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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2509.17171 |
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| _version_ | 1866914057907339264 |
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| author | Lin, Y. -X. Wang, Y. -G. |
| author_facet | Lin, Y. -X. Wang, Y. -G. |
| contents | In this paper, we consider the initial value problem of the incompressible generalized Navier-Stokes equations with initial data being in negative order Sobolev spaces, in the whole space $\mathbb{R}^d$ with $d \geq 2$. The generalized Navier-Stokes equations studied here is obtained by replacing the standard Laplacian in the classical Navier-Stokes equations by the fractional order Laplacian $-(-Δ)^\al$ with $\al \in \left( \frac{1}{2},\frac{d+2}{4} \right]$. After an appropriate randomization on the initial data, we obtain the almost sure existence and optimal decay rate of global weak solutions when the initial data belongs to $\Dot{H}^s(\mathbb{R}^d)$ with $s\in (-\al+(1-\al)_+,0)$. Moreover, we show that the weak solutions are unique when $\al=\frac{d+2}{4}$ with $d \geq 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_17171 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Almost sure global weak solutions and optimal decay for the incompressible generalized Navier-Stokes equations Lin, Y. -X. Wang, Y. -G. Analysis of PDEs In this paper, we consider the initial value problem of the incompressible generalized Navier-Stokes equations with initial data being in negative order Sobolev spaces, in the whole space $\mathbb{R}^d$ with $d \geq 2$. The generalized Navier-Stokes equations studied here is obtained by replacing the standard Laplacian in the classical Navier-Stokes equations by the fractional order Laplacian $-(-Δ)^\al$ with $\al \in \left( \frac{1}{2},\frac{d+2}{4} \right]$. After an appropriate randomization on the initial data, we obtain the almost sure existence and optimal decay rate of global weak solutions when the initial data belongs to $\Dot{H}^s(\mathbb{R}^d)$ with $s\in (-\al+(1-\al)_+,0)$. Moreover, we show that the weak solutions are unique when $\al=\frac{d+2}{4}$ with $d \geq 2$. |
| title | Almost sure global weak solutions and optimal decay for the incompressible generalized Navier-Stokes equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2509.17171 |