The Pressureless Damped Euler-Riesz System in the Critical Regularity Framework
We are concerned with a system governing the evolution of the pressureless compressible Euler equations with Riesz interaction and damping in Rd (d≥1), where the interaction force is given by ∇(-Δ)(α-d)/2ρ with d-2<α<d. It is observed by the eigenvalue analysis that the density exhibits fracti...
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| Published in | Journal of mathematical fluid mechanics Vol. 27; no. 4 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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Springer Nature B.V
01.11.2025
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| ISSN | 1422-6928 1422-6952 |
| DOI | 10.1007/s00021-025-00964-w |
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| Abstract | We are concerned with a system governing the evolution of the pressureless compressible Euler equations with Riesz interaction and damping in Rd (d≥1), where the interaction force is given by ∇(-Δ)(α-d)/2ρ with d-2<α<d. It is observed by the eigenvalue analysis that the density exhibits fractional heat diffusion behavior at low frequencies, which enables us to establish the global existence and large-time behavior of solutions to the Cauchy problem in the critical Lp framework. Precisely, the density and its σ-order derivative converge to the equilibrium at the Lp-rate (1+t)-(σ-σ1)/(α-d+2) with -d/p-1≤σ1<d/p-1, consistent with the rate of solutions for the frictional heat equation. A non-local hypercoercivity argument and the effective unknown z=u+∇Λα-dρ associated with the Darcy law are introduced to overcome the difficulty from the absence of hyperbolic symmetrization for first-order dissipative systems. |
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| AbstractList | We are concerned with a system governing the evolution of the pressureless compressible Euler equations with Riesz interaction and damping in Rd (d≥1), where the interaction force is given by ∇(-Δ)(α-d)/2ρ with d-2<α<d. It is observed by the eigenvalue analysis that the density exhibits fractional heat diffusion behavior at low frequencies, which enables us to establish the global existence and large-time behavior of solutions to the Cauchy problem in the critical Lp framework. Precisely, the density and its σ-order derivative converge to the equilibrium at the Lp-rate (1+t)-(σ-σ1)/(α-d+2) with -d/p-1≤σ1<d/p-1, consistent with the rate of solutions for the frictional heat equation. A non-local hypercoercivity argument and the effective unknown z=u+∇Λα-dρ associated with the Darcy law are introduced to overcome the difficulty from the absence of hyperbolic symmetrization for first-order dissipative systems. |
| ArticleNumber | 60 |
| Author | Shou, Ling-Yun Chi, Meiling Xu, Jiang |
| Author_xml | – sequence: 1 givenname: Meiling surname: Chi fullname: Chi, Meiling – sequence: 2 givenname: Ling-Yun surname: Shou fullname: Shou, Ling-Yun – sequence: 3 givenname: Jiang orcidid: 0000-0001-8125-5432 surname: Xu fullname: Xu, Jiang |
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| SubjectTerms | Cauchy problems Compressibility Damping Darcys law Density Eigenvalues Euler-Lagrange equation Thermodynamics |
| Title | The Pressureless Damped Euler-Riesz System in the Critical Regularity Framework |
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