The pressureless damped Euler-Riesz system in the critical regularity framework
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arXiv
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| Natura: | Preprint |
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2024
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| author | Chi, Meiling Shou, Ling-Yun Xu, Jiang |
| author_facet | Chi, Meiling Shou, Ling-Yun Xu, Jiang |
| contents | We are concerned with a system governing the evolution of the pressureless compressible Euler equations with Riesz interaction and damping in $\mathbb{R}^{d}$ ($d\geq1$), where the interaction force is given by $\nabla(-Δ)^{\smash{\frac{α-d}{2}}}(ρ-\barρ)$ with $d-2<α<d$. Referring to the standard dissipative structure of first-order hyperbolic systems, the purpose of this paper is to investigate the weaker dissipation effect arising from the interaction force and to establish the global existence and large-time behavior of solutions to the Cauchy problem in the critical $L^p$ framework. More precisely, it is observed by the spectral analysis that the density behaves like fractional heat diffusion at low frequencies. Furthermore, if the low-frequency part of the initial perturbation is bounded in some Besov space $\dot{B}^{σ_1}_{p,\infty}$ with $-d/p-1\leq σ_1<d/p-1$, it is shown that the $L^p$-norm of the $σ$-order derivative for the density converges to its equilibrium at the rate $(1+t)^{-\smash{\frac{σ-σ_1}{α-d+2}}}$, which coincides with that of the fractional heat kernel. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_19955 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The pressureless damped Euler-Riesz system in the critical regularity framework Chi, Meiling Shou, Ling-Yun Xu, Jiang Analysis of PDEs We are concerned with a system governing the evolution of the pressureless compressible Euler equations with Riesz interaction and damping in $\mathbb{R}^{d}$ ($d\geq1$), where the interaction force is given by $\nabla(-Δ)^{\smash{\frac{α-d}{2}}}(ρ-\barρ)$ with $d-2<α<d$. Referring to the standard dissipative structure of first-order hyperbolic systems, the purpose of this paper is to investigate the weaker dissipation effect arising from the interaction force and to establish the global existence and large-time behavior of solutions to the Cauchy problem in the critical $L^p$ framework. More precisely, it is observed by the spectral analysis that the density behaves like fractional heat diffusion at low frequencies. Furthermore, if the low-frequency part of the initial perturbation is bounded in some Besov space $\dot{B}^{σ_1}_{p,\infty}$ with $-d/p-1\leq σ_1<d/p-1$, it is shown that the $L^p$-norm of the $σ$-order derivative for the density converges to its equilibrium at the rate $(1+t)^{-\smash{\frac{σ-σ_1}{α-d+2}}}$, which coincides with that of the fractional heat kernel. |
| title | The pressureless damped Euler-Riesz system in the critical regularity framework |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2406.19955 |