Profile
Profile

Prof. Dr. Herbert Koch

E-mail: koch(at)math.uni-bonn.de
Phone: +49 228 73 3787
Fax: +49 228 737658
Homepage: http://www.math.uni-bonn.de/people/koch/
Room: 2.011
Location: Mathematics Center
Institute: Mathematical Institute
Research Areas: Research Area A (Leader)
Research Area B
Date of birth: 14.Sep 1962
Mathscinet-Number: 340038

Academic Career

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

Research Profile

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 [3] 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 [5]. Global existence and scattering for dispersive equations and the construction of conserved energies for the Korteweg-de Vries and the cubic nonlinear Schrödinger equation [1] are about global consequences of local properties. The self-similar solution to the generalized KdV equations [6] 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.

Research Projects and Activities

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

Contribution to Research Areas

Research Area A
The cubic nonlinear Schrödinger equation describes interacting nonlinear waves for nonlinear optics and water waves in certain asymptotic regimes. In [1] we construct a continuous family of conserved energies leading to uniform in time estimates in a large range of fractional Sobolev and Besov spaces. This construction relies on the Lax pair, without decay assumptions on the initial data, which however would be needed for the inverse scattering machinery.

A partial hodograph transform changes the thin obstacle problem at a regular point of the codimension two free boundary into a degenerate Monge-Ampere type equation modeled after the Baoundi-Grushin operator. In a series of papers we explored this connection and combined it with Carleman estimates to obtain smoothness of the free boundary of codimension two.

Multiple length and frequency scale, and their interaction are behind the estimates of eigenfunctions for Lipschitz coefficients in [2]: For a fixed frequency there is the small dispersive length scale, and the much larger length scale for energy estimates.
Research Area B
The flow of gas through a porous medium described by the porous medium equation is a long studied test problem. In [3] we studied smoothness of flat weak solutions, with large time regularity and asymptotics as main consequence, for the first time without regularity or nondegeneracy assumptions on the initial data. With Denzler and McCann [4] we obtained the detailed asymptotics for the fast diffusion equation, again with the suprising geometry of the linearization playing a decisive role.

Selected Publications

[1] Herbert Koch, Daniel Tataru
Conserved energies for cubic NLS in 1-d
arXiv , 1609.02534:
2016
[2] 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
[3] Clemens Kienzler, Herbert Koch, Juan Luis Vazquez
Flatness implies smoothness for solutions of the porous medium equation
arXiv , 1609.09048:
2016
[4] 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
[5] 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
[6] Herbert Koch
Self-similar solutions to super-critical gKdV
Nonlinearity , 28: (3): 545--575
2015
DOI: 10.1088/0951-7715/28/3/545
[7] 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
[8] 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
[9] 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
[10] 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

Publication List

Editorships

• Mathematische Annalen (2006 - 2014)
• Analysis and PDE (since 2008)
• SIAM Journal Mathematical Analysis (since 2012)

Awards

2005

Miller professorship, Miller Institute, Berkeley, CA, USA

Selected Invited Lectures

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

Habilitations

Flavius Guias (2006), now Professor, Fachhochschule Dortmund (University of Applied Sciences and Arts)

Axel Grünrock (2011), now Priv.-Doz., University of Düsseldorf

Selected PhD students

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”

Clemens Kienzler (2013): “Flat fronts and stability for the porous medium equation”,
now 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 Postdoc, Yale University, CT, USA

Angkana Rüland (2014): “On some rigidity properties in PDEs”,
now Postdoc, Oxford University, England, UK

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

Supervised Theses

  • Master theses currently: 2
  • Diplom theses: 4, currently 2
  • PhD theses: 10, currently 7
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