Suppression of blow-up in Patlak-Keller-Segel system coupled with linearized Navier-Stokes equations via the 3D Couette flow
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| Format: | Preprint |
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2024
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| _version_ | 1866929346860548096 |
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| author | Cui, Shikun Wang, Lili Wang, Wendong |
| author_facet | Cui, Shikun Wang, Lili Wang, Wendong |
| contents | It is known that finite-time blow-up in the 3D Patlak-Keller-Segel system may occur for arbitrarily small values of the initial mass. It's interesting whether one can prevent the finite-time blow-up via the stabilizing effect of the moving fluid. Consider the three-dimensional Patlak-Keller-Segel system coupled with the linearized Navier-Stokes equations near the Couette flow $(\ Ay, 0, 0 \ )$ in a finite channel $\mathbb{T}\times\mathbb{I}\times\mathbb{T}$ with $ \mathbb{T}=[0,2π) $ and $ \mathbb{I}=[-1,1] $, with the non-slip boundary condition, and 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 as long as the initial cell mass is sufficiently small (for example, $M<\frac49$) and $ A\left(\|u_{2,0}(0)\|_{L^{2}}+\|u_{3,0}(0)\|_{L^{2}} \right)\leq C_{0} $, which seems to be the first result of considering the suppression effect of Couette flow in the 3D Patlak-Keller-Segel-Navier-Stokes model, and also the first time considering the non-slip boundary condition. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_10337 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Suppression of blow-up in Patlak-Keller-Segel system coupled with linearized Navier-Stokes equations via the 3D Couette flow Cui, Shikun Wang, Lili Wang, Wendong Analysis of PDEs It is known that finite-time blow-up in the 3D Patlak-Keller-Segel system may occur for arbitrarily small values of the initial mass. It's interesting whether one can prevent the finite-time blow-up via the stabilizing effect of the moving fluid. Consider the three-dimensional Patlak-Keller-Segel system coupled with the linearized Navier-Stokes equations near the Couette flow $(\ Ay, 0, 0 \ )$ in a finite channel $\mathbb{T}\times\mathbb{I}\times\mathbb{T}$ with $ \mathbb{T}=[0,2π) $ and $ \mathbb{I}=[-1,1] $, with the non-slip boundary condition, and 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 as long as the initial cell mass is sufficiently small (for example, $M<\frac49$) and $ A\left(\|u_{2,0}(0)\|_{L^{2}}+\|u_{3,0}(0)\|_{L^{2}} \right)\leq C_{0} $, which seems to be the first result of considering the suppression effect of Couette flow in the 3D Patlak-Keller-Segel-Navier-Stokes model, and also the first time considering the non-slip boundary condition. |
| title | Suppression of blow-up in Patlak-Keller-Segel system coupled with linearized Navier-Stokes equations via the 3D Couette flow |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2405.10337 |