Suppression of blow-up in the 3D Patlak-Keller-Segel-Navier-Stokes system via non-parallel shear flows
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916423955120128 |
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| author | Cui, Shikun Wang, Lili Wang, Wendong |
| author_facet | Cui, Shikun Wang, Lili Wang, Wendong |
| contents | In this paper, we consider the three-dimensional Patlak-Keller-Segel system coupled with the Navier-Stokes equations near the non-parallel shear flow $( Ay, 0, Ay )$ in $\mathbb{T}\times\mathbb{R}\times\mathbb{T}$. We show that if the shear flow is sufficiently strong (A is large enough), then the solutions to Patlak-Keller-Segel-Navier-Stokes system are global in time without any smallness restriction on the initial cell mass as long as the initial velocity satisfies $A^{\frac{2}{3}}\|u_{\rm in}\|_{H^2}\leq C_0$, which seems to be the first result of studying the suppression effect of shear flows for the 3D Patlak-Keller-Segel-Navier-Stokes system. Moreover, it implies that the solutions of the 3D Navier-Stokes equations are global in time if the initial velocity satisfies $A^{\frac{2}{3}}\|v_{\rm in}-(Ay,0,Ay)\|_{H^2}\leq C_0,$ which also shows the transition threshold for the shear flow $(Ay,0,Ay)$ in $\mathbb{T}\times\mathbb{R}\times\mathbb{T}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_15982 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Suppression of blow-up in the 3D Patlak-Keller-Segel-Navier-Stokes system via non-parallel shear flows Cui, Shikun Wang, Lili Wang, Wendong Analysis of PDEs In this paper, we consider the three-dimensional Patlak-Keller-Segel system coupled with the Navier-Stokes equations near the non-parallel shear flow $( Ay, 0, Ay )$ in $\mathbb{T}\times\mathbb{R}\times\mathbb{T}$. We show that if the shear flow is sufficiently strong (A is large enough), then the solutions to Patlak-Keller-Segel-Navier-Stokes system are global in time without any smallness restriction on the initial cell mass as long as the initial velocity satisfies $A^{\frac{2}{3}}\|u_{\rm in}\|_{H^2}\leq C_0$, which seems to be the first result of studying the suppression effect of shear flows for the 3D Patlak-Keller-Segel-Navier-Stokes system. Moreover, it implies that the solutions of the 3D Navier-Stokes equations are global in time if the initial velocity satisfies $A^{\frac{2}{3}}\|v_{\rm in}-(Ay,0,Ay)\|_{H^2}\leq C_0,$ which also shows the transition threshold for the shear flow $(Ay,0,Ay)$ in $\mathbb{T}\times\mathbb{R}\times\mathbb{T}$. |
| title | Suppression of blow-up in the 3D Patlak-Keller-Segel-Navier-Stokes system via non-parallel shear flows |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2401.15982 |