The chemotaxis–fluid system in the whole space ℝ^N ( N ≥ 2 ) is considered. By choosing scaling invariant Besov–Morrey spaces as the spaces of the given data, we construct mild solutions to the system. If the given data are small, then our mild solutions exist globally in time. We also investigate time-decay properties of global mild solutions. Although the corresponding result has already been obtained by Yang, Fu, and Sun (J. Differential Equations 266 (2019), no. 9, 5867–5894) and Ferreira and Postigo (J. Math. Phys 60 (2019), no. 6, 061502, 19), we remove some smallness conditions of the given data. Furthermore, our result includes the case where the index of the Morrey space is 1. Such a result is regarded as the result corresponding to that given by Kozono and Yamazaki (Comm. Partial Differential Equations 19 (1994), no. 5-6, 959–1014) for the Navier–Stokes system.
In this paper, we introduce a new compressible fluid-particle interaction model which comprises the full compressible Navier–Stokes equations and the Vlasov–Fokker–Planck equation. These two components couple together via the energy exchanges and the density-dependent friction force. In the whole space ℝ^3 case, for the initial data that constitute a small perturbation of the given equilibrium in H^3(ℝ^3) ∩ L^1(ℝ^3) , we establish the global well-posedness and optimal time-decay rates of classical solutions and their gradients by employing the delicate energy method and Fourier analysis techniques. We develop a new energy method to address the zero-order nonlinear terms where the conventional energy methods fail to handle effectively. For the periodic domain 𝕋^3 case, our global classical solutions decay exponentially. Our paper provides a first rigorous result on the global well-posedness of the compressible fluid-particle interaction model incorporating energy exchanges and density-dependent friction forces.
We study rates of decay for (possibly unbounded) C_0 -semigroups on Banach spaces under the assumption that the norm of the resolvent of the respective semigroup generator grows as a regularly varying function of type β >0 , that is, as |s|^βℓ (1+|s|) or |s|^β/κ (1+|s|) , where ℓ ,κ are arbitrary monotone and slowly varying functions. The main result extends the estimates obtained by Deng, Rozendaal and Veraar (J. Evol. Equ. 24, 99 (2024)) to this setting of regularly varying functions and improves the estimates obtained by Santana and Carvalho (J. Evol. Equ. 24, 28 (2024)) in case |s|^βlog (1+|s|)^b , with b≥ 0 .
This paper explores the Keller–Segel–Stokes system with indirect signal production in a smoothly bounded domain Ω⊂ℝ^3 , where ϕ∈ W^2,∞(Ω ) and |𝒮(x,n,v,w)|≤ C_𝒮 . Then given any sufficiently regular nonnegative initial configuration, alongside the time evolution for an energy-like functional ∫ _Ωn_ε^p+ ∫ _Ω|∇ v_ε|^2q, the associated problem (*) possesses at least one globally bounded weak solution if m>1, therefore not only covering the previous results (Remark 2.1), but also providing evidence for the advantageous influence of indirect signal production on the global solvability in 3D Keller–Segel–Stokes system (Remark 2.2).
We investigate a two-species Keller–Segel system with linear diffusion and nonlocal attractive power-law interaction in ℝ^d(d ≥ 3) . Specifically, we analyze the initial data conditions that lead to the regime where attractive forces prevail over diffusion, and further classify the global existence and finite-time blow-up behaviors of solutions. It is shown that this system exhibits a scaling invariance, which preserves the L^d/2(ℝ^d) norm of solutions, and solutions exist globally when the initial data are small in L^d/2(ℝ^d) norm, whereas finite-time blowup of solutions occurs when the initial data are large in L^d/2(ℝ^d) norm.
