# Daniel Tataru

 Department of Mathematics University  of California, Berkeley Berkeley, CA 94720, USA

Office: Evans Hall 841
Phone: (510) 643-1284
Email: tataru@math.berkeley.edu

Current and upcoming events:
 - Mondays 4:00-5:00: Analysis and PDE seminar - Tuesdays 3:30-5:00: HADES graduate seminar

Research Page

This page contains  a list of selected publications with some brief abstracts and also preprints that you can download in dvi, postscript or pdf format. Comments are greatly appreciated. This material is based upon work supported by the National Science Foundation, as well as by the Simons Foundation under various grants.

One of my interests earlier on was in L2 Carleman estimates  and unique continuation for pde's, as well as their applications in control theory. I have collected together those papers in the Unique  Continuation Page. Other more recent papers on unique continuation dealing with Lp Carleman estimates are listed below.

The notes from the Oct 2012 Oberwolfach Seminaire are here, or you can get the entire book at Amazon.

Following are some selected published papers, as well as all the new preprints. I have tried to loosely and imperfectly organize them by topic, though several would fit in more than one class. The topics are as follows:

Semilinear dispersive problems.

 On the optimal local regularity for Yang-Mills equations in  R4+1 with Sergiu Klainerman J. Amer. Math. Soc. 12 (1999), no. 1, 93--116. pdf On the equation $Box$u=| ∇ u|2 in 5+1 dimensions. Math. Res. Lett. 6 (1999), no. 5-6, 469--485 pdf Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation. Trans. Amer. Math. Soc. 353 (2001), no. 2, 795--807 Journal A-Priori Bounds for the 1-D Cubic NLS in Negative Sobolev Spaces with Herbert Koch IMRN 2007, no. 16, Art. ID rnm053, 36 pp. arXiv:math/0612717v2 Global well-posedness of the KP-I initial-value problem in the energy space with Alexandru Ionescu and Carlos Kenig Invent. Math. 173 (2008), no. 2, 265--304. arXiv:0705.4239 Global wellposedness in the energy space for the Maxwell-Schrödinger system with Ioan Bejenaru Comm. Math. Phys. 288 (2009), no. 1, 145--198. On the 2D Zakharov system with L2 Schrödinger data. with Ioan Bejenaru, Sebastian Herr and Justin Holmer Nonlinearity 22 (2009), no. 5, 1063--1089. A convolution estimate for two-dimensional hypersurfaces with Ioan Bejenaru and Sebastian Herr Rev. Mat. Iberoam. 26 (2010), no. 2, 707-728 Global well-posedness of the energy critical Nonlinear Schroedinger equation with small initial data in H^1(T^3) with Sebastian Herr, Nikolay Tzvetkov (2010) Duke Math. J. 159 (2011), no. 2, 329-349. Strichartz estimates for partially periodic solutions to Schroedinger equations in 4d and applications with Sebastian Herr, Nikolay Tzvetkov (2010) J. Reine Angew. Math. 690 (2014) Energy and local energy bounds for the 1-D cubic NLS equation in H^{-1/4} with Herbert Koch (2010) Ann. Inst. H. Poincare Anal. Non Lineaire 29 (2012), no. 6, 955-988. Low regularity bounds for mKdV with Michael Christ and Justin Holmer (2012) Lib. Math. (N.S.) 32 (2012), no. 1, 51-75. Local wellposedness of Chern-Simons-Schroedinger with Baoping Liu and Paul Smith (2012) Int. Math. Res. Not. IMRN 2014, no. 23, 6341-6398.

Carleman estimates and unique continuation

 Unique continuation for operators with partially analytic coefficients J. Math. Pures Appl. (9) 78 (1999), no. 5, 505--521. pdf Carleman estimates and unique continuation for second-order elliptic equations with nonsmooth coefficients with Herbert Koch Comm. Pure Appl. Math. 54 (2001), no. 3, 339--360. pdf Sharp counterexamples in unique continuation for second order elliptic equations with Herbert Koch J. Reine Angew. Math. 542 (2002), 133--146. pdf Dispersive estimates for principally normal pseudodifferential operators. Comm. Pure Appl. Math. 58 (2005), no. 2, 217--284. Lp eigenfunction bounds for the Hermite operator. with Herbert Koch Duke Math. J. 128 (2005), no. 2, 369--392. arXiv:math/0402261 Carleman estimates and absence of embedded eigenvalues. with Herbert Koch Comm. Math. Phys. 267 (2006), no. 2, 419--449. arXiv:math-ph/0508052 Carleman estimates and unique continuation for second order parabolic equations with nonsmooth coefficients with Herbert Koch Comm. Partial Differential Equations 34 (2009), no. 4-6, 305--366. arXiv:0704.1349 Uniqueness in Calderon's problem with Lipschitz conductivities with Boaz Haberman (2011) Duke Math. J. 162 (2013), no. 3, 496-516.

Wave packets and nonlinear waves

 Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation Amer. J. Math. 122 (2000), no. 2, 349--376 pdf Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. III. J. Amer. Math. Soc. 15 (2002), no. 2, 419--442 Journal Sharp counterexamples for Strichartz estimates for low regularity metrics. with Hart Smith Math. Res. Lett. 9 (2002), no. 2-3, 199--204. pdf Strichartz estimates for a Schrödinger operator with nonsmooth coefficients. with Gigliola Staffilani Comm. Partial Differential Equations 27 (2002), no. 7-8, 1337--1372. pdf On the Fefferman-Phong inequality and related problems. Comm. Partial Differential Equations 27 (2002), no. 11-12, 2101--2138 pdf Nonlinear wave equations. Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), 209--220 arXiv:math/0304397 Null form estimates for second order hyperbolic operators with rough coefficients Harmonic analysis at Mount Holyoke (South Hadley, MA, 2001), 383--409, Contemp. Math., 320 pdf Phase space transforms and microlocal analysis. Phase space analysis of partial differential equations. Vol. II, 505--524, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, 2004. pdf Dispersive estimates for wave equations with Dan Geba Comm. Partial Differential Equations 30 (2005), no. 4-6, 849--880. pdf Sharp local well-posedness results for the nonlinear wave equation. with Hart Smith Ann. of Math. (2) 162 (2005), no. 1, 291--366. dvi-letter, ps-letter Sharp Lq bounds on special clusters for Holder metrics. with Herbert Koch and Hart Smith Math. Res. Lett. 14 (2007), no. 1, 77--85 pdf Semiclassical Lp estimates with Herbert Koch and Maciej Zworski Ann. Henri Poincaré 8 (2007), no. 5, 885--916 arXiv:math-ph/0603080 A phase space transform adapted to the wave equation with Dan Geba Comm. PDE 32 (2007), no. 7-9, 1065--1101. pdf Large data  local solutions for the derivative NLS equation with Ioan Bejenaru J. Eur. Math. Soc. (JEMS) 10 (2008), no. 4, 957--985 Wave packet parametrices for evolutions governed by pdo's with rough symbols with Jeremy Marzuola and Jason Metcalfe Proc. Amer. Math. Soc. 136 (2008), no. 2, 597--604 Subcritical Lp bounds on spectral clusters for Lipschitz metrics with Herbert Koch and Hart Smith Math. Res. Lett. 14 (2007), no. 1, 77--85. arXiv:0709.2764 Gradient NLW on curved background in 4 + 1 dimensions with Dan Geba Int. Math. Res. Not. IMRN 2008, Art. ID rnn 108, 58 pp. arXiv:0802.3870 Quasilinear Schroedinger equations I: Small data and quadratic interactions with Jeremy L. Marzuola and Jason Metcalfe (2011) Adv. Math. 231 (2012), no. 2, 1151-1172. Sharp L^p bounds on spectral clusters for Lipschitz metrics with Herbert Koch and Hart Smith (2012) Amer. J. Math. 136 (2014), no. 6, 1629-1663. Quasilinear Schroedinger equations II: Small data and cubic nonlinearities with Jeremy Marzuola and Jason Metcalfe (2012) Kyoto J. Math. 54 (2014), no. 3, 529-546.

Wave Maps

 Local and global results for wave maps. I Comm. Partial Differential Equations 23 (1998), no. 9-10, 1781--1793 pdf On global existence and scattering for the wave maps equation Amer. J. Math. 123 (2001), no. 3, 385--423. pdf The wave maps equation Bull. Amer. Math. Soc. (N.S.) 41 (2004), no. 2, 185--204 pdf Journal Rough solutions for the wave maps equation Amer. J. Math. 127 (2005), no. 2, 293--377. pdf Energy dispersed large data wave maps in 2+1 dimensions with Jacob Sterbenz Comm. Math. Phys. 298 (2010), no. 1, 139-230 Regularity of Wave-Maps in dimension 2+1 with Jacob Sterbenz Comm. Math. Phys. 298 (2010), no. 1, 231-264

Schroedinger Maps.

 Global Schrödinger maps in dimension two and higher: small data in the critical Sobolev spaces with Ioan Bejenaru, Alexandru Ionescu and Carlos Kenig (2008) Ann. of Math. (2) 173 (2011), no. 3, 1443-1506 Equivariant Schroedinger Maps in two spatial dimensions with Ioan Bejenaru, Alexandru Ionescu and Carlos Kenig (2011) Duke Math. J. 162 (2013), no. 11 Equivariant Schroedinger Maps in two spatial dimensions: the H^2 target with Ioan Bejenaru, Alexandru Ionescu and Carlos Kenig (2012) Kyoto J. Math. 56 (2016), no. 2, 283-323.

Near soliton evolutions in geometric dispersive flows.

 Renormalization and blow up for charge one equivariant critical wave maps with Joachim Krieger and Wilhelm Schlag Invent. Math. 171 (2008), no. 3, 543--615. arXiv:math/0610248 Slow blow-up solutions for the H1(R3) critical focusing semi-linear wave equation with Joachim Krieger and Wilhelm Schlag Duke Math. J. 147 (2009), no. 1, 1--53. arXiv:math/0702033 Renormalization and blow up for the critical Yang-Mills problem. with Joachim Krieger and Wilhelm Schlag Adv. Math. 221 (2009), no. 5, 1445--1521 Near soliton evolution for equivariant Schroedinger Maps in two spatial dimensions with Ioan Bejenaru (2010) Mem. Amer. Math. Soc. 228 (2010), no. 1069 A codimension two stable manifold of near soliton equivariant wave maps with Ioan Bejenaru and Joachim Krieger (2011) Anal. PDE 6 (2013), no. 4, 829-857.

Decay of linear waves on asymptotically flat space times. Some of this is GR based/motivated.

 Parametrices and dispersive estimates for Schrödinger operators with variable coefficients Amer. J. Math. 130 (2008), no. 3, 571--634. pdf Local decay of waves on asymptotically flat stationary space-times (2009) Amer. J. Math. 135 (2013), no. 2, 361-401. Strichartz estimates and local smoothing estimates for asymptotically flat Schrödinger equations with Jeremy Marzuola and Jason Metcalfe J. Funct. Anal. 255 (2008), no. 6, 1497--1553. arXiv:0706.0544 Global  parametrices and dispersive estimates for variable coefficients wave equations with Jason Metcalfe Math. Ann. 353 (2012), no. 4, 1183-1237 arXiv:0707.1191 Decay estimates for variable coefficient wave equations in exterior domains with Jason Metcalfe Advances in phase space analysis of partial differential equations, 201-216, Progr. Nonlinear Differential Equations Appl., 78 Strichartz estimates on Schwarzschild black hole backgrounds with Jeremy Marzuola, Jason Metcalfe and Mihai Tohaneanu Comm. Math. Phys. 293 (2010), no. 1, 37--83. arXiv:0802.3942 A local energy estimate on Kerr black hole backgrounds with Mihai Tohaneanu (2008) IMRN 2011, no. 2, 248â€“292 Price's Law on Nonstationary Spacetimes with Jason Metcalfe and Mihai Tohaneanu (2011) Adv. Math. 230 (2012), no. 3, 995-1028. Local energy decay for Maxwell fields part I: Spherically symmetric black-hole backgrounds with Jacob Sterbenz (2013) Int. Math. Res. Not. IMRN 2015, no. 11, 3298-3342. Pointwise decay for the Maxwell field on black hole space-times with Jason Metcalfe and Mihai Tohaneanu (2014) Adv. Math. 316 (2017), 53-93. Local energy decay for scalar fields on time dependent non-trapping backgrounds with Jason Metcalfe, Jacob Sterbenz (2017) preprint

Geometric nonlinear waves. Primarily MKG and YM.

 Global well-posedness for the Maxwell-Klein Gordon equation in 4+1 dimensions. Small energy with Joachim Krieger and Jacob Sterbenz (2012) Duke Math. J. 164 (2015), no. 6, 973-1040. Local well-posedness of the (4+1)-dimensional Maxwell-Klein-Gordon equation at energy regularity with Sung-Jin Oh (2015) Ann. PDE 2 (2016), no. 1, Art. 2, 70 Energy dispersed solutions for the (4+1)-dimensional Maxwell-Klein-Gordon equation with Sung-Jin Oh (2015) AJM, to appear Global well-posedness and scattering of the (4+1)-dimensional Maxwell-Klein-Gordon equation with Sung-Jin Oh (2015) Invent. Math. 205 (2016), no. 3, 781-877. Global well-posedness for the Yang-Mills equation in 4+1 dimensions. Small energy with Joachim Krieger Ann. of Math. (2) 185 (2017), no. 3, 831-893 (2015) The Yang--Mills heat flow and the caloric gauge with Sung-Jin Oh (2017) preprint The hyperbolic Yang--Mills equation in the caloric gauge. Local well-posedness and control of energy dispersed solutions with Sung-Jin Oh (2017) preprint The hyperbolic Yang--Mills equation for connections in an arbitrary topological class with Sung-Jin Oh (2017) preprint The threshold conjecture for the energy critical hyperbolic Yang--Mills equation with Sung-Jin Oh (2017) preprint The Threshold Theorem for the (4+1)-dimensional Yang--Mills equation: an overview of the proof with Sung-Jin Oh (2017)

Fluid dynamics. This also includes model problems arising in the study of various fluid models.

 Well-posedness for the Navier-Stokes equations. with Herbert Koch Adv. Math. 157 (2001), no. 1, 22--35. pdf Long time Solutions for a Burgers-Hilbert Equation via a Modified Energy Method with John K. Hunter, Mihaela Ifrim and Tak Kwong Wong (2013) Proc. Amer. Math. Soc. 143 (2015), no. 8, 3407-3412. Two dimensional water waves in holomorphic coordinates with John K. Hunter and Mihaela Ifrim (2013) Comm. Math. Phys. 346 (2016), no. 2, 483-552. Global bounds for the cubic nonlinear Schrödinger equation (NLS) in one space dimension with Mihaela Ifrim (2014) Nonlinearity 28 (2015), no. 8, 2661-2675. Two dimensional water waves in holomorphic coordinates II: global solutions with Mihaela Ifrim (2014) Bull. Soc. Math. France 144 (2016), no. 2, 369-394. The lifespan of small data solutions in two dimensional capillary water waves with Mihaela Ifrim (2014) Arch. Ration. Mech. Anal. 225 (2017), no. 3, 1279-1346. The lifespan of small data solutions to the KP-I with Benjamin Harrop-Griffiths and Mihaela Ifrim (2014) Int. Math. Res. Not. IMRN 2017, no. 1, 1-28. Two dimensional gravity water waves with constant vorticity: I. Cubic lifespan with Mihaela Ifrim (2015) preprint Finite depth gravity water waves in holomorphic coordinates with Mihaela Ifrim (2016) Ann. PDE 3 (2017), no. 1, Art. 4, 102 pp. Well-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation with Mihaela Ifrim (2017) Ann. Sci. Ecole Norm. S, to appear;

Integrable models and inverse scattering. This portion only includes work on integrable models which involves inverse scattering in one way or another. However, integrable models occur also as part of the other projects, in particular as water wave models in fluids.

 Conserved energies for the cubic NLS in 1-d with Herbert Koch (2016) preprint A Nonlinear Plancherel Theorem with Applications to Global Well-Posedness for the Defocusing Davewy-Stewartson Equation and to the Inverse Boundary Value Problem of Calderon with Adrian I. Nachman, Idan Regev (2017) preprint;