Paper Info
Title
Large-Data Global Generalized Solutions in a Chemotaxis System with Tensor-Valued Sensitivities
Abstract
The chemotaxis system u(t) = Delta u - del. (uS(x, u, v) . del v); v(t) = Delta v - uf(v) (referred to as (star) in this abstract), for the density u = u(x, t) of a cell population and the concentration v = v(x, t) of an attractive chemical consumed by the former, is considered under no-flux boundary conditions in a bounded domain Omega subset of R-n, n >= 1, with smooth boundary, where f is an element of C-1([0,infinity); [0,infinity)) and S is an element of C-2((Omega) over bar x [0,infinity)(2); R-nxn) are given functions such that f(0) = 0. In contrast to related Keller-Segel-type problems with scalar sensitivities, in the presence of such matrix-valued S the system (star) in general apparently does not possess any useful gradient-like structure. Accordingly, its analysis needs to be based on new types of a priori bounds. Using a spatio-temporal L-2 estimate for del ln(u + 1) as a starting point, we derive a series of compactness properties of solutions to suitably regularized versions of (star). Motivated by these, we develop a generalized solution concept which requires solutions to satisfy very mild regularity hypotheses only, especially for the component u; in particular, the chemotactic flux uS(x, u, v) . del v need not be integrable in this context. On the basis of the above compactness properties, it is finally shown that within this framework, under a mild growth assumption on S and for all sufficiently regular nonnegative initial data, the corresponding initial-boundary value problem for (star) possesses at least one global generalized solution. This extends known results which in the case of such general matrix-valued S provide statements on global existence only in the two-dimensional setting and under the additional restriction that parallel to v(0)parallel to(L infinity(Omega)) be small.
Year
DOI
Venue
2015
10.1137/140979708
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
Keywords
Field
DocType
chemotaxis,global existence,generalized solution
Population,Boundary value problem,Mathematical optimization,Nabla symbol,Tensor,Mathematical analysis,Weak solution,Compact space,Omega,Mathematics,Bounded function
Journal
Volume
Issue
ISSN
47
4
0036-1410
Citations 
PageRank 
References 
3
0.63
2
Authors
1
Name
Order
Citations
PageRank
Michael Winkler1235.52