Project B6 • Stability of patterns for hyperbolic-parabolic equations (7/2015 - 6/2019)
Principal investigators
Prof. Dr. Michael Plum
(7/2015 - 6/2019)
JProf. Dr. Jens Rottmann-Matthes
(7/2015 - 12/2017)
Project summary
Introduction
Chemotaxis is the mechanism behind bacteria and cell movement, in response to a chemical substance. This biological phenomenon, draws a lot of attention due to its correlation to many significant topics of research (e.g. in Immune response, Embryogenesis, Alzheimer, Metastatic Cancer etc).
In this project, we study a model, introduced by Dolak and Hillen (derived from the classical model for Chemotactic movement of Patlak, Keller and Segel ) that applies Cattaneo’s law of heat propagation with finite speed, based on the individual movement patterns of the species. This alternative mode reads as follows \[\begin{equation}\label{eq:B6:01} u_t+q_x=\rho u\textstyle(1-\frac{u}{\lambda})(u-\frac{\lambda}{4}), \quad \displaystyle q_t+D\tau u_x=\frac{\alpha}{\tau}\frac{S_xu}{(1+bS)^2}-\frac{1}{\tau}q, \quad S_t-\frac{d}{\epsilon}S_{xx}=\frac{1}{\epsilon}\frac{\lambda u}{(1+\gamma u)}-\frac{1}{\epsilon}S\end{equation}.\]
The advantage of the Cattaneo-Chemotaxis model, is that the undesired feature of infinite fast propagation of information is omitted. Therefore, the model is much more descriptive with respect to the movement of the population studied.
Numerical evidence shows the existence of Traveling wave solutions of system \eqref{eq:B6:01} on a bounded domain \(\Omega = [−R, R]\). A numerical approximation for the solution components \(u\), \(q\) and \(S\) is shown in Figure 1 below.
Figure 1: Traveling wave solution for \(u\), \(q\), \(S\) on a domain \(\Omega=[-10,10]\)
Goals
Aim of the project is to rigorously proof existence of traveling wave solutions of system \eqref{eq:B6:01} on \(\Omega=\mathbb{R}\) as Connecting Orbits between two stationary points, using a Computer-Assisted Proof (CAP) method. Later on, stability of those solutions shall be investigated.
Method
In recent past, computer-assisted proofs have solved a long list of mathematical problems with some of them being the Kepler conjecture, the existence of the Lorenz attractor, the existence of chaos and the four-colour theorem. In comparison to a ”theoretical proof”, CAP give us the advantage of providing accurate quantitative information. In our problem, this translates to tight and explicit bounds for the solution. Continue reading.Collapse content.
Principal of the method / Sketch of the proof:
Formulate \eqref{eq:B6:01} as a zero-finding problem. Then, solving \eqref{eq:B6:01} amounts to \[\begin{equation}\label{eq:B6:02}finding\quad u\in X\quad satisfying\quad \mathcal{F}(u)=0\end{equation}\] with \(\mathcal{F}:X\to Y\) denoting some Fréchet differentiable mapping and \((X, h\left\langle\cdot,\cdot\right\rangle_X)\) and \((Y,h\left\langle\cdot,\cdot\right\rangle_Y)\) two real Hilbert spaces. Let \(\omega\in X\) denote some approximate solution to \eqref{eq:B6:02} (computed by numerical means), and \(L:=\mathcal{F}'(\omega):X\to Y\) the Fréchet derivative of \(\mathcal{F}\) at \(\omega\) i.e., \(L\in B(X,Y)\) (the Banach space of all linear bounded operators from X to Y ). Then one has to explicitly compute constants \(\delta\) and \(K\) and a non-decreasing function \(g:[0,\infty]\to[0,\infty]\) such that \begin{equation}\label{eq:B6:03}||\mathcal{F}(\omega)||_Y\leq\delta\end{equation} i.e., \(\delta\) bounds the defect (residual) of the approximate solution \(\omega\) to \eqref{eq:B6:02}, i.e.,\begin{equation}\label{eq:B6:04}||u||_X\leq K\,||L[u]||_Y\end{equation} i.e., \(\forall u\in X\), \(K\) bounds the inverse of the linearization \(L\),\begin{equation}\label{eq:B6:05}||\mathcal{F}'(\omega+u)-\mathcal{F}'(\omega)||_{B(X,Y)}\leq g(||u||_X)\end{equation} ie., \(\forall u\in X\), \(g\) majorizes the modulus of continuity of \(\mathcal{F}'\) at \(\omega\) and \begin{equation}\label{eq:B6:06}g(t)\to0\text{ as }t\to0\end{equation} which in particular requires \(\mathcal{F}'\) to be continuous at \(\omega\).
Having those explicit bounds at hand will enable us to formulate and prove the main theorem (of Newton-Kantorovich type) stating the following:
Statement: Let \(\delta\), \(K\), \(g\) satisfy conditions \((3)-(6)\). Then there exists some \(\alpha\geq0\) such that \[\delta\leq\frac{\alpha}{K}-G(\alpha)\quad\text{and}\quad Kg(\alpha)<1\] where \(G(t):=\int_0^tg(s)\,ds\). Then, there exists a solution \(u\in X\) of the equation \(\mathcal{F}(u)=0\) satisfying \[||u-\omega||_X\leq\alpha.\]
The explicit computation of such \(\delta\), \(K\) and \(g\) the main challenge of this method. For that, an interval arithmetic environment will be used, aiming to give us results that take into account truncation and round-off errors. Continue reading.Collapse content.
R. Flohr and J. Rottmann-Matthes. A splitting approach for freezing waves. In C. Klingenberg and M. Westdickenberg, editors, Theory, numerics and applications of hyperbolic problems. I, volume 236 of Springer Proc. Math. Stat., pages 539–550. Springer, Cham, June2018.[preprint][bibtex]
