2024/11/27 by T. Howard Black, Black, Tobias
Computer Science · Mathematics · Physics and Astronomy · #35A01 #35D30 #35K65 #35Q35 #35Q92 #92C17 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Micro and Nano Robotics
paper · pdf · doi:10.48550/arxiv.2411.18336
openalex publication_date 2024/11/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider an initial-boundary value problem for the chemotaxis-Navier--Stokes system \ c@ l@ l@ c nt+u⋅∇ n=∇⋅(D(n)∇ n-nS(x,n,c)⋅∇ c), · amp;x∈Ω, · amp; t · gt;0,
ct+u⋅∇ c=Δc-cn, · amp;x∈Ω, · amp; t · gt;0,
ut+(u⋅∇)u=Δu+∇ P+n∇Φ, ∇⋅ u=0, · amp;x∈Ω, · amp; t · gt;0,
(D(n)∇ n-nS(x,n,c)⋅∇ c)⋅ν=∇ c⋅ν=0, u=0, · amp;x∈∂Ω, · amp; t · gt;0,
n(⋅,0)=n0, c(⋅,0)=c0, u(⋅,0)=u0, · amp;x∈Ω.. in a smoothly bounded domain Ω⊂ℝ2. Assuming S:Ω×[0,∞)×(0,∞)→ ℝ2× 2 to be sufficiently regular and such that with γ∈[0,\frac56] and some non-decreasing S0:(0,∞)→(0,∞), we have |S(x,n,c)|≤ (S0(c))/(cγ)\quadfor all (x,n,c)∈Ω×[0,∞)×(0,∞), we show that if D:[0,∞)→[0,∞) is suitably regular and positive throughout (0,∞), then for all M>0 one can find L(M)>0 such that whenever \liminfn→∞ D(n)gt;L\quadand \liminfn\searrow0(D(n))/(n)gt;0 are satisfied and the initial data (n0,c0,u0) are suitably regular and satisfy ‖c0‖L∞(Ω)≤ M there is a global and bounded weak solution for the initial-boundary value problem above. Under the additional assumption of D(0)>0 this solution is moreover a classical solution of the same problem.