The seminars take place on Wednesdays from 4.30 - 5.30 pm in MALL 2 unless stated otherwise.
Zero-one laws for operator semigroups
The classical 0-1 law for one-parameter semigroups (T(t)) of operators (Hille, 1950) says that either the lim sup of the norm of T(t)-I (as t goes to 0) is at least 1, or it is 0, in which case the semigroup is just exp(At) for some bounded operator A. Using techniques of complex analysis and Fourier-Borel transforms, we look at generalisations of this, including the 0-1/4 law, and obtain estimates for a functional calculus for unbounded operators. This is joint work with I. Chalendar (Paris) and J. Esterle (Bordeaux).
Unique continuation for waves, Carleman estimates, and applications
In this talk, we survey some recent unique continuation results for wave equations in "degenerate" settings for which the classical theory fails to apply. Examples include unique continuation results from portions of "infinity". We also discuss the novel geometric Carleman estimates used to prove these results, and we discuss further applications of these estimates to other problems in PDEs, such as in control theory and inverse problems.
Recent results on quasineutral limit for Vlasov-Poisson via Wasserstein stability estimates
The Vlasov-Poisson system is a kinetic equation that models collisionless plasma. A plasma has a characteristic scale called the Debye length, which is typically much shorter than the scale of observation. In this case the plasma is called â€˜quasineutralâ€™. This motivates studying the limit in which the ratio between the Debye length and the observation scale tends to zero. Under this scaling, the formal limit of the Vlasov-Poisson system is the Kinetic Isothermal Euler system.
The Vlasov-Poisson system itself can formally be derived as the limit of a system of ODEs describing the dynamics of a system of N interacting particles, as the number of particles approaches infinity. The rigorous justification of this mean field limit remains a fundamental open problem.
In this talk we present the rigorous justification of the quasineutral limit for very small but rough perturbations of analytic initial data for the Vlasov-Poisson equation in dimensions 1, 2, and 3. Also, we discuss a recent result in which we derive the Kinetic Isothermal Euler system from a regularised particle model. Our approach uses a combined mean field and quasineutral limit.
Existence of entropy solutions for compressible, isentropic Euler equations with geometric effects
I will discuss the application of the method of compensated compactness to the compressible, isentropic Euler equations under certain geometric assumptions, e.g. the case of fluid flow in a nozzle of varying cross-sectional area or the assumption of planar symmetry under special relativity. Under these assumptions, the equations reduce to the classical (or relativistic) one-dimensional isentropic Euler equations with additional geometric source terms. In this talk, I will explain how the classical strategy of DiPerna, Chen et. al. can be adapted to handle these more complicated systems and will highlight some of the difficulties involved in extending the techniques to the relativistic setting.
Approximations by Gaussians with Applications to Signal Theory