Daniel Tataru



My picture
Address:
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


 Teaching        Biographical Information       Links

 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.
arXiv:0712.0098
On the 2D Zakharov system with L2 Schrödinger data. with Ioan Bejenaru, Sebastian Herr and Justin Holmer
Nonlinearity 22 (2009), no. 5, 1063--1089.
arXiv:0811.3047
A convolution estimate for two-dimensional hypersurfaces with Ioan Bejenaru and Sebastian Herr
Rev. Mat. Iberoam. 26 (2010), no. 2, 707-728
arXiv:0809.5091
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.
arXiv:1005.2832
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)
arXiv:1011.0591
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.
arXiv:1012.0148
Low regularity bounds for mKdV with Michael Christ and Justin Holmer
(2012) Lib. Math. (N.S.) 32 (2012), no. 1, 51-75.
arXiv:1207.6738
Local wellposedness of Chern-Simons-Schroedinger with Baoping Liu and Paul Smith
(2012) Int. Math. Res. Not. IMRN 2014, no. 23, 6341-6398.
arXiv:1212.1476

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. arXiv:math/0401234
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.
arXiv:1108.6068

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
arXiv:math/0610092
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
arXiv:math/0611252
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.
arXiv:1106.0490
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.
arXiv:1207.2417
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.
arXiv:1208.0544

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
arXiv:0810.5766
Regularity of Wave-Maps in dimension 2+1 with Jacob Sterbenz
Comm. Math. Phys. 298 (2010), no. 1, 231-264
arXiv:0907.3148

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
arXiv:0807.0265v1
Equivariant Schroedinger Maps in two spatial dimensions with Ioan Bejenaru, Alexandru Ionescu and Carlos Kenig
(2011) Duke Math. J. 162 (2013), no. 11
arXiv:1112.6122
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.
arXiv:1212.2566

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
arXiv:0809.2114
Near soliton evolution for equivariant Schroedinger Maps in two spatial dimensions with Ioan Bejenaru
(2010) Mem. Amer. Math. Soc. 228 (2010), no. 1069
arXiv:1009.1608
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.
arXiv:1109.3129

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.

arXiv:0910.5290
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
arXiv:0806.3409
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, 248–292
arXiv:0810.5766
Price's Law on Nonstationary Spacetimes with Jason Metcalfe and Mihai Tohaneanu
(2011) Adv. Math. 230 (2012), no. 3, 995-1028.
arXiv:1104.5437
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.
arXiv:1305.5261
Pointwise decay for the Maxwell field on black hole space-times with Jason Metcalfe and Mihai Tohaneanu
(2014) Adv. Math. 316 (2017), 53-93.
arXiv:1411.3693
Local energy decay for scalar fields on time dependent non-trapping backgrounds with Jason Metcalfe, Jacob Sterbenz
(2017) preprint
arXiv:1703.08064

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.
arXiv:1211.3527
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
arXiv:1503.01560
Energy dispersed solutions for the (4+1)-dimensional Maxwell-Klein-Gordon equation with Sung-Jin Oh
(2015) AJM, to appear
arXiv:1503.01561
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.
arXiv:1503.01562
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)
arXiv:1509.00751
The Yang--Mills heat flow and the caloric gauge with Sung-Jin Oh
(2017) preprint
arXiv:1709.08599
The hyperbolic Yang--Mills equation in the caloric gauge. Local well-posedness and control of energy dispersed solutions with Sung-Jin Oh
(2017) preprint
arXiv:1709.09332
The hyperbolic Yang--Mills equation for connections in an arbitrary topological class with Sung-Jin Oh
(2017) preprint
arXiv:1709.08604
The threshold conjecture for the energy critical hyperbolic Yang--Mills equation with Sung-Jin Oh
(2017) preprint
arXiv:1709.08606
The Threshold Theorem for the (4+1)-dimensional Yang--Mills equation: an overview of the proof with Sung-Jin Oh
(2017)
arXiv:1709.09088

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.
arXiv:1301.1947
Two dimensional water waves in holomorphic coordinates with John K. Hunter and Mihaela Ifrim
(2013) Comm. Math. Phys. 346 (2016), no. 2, 483-552.
arXiv:1401.1252
Global bounds for the cubic nonlinear Schrödinger equation (NLS) in one space dimension with Mihaela Ifrim
(2014) Nonlinearity 28 (2015), no. 8, 2661-2675.
arXiv:1404.7581
Two dimensional water waves in holomorphic coordinates II: global solutions with Mihaela Ifrim
(2014) Bull. Soc. Math. France 144 (2016), no. 2, 369-394.
arXiv:1404.7583
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.
arXiv:1406.5471
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.
arXiv:1409.4487
Two dimensional gravity water waves with constant vorticity: I. Cubic lifespan with Mihaela Ifrim
(2015) preprint
arXiv:1510.07732
Finite depth gravity water waves in holomorphic coordinates with Mihaela Ifrim
(2016) Ann. PDE 3 (2017), no. 1, Art. 4, 102 pp.
arXiv:1607.02409
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;
arXiv:1701.08476

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
arXiv:1607.02534
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;
arXiv:1708.04759