WK Differential Equations - Student Member:
Sabine Hittmeir

Address:

Faculty for Mathematics
University of Vienna
Nordbergstr. 15
A-1090 Wien, Austria

Telephone: (+43) (1) 4277-50674
Fax: (+43) (1) 4277-50665
Email: sabine.hittmeir@univie.ac.at

Advisor: Christian Schmeiser


Research: Nonlinear waves in kinetic and macroscopic models

Weak kinetic shock profiles for systems of nonlinear hyperbolic conservation laws

A connection between kinetic transport equations and hyperbolic systems of conservation laws can be established by the macroscopic limit. Shock waves are basic weak solutions of nonlinear hyperbolic conservation laws featuring a discontinuity. The main question we consider is the existence and dynamic stability of kinetic shock profiles, i.e. smooth travelling wave solutions of the kinetic equation, sharing the far-field states with the shock wave. For systems of nonlinear conservation laws, in contrast to the scalar conservation law, only results for small amplitude shock waves are available. In this case, the Chapman-Enskog approximation, i.e. a diffusive regularisation of the conservation laws, can be expected to provide a good approximation for solutions of the kinetic equation. Kinetic shock profiles can be constructed close to the viscous shock profiles. The classical result on the existence of small amplitude kinetic shock profiles for the gas dynamics Boltzmann equation is due to Caflisch and Nicolaenko. We give a modified and generalised version of their approach leading to more accurate approximation results and carry out the details for the kinetic BGK model for the isentropic, isothermal and Euler system of gas dynamics. The approach for the stability of small amplitude kinetic shock profiles is based on energy (actually entropy) estimates in the spirit of the work of Liu and Yu. The main idea is again to start from an approach for the system with diffusive regularisation.

Travelling waves of a kinetic reaction model for the Fisher-KPP equation

The Fisher or KPP equation is a nonlinear reaction diffusion equation describing e.g. chemical reactions. In a kinetic model of the same situation the diffusion can be replaced by collisions with a nonmoving background medium. This kinetic reaction equation is in terms of a small parameter scaled such that we recover a reaction-diffusion equation at the macroscopic limit. Following the procedure from above we construct kinetic profiles close to travelling waves of the Fisher equation. The major difficulty in this work causes the fact, that in contrary to the previous works we now consider a nonconserved problem. The existence result can be extended to also give strict monotonicity and therefore positivity for the density of the kinetic profile. The stability analysis of the kinetic waves is strongly related to the behaviour of the macroscopic equation. Since the spectrum of the linear operator associated to the Fisher equation also contains 0, we cannot expect stability in the strict sense. This problem is in the literature overcome by introducing norms with appropriated weights. Our aim is to adapt a similar functional analytical setting, in which energy estimates for the full kinetic problem can be derived.

On a Burgers-type equation with a nonlocal diffusion term

This nonlinear equation under consideration is of Burgers-type with a nonlocal diffusive term, which arises from a problem in fluid mechanics associated with the triple-deck theory. This diffusion term corresponds to a fractional derivative of order 4/3, which can also be represented as a Volterra integral operator with a singular kernel. First results show the global existence of smooth solutions for the Cauchy problem with small smooth intial data. In the case of a convex or concave flux function smooth travelling wave solutions exist, which are monotone and satisfy the entropy condition for the shock solution of the corresponding conservation law. Therefore this fractional derivative has a similar regularising effect to the conservation law as the classically used second order derivative. The aim is to derive similar results for a cubic flux function and also to include a dispersive term into the equation.

Master's Thesis:

[1] S. Hittmeir: Dynamic Optimization - Pontryagin's minimum principle and functional analytical methods, University of Salzburg, 2006.

Publications:

[2] C. Cuesta, S. Hittmeir, C. Schmeiser: Kinetic shock profiles for nonlinear hyperbolic conservation laws, to appear in Riv. Mat. Univ. Parma.
[3] C. Cuesta, S. Hittmeir, C. Schmeiser: Weak shocks of a BGK kinetic model for the isentropic system of gas dynamics, to be published.

Work in progress:

[4] C. Cuesta, S. Hittmeir, C. Schmeiser: Travelling waves of a reactive kinetic model for the Fisher-KPP equation.

[5] S. Hittmeir, C. Schmeiser: On a Burgers-type equation with a nonlocal diffusion term.

[6] F. Achleitner, S. Hittmeir, C. Schmeiser: Weak shocks of a BGK kinetic model for the Euler system of gas dynamics.

Some professional Activities:

•  2007, May 7 - 16: Minicourse on Optimal Transportation, gradient flows and entropy methods, WPI, University of Vienna
•  2007, April 23 - 29: Short course on Finite Volume and Finite Element Methods in CFD, WPI, University of Vienna
•  2007, July 22 - 27: Summer School Topics in Nonlinear PDEs, CIM/UC Coimbra, Portugal
•  2008, April 7 - 11: Summer School Topics and PDE's and applications 2008, FisyMat Granada, Spain
•  2008, May 5 - 9: Short course Multiscale problems and models in traffic flow, WPI, University of Vienna
• 2008, June 9 - 14: Summer School Methods and Models of kinetic theory, Porto Ercole, Italy. Poster presentation: Weak shocks of a BGK model for the isentropic gas dynamics.
•  2009, March 9 – June 12: Long-term program Quantum and Kinetic Transport: Analysis, Computations, and New Applications. Scholarship-supported research stay at IPAM, UCLA, Los Angeles. Seminar talk: Weak shocks of a BGK model for the isothermal gas dynamics, April 7.