Almost sure existence of global weak solutions for incompressible generalized Navier-Stokes equations
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915164844982272 |
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| author | Lin, Yuan-Xin Wang, Ya-Guang |
| author_facet | Lin, Yuan-Xin Wang, Ya-Guang |
| contents | In this paper we consider the initial value problem of the incompressible generalized Navier-Stokes equations in torus $\mathbb{T}^d$ with $d \geq 2$. The generalized Navier-Stokes equations is obtained by replacing the standard Laplacian in the classical Navier-Stokes equations by the fractional order Laplacian $-(-Δ)^\al$ with $\al \in \left( \frac{2}{3},1 \right]$. After an appropriate randomization on the initial data, we obtain the almost sure existence of global weak solutions for initial data being in $\Dot{H}^s(\mathbb{T}^d)$ with $s\in (1-2\al,0)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_15273 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Almost sure existence of global weak solutions for incompressible generalized Navier-Stokes equations Lin, Yuan-Xin Wang, Ya-Guang Analysis of PDEs In this paper we consider the initial value problem of the incompressible generalized Navier-Stokes equations in torus $\mathbb{T}^d$ with $d \geq 2$. The generalized Navier-Stokes equations is obtained by replacing the standard Laplacian in the classical Navier-Stokes equations by the fractional order Laplacian $-(-Δ)^\al$ with $\al \in \left( \frac{2}{3},1 \right]$. After an appropriate randomization on the initial data, we obtain the almost sure existence of global weak solutions for initial data being in $\Dot{H}^s(\mathbb{T}^d)$ with $s\in (1-2\al,0)$. |
| title | Almost sure existence of global weak solutions for incompressible generalized Navier-Stokes equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2502.15273 |