The linear transport equation is the possibly simplest of all PDE, describing the evolution of a density under the flow of a given vectorfield. It is well-known that, as long as the vectorfield is Lipschitz continuous, solutions of the linear transport equation are closely related, via the method of characteristics, to the Lagrangian flow map generated by the vectorfield. However, this link breaks down if the vectorfield is merely Sobolev, since in this case the ODE does not make sense classically. Of course, formulation of the PDE poses no problems even without differentiability assumptions. On the other hand there are numerous applications in fluid mechanics and kinetic theory, where the transport equation appears with a vectorfield which is in some Sobolev space, possibly even continuous, but not Lipschitz. For such cases DiPerna and Lions developed in the late 1980s a theory of renormalization, leading to a well-posed solution concept for both the PDE and the ODE. In the talk we discuss the limits of this theory and present examples showing that unless certain additional integrability conditions are imposed, the theory of renormalization does not lead to a unique solution.
In nonlinear dynamical system self-excited vibrations frequently occur where an equilibrium undergoes a Hopf bifurcation and limit-cycle oscillations develop.
The Hopf bifurcation has two types, supercritical (soft) and subcritical (dangerous). The type of the bifurcation depends on whether the nonhyperbolic equilibrium is weakly stable or unstable. The stability of the equilibrium (and thus the type of the Hopf bifurcation) is determined by the sign of the so-called Poincaré-Lyapunov constant.
This talk discusses three short topics centered around Poincaré-Lyapunov constants:
1, We pose and affirmatively answer the question whether the stability of a nonlinear center can be determined from the eigenvalues of the Jacobian matrix AWAY from the equilibrium point.
2, We recognize that the Poincaré-Liapunov constant is a quadratic form in a 10-dimensional space of the coefficients associated with the normal form of a Hopf bifurcation. This real manifold (the "Hopf quadric") separates regions of the parameter space corresponding to supercritical and subcritical bifurcations.
The stationary points of the squared distance function from a parameter point to the Hopf quadric are the real zeros of a univariate algebraic equation. The distance to the quadric is the minimal positive zero of this equation. This distance can be used as a measure of the "criticality" of the bifurcation. Joint work with Alexei Yu. Uteshev.
3, We use the so-called Carleman embedding technique to recast the normal form of a Hopf bifurcation as an infinite-dimensional linear system. We describe the connection between the Poincaré-Lyapunov constants and the linear algebraic properties of the Carleman matrices. This connection provides a new algorithm to compute Poincaré-Lyapunov constants. Joint work with Csanád Hubay.
Systems of non-linear partial differential equations (PDEs) are often used to describe mathematically the long-range transport of air pollutants. The discretization of the spatial derivatives involved in these systems of PDEs leads to the solution of large systems of non-linear ordinary differential equations (ODEs), which are very stiff and, therefore, must be handled by applying implicit numerical methods for solving systems of ODEs. That leads to the solution of systems of non-linear algebraic equations, which have to be treated, at every time-step, by suitable iterative methods. Some version of the well-known Newton Iterative Method is normally used and systems of linear algebraic equations (LAEs) are to be solved many times in the inner loop of the Newton procedure. The systems of LAEs are huge when fine spatial resolution is used, which is nearly always highly desirable. Moreover, many such systems are to be treated, because the time-interval is nearly always very long. Handling many millions of systems of LAE’s, each of which contain several hundred million equations, is not unusual. Therefore, such complex models have necessarily to be run on high-performance computers by applying special techniques; see, for example, Z. Zlatev and I. Dimov: “Computational and Numerical Challenges in Environmental Modelling”, Studies in Computational Mathematics, Vol. 13, Elsevier, Amsterdam, 2006. The problems are becoming much more difficult and time-consuming when large-scale air pollution models (a) are used to study the sensitivity of the pollution levels to variations of some key parameters as, for example, the emissions and (b) are combined with different climatic scenarios in the efforts to investigate the influence of climatic changes on some high and harmful pollution levels. The treatment of the air pollution models in this extremely difficult situation will be discussed in this talk. It will, furthermore, be shown that the climatic changes are normally resulting in increased levels of some pollutants. However, the major aim will be to demonstrate the fact that some of these enormous computational tasks cannot be handled directly even on the fastest parallel computers. Therefore, some special techniques, fast numerical methods and appropriate splitting procedures must necessarily be used.