A Convex Approach to Hydrodynamic Analysis

We study stability and input-state analysis of three dimensional (3D) incompressible, viscous flows with invariance in one direction. By taking advantage of this invariance property, we propose a class of Lyapunov and storage functionals. We then consider exponential stability, induced L2-norms, and input-to-state stability (ISS). For streamwise constant flows, we formulate conditions based on matrix inequalities. We show that in the case of polynomial laminar flow profiles the matrix inequalities can be checked via convex optimization. The proposed method is illustrated by an example of rotating Couette flow.Comment: Preliminary version submitted to 54rd IEEE Conference on Decision and Control, Dec. 15-18, 2015, Osaka, Japa

Constructing solutions for a kinetic model of angiogenesis in annular domains

We prove existence and stability of solutions for a model of angiogenesis set in an annular region. Branching, anastomosis and extension of blood vessel tips are described by an integrodifferential kinetic equation of Fokker-Planck type supplemented with nonlocal boundary conditions and coupled to a diffusion problem with Neumann boundary conditions through the force field created by the tumor induced angiogenic factor and the flux of vessel tips. Our technique exploits balance equations, estimates of velocity decay and compactness results for kinetic operators, combined with gradient estimates of heat kernels for Neumann problems in non convex domains.Comment: to appear in Applied Mathematical Modellin