The Navier-Stokes problem. Solution of a millennium problem related to the Navier-Stokes equations
The goal of this paper is to present the author's results concerning the Navier-Stokes problem (NSP) in \(\mathbb{R}^3\) without boundaries. It is proved that the NSP is contradictory in the following sense: Assume (for simplicity only) that the exterior force \(f=f(x,t)=0\). If one assumes t...
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Tuncer Acar
2024-01-01
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| author | Alexander Ramm |
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| description | The goal of this paper is to present the author's results concerning the Navier-Stokes problem (NSP) in \(\mathbb{R}^3\) without boundaries. It is proved that the NSP is contradictory in the following sense:
Assume (for simplicity only) that the exterior force \(f=f(x,t)=0\). If one assumes that the initial data \(v(x,0)\not\equiv 0\), \(v(x,0)\) is a smooth and rapidly decaying at infinity vector function, \(\nabla \cdot v(x,0)=0\), and the solution to the NSP exists for all \(t\ge 0\), then one proves that the solution \(v(x,t)\) to the NSP has the property \(v(x,0)=0\).
This paradox (the NSP paradox) shows that the NSP is not a correct description of the fluid mechanics problem and the NSP does not have a solution defined for all times \(t>0\). This solves the millennium problem concerning the Navier-Stokes equations: the solution does not exist for all \(t>0\) if \(v(x,0)\not\equiv 0\), \(v(x,0)\) is a smooth and rapidly decaying at infinity vector function, \(\nabla \cdot v(x,0)=0\). In the exceptional case, when the data are equal to zero, the solution \(v(x,t)\) to the NSP exists for all \(t\ge 0\) and is equal to zero, \(v(x,t)\equiv 0\). |
| format | Article |
| id | doaj-art-352e1ea6578e42568cd868e540096329 |
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| issn | 3023-5294 |
| language | English |
| publishDate | 2024-01-01 |
| publisher | Tuncer Acar |
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| series | Modern Mathematical Methods |
| spelling | doaj-art-352e1ea6578e42568cd868e5400963292025-08-20T02:09:47ZengTuncer AcarModern Mathematical Methods3023-52942024-01-012119268The Navier-Stokes problem. Solution of a millennium problem related to the Navier-Stokes equationsAlexander Ramm0https://orcid.org/0000-0002-5160-4248Kansas State University, Department of Mathematics, Manhattan, Ks 66506, USAThe goal of this paper is to present the author's results concerning the Navier-Stokes problem (NSP) in \(\mathbb{R}^3\) without boundaries. It is proved that the NSP is contradictory in the following sense: Assume (for simplicity only) that the exterior force \(f=f(x,t)=0\). If one assumes that the initial data \(v(x,0)\not\equiv 0\), \(v(x,0)\) is a smooth and rapidly decaying at infinity vector function, \(\nabla \cdot v(x,0)=0\), and the solution to the NSP exists for all \(t\ge 0\), then one proves that the solution \(v(x,t)\) to the NSP has the property \(v(x,0)=0\). This paradox (the NSP paradox) shows that the NSP is not a correct description of the fluid mechanics problem and the NSP does not have a solution defined for all times \(t>0\). This solves the millennium problem concerning the Navier-Stokes equations: the solution does not exist for all \(t>0\) if \(v(x,0)\not\equiv 0\), \(v(x,0)\) is a smooth and rapidly decaying at infinity vector function, \(\nabla \cdot v(x,0)=0\). In the exceptional case, when the data are equal to zero, the solution \(v(x,t)\) to the NSP exists for all \(t\ge 0\) and is equal to zero, \(v(x,t)\equiv 0\).https://modernmathmeth.com/index.php/pub/article/view/8the navier-stokes problemthe paradox the solution to the millennium problem related to the navier-stokes equations |
| spellingShingle | Alexander Ramm The Navier-Stokes problem. Solution of a millennium problem related to the Navier-Stokes equations Modern Mathematical Methods the navier-stokes problem the paradox the solution to the millennium problem related to the navier-stokes equations |
| title | The Navier-Stokes problem. Solution of a millennium problem related to the Navier-Stokes equations |
| title_full | The Navier-Stokes problem. Solution of a millennium problem related to the Navier-Stokes equations |
| title_fullStr | The Navier-Stokes problem. Solution of a millennium problem related to the Navier-Stokes equations |
| title_full_unstemmed | The Navier-Stokes problem. Solution of a millennium problem related to the Navier-Stokes equations |
| title_short | The Navier-Stokes problem. Solution of a millennium problem related to the Navier-Stokes equations |
| title_sort | navier stokes problem solution of a millennium problem related to the navier stokes equations |
| topic | the navier-stokes problem the paradox the solution to the millennium problem related to the navier-stokes equations |
| url | https://modernmathmeth.com/index.php/pub/article/view/8 |
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