Analysis of taxis processes in markedly irregular frameworks / Frederic Heihoff ; Researched and written at the Institute of Mathematics of Paderborn University under the supervision of Herr Univ.-Prof. Dr. Michael Winkler. Paderborn, 2025
Inhalt
- Introduction
- On the modeling and dynamics of chemotaxis
- Framework 1: A chemotaxis-fluid model with non-rotational flux
- Framework 2: Haptotaxis with a potentially degenerate diffusion and taxis tensor
- Framework 3: Measure-valued initial data in attractive-repulsive chemotaxis as well as chemotaxis-consumption models
- Notation
- Two new functional inequalities based on the Trudinger–Moser inequality
- Main result
- Approach
- The Trudinger–Moser inequality
- A variational approach to minimizing C_G
- Proving our new functional inequalities
- Eventually smooth generalized solutions to a chemotaxis-fluid system with rotational flux
- Main result
- Approach
- Generalized solution concept
- Approximate solutions
- Construction of generalized solutions
- Proving the eventual smoothness and stabilization properties of (n,c,u)
- Eventual uniform smallness of the family (c)(0,1) in Lp() for all p [1,)
- Eventual uniform smallness of a key functional and its associated norms
- Eventual uniform bounds for nL(), cLp(), AuLp() for p [1,) and (12, 1) via bootstrap arguments
- Establishing baseline uniform parabolic Hölder bounds for the families (n)(0,1), (c)(0,1) and (u)(0,1)
- Deriving C2+, 1+2-type parabolic Hölder regularity properties for (n,c,u)
- Stabilization of (n,c,u)
- Proof of the main theorem
- Weak solutions to a haptotaxis system with a potentially degenerate diffusion and taxis tensor
- Smooth solutions to an attractive-repulsive chemotaxis system with measure-valued initial data
- Main result
- Approach
- Approximate solutions
- A priori estimates degrading toward zero
- Estimates away from t = 0 and the construction of our solution candidates
- Continuity at t = 0
- Proof of the main theorem
- Immediate smoothing in a chemotaxis-consumption system starting from measure-type initial data in arbitrary dimension
- Main result
- Approach
- Approximate solutions
- A priori estimates up to t = 0
- Construction of a solution candidate
- Continuity in t = 0
- Proof of the main theorem
- Bibliography