Uniform stability and optimal time decay rates of the compressible pressureless Navier-Stokes system in the critical regularity framework
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| Format: | Preprint |
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2025
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| _version_ | 1866911248093806592 |
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| author | Li, Fucai Ni, Jinkai Zhang, Zhipeng |
| author_facet | Li, Fucai Ni, Jinkai Zhang, Zhipeng |
| contents | This paper investigates the Cauchy problem for the compressible pressureless Navier-Stokes system in $\mathbb{R}^d$ with $d \geq 2$. Unlike the standard isentropic compressible Navier-Stokes system, the density in the pressureless model lacks a dissipative mechanism, leading to significant coupling effects from nonlinear terms in the momentum equations. We first prove the global well-posedness and uniform stability of strong solutions to the compressible pressureless Navier-Stokes system in the critical Besov space $\dot{B}_{2,1}^{\frac{d}{2}} \times \dot{B}_{2,1}^{\frac{d}{2}-1}$. Then, under the additional assumption that the low-frequency component of the initial density belongs to $\dot{B}_{2,\infty}^{σ_0+1}$ and that the initial velocity is sufficiently small in $\dot{B}_{2,\infty}^{σ_0}$ with $σ_0 \in (-\frac{d}{2}, \frac{d}{2}-1]$, we overcome the challenge of derivative loss caused by nonlinearity and establish optimal decay estimates for $u$ in $\dot{B}_{2,1}^σ$ with $σ\in (σ_0, \frac{d}{2}+1]$. In particular, it is shown that the density remains uniformly bounded in time which reveals a new asymptotic behavior in contrast to the isentropic compressible Navier-Stokes system where the density exhibits a dissipative structure and decays over time. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_02321 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniform stability and optimal time decay rates of the compressible pressureless Navier-Stokes system in the critical regularity framework Li, Fucai Ni, Jinkai Zhang, Zhipeng Analysis of PDEs This paper investigates the Cauchy problem for the compressible pressureless Navier-Stokes system in $\mathbb{R}^d$ with $d \geq 2$. Unlike the standard isentropic compressible Navier-Stokes system, the density in the pressureless model lacks a dissipative mechanism, leading to significant coupling effects from nonlinear terms in the momentum equations. We first prove the global well-posedness and uniform stability of strong solutions to the compressible pressureless Navier-Stokes system in the critical Besov space $\dot{B}_{2,1}^{\frac{d}{2}} \times \dot{B}_{2,1}^{\frac{d}{2}-1}$. Then, under the additional assumption that the low-frequency component of the initial density belongs to $\dot{B}_{2,\infty}^{σ_0+1}$ and that the initial velocity is sufficiently small in $\dot{B}_{2,\infty}^{σ_0}$ with $σ_0 \in (-\frac{d}{2}, \frac{d}{2}-1]$, we overcome the challenge of derivative loss caused by nonlinearity and establish optimal decay estimates for $u$ in $\dot{B}_{2,1}^σ$ with $σ\in (σ_0, \frac{d}{2}+1]$. In particular, it is shown that the density remains uniformly bounded in time which reveals a new asymptotic behavior in contrast to the isentropic compressible Navier-Stokes system where the density exhibits a dissipative structure and decays over time. |
| title | Uniform stability and optimal time decay rates of the compressible pressureless Navier-Stokes system in the critical regularity framework |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2511.02321 |