|
|
|
| 1990 | PhD, University of Heidelberg | | 1988 - 1992 | Postdoc, University of Heidelberg | | 1992 - 1994 | Visiting Assistant Professor, Northwestern University, Evanston, IL, USA | | 1994 - 2000 | Postdoc, University of Heidelberg | | 2000 | Habilitation | | 2000 - 2006 | Professor (C4), University of Dortmund | | 2005 - 2006 | Visiting Miller Professor, Mathematical Sciences Research Institute (MSRI), University of California, Berkeley, CA, USA | | Since 2006 | Professor (W3), University of Bonn |
|
|
Partial differential equations provide a ‘language’ for describing phenomena ranging from geometry and analysis, physics and chemistry to engineering and economy. Central themes are the study of local properties of solutions and the passage from local considerations to global conclusions. The local regularity considerations of [1] imply regularity of solutions to the porous medium equation for large time without nondegeneracy conditions on the initial data. Carleman inequalities provide a robust alternative to monotonicity formulas. They provide an essential tool for the study of thin obstacles in the variable coefficient case [2]. Global existence and scattering for dispersive equations and the construction of conserved energies for the Korteweg-de Vries and the cubic nonlinear Schrödinger equation [3] are about global consequences of local properties. The self-similar solution to the generalized KdV equations [4] describe the structure of the blow-up.
The local analysis of three phase problems with triple lines of codimension 2 is challenging but it became accessible for the simplest model problems like thin obstacles. A first step consists in the proper linearization, and its connection to Calderón-Zygmund estimates. The quest for a more global understanding of dispersive waves is the driving motivation for many recent questions: Soliton resolution is a vague imprecise conjecture which I want to attack for the Korteweg-de Vries equation with PDE techniques. The Korteweg-de Vries equation is an asymptotic equation for water waves with finite depth. Global existence for small data may be within reach, with a study of the dynamics of the Gross-Pitaeviskii being an intermediate step.
|
|
DFG Collaborative Research Center SFB 611 “Singular Phenomena and Scaling in Mathematical Models”
Project leader, 2007 - 2012
DFG Collaborative Research Center SFB 1060 “The Mathematics of Emergent Effects”
Project leader, since 2013
|
|
[ 1] Clemens Kienzler, Herbert Koch, Juan Luis Vazquez
Flatness implies smoothness for solutions of the porous medium equation
arXiv , 1609.09048:
2016[ 2] Herbert Koch, Angkana Rüland, Wenhui Shi
The variable coefficient thin obstacle problem: Carleman inequalities
Adv. Math. , 301: : 820--866
2016
DOI: 10.1016/j.aim.2016.06.023[ 3] Herbert Koch, Daniel Tataru
Conserved energies for cubic NLS in 1-d
arXiv , 1607.02534:
2016[ 6] Herbert Koch, Hart F. Smith, Daniel Tataru
Sharp Lp bounds on spectral clusters for Lipschitz metrics
Amer. J. Math. , 136: (6): 1629--1663
2014
DOI: 10.1353/ajm.2014.0039[ 7] Jochen Denzler, Herbert Koch, Robert J. McCann
Higher-order time asymptotics of fast diffusion in Euclidean space: a dynamical systems approach
Mem. Amer. Math. Soc. , 234: (1101): vi+81
2015
ISBN: 978-1-4704-1408-5
DOI: 10.1090/memo/1101[ 8] Herbert Koch, Daniel Tataru
Energy and local energy bounds for the 1-d cubic NLS equation in H-\frac14
Ann. Inst. H. Poincaré Anal. Non Linéaire , 29: (6): 955--988
2012
DOI: 10.1016/j.anihpc.2012.05.006[ 9] Herbert Koch, Tobias Lamm
Geometric flows with rough initial data
Asian J. Math. , 16: (2): 209--235
2012
DOI: 10.4310/AJM.2012.v16.n2.a3[ 10] Martin Hadac, Sebastian Herr, Herbert Koch
Well-posedness and scattering for the KP-II equation in a critical space
Ann. Inst. H. Poincaré Anal. Non Linéaire , 26: (3): 917--941
2009
DOI: 10.1016/j.anihpc.2008.04.002[ 11] Herbert Koch, Daniel Tataru
Dispersive estimates for principally normal pseudodifferential operators
Comm. Pure Appl. Math. , 58: (2): 217--284
2005
DOI: 10.1002/cpa.20067 |
|
|
|
|
|
|
|
• Mathematische Annalen (2006 - 2014)
• Analysis and PDE (since 2008)
• SIAM Journal Mathematical Analysis (since 2012)
|
|
| 2005 | Miller professorship, Miller Institute, Berkeley, CA, USA |
|
|
| 2004 | International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, Madrid, Spain | | 2005 | Analyse des Equations aux Dérivées Partielles, Forges-Les-Eaux, France | | 2017 | DiPerna Lecture Berkeley, CA, USA |
|
|
Flavius Guias (2006), now Professor, Fachhochschule Dortmund (University of Applied Sciences and Arts)
Axel Grünrock (2011), now apl. Professor, University of Düsseldorf
|
|
Adina Guias (2005): “Eine analytische Methode zur Punktereduktion und Flächenrekonstruktion”,
now Teacher, Phoenix Gymnasium
Sebastian Herr (2006): “Well-Posedness Results for Nonlinear Dispersive Equations with Derivative Nonlinearities”,
now Professor, University of Bielefeld
Martin Hadac (2007): “On the Local Well-Posedness of the Kadomtsev-Petviashvili II Equation”,
now in Consulting
Tobias Schottdorf (2013): “Global Existence without decay”,
now working at Cockroach Labs
Clemens Kienzler (2013): “Flat fronts and stability for the porous medium equation”,
now working at McKinsey
Dominik John (2013): “Uniqueness and Stability near Stationary Solutions for the Thin-Film Equation in Multiple Space Dimensions with Small Initial Lipschitz Perturbations”,
now in Consulting
Stefan Steinerberger (2013): “Geometric structures arising from partial differential equations”,
now Assistant Professor, Yale University, CT, USA
Angkana Rüland (2014): “On some rigidity properties in PDEs”,
now leader of Max Planck Research Group at the Max Planck Institute for Mathematics in the Sciences, Leipzig
Habiba Kalantarova (2015): “Local Smoothing and Well-Posedness Results for KP-II Type Equations”,
now Associate Professor, Baku, Azerbaijan
Christian Zillinger (2015): “Linear Inviscid Damping for Monotone Shear Flows, Boundary Effects and Sharp Sobolev Regularity”,
now Postdoc, University of Southern California, Los Angeles, USA
|
|
- Master theses currently: 2
- Diplom theses: 4, currently 2
- PhD theses: 10, currently 7
|
|
Download Profile  |
