

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 [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 Kortewegde Vries and the cubic nonlinear Schrödinger equation [1] are about global consequences of local properties. The selfsimilar solution to the generalized KdV equations [6] describe the structure of the blowup.
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 CalderónZygmund 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 Kortewegde Vries equation with PDE techniques. The Kortewegde 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 GrossPitaeviskii 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


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 MongeAmpere type equation modeled after the BaoundiGrushin 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. 


[ 1] Herbert Koch, Daniel Tataru
Conserved energies for cubic NLS in 1d arXiv , 1609.02534: 2016[ 2] Herbert Koch, Hart F. Smith, Daniel Tataru
Sharp L^{p} bounds on spectral clusters for Lipschitz metrics Amer. J. Math. , 136: (6): 16291663 2014[ 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
Higherorder time asymptotics of fast diffusion in Euclidean space: a dynamical systems approach Mem. Amer. Math. Soc. , 234: (1101): vi+81 2015 ISBN: 9781470414085[ 5] Herbert Koch, Angkana Rüland, Wenhui Shi
The variable coefficient thin obstacle problem: Carleman inequalities Adv. Math. , 301: : 820866 2016[ 6] Herbert Koch
Selfsimilar solutions to supercritical gKdV Nonlinearity , 28: (3): 545575 2015[ 7] Herbert Koch, Daniel Tataru
Energy and local energy bounds for the 1d cubic NLS equation in H^{\frac14} Ann. Inst. H. Poincaré Anal. Non Linéaire , 29: (6): 955988 2012[ 8] Herbert Koch, Tobias Lamm
Geometric flows with rough initial data Asian J. Math. , 16: (2): 209235 2012[ 9] Martin Hadac, Sebastian Herr, Herbert Koch
Wellposedness and scattering for the KPII equation in a critical space Ann. Inst. H. Poincaré Anal. Non Linéaire , 26: (3): 917941 2009[ 10] Herbert Koch, Daniel Tataru
Dispersive estimates for principally normal pseudodifferential operators Comm. Pure Appl. Math. , 58: (2): 217284 2005





• 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, ForgesLesEaux, 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): “WellPosedness Results for Nonlinear Dispersive Equations with Derivative Nonlinearities”,
now Professor, University of Bielefeld
Martin Hadac (2007): “On the Local WellPosedness of the KadomtsevPetviashvili 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 ThinFilm 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 WellPosedness Results for KPII 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


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