Weak Singularity of Navier-Stokes Equations Based on Energy Estimation in Sobolev Space
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866918378821648384 |
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| author | Kit, Chio Chon |
| author_facet | Kit, Chio Chon |
| contents | Based on Dou Huashu's energy gradient theory, this paper focuses on the weak singularity of the incompressible Navier-Stokes (NS) equations in steady, fully developed flows. When the gradient of total mechanical energy is perpendicular to the streamline (i.e., $ u_j \frac{\partial E}{\partial x_j} = 0 $), substituting this critical condition into the NS equations with no-slip boundary conditions leads to the viscous term $ ν\to 0 $. To rigorously analyze the regularity of the solution, Sobolev space $ H_0^1(Ω) $ is introduced for energy estimation. The results show that the velocity field loses $ H^1 $-regularity, and the NS equations degenerate into Euler equations, which admit discontinuous weak solutions. Thus, the position where the mechanical energy gradient is perpendicular to the streamline becomes a weak singularity of the NS equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_07574 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Weak Singularity of Navier-Stokes Equations Based on Energy Estimation in Sobolev Space Kit, Chio Chon Fluid Dynamics Analysis of PDEs Based on Dou Huashu's energy gradient theory, this paper focuses on the weak singularity of the incompressible Navier-Stokes (NS) equations in steady, fully developed flows. When the gradient of total mechanical energy is perpendicular to the streamline (i.e., $ u_j \frac{\partial E}{\partial x_j} = 0 $), substituting this critical condition into the NS equations with no-slip boundary conditions leads to the viscous term $ ν\to 0 $. To rigorously analyze the regularity of the solution, Sobolev space $ H_0^1(Ω) $ is introduced for energy estimation. The results show that the velocity field loses $ H^1 $-regularity, and the NS equations degenerate into Euler equations, which admit discontinuous weak solutions. Thus, the position where the mechanical energy gradient is perpendicular to the streamline becomes a weak singularity of the NS equations. |
| title | Weak Singularity of Navier-Stokes Equations Based on Energy Estimation in Sobolev Space |
| topic | Fluid Dynamics Analysis of PDEs |
| url | https://arxiv.org/abs/2603.07574 |