2022/11/01 by Frederic Heihoff, Heihoff, Frederic · 2 citations
Immunology and Microbiology · Mathematics · #35A15 #35A23 #35D30 #35J20 #35K55 (primary) #35Q35 #35Q92 #92C17 (secondary) #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Navier-Stokes equation solutions #Phagocytosis and Immune Regulation
paper · pdf · doi:10.48550/arxiv.2211.00624
openalex publication_date 2022/11/01 · openalex created_date 2022/11/07 · openalex updated_date 2026/07/28
We prove two new functional inequalities of the forms∫G φ(ψ- ψ) ≤ (1)/(a)∫G ψln (( ψ )/( ψ)) + (a)/(4β0) \ ∫G ψ\∫G|∇ φ|2 and ∫G ψln (( ψ )/( ψ)) ≤ (1)/(β0)\ ∫G ψ\∫G |∇ ln(ψ)|2 for any finitely connected, bounded C2-domain G ⊆ ℝ2, a constant β0 > 0, any a > 0 and sufficiently regular functions φ, ψ. We then illustrate their usefulness by proving long time stabilization and eventual smoothness properties for certain generalized solutions to the chemotaxis-Navier-Stokes system\ \beginaligned nt + u ⋅ ∇ n amp; = Δn - ∇ ⋅ (nS(x,n,c) ∇ c),
ct + u⋅ ∇ c amp; = Δc - n f(c),
ut + (u⋅ ∇) u amp; = Δu + ∇ P + n ∇ ϕ, ∇ ⋅ u = 0, \endaligned . on a smooth, bounded, convex domain Ω⊆ ℝ2 with no-flux boundary conditions for n and c as well as a Dirichlet boundary condition for u. We further allow for a general chemotactic sensitivity S attaining values in ℝ2× 2 as opposed to a scalar one.