2017/04/19 by Michael Winkler, Winkler, Michael · 2 citations
Mathematics · #35B40 #35K55 (primary) #35Q35 #35Q92 #92C17 (secondary) #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth
paper · pdf · doi:10.48550/arxiv.1704.05648
openalex publication_date 2017/04/19 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
A class of chemotaxis-Stokes systems generalizing the prototype \
left
\n\nt + u
cdot
nabla n · amp;= · amp;
nabla
cdot
big(nm-1
nabla\nn
big) -
nabla
cdot
big(n
nabla c
big), ct + u
cdot
nabla c · amp;= · amp;
Delta\nc-nc, ut +
nabla P · amp;= · amp;
Delta u + n
nabla
phi,
qquad
nabla
cdot u =0,\n\
right. is considered in bounded convex three-dimensional\ndomains, where \φ\∈ W2,\∞(\Ω) is given. The paper develops an\nanalytical approach which consists in a combination of energy-based arguments\nand maximal Sobolev regularity theory, and which allows for the construction of\nglobal bounded weak solutions to an associated initial-boundary value problem\nunder the assumption that \mgt;
frac98.
qquad (
star) Moreover, the\nobtained solutions are shown to approach the spatially homogeneous steady state\n(\(1)/(|\Ω|) \∫_\Ω n0,0,0) in the large time limit. This\nextends previous results which either relied on different and apparently less\nsignificant energy-type structures, or on completely alternative approaches,\nand thereby exclusively achieved comparable results under hypotheses stronger\nthan (\⋆).\n