-8\sqrt7}{9} > 0$). We construct solutions in $\mathbb{R}^3\times [0,T]$ with a finite $T>0$ and with an external forcing which is in $L^1_t([0, T]) C_x^{1,ε}\cap L^{\infty}_{t}L_{x}^2$, such that on the time interval $0 \le t < T$, the velocity $u$ is in the space $C^\infty\cap L^2$ and such that as the time $t$ approaches the blow-up moment $T$, the integral $\int_0^t |\nabla u| ds$ tends to infinity. |
| title | Finite time blow-up for the hypodissipative Navier Stokes equations with a force in $L^1_t C_x^{1,ε}\cap L^{\infty}_{t}L_{x}^2$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2407.06776 |