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Current and upcoming events:

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.
The notes from the Oct 2022 Oberwolfach Seminaire are available from Springer. The lectures from the 2020 summer school "Introduction to water waves", jointly coorganized with Mihaela Ifrim, are available from SLMATH (MSRI). Also more MSRI lectures can be found here.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 YangMills equations in R^{4+1}  with Sergiu Klainerman J. Amer. Math. Soc. 12 (1999), no. 1, 93116. 

On the equation $Box$u= ∇ u^{2} in 5+1 dimensions.  Math. Res. Lett. 6 (1999), no. 56, 469485  
Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation.  Trans. Amer. Math. Soc. 353 (2001), no. 2, 795807  Journal 
APriori Bounds for the 1D
Cubic NLS in Negative Sobolev Spaces 
with Herbert Koch IMRN 2007, no. 16, Art. ID rnm053, 36 pp. 
arXiv:math/0612717v2 
Global wellposedness of the
KPI initialvalue problem in the energy space 
with Alexandru Ionescu and
Carlos Kenig Invent. Math. 173 (2008), no. 2, 265304. 
arXiv:0705.4239 
Global wellposedness in the energy space for the MaxwellSchrödinger system  with Ioan Bejenaru Comm. Math. Phys. 288 (2009), no. 1, 145198. 
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, 10631089. 
arXiv:0811.3047 
A convolution estimate for twodimensional hypersurfaces  with Ioan Bejenaru and Sebastian Herr Rev. Mat. Iberoam. 26 (2010), no. 2, 707728 
arXiv:0809.5091 
Global wellposedness 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, 329349. 
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 1D cubic NLS equation in H^{1/4}  with Herbert Koch (2010) Ann. Inst. H. Poincare Anal. Non Lineaire 29 (2012), no. 6, 955988. 
arXiv:1012.0148 
Low regularity bounds for mKdV  with Michael Christ and Justin Holmer (2012) Lib. Math. (N.S.) 32 (2012), no. 1, 5175. 
arXiv:1207.6738 
Local wellposedness of ChernSimonsSchroedinger  with Baoping Liu and Paul Smith (2012) Int. Math. Res. Not. IMRN 2014, no. 23, 63416398. 
arXiv:1212.1476 
Null structures and degenerate dispersion relations in two space dimensions  with Yuqui Fu (2012) Int. Math. Res. Not. IMRN, (10):72997338, 2021. 
arXiv:1801.00099 
Carleman estimates and unique continuation
Unique continuation for operators with partially analytic coefficients  J. Math. Pures Appl. (9) 78 (1999), no. 5, 505521.  
Carleman estimates and unique continuation for secondorder elliptic equations with nonsmooth coefficients  with Herbert Koch Comm. Pure Appl. Math. 54 (2001), no. 3, 339360. 
pdf 
Sharp counterexamples in unique continuation for second order elliptic equations  with Herbert Koch J. Reine Angew. Math. 542 (2002), 133146. 
pdf 
Dispersive estimates for principally normal pseudodifferential operators.  Comm. Pure Appl. Math. 58 (2005), no. 2, 217284.  arXiv:math/0401234 
L^{p} eigenfunction bounds for the Hermite operator.  with Herbert Koch Duke Math. J. 128 (2005), no. 2, 369392. 
arXiv:math/0402261 
Carleman estimates and absence of embedded eigenvalues.  with Herbert Koch Comm. Math. Phys. 267 (2006), no. 2, 419449. 
arXiv:mathph/0508052 
Carleman estimates and unique
continuation for second order parabolic equations with nonsmooth
coefficients 
with Herbert Koch Comm. Partial Differential Equations 34 (2009), no. 46, 305366. 
arXiv:0704.1349 
Uniqueness in Calderon's problem with Lipschitz conductivities  with Boaz Haberman (2011) Duke Math. J. 162 (2013), no. 3, 496516. 
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, 349376  
Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. III.  J. Amer. Math. Soc. 15 (2002), no. 2, 419442  Journal 
Sharp counterexamples for Strichartz estimates for low regularity metrics.  with Hart Smith Math. Res. Lett. 9 (2002), no. 23, 199204. 

Strichartz estimates for a Schrödinger operator with nonsmooth coefficients.  with Gigliola Staffilani Comm. Partial Differential Equations 27 (2002), no. 78, 13371372. 

On the FeffermanPhong inequality and related problems.  Comm. Partial Differential Equations 27 (2002), no. 1112, 21012138  
Nonlinear wave equations.  Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), 209220  arXiv:math/0304397 
Null form estimates for second order hyperbolic operators with rough coefficients  Harmonic analysis at Mount Holyoke (South Hadley, MA, 2001), 383409, Contemp. Math., 320  
Phase space transforms and microlocal analysis.  Phase space analysis of partial differential equations. Vol. II, 505524, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, 2004.  
Dispersive estimates for wave equations  with Dan Geba Comm. Partial Differential Equations 30 (2005), no. 46, 849880. 

Sharp local wellposedness results for the nonlinear wave equation.  with Hart Smith Ann. of Math. (2) 162 (2005), no. 1, 291366. 
dviletter, psletter 
Sharp L^{q}
bounds on special clusters for Holder metrics. 
with Herbert Koch and Hart Smith Math. Res. Lett. 14 (2007), no. 1, 7785 

Semiclassical L^{p} estimates  with Herbert Koch and Maciej
Zworski Ann. Henri Poincaré 8 (2007), no. 5, 885916 
arXiv:mathph/0603080 
A phase space transform adapted
to the wave equation 
with Dan Geba Comm. PDE 32 (2007), no. 79, 10651101. 

Large data local solutions
for the derivative NLS equation 
with Ioan Bejenaru J. Eur. Math. Soc. (JEMS) 10 (2008), no. 4, 957985 
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, 597604 
arXiv:math/0611252 
Subcritical L^{p} bounds
on spectral clusters for Lipschitz metrics 
with Herbert Koch and Hart Smith Math. Res. Lett. 14 (2007), no. 1, 7785. 
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, 11511172. 
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, 16291663. 
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, 529546. 
arXiv:1208.0544 
Quasilinear Schrödinger equations III: Large Data and Short Time  with Jeremy Marzuola and Jason Metcalfe (2020) Arch. Ration. Mech. Anal., 242(2):1119 1175, 2021. 
arXiv:2001.01014 
Local wellposedness for quasilinear problems: a primer  with Mihaela Ifrim (2020) Bull. Amer. Math. Soc. (N.S.), 60(2):167194, 2023 
arXiv:2008.05684 
The timelike minimal surface equation in Minkowski space: low regularity solutions  with Albert Ai and Mihaela Ifrim (2021) Invent. Math., 235(3):745891, 2024. 
arXiv:2110.15296 
Testing by wave packets and modified scattering in nonlinear dispersive pde's  with Mihaela Ifrim (2020) Trans. Amer. Math. Soc. Ser. B, 11:164214, 2024. 
arXiv:arXiv:2204.13285 
Wave Maps
Local and global results for wave maps. I  Comm. Partial Differential Equations 23 (1998), no. 910, 17811793  
On global existence and scattering for the wave maps equation  Amer. J. Math. 123 (2001), no. 3, 385423.  pdf 
The wave maps equation  Bull. Amer. Math. Soc. (N.S.) 41 (2004), no. 2, 185204  pdf Journal 
Rough solutions for the wave maps equation  Amer. J. Math. 127 (2005), no. 2, 293377.  
Energy dispersed large data wave maps in 2+1 dimensions  with Jacob Sterbenz Comm. Math. Phys. 298 (2010), no. 1, 139230 
arXiv:0810.5766 
Regularity of WaveMaps in dimension 2+1  with Jacob Sterbenz Comm. Math. Phys. 298 (2010), no. 1, 231264 
arXiv:0907.3148 
Wave maps on (1+2)dimensional curved spacetimes  with Cristian Gavrus and Casey Jao Anal. PDE, 14(4):9851084, 2021 
arXiv:1810.05632 
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, 14431506 
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, 283323. 
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, 543615. 
arXiv:math/0610248 
Slow blowup solutions for the H^{1}(R^{3})
critical focusing semilinear wave equation 
with Joachim Krieger and Wilhelm
Schlag Duke Math. J. 147 (2009), no. 1, 153. 
arXiv:math/0702033 
Renormalization and blow up for the critical YangMills problem.  with Joachim Krieger and Wilhelm Schlag Adv. Math. 221 (2009), no. 5, 14451521 
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, 829857. 
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, 571634. 

Local decay of waves on asymptotically flat stationary spacetimes  (2009) Amer. J. Math. 135 (2013), no. 2, 361401. 
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, 14971553. 
arXiv:0706.0544 
Global parametrices and
dispersive estimates for variable coefficients wave equations 
with Jason Metcalfe Math. Ann. 353 (2012), no. 4, 11831237 
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, 201216, 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, 3783. 
arXiv:0802.3942 
A local energy estimate on Kerr black hole backgrounds  with Mihai Tohaneanu (2008) IMRN 2011, no. 2, 248292 
arXiv:0810.5766 
Price's Law on Nonstationary Spacetimes  with Jason Metcalfe and Mihai Tohaneanu (2011) Adv. Math. 230 (2012), no. 3, 9951028. 
arXiv:1104.5437 
Local energy decay for Maxwell fields part I: Spherically symmetric blackhole backgrounds  with Jacob Sterbenz (2013) Int. Math. Res. Not. IMRN 2015, no. 11, 32983342. 
arXiv:1305.5261 
Pointwise decay for the Maxwell field on black hole spacetimes 
with Jason Metcalfe and Mihai Tohaneanu
(2014) Adv. Math. 316 (2017), 5393. 
arXiv:1411.3693 
Local energy decay for scalar fields on time dependent nontrapping backgrounds 
with Jason Metcalfe, Jacob Sterbenz
(2017) Amer. J. Math., 142(3):821883, 2020 
arXiv:1703.08064 
Geometric nonlinear waves. Primarily MKG and YM.
Global wellposedness for the MaxwellKlein Gordon equation in 4+1 dimensions. Small energy  with Joachim Krieger and Jacob Sterbenz (2012) Duke Math. J. 164 (2015), no. 6, 9731040. 
arXiv:1211.3527 
Local wellposedness of the (4+1)dimensional MaxwellKleinGordon equation at energy regularity 
with SungJin Oh
(2015) Ann. PDE 2 (2016), no. 1, Art. 2, 70 
arXiv:1503.01560 
Energy dispersed solutions for the (4+1)dimensional MaxwellKleinGordon equation 
with SungJin Oh
(2015) Amer. J. Math., 140(1):182, 2018 
arXiv:1503.01561 
Global wellposedness and scattering of the (4+1)dimensional MaxwellKleinGordon equation 
with SungJin Oh
(2015) Invent. Math. 205 (2016), no. 3, 781877. 
arXiv:1503.01562 
Global wellposedness for the YangMills equation in 4+1 dimensions. Small energy 
with Joachim Krieger
(2015) Ann. of Math. (2) 185 (2017), no. 3, 831893 
arXiv:1509.00751 
The YangMills heat flow and the caloric gauge 
with SungJin Oh
(2017) Ast ́erisque, (436):viii+128, 2022 
arXiv:1709.08599 
The hyperbolic YangMills equation in the caloric gauge. Local wellposedness and control of energy dispersed solutions 
with SungJin Oh
(2017) Pure Appl. Anal., 2(2):233–384, 2020 
arXiv:1709.09332 
The hyperbolic YangMills equation for connections in an arbitrary topological class 
with SungJin Oh
(2017) Comm. Math. Phys., 365(2):685739, 2019 
arXiv:1709.08604 
The threshold conjecture for the energy critical hyperbolic YangMills equation 
with SungJin Oh
(2017) Ann. of Math. (2), 194(2):393473, 2021 
arXiv:1709.08606 
The Threshold Theorem for the (4+1)dimensional YangMills equation: an overview of the proof 
with SungJin Oh
(2017) Bull. Amer. Math. Soc. (N.S.), 56(2):171210, 2019 
arXiv:1709.09088 
Fluid dynamics. This also includes model problems arising in the study of various fluid models.
Wellposedness for the NavierStokes equations.  with Herbert Koch Adv. Math. 157 (2001), no. 1, 2235. 

Long time Solutions for a BurgersHilbert 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, 34073412. 
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, 483552. 
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, 26612675. 
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, 369394. 
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, 12791346. 
arXiv:1406.5471 
The lifespan of small data solutions to the KPI 
with Benjamin HarropGriffiths and Mihaela Ifrim
(2014) Int. Math. Res. Not. IMRN 2017, no. 1, 128. 
arXiv:1409.4487 
Two dimensional gravity water waves with constant vorticity: I. Cubic lifespan 
with Mihaela Ifrim
(2015) Anal. PDE, 12(4):903–967, 2019 
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 
Wellposedness and dispersive decay of small data solutions for the BenjaminOno equation 
with Mihaela Ifrim
(2017) Ann. Sci. ́Ec. Norm. Sup ́er. (4), 52(2):297 335, 2019 
arXiv:1701.08476 
A Morawetz inequality for water waves  with Thomas Alazard and Mihaela Ifrim (2018) Amer. J. Math., 144(3):607699, 2022 
arXiv:1806.08443 
No solitary waves in 2D gravity and capillary waves in deep water  with Mihaela Ifrim (2018) Nonlinearity, 33(10):54575476, 2020 
arXiv:1808.07916 
The NLS approximation for two dimensional deep gravity waves  with Mihaela Ifrim (2018) Sci. China Math., 62(6):1101–1120, 2019. 
arXiv:1809.05060 
Dispersive decay of small data solutions for the KdV equation  with Mihaela Ifrim and Herbert Koch (2019) Ann. Sci. ́Ec. Norm. Sup ́er. (4), 56(6):17091746, 2023 
arXiv:1901.05934 
A Morawetz inequality for gravitycapillary water waves at low Bond number  with Thomas Alazard and Mihaela Ifrim (2019) Water Waves, 3(3):429472, 2021. 
arXiv:1910.02529 
Two dimensional gravity waves at low regularity I: Energy estimates  with Albert Ai and Mihaela Ifrim (2019) preprint 
arXiv:1910.05323 
Two dimensional gravity waves at low regularity II: Global solutions  with Albert Ai and Mihaela Ifrim (2019) Ann. Inst. H. Poincare C Anal. Non Lineaire, 39(4):819 884, 2022. 
arXiv:2009.11513 
No pure capillary solitary waves exist in 2D finite depth  with Mihaela Ifrim, Ben Pineau and Mitchell Taylor (2021) SIAM J. Math. Anal., 54(4):44524464, 2022. 
arXiv:2104.07845 
The BenjaminOno approximation for 2D gravity water waves with constant vorticity  with Mihaela Ifrim, James Rowan and Lizhe Wan (2021) Ars Inven. Anal., pages Paper No. 3, 33, 2022. 
arXiv:2108.08964 
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 1d 
with Herbert Koch
(2016) Duke Math. J., 167(17):32073313, 2018. 
arXiv:1607.02534 
A Nonlinear Plancherel Theorem with Applications to Global WellPosedness for the Defocusing DavewyStewartson Equation and to the Inverse Boundary Value Problem of Calderon 
with Adrian I. Nachman, Idan Regev
(2017) Invent. Math., 220(2):395451, 2020. 
arXiv:1708.04759 
Multisolitons for the cubic NLS in 1d and their stability  with Herbert Koch (2012) Publications Math ́ematiques de l'IHES, to appear. 
arXiv:2008.13352 
Free boundary problems.
This work is primarily about free boundary problems arising in fluid dynamics, either of
the compressible or incompressible variety.
The compressible Euler equations in a physical vacuum: a comprehensive Eulerian approach  with Mihaela Ifrim (2020) Ann. Inst. H. Poincar ́e C Anal. Non Lin ́eaire, 41(2):405495, 2024. 
arXiv:2007.05668 
The relativistic Euler equations with a physical vacuum boundary: Hadamard local wellposedness, rough solutions, and continuation criterion  with Marcelo Disconzi and Mihaela Ifrim (2020) Arch. Ration. Mech. Anal., 245(1):127182, 2022. 
arXiv:2007.05787 
Sharp Hadamard local wellposedness, enhanced uniqueness and pointwise continuation criterion for the incompressible free boundary Euler equations  with Mihaela Ifrim, Ben Pineau and Mitchell Taylor (2023) preprint 
arXiv:2309.05625 
Skew Mean Curvature Flow.
This is the Schroedinger counterpart of the mean curvature flow, for codimension two embedded submanifolds.
Local wellposedness of skew mean curvature flow for small data in d≥4 dimensions  with Jiaxi Huang (2021) Comm. Math. Phys., 389(3):15691645, 2022. 
arXiv:2101.00358 
Local wellposedness of the Skew mean curvature flow for small data in d≥2 dimensions  with Jiaxi Huang (2022) Arch. Ration. Mech. Anal., 248(1):10, 2024. 
arXiv:2202.10632 
Global regularity of Skew mean curvature flow for small data in d≥4 dimensions  with Jiaxi Huang and Ze Li (2022) nt. Math. Res. Not. IMRN, (5):37483798, 2024 
arXiv:2209.08941 
The global wellposedness conjectures.
These conjectures are concerned with global wellposedness for dispersive pde's in strongly nonlinear regimes
Global solutions for 1D cubic defocusing dispersive equations: Part I  with Mihaela Ifrim (2022) Forum Math. Pi, 11:Paper No. e31, 46, 2023. 
arXiv:2205.12212 
Long time solutions for 1D cubic dispersive equations, Part II: the focusing case  with Mihaela Ifrim (2022) Vietnam Journal of Mathematics, Special issue dedicated to Carlos Kenig, to appear 
arXiv:2210.17007 
Global solutions for 1D cubic dispersive equations, Part III: the quasilinear Schrödinger flow  with Mihaela Ifrim (2022) preprint. 
arXiv:2306.00570 
The global wellposedness conjecture for 1D cubic dispersive equations  with Mihaela Ifrim (2023) preprint 
arXiv:2311.15076 
Global solutions for cubic quasilinear Schroedinger flows in two and higher dimensions  with Mihaela Ifrim (2024) preprint 
arXiv:2404.09970 