Dissertation Presentation | Fabian Gabel | Spectral Theory, Control, and Higher Regularity
Welcome to the presentation of my #dissertation topic at #tuhh. Here, we will visit the three chapters of my dissertation, titled ===== On Spectral Theory, Control, and Higher Regularity of Infinite-dimensional Operator Equations ===== Slides are available at: https://bit.ly/3MFGuG6 This is *not* the presentation held before the doctoral defense. PhD applicants at TUHH need to present their topic publicly *before* handing in the dissertation. Chapters 0:00 Intro: The three elements of the talk: **earth**, **water**, and **air** 1:02 ======= EARTH ======= 1:03 Applicability of the finite Section Method for Discrete Schrödinger Operators with (0,λ)-valued Periodic Potentials 5:35 Applicability of the FSM is Characterized by Trace Condition (recursion formula and monodromy matrices) 10:06 FSM-applicability by Checking Shifted One-sided Compressions (the limit operator technique) 12:22 4-periodic Example: FSM Applicable if |tr(M)| gt 2! 13:52 5-periodic Example: FSM Applicable if |tr(M)| gt and λ not in Q! 14:16 9-periodic Example: FSM Applicable even though |tr(M)| gt 2! 14:39 ======== WATER ======== 14:47 Final-state Observability for Non-autonomous Cauchy Problems with →Moving→ Sensor Sets 17:35 Measuring concentration on sensor sets despite moving blind spots 21:07 Non-autonomous Diffusion Problems with Moving Sensor Sets (Thickness condition on sensor sets as a sufficient condition for final-state observability) 25:20 Looking for necessary geometric conditions on the sensor sets: Mean-thick Families of Sensor Sets are Necessary for Final-state Observability 27:20 Examples of families of mean-thick sets 27:59 Non-autonomous Generalization of the Lebeau-Robbiano Strategy 29:39 ======= AIR ======= 29:45 Higher Regularity for Leray-Hopf Weak Solutions of Navier-Stokes Equations on Planar Lipschitz Domains 31:12 Example for a 2-dimensional problem: how does an airplane's wing generate lift? 34:33 Higher Regularity for Leray-Hopf solutions in Lp-spaces 39:08 Higher Regularity for Leray-Hopf solutions in Spaces of Distributions 40:06 Use Linear Theory to Solve Non-linear Problem 45:30 ===== OUTRO ====== 45:31 Take Home Images and Messages 46:15 References 46:26 Outro ===== Links and Further Information ===== Find more about my research at https://math.fabian-gabel.de/research/ ==== References ==== F. Gabel, D. Gallaun, J. Großmann, M. Lindner, R. Ukena, Finite sections of periodic Schrödinger operators, 2022. arXiv:2110.09339 [math.SP]. https://arxiv.org/abs/2110.09339 To appear in Operator Theory: Advances and Applications. F. Gabel, D. Gallaun, J. Großmann, M. Lindner, R. Ukena, Analysis Code for Finite Sections of Periodic Schrödinger Operators, TUHH Universitätsbibliothek, 2021. https://doi.org/10.15480/336.3828 C. Bombach, F. Gabel, C. Seifert, M. Tautenhahn, Observability for non-autonomous systems, 2022. arXiv:2203.08469 [math.FA]. https://arxiv.org/abs/2203.08469 To appear in SIAM Journal on Control and Optimization. F. Gabel, P. Tolksdorf, The Stokes operator in two-dimensional bounded Lipschitz domains, Journal of Differential Equations 340, 2022. https://doi.org/10.1016/j.jde.2022.09.001 #math #spectraltheory #controltheory #schrödinger #diffusion #navier-stokes #leray-hopf #weak-solution #higher-regularity #maximal-regularity #seaweed #coffee #air #water #4elements
Welcome to the presentation of my #dissertation topic at #tuhh. Here, we will visit the three chapters of my dissertation, titled ===== On Spectral Theory, Control, and Higher Regularity of Infinite-dimensional Operator Equations ===== Slides are available at: https://bit.ly/3MFGuG6 This is *not* the presentation held before the doctoral defense. PhD applicants at TUHH need to present their topic publicly *before* handing in the dissertation. Chapters 0:00 Intro: The three elements of the talk: **earth**, **water**, and **air** 1:02 ======= EARTH ======= 1:03 Applicability of the finite Section Method for Discrete Schrödinger Operators with (0,λ)-valued Periodic Potentials 5:35 Applicability of the FSM is Characterized by Trace Condition (recursion formula and monodromy matrices) 10:06 FSM-applicability by Checking Shifted One-sided Compressions (the limit operator technique) 12:22 4-periodic Example: FSM Applicable if |tr(M)| gt 2! 13:52 5-periodic Example: FSM Applicable if |tr(M)| gt and λ not in Q! 14:16 9-periodic Example: FSM Applicable even though |tr(M)| gt 2! 14:39 ======== WATER ======== 14:47 Final-state Observability for Non-autonomous Cauchy Problems with →Moving→ Sensor Sets 17:35 Measuring concentration on sensor sets despite moving blind spots 21:07 Non-autonomous Diffusion Problems with Moving Sensor Sets (Thickness condition on sensor sets as a sufficient condition for final-state observability) 25:20 Looking for necessary geometric conditions on the sensor sets: Mean-thick Families of Sensor Sets are Necessary for Final-state Observability 27:20 Examples of families of mean-thick sets 27:59 Non-autonomous Generalization of the Lebeau-Robbiano Strategy 29:39 ======= AIR ======= 29:45 Higher Regularity for Leray-Hopf Weak Solutions of Navier-Stokes Equations on Planar Lipschitz Domains 31:12 Example for a 2-dimensional problem: how does an airplane's wing generate lift? 34:33 Higher Regularity for Leray-Hopf solutions in Lp-spaces 39:08 Higher Regularity for Leray-Hopf solutions in Spaces of Distributions 40:06 Use Linear Theory to Solve Non-linear Problem 45:30 ===== OUTRO ====== 45:31 Take Home Images and Messages 46:15 References 46:26 Outro ===== Links and Further Information ===== Find more about my research at https://math.fabian-gabel.de/research/ ==== References ==== F. Gabel, D. Gallaun, J. Großmann, M. Lindner, R. Ukena, Finite sections of periodic Schrödinger operators, 2022. arXiv:2110.09339 [math.SP]. https://arxiv.org/abs/2110.09339 To appear in Operator Theory: Advances and Applications. F. Gabel, D. Gallaun, J. Großmann, M. Lindner, R. Ukena, Analysis Code for Finite Sections of Periodic Schrödinger Operators, TUHH Universitätsbibliothek, 2021. https://doi.org/10.15480/336.3828 C. Bombach, F. Gabel, C. Seifert, M. Tautenhahn, Observability for non-autonomous systems, 2022. arXiv:2203.08469 [math.FA]. https://arxiv.org/abs/2203.08469 To appear in SIAM Journal on Control and Optimization. F. Gabel, P. Tolksdorf, The Stokes operator in two-dimensional bounded Lipschitz domains, Journal of Differential Equations 340, 2022. https://doi.org/10.1016/j.jde.2022.09.001 #math #spectraltheory #controltheory #schrödinger #diffusion #navier-stokes #leray-hopf #weak-solution #higher-regularity #maximal-regularity #seaweed #coffee #air #water #4elements