A three-dimensional Keller–Segel–Navier–Stokes system with logistic source: Global weak solutions and asymptotic stabilization
A three-dimensional Keller–Segel–Navier–Stokes system with logistic source: Global weak solutions and asymptotic stabilization
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DOI：
10.1016/j.jfa.2018.12.009
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发表时间：
2019-03
影响因子：
1.7
通讯作者：
M. Winkler
中科院分区：
文献类型：
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作者：
M. Winkler
考虑Keller-Segel-Navier-Stokes方程组（<$）{nt + u <$<$n= Δ n− χ <$p（n <$c）+ ρ n− μ n2，ct + u <$p = Δ c− c+ n，ut+（u <$p）u= Δ u+<$P+ n <$p + f（x，t），<$p u= 0，在光滑边界的有界凸区域Ω <$R3中，其中<$p ∈ W1，∞（Ω），f∈ C1（Ω <$×[0，∞）），且其中χ> 0，ρ∈ R和μ> 0是给定的参数。证明了在假设supt> 0 <$t t+ 1 <$f（Ω，s）<$L65（Ω）ds有限的条件下，对任意充分正则的初值，（n 0，c 0，u 0）满足n 0≥ 0和c 0≥ 0，在无条件下，（？）的初值问题n和c的通量边界条件以及u的齐次Dirichlet边界条件在适当的广义意义上至少拥有一个全局定义的解，并且该解关于L1（Ω）× L 6（Ω）× L2（Ω; R3）中的范数是一致有界的.在μ> χ ρ+ 4的明确假设下，当t→∞时，当p∈[1，6]时，通过满足L1（Ω）× Lp（Ω）中的（n（n，t），c（n，t））→（ρ+ μ，ρ+ μ），证明了这些解在前两个分量中稳定于空间齐次态.最后，在f随时间衰减的附加条件下，证明了当t→∞时，第三个解分量在L2（Ω; R3）中u（t，t）→ 0也是平衡的.
Abstract The Keller–Segel–Navier–Stokes system (⋆){n t+ u⋅∇ n= Δ n− χ∇⋅(n∇ c)+ ρ n− μ n 2, c t+ u⋅∇ c= Δ c− c+ n, u t+(u⋅∇) u= Δ u+∇ P+ n∇ ϕ+ f (x, t),∇⋅ u= 0, is considered in a bounded convex domain Ω⊂ R 3 with smooth boundary, where ϕ∈ W 1,∞(Ω) and f∈ C 1 (Ω¯×[0,∞)), and where χ> 0, ρ∈ R and μ> 0 are given parameters. It is proved that under the assumption that sup t> 0∫ t t+ 1‖ f (⋅, s)‖ L 6 5 (Ω) d s be finite, for any sufficiently regular initial data (n 0, c 0, u 0) satisfying n 0≥ 0 and c 0≥ 0, the initial-value problem for (⋆) under no-flux boundary conditions for n and c and homogeneous Dirichlet boundary conditions for u possesses at least one globally defined solution in an appropriate generalized sense, and that this solution is uniformly bounded in with respect to the norm in L 1 (Ω)× L 6 (Ω)× L 2 (Ω; R 3). Moreover, under the explicit hypothesis that μ> χ ρ+ 4, these solutions are shown to stabilize toward a spatially homogeneous state in their first two components by satisfying (n (⋅, t), c (⋅, t))→(ρ+ μ, ρ+ μ) in L 1 (Ω)× L p (Ω) for all p∈[1, 6) as t→∞. Finally, under an additional condition on temporal decay of f it is shown that also the third solution component equilibrates in that u (⋅, t)→ 0 in L 2 (Ω; R 3) as t→∞.