| Thursday 25. June, 13:15-14:00 in room 734 |
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| Samuel Walsh (University of Missouri) Title: Mathematical theory of wind-driven water waves |
| Abstract: It is easy to see that wind blowing over a body of water can create waves. But this simple observation leads to a more fundamental question: Under what conditions on the velocity profile of the wind will persistent surface water waves be generated? This problem has been studied intensively in the applied fluid dynamics community since the first efforts of Kelvin in 1871. In this talk, we will present a mathematical treatment of the predominant model for wind-wave generation, the so-called quasi-laminar model of J. Miles. Essentially, this entails determining the (linear) stability properties of the family of laminar flow solutions to the two-phase interface Euler equation. We give a rigorous derivation of the linearized evolution equations about an arbitrary steady solution, and, using this, a complete proof of the celebrated instability criterion of Miles. In particular, our analysis incorporates both the effects of surface tension and a vortex sheet on the air–sea interface. We are thus able to give a unified equation connecting the Kelvin–Helmholtz and quasi-laminar models of wave generation. |
| Tuesday 16. June, 14:15-15:00 in room 734 |
| Anders Samuelsen Nordli Title: Traffic flow close to an intersection |
| Abstract: Traffic flow in the presence of an intersection can be modeled by conservation of cars in each road and rules governing the flow in and out of the intersection. We will look at the Riemann problem, which can be reduced to a system of ODEs with discontinuous right hand side. Some special cases will be considered. This work in progress is a joint work with Alberto Bressan. |
| Thursday 18. June, 13:15-14:00 in room 734 |
| Mathew Johnson (University of Kansas, Lawrence) Title: Stability of Wave Trains in Whitham Equations |
| Abstract: In this talk, we will discuss recent advances in the existence and dynamics of periodic traveling waves in Whitham type equations for water waves. These nonlocal dispersive equations incorporate a canonical shallow water nonlinearity and the full dispersion relation for unidirectional surface water waves. Specifically, we will be interested in the impact of various physical effects (surface tension, constant vorticity, etc.) on the modulational instability of wave trains with sufficiently small amplitudes. Time permitting, we may also discuss the effect of incorporating bidirectionally in the model. This is joint work with Vera Mikyoung Hur from University of Illinois at Urbana-Champaign (USA). |
| Thursday 11. June, 13:15-14:00 in room 734 |
| Peter Kunkel (Universität Leipzig) Title: Self-adjoint differential-algebraic equations, symplectic flows, and geometric integration |
| Abstract: Self-adjoint linear DAEs arise, e. g., in the necessary conditions for linear-quadratic optimal control problems with constraining linear DAEs or by linearization of DAEs from the modelling of multibody systems. Starting from local and global canonical forms for such structured problems, we show that under a suitable restricted class of transformations we are able to separate a hamiltonian system of differential equations. In this sense, we may say that a self-adjoint linear DAE exhibits a symplectic flow. Based on these observations, we will discuss a possibility for the geometric integration of self-adjoint linear DAEs. Techniques include structured index reduction, time-dependent transformations and automatic differentiation. |
| 2–5 June |
| Workshop: Complex and Harmonic Analysis, Differential Equations, Numerical Methods |
| Wednesday 20.05.2015, 11:15 in room 734. Joint DNA and Analysis seminar. |
| Anton Shiriaev (NTNU) Title: On Motion Planning, Motion Representation and its Orbital Stabilization for Mechanical System |
| Abstract This talk is about motion planning, motion representation and steps in orbital stabilization of motions of mechanical systems, which might be redundant or have one or several passive degrees of freedom. Given a motion, we suggest to search for its representation without explicit time dependence: an evolution of one of degrees of freedom is defined by certain differential equation (a motion generator); while other degrees of freedom are found through relations valid between coordinates on the motion. Such representation of a motion becomes compact and, as shown, it is often useful in analysis of dynamics in vicinity of its orbit and controller design. In particular, we show steps in a feedback control design that are based on construction of a transverse linearization. Roughly speaking, the transverse linearization is a linear system of dimension one less than the nonlinear system such that stabilization of this system is in certain sense equivalent to exponential orbital stabilization of a desired (periodic) motion of the original nonlinear system. The proposed approach is illustrated on popular research benchmark set-ups (the Furuta pendulum, the Acrobot, a pendulum on a cart, a spherical pendulum on a puck) and applications (design stable gaits for bipeds, quadrupeds; analysis of recorded motions of humans). Remarkably, for mechanical systems the transverse linearization of any feasible (forced) orbit, which in general is related to defining moving Poincaré sections, can be introduced analytically. This fact opens a broad range of opportunities. |
| Tuesday 28. April, 13:15-14:00 in room F4 (Gamle fysikk) |
| Tore Halvorsen (UiO) Title: Upwinding in finite element methods for a convection-diffusion equation |
| Abstract: In this talk we will introduce a convection-diffusion equation, and present a numerical method in the regime of small viscosity. We identify a norm for which we have both continuity and an inf-sup condition, which are uniform in mesh-width and viscosity, up to logarithmic terms, as long as the viscosity is smaller than the mesh-width. The analysis allows for the formation of a boundary layer. |
| Thursday 23. April, 13:15-14:00 in room 734 |
| Ulrik Skre Fjordholm Title: Well-posedness of the uncertain Cauchy problem for conservation laws |
| Abstract: We consider the hyperbolic conservation law \(u_t + f(u)_x = 0\) with initial data \(u(x,0) = \bar{u}(x)\). In practically all real-world applications, the initial data \(\bar{u}\) will not be known exactly due to measurement error, low measurement resolution, inherent physical uncertainties, etc. However, we may assume that we know certain statistics about the data (mean, variance, etc.). The goal of Uncertainty Quantification is to propagate the uncertain data and give statistics of the solution at a later time, given initial uncertainty. We argue that the correct way of representing uncertain data is via one of two equivalent representations: either as a probability measure on the underlying function space (\(L^p\)-space), or as a family of Young measures giving correlations of the solution at different spatial points. We derive the notion of statistical solutions for conservation laws, and show existence and uniqueness in certain cases. |
| Thursday 16. April, 13:15-14:00 in room 734 |
| Erik Lindgren (KTH) Title: Nonlinear nonlocal equations and regularity |
| Abstract: In this talk, I will discuss some classes of nonlocal equations. In particular, I will describe certain nonlocal versions of quasilinear equations such as the p-Laplace equation. For certain equations, it is actually easier prove some regularity results in the non-local setting than in the local setting. I will present a very simple method due to Luis Silvestre that applies in great generality to nonlocal equations. |
| Thursday 26. March, 13:15-14:00 in room 734 |
| Mathias Nikolai Arnesen Title: Existence and stability of solitary wave solutions to non-local equations |
| Abstract: We prove existence and conditional energetic stability of solitary wave solutions for the two classes of pseudodifferential equations \(u_t+\left(f(u)\right)_x-\left(L u\right)_x=0\) and \(u_t+\left(f(u)\right)_x+\left(L u\right)_t=0\), where \(f\) is a nonlinear term, typically of the form \(c|u|^p\) or \(cu|u|^{p-1}\), and \(L\) is a Fourier multiplier operator of positive order. The former class includes for instance the Whitham equation with capillary effects and the generalized Korteweg—de Vries equation, and the latter the Benjamin—Bona—Mahony equation. Existence and conditional energetic stability results have earlier been established using the method of concentration–compactness for a class of operators with symbol of order \(s\geq 1\). We extend these results to symbols of order \(0<s<1\), thereby improving upon the results for general operators with symbol of order \(s\geq 1\) by enlarging both the class of linear operators and nonlinearities admitting existence of solitary waves. Instead of using abstract operator theory, the new results are obtained by direct calculations involving the non-local operator \(L\), something that gives us the bounds and estimates needed for the method of concentration–compactness. |
| Thursday 12. March, 13:15-14:00 in room 734 |
| Jørgen Endal Title: \(L^1\) Contraction for Bounded (Nonintegrable) Solutions of Degenerate Parabolic Equations |
| Abstract: We obtain new \(L^1\) contraction results for bounded entropy solutions of Cauchy problems for degenerate parabolic equations. The equations we consider have possibly strongly degenerate local or nonlocal diffusion terms. As opposed to previous results, our results apply without any integrability assumption on the solutions. They take the form of partial Duhamel formulas and can be seen as quantitative extensions of finite speed of propagation local \(L^1\) contraction results for scalar conservation laws. A key ingredient in the proofs is a new and nontrivial construction of a subsolution of a fully nonlinear (dual) equation. Consequences of our results are maximum and comparison principles, new a priori estimates, and, in the nonlocal case, new existence and uniqueness results. Joint work with Espen R. Jakobsen. |
| Thursday 26. February, 13:15-14:00 in room 734 |
| Espen R. Jakobsen Title: On non-local quasi-linear PDEs and their local limits |
| Abstract: We will introduce and study new non-local quasi-linear equations. Many examples will be discussed along with the relationships to local quasi-linear equations and to stochastic control theory. We will then outline our main results in for a model equation: (i) comparison, uniqueness and existence results for weak (viscosity) solutions, and (ii) sufficient conditions and convergence results for limit problems where non-local equations approximate local ones. Our equations involve new non-local quasi-linear operators, special cases are new p and infinity Laplace type operators. Our operators are however non-variational and different from the many other non-local versions of p and infinity Laplacians that recently have appeared in the literature.Finally we note that, for "any" fractional diffusion and "any" local quasi-linear operator, by our results we can define a corresponding fractional quasi-linear version. This is joint work with Emmanuel Chasseigne (University of Tours, France). |
| Tuesday 10. February, 13:15-14:00 in room 734 |
| Christoph Walker (Leibniz Universität Hannover) Title: A Free Boundary Problem for MEMS |
| Abstract: Idealized microelectromechanical systems (MEMS) consist of a fixed ground plate above which an elastic plate (or membrane) is suspended that deforms due to a voltage difference that is applied between the two components. The mathematical model involves the harmonic electrostatic potential in the free domain between the two plates along with a singular evolution equation for the displacement of the elastic plate, the coupling term being the trace of the potential gradient on the deformed plate. The number of stationary solutions and the possible occurrence of blow-up solutions (corresponding to a touchdown of the elastic plate on the ground plate) are studied. |
| Thursday 22. January, 13:15-14:00 in room 734 |
| Peter Lindqvist Title: Infinite initial values for a quasilinear parabolic equation |
| Abstract |
| Tuesday 13. January, 11:15-12:00 in room 734 |
| Johannes Keller (Zentrum Mathematik Technische Universität München) Title: Computational molecular quantum dynamics with classical trajectories |
| Abstract: The governing equation for the quantum evolution of the nuclei in a molecule is the time-dependent semiclassical Schrödinger equation, which is a partial differential equation on a typically high dimensional space. Its solutions are highly oscillatory in space and time, and consequently direct numerical discretizations are often very expensive, or even unfeasible. In this talk we introduce approximations for the evolution of quantum expectations which are based on the classical trajectories of the underlying Hamiltonian system, and hence are much less oscillatory. We present algorithmic discretizations and illustrate their applicability by means of various numerical experiments. |