Prof. Dr. Herbert Koch - Publications



  1. M. Ifrim, H. Koch, D. Tataru. Dipersive decay of small data solutions for the KdV equation. (2019) arXiv:1901.05934
  2. H. Koch, X. Liao Conserved energies for the one dimensional Gross-Pitaevskii equation: small energy case. (2018) arXiv:1801.08386
  3. H. Koch, D. Tataru Conserved energies for the cubic NLS in 1-d Duke Math. J. 167 no. 17, 3207-3313 (2018)
  4. H. Koch, M. Gubinelli, T. Oh. Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity. arXiv:1811.07808
  5. H. Koch, M. Gubinelli, T. Oh. Renormalization of the two-dimensional stochastic nonlinear wave equations. Trans. Amer. Math. Soc. 370, no 10, 7335–7359 (2018).
  6. Kienzler, C., Koch, H., Vazquez, J. L. Flatness implies smoothness for solutions of the porous medium equation Calculus of Variations and PDE's 57, no 1, Art. 18, 42pp. (2018).
  7. H. Koch, P. Gérard. The cubic Szegő flow at low regularity. Séminaire Laurent Schwartz—Équations aux dérivées partielles et applications. Année 2016–2017, Exp. No. XIV, 14 pp., Ed. Éc. Polytech., Palaiseau (2017).
  8. H. Koch, P. Raphaël, D. Tataru, M. Visan Nonlinear Waves and Dispersive Equations. Oberwolfach Rep. 14, no. 2, 1681–1745 (2017).
  9. H. Koch, A. Rüland, W. Shi The Variable Coefficient Thin Obstacle Problem: Higher Regularity Advances in Differential Equations Volume 22, Numbers 11-12, 793-866 (2017).
  10. H. Koch, J. Li Global well posedness and scattering for small data for the 3-d KP-II Cauchy problem. Comm. PDE 42, no. 6, 950-976 (2017)
  11. H. Koch, A. Rüland, W. Shi The Variable Coefficient Thin Obstacle Problem: Optimal Regularity and Regularity of the Regular Free Boundary. Annales de l'Institut Henri Poincare 34, no. 4, 845-897 (2017)
  12. H. Koch, C.-L. Lin, J.-N. Wang Doubling inequalities for the Lamé system with rough coefficients. Proc. Amer. Math. Soc. 144, no. 12, 5309-5318 (2016)
  13. Denzler, J., H. Koch, McCann, R. J. Long-time asymptotic expansions for nonlinear diffusions in Euclidean space. Mathematical Congress of the Americas, 85–94, Contemp. Math., 656, Amer. Math. Soc., Providence, RI. (2016)
  14. H. Koch, N. Nadirashvili Partial analyticity and nodal sets for nonlinear elliptic systems. arXiv:1506.06224 (2015)
  15. H. Koch Convexity and concavity of the ground state energy. New York Journal of Mathematics 21, 1003-1005 (2015)
  16. H. Koch Self-similar solutions to super-critical gKdV. Nonlinearity 28, 545-575 (2015) http://stacks.iop.org/0951-7715/28/545
  17. H. Koch, A. Rüland, W. Shi The Variable Coefficient Thin Obstacle Problem: Carleman Inequalities. Adv. Math. 301 , 820–866 (2016)
  18. H. Koch, T. Lamm Parabolic equations with rough data. Mathematica Bohemica, vol 140, no. 4, pp. 457-477 (2015)
  19. H. Koch, D. Tataru, M. Visan Nonlinear Dispersive equations in Dispersive equations and nonlinear waves. Oberwolfach Seminars, Birkhäuser (2014)
  20. H. Koch Adapted Function Spaces for Dispersive Equations. Singular Phenomena and Scaling in Mathematical Models, Springer, 49-67 (2014)
  21. H. Koch, A. Petrosyan, W. Shi Higher regularity of the free bouondary in the elliptic Signorini problem. Nonlinear Analysis 126, 3-44 (2015) (2014)
  22. H. Koch, S. Steinerberger Convolution Estimates for Singular Measures and Some Global Nonlinear Brascamp-Lieb Inequalities. Procedings A of the Royal Sociecty of Edinburgh 145, no. 6, 1223-1237 (2015)
  23. H. Koch, P. Koskela, E. Saksman, T. Soto Bounded compositions on scaling invariant Besov spaces. J. Funct. Anal. 266.5, 2765-2788 (2014)
  24. H. Koch, W. Lück On the spectral density function of the Laplacian of a graph. Expositiones Mathematicae 32.2, 178-189 (2014)
  25. H. Koch, H. Smith, D. Tataru Sharp L^p bounds on spectral clusters for Lipschitz metrics. American Journal of Math 136, no. 6, 1629-1663 (2014)
  26. T. Buckmaster, H. Koch The Korteweg-de-Vries equation at H-1 regularity. Ann. Inst. H. Poincaré Anal. Non Linéaire 32, no. 5, 1071-1098 (2015)
  27. J. Denzler, H. Koch, R. McCann Higher-order time asymptotics of fast diffusion in Euclidean space: a dynamical systems approach. Memoirs of the AMS 234, no. 1101, vi+81 pp., ISBN 978-1-4704-1408-5 (2015) pdf
  28. H. Koch, D. Tataru. Energy and local energy bounds for the 1-D cubic NLS equation in H-1/4. Ann. Inst. H. Poincaré Anal. Non Linéaire 29, no. 6, 955-988 (2012)
  29. H. Koch, J. Marzuola. Small Data Scattering and Soliton Stability In H-1/6 for the Quartic KDV Equation. Anal. PDE 5, no. 1, 145-198 (2012)
  30. H. Koch, T. Lamm. Geometric Flows with Rough Initial Data. Asian J. Math. 16, no. 2, 209-235 (2012)
  31. S. Herr, A. Ionescu, D. Alexandru, C. E. Kenig and H. Koch. A para-differential renormalization technique for nonlinear dispersive equations. Comm. Partial Differential Equations 35, no. 10, 1827-1875 (2010)
  32. F. Colombini and H. Koch. Strong unique continuation for products of elliptic operators of second order. Trans. Amer. Math. Soc. 362, no. 1, 345-355 (2010)
  33. M. Hadac, S. Herr and H. Koch. Well-posedness and scattering for the KP-II equation in a critical space. Ann. Inst. H. Poincaré Anal. Non Linéaire 26, no. 3, 917-941 (2009). Erratum in Ann. Inst. H. Poincaré Anal. Non Linéaire 27, no. 3, 971-972 (2010)
  34. Nonlinear waves and dispersive equations. Abstracts from the workshop held September 12-18, 2010. Organized by Carlos E. Kenig, Herbert Koch and Daniel Tataru. Oberwolfach Reports. Vol. 7, no 3. Oberwolfach Rep. 7, no. 3, 2393-2463 (2010).
  35. N. Anantharaman, H. Koch and S. Nonnenmacher. Entropy of eigenfunctions. In New Trends in Mathematical Physics. Selected contributions of the XVth International Congress on Mathematical Physics. Sidoravicius, Vladas (Ed.), 1-22 (2009)
  36. H. Koch and D. Tataru. Carleman estimates and unique continuation for second order parabolic equations with nonsmooth coefficients. Comm. Partial Differential Equations 34, no 4-6, 305-366 (2009)
  37. H. Koch. Partial Differential Equations with Non-Euclidean Geometries. Discrete Contin. Dyn. Syst. Ser. S 1, no 3, 481-504 (2008).
  38. H. Koch, H. F. Smith and D. Tataru. Subcritical Lp bounds on spectral clusters for Lipschitz metrics. Math. Res. Lett. 15, no. 5, 993-1002 (2008).
  39. H. Koch and I. Lasiecka. Backward uniqueness in linear thermoelasticity with time and space variable coefficients. Functional analysis and evolution equations, 389-403, Birkhäuser, Basel (2008).
  40. H. Koch and N. Tzvetkov. On finite energy solutions of the KP-I equation. Math. Z. 258,no. 1, 55-68 (2008).
  41. Nonlinear waves and dispersive equations. Abstracts from the workshop held September 09-15, 2007. Organized by Carlos E. Kenig, Herbert Koch and Daniel Tataru. Oberwolfach Reports. Vol. 4, no 4. Oberwolfach Rep. 4, no. 4, 2609-2669 (2007).
  42. H. Koch, D. Tataru and M. Zworski. Semiclassical Lp estimates. Ann. Henri Poincaré 8, no. 5, 885-916 (2007)
  43. H. Koch and D. Tataru. A-priori bounds for the 1-D cubic NLS in negative Sobolev spaces. Int. Math. Res. Not., IMRN, Vol. 2007, no. 16, Art. ID rnm053, 36 pp (2007)
  44. J. Kato and H. Koch. Uniqueness of the modified Schrödinger map in H^{3/4+e}(R^2). Comm. Partial Differential Equations, 32 (1-3): 415-429 (2007).
  45. H. Koch and F. Ricci. Spectral projections for the twisted Laplacian. Studia Math, 180 (2): 103-110 (2007).
  46. H. Koch, H. F. Smith and D. Tataru. Sharp Lq bounds on spectral clusters for Hölder metrics. Math. Res. Lett. 14, No. 1, 77-85 (2007)
  47. H. Koch and J.-C. Saut. Local smoothing and local solvability for third order dispersive equations. SIAM J. Math. Anal. 38, No. 5, 1528-1541 (2006/07).
  48. H. Koch, E. Zuazua. A hybrid system of PDE's arising in multi-structure interaction: coupling of wave equations in n and n-1 space dimensions. Recent trends in partial differential equations, 55--77, Contemp. Math., 409, Amer. Math. Soc., Providence, RI, (2006).
  49. H. Koch and D. Tataru. Carleman estimates and absence of embedded eigenvalues. Comm. Math. Phys., 267, 419-449 (2006).
  50. H. Koch and D. Tataru. Dispersive estimates and absence of embedded eigenvalues. Journées "Équations aux Dérivées Partielles", Exp. No. VI, 10 pp., École Polytech., Palaiseau (2005).
  51. H. Koch, G. Leoni and M. Morini. On optimal regularity of free boundary problems and a conjecture of De Giorgi. Comm. Pure Appl. Math 58, No 8, 1051-1076 (2005).
  52. H. Koch and D. Tataru. Dispersive estimates for principally normal pseudodifferential operators. Comm. Pure Appl. Math. 58, No 2, 217-284 (2005).
  53. H. Koch and D. Tataru. Lp eigenfunction bounds for the Hermite operator. Duke Math. J. 128, No 2, 369-392 (2005).
  54. H. Koch and N. Tzvetkov. Nonlinear wave interactions for the Benjamin-Ono equation. Int. Math. Res. Not. 2005, no. 30, 1833-1847 (2005).
  55. Nonlinear waves and dispersive equations. Abstracts from the workshop held October 24-30, 2004. Organized by Carlos E. Kenig, Herbert Koch and Daniel Tataru. Oberwolfach Reports. Vol. 1, no 4. Oberwolfach Rep. 1, no. 4, 2653-2728 (2004).
  56. S. Benachour, H. Koch and P. Laurencot. Very singular solutions to a nonlinear parabolic equation with absorption. II: Uniqueness. Proc. R. Soc. Edinb., Sect. A, Math. 134, No 1, 39-54 (2004).
  57. H. Koch. Partial differential equations and singular integrals. Dispersive nonlinear problems in mathematical physics, 59-122, Quad. Mat., 15, Dept. Math., Seconda Univ. Napoli, Caserta (2004).
  58. M. Ben-Artzi, H. Koch and J.-C. Saut. Dispersion estimates for third order equations in two dimensions. Comm. Partial Differential Equations 28, No 11-12, 1943-1974 (2003).
  59. H. Koch and D. Tataru. Dispersive estimates for principally normal operators and applications to unique continuation. F. Colombini (ed.) et al., Hyperbolic problems and related topics. Proceedings of the conference, Cortona, Italy, September 10-14, 2002. Somerville, MA: International Press. Grad. Ser. Anal., 201-217 (2003).
  60. H. Koch and N. Tzvetkov. On the local well-posedness of the Benjamin-Ono equation in Hs(R), Int. Math. Res. Not. 2003, no. 26, 1449-1464, (2003).
  61. H. Koch and V. A. Solonnikov. Lq-estimates of the first-order derivatives of solutions to the nonstationary Stokes problem. M. Sh. Birman (ed.) et al., Nonlinear problems in mathematical physics and related topics I. In honor of Professor O. A. Ladyzhenskaya. New York, NY: Kluwer Academic/Plenum Publishers. Int. Math. Ser., N.Y. 1, 203-218 (2002).
  62. H. Koch and W. Sickel: Pointwise multipliers of Besov spaces of smoothness zero and spaces of continuous functions. Rev. Mat. Iberoamericana 18, No 3, 587-626 (2002).
  63. H. Koch: Transport and instability for perfect fluids. Math. Ann. 323, No 3, 491-523 (2002).
  64. H. Koch and D. Tataru: Sharp counterexamples in unique countinuation for second order elliptic equations. J. Reine Angew. Math. 542, 133-146 (2002).
  65. H. Koch and I. Lasiecka. Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems. A. Lorenzi (ed.) et al., Evolution equations, semigroups and functional analysis. In memory of B. Terreni. Containing papers of the conference, Milano, Italy, September 27-28, 2000. Basel: Birkhäuser. Progr. Nonlinear Differential Equations Appl. 50, 197-216 (2002).
  66. H. Koch and V. A. Solonnikov. Lq-Estimates for a solution to the nonstationary Stokes equation. Function theory and phase transitions. J. Math. Sci. (New York) 106, No 3, 3042-3072 (2001).
  67. H. Koch and D. Tataru. Carleman estimates and unique continuation for second-order elliptic equations with nonsmooth coefficients. Comm. Pure Appl. Math. 54, No 3, 339-360 (2001).
  68. H. Koch and D. Tataru: Well-posedness for the Navier-Stokes equations. Adv. Math. 157, No 1, 22-35 (2001) .
  69. H. Koch and D. Tataru: Recent results on unique continuation for second order elliptic equations. F. Colombini (ed.) et al., Carleman estimates and applications to uniqueness and control theory, Cortona (1999). Birkhäuser. Boston, Boston, MA, Prog. Nonlinear Differential Equations Appl., 46, 73-84 (2001).
  70. H. Koch: Differentiability of parabolic semi-flows in Lp-spaces and inertial manifolds J. Dynam. Differential Equations 12, No 3, 511-531 (2000).
  71. H. Koch: Slow decay in linear thermoelasticity. Quart. Appl. Math., 58, No 4, 601-612 (2000).
  72. C. Kenig, H. Koch, J. Pipher and T. Toro: A new approach to absolute continuity of elliptic measure, with applications to non-symmetric equations. Adv. Math. 153, No 2, 231-298 (2000).
  73. H. Koch and S. S. Antman: Stability and Hopf bifurcation for fully nonlinear parabolic-hyperbolic equations. SIAM J. Math. Anal. 32, No 2, 360-384 (2000).
  74. S. S. Antman and H. Koch: Self-sustained oscillations of nonlinearly viscoelastic layers. SIAM J. Appl. Math. 60, No 4, 1357-1387 (2000) (electronic).
  75. M. Ben-Artzi, J.-C. Saut and H. Koch: Dispersion estimates for fourth order Schrödinger equations. C.R. Acad. Sci. Paris, Sér. I, Math. 330, No 2, 87-92, 2000.
  76. H. Koch: Non-Euclidean singular integrals and the porous medium equation. Habilitation thesis (1999).
  77. M. Ben-Artzi and H. Koch: Decay of mass for a semilinear parabolic equation. Comm. Partial Differential Equations, 24, No 5-6, 869-881, (1999).
  78. H. Koch: Instability for incompressible and inviscid fluids. W. Jäger. (ed.) et al., Partial differential equations: theory and numerical solution. Proceedings of the ICM'98 satellite conference, Prague, Czech Republic, August 10-16, 1998. Boca Raton, FL: Chapman & Hall/CRC. Chapman Hall/CRC Res. Notes Math. 406, 240-247 (2000).
  79. H. Koch: Classical solutions to phase transition problems are smooth. Comm. Partial Differential Equations 23, No 3-4, 389-437 (1998).
  80. U. Bunke, H. Koch: The Etaform and a generalized Maslov index. Manuscripta Math. 95, No 2, 189-212 (1998)
  81. H. Koch: On center manifolds. Nonlinear Anal., Theory Methods Appl. 28, No 7, 1227-1248 (1997).
  82. H. Koch: Global classical solutions to a two phase Stefan problem. M. Demuth (ed.) et al., Differential equations, asymptotic analysis, and mathematical physics. Papers associated with the International Conference on Partial Differential Equations, Potsdam, Germany, June 29-July2, 1996. Akademie Verlag Math. Res 100, 181-183 (1997).
  83. H. Koch: Finite dimensional aspects of semilinear parabolic equations. J. Dynamics Diff. Equations 8, No 2, 177-202 (1996).
  84. H. Koch: On a fully nonlinear mixed parabolic problem with oblique boundary condition. Preprint, SFB 359, Heidelberg, 1995.
  85. J. Cooper and H. Koch: Remarks on the spectrum of a linear wave operator with time periodic boundary condition. A. C. McBride (ed.) et al., Recent developments in evolution equations. Proceedings of a meeting held at the University of Strathclyde, UK, 25-29 July, 1994. Harlow: Longman Scientific & Technical. Pitman Res. Notes Math. Ser. 324, 94-99 (1995).
  86. J. Cooper and H. Koch: The spectrum of a hyperbolic evolution operator. J. Funct. Anal., 133, No 2, 301-328,(1995).
  87. H. Koch and D. Tataru: On the spectrum of hyperbolic semigroups. Comm. Partial Differential Equations, 20, No 5-6, 901-937 (1995).
  88. H. Koch and A. Stahel: Global existence of classical solutions to the dynamical von Kármán equations. Math. Methods Appl. Sci. 16, No 8, 581-586 (1993).
  89. H. Koch: Mixed problems for fully nonlinear hyperbolic equations. Math. Z., 214, No 1, 9-42 (1993).
  90. H. Koch: Small periodic solutions of quasilinear hyperbolic equations. C. Perello (ed.) et al., International Conference on Differential Equations. Vol. 1, 2. Proceedings of the conference, EQUADIFF 91, Barcelona, Spain, August 26-31 (1991), World Sci. Publ., River Edge, NJ, 638-644 (1993)
  91. H. Koch: Hyperbolic equations of second order. Heidelberg: Univ. Heidelberg, Naturwiss.-Math. Fak., Thesis. 104p. (1990).

Lecture notes




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