On the Cauchy problem to the axially-symmetric solutions to the Navier-Stokes equations
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2026
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| author | Grygierzec, Wiesław J. Zajączkowski, Wojciech M. |
| author_facet | Grygierzec, Wiesław J. Zajączkowski, Wojciech M. |
| contents | We consider the Cauchy problem to the axisymmetric Navier-Stokes equations. To prove an existence of global regular solutions we examine the Navier-Stokes equations near the axis of symmetry and far from it separately. We derive only a global a priori estimate. To show it near the axis of symmetry we need the energy estimate, $L_\infty$-estimate for swirl, $H^2$ and $H^3$ estimates for the modified stream function (stream function divided by radius) and also expansions of velocity and modified stream function found by Liu-Wang. The estimate for solutions far from the axis of symmetry follows easily. Hence, having so regular solutions that Liu-Wang expansions hold we have the global a priori estimate $(Ω=\mathbb{R}^3)$ $$ \|ω_{r/r} \|_{V(Ω^t)} + \|ω_{φ/r}\|_{V(Ω^t)}\leϕ({\rm data}),\ \ t<\infty, \qquad(*) $$ where $ω_r$ is the radiar component of vorticity, $ω_φ$ the angular, $V(Ω^t)$ is the energy norm. Estimate $(*)$ can be treated as an a priori estimate derived on sufficiently regular solutions. Increasing regularity of solutions $(*)$ we derive the estimate $$\eqalign{ &\|v\|_{W_3^{3,3/2}(Ω^t)}+\|\nabla p\|_{W_3^{1,1/2}(Ω^t)}\cr &\leϕ(ϕ({\rm data}),\|f\|_{W_3^{1,1/2}(Ω^t)},\|v(0)\|_{W_3^{3-2/3}(Ω)}),\cr} \qquad(**) $$ where $ϕ$ is an increasing positive function. The estimate is proved on the local solution. Estimate $(**)$ plus existence of local solutions imply the existence of global regular solutions to the Cauchy problem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_04377 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Cauchy problem to the axially-symmetric solutions to the Navier-Stokes equations Grygierzec, Wiesław J. Zajączkowski, Wojciech M. Analysis of PDEs 35A01, 35B01, 35Q30, 76D03, 76D05 We consider the Cauchy problem to the axisymmetric Navier-Stokes equations. To prove an existence of global regular solutions we examine the Navier-Stokes equations near the axis of symmetry and far from it separately. We derive only a global a priori estimate. To show it near the axis of symmetry we need the energy estimate, $L_\infty$-estimate for swirl, $H^2$ and $H^3$ estimates for the modified stream function (stream function divided by radius) and also expansions of velocity and modified stream function found by Liu-Wang. The estimate for solutions far from the axis of symmetry follows easily. Hence, having so regular solutions that Liu-Wang expansions hold we have the global a priori estimate $(Ω=\mathbb{R}^3)$ $$ \|ω_{r/r} \|_{V(Ω^t)} + \|ω_{φ/r}\|_{V(Ω^t)}\leϕ({\rm data}),\ \ t<\infty, \qquad(*) $$ where $ω_r$ is the radiar component of vorticity, $ω_φ$ the angular, $V(Ω^t)$ is the energy norm. Estimate $(*)$ can be treated as an a priori estimate derived on sufficiently regular solutions. Increasing regularity of solutions $(*)$ we derive the estimate $$\eqalign{ &\|v\|_{W_3^{3,3/2}(Ω^t)}+\|\nabla p\|_{W_3^{1,1/2}(Ω^t)}\cr &\leϕ(ϕ({\rm data}),\|f\|_{W_3^{1,1/2}(Ω^t)},\|v(0)\|_{W_3^{3-2/3}(Ω)}),\cr} \qquad(**) $$ where $ϕ$ is an increasing positive function. The estimate is proved on the local solution. Estimate $(**)$ plus existence of local solutions imply the existence of global regular solutions to the Cauchy problem. |
| title | On the Cauchy problem to the axially-symmetric solutions to the Navier-Stokes equations |
| topic | Analysis of PDEs 35A01, 35B01, 35Q30, 76D03, 76D05 |
| url | https://arxiv.org/abs/2602.04377 |