Mild solutions and logarithmic decay for the time-fractional Navier-Stokes equations with Caputo-Hadamard derivative

Xiaoting Li; Zhen Wang; Xiaoting Li; Zhen Wang

doi:10.3934/nhm.2026037

Networks and Heterogeneous Media

2026, Volume 21, Issue 3: 894-914. doi: 10.3934/nhm.2026037

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Mild solutions and logarithmic decay for the time-fractional Navier-Stokes equations with Caputo-Hadamard derivative

Xiaoting Li ¹,
Zhen Wang ^{2
,
,}

1.
Jingjiang College, Jiangsu University, Zhenjiang 212013, China
2.
School of Mathematical Sciences, Jiangsu University, Zhenjiang 212013, China

Received: 04 November 2025 Revised: 16 April 2026 Accepted: 27 April 2026 Published: 07 May 2026

This paper is devoted to the study of mild solutions for the time-fractional Navier-Stokes equations, where the time derivative is interpreted in the Caputo-Hadamard sense. The Caputo-Hadamard derivative, which involves a logarithmic kernel, is particularly suitable for describing anomalous diffusion processes with ultra-slow dynamics. The main contribution of this work is threefold. First, by reformulating the equations as an abstract Cauchy problem and employing the Mittag-Leffler operator representations, we derived an integral formulation of mild solutions. Second, using Banach fixed point theorem with suitable decay estimates of the linear semigroup generated by the Stokes operator, we established the existence and uniqueness of both global (for sufficiently small initial data) and local (for arbitrarily large initial data) mild solutions in critical Lebesgue spaces. Third, we provided explicit logarithmic decay estimates for these solutions, which reflect the ultra-slow dissipation mechanism induced by the Caputo-Hadamard derivative.
- Caputo-Hadamard derivative,
- Navier-Stokes equations,
- mild solution,
- existence and uniqueness,
- logarithmic decay
Citation: Xiaoting Li, Zhen Wang. Mild solutions and logarithmic decay for the time-fractional Navier-Stokes equations with Caputo-Hadamard derivative[J]. Networks and Heterogeneous Media, 2026, 21(3): 894-914. doi: 10.3934/nhm.2026037

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Abstract

This paper is devoted to the study of mild solutions for the time-fractional Navier-Stokes equations, where the time derivative is interpreted in the Caputo-Hadamard sense. The Caputo-Hadamard derivative, which involves a logarithmic kernel, is particularly suitable for describing anomalous diffusion processes with ultra-slow dynamics. The main contribution of this work is threefold. First, by reformulating the equations as an abstract Cauchy problem and employing the Mittag-Leffler operator representations, we derived an integral formulation of mild solutions. Second, using Banach fixed point theorem with suitable decay estimates of the linear semigroup generated by the Stokes operator, we established the existence and uniqueness of both global (for sufficiently small initial data) and local (for arbitrarily large initial data) mild solutions in critical Lebesgue spaces. Third, we provided explicit logarithmic decay estimates for these solutions, which reflect the ultra-slow dissipation mechanism induced by the Caputo-Hadamard derivative.

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