arXiv subjects

Daniel Tataru

Publications and source records attributed to Daniel Tataru.

At least 19 recordsLinked to original sources

arXiv2026

Global solutions for 1D cubic defocusing dispersive equations, Part V: low regularity NLS

This article is motivated by a broad conjecture, formulated by the first and last authors in earlier work, asserting that one-dimensional cubic defocusing dispersive flows with small initial data have global, dispersive solutions. The conjecture was first established for a class of semilinear Schr\"odinger-type models at $L^2$ regularity, the classical cubic NLS among them. In a complementary direction, Harrop-Griffiths, Killip and Vi\c{s}an have recently shown, using the completely integrable structure, that the cubic NLS is globally well-posed in $H^s$ for every $-\tfrac12 < s < 0$. Our aim here is to extend the reach of the global well-posedness conjecture for one dimensional cubic NLS problems to data which is small in negative Sobolev spaces, and to show that global dispersive bounds persist there. We do so for a broad class of nonlinearities which includes the cubic NLS but which in general generates flows that are not completely integrable. Our method is correspondingly robust, resting on density-flux identities, interaction Morawetz estimates and an implicit normal form transformation rather than on integrability, and it reaches all the way to the scaling-critical threshold, namely $s > -\tfrac12$. As in the earlier work, the global bounds we obtain include both $L^6_{t,x}$ Strichartz estimates and bilinear $L^2_{t,x}$ estimates; these are new even for the classical defocusing cubic NLS at negative Sobolev regularity. There, by scaling, our dispersive bounds also extend to the large data case.

math.AP↗

arXiv2026

Enhanced lifespan bounds for 1D quasilinear Klein-Gordon flows

In this article we consider one-dimensional scalar quasilinear Klein--Gordon equations with general nonlinearities, on both $\mathbb{R}$ and $\mathbb{T}$. By employing a refined modified-energy framework of Ifrim and Tataru, we investigate long time lifespan bounds for small data solutions. Our main result asserts that solutions with small initial data of size $\epsilon$ persist on the improved cubic timescale $|t| \lesssim \epsilon^{-2}$ and satisfy sharp cubic energy estimates throughout this interval. We also establish difference bounds on the same time scale. In the case of $\mathbb{R}$, we are further able to use dispersion in order to extend the lifespan to $\epsilon^{-4}$. This generalizes earlier results obtained by Delort in the semilinear case.

math.AP↗

arXiv2025

Local well-posedness of the skew mean curvature flow for large data

The skew mean curvature flow is an evolution equation for $d$ dimensional ma\-nifolds embedded in $\mathbb{R}^{d+2}$ (or more generally, in a Riemannian manifold). It can be viewed as a Schr\"odinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schr\"odinger Map equation. In this article, we prove large data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension $d\geq 2$. This is achieved by introducing several new ideas: (i) a time discretization method to establish the existence of smooth solutions, (ii) constructing the orthonormal frame by a parallel transport method and a lifting criterion, (iii) introducing intrinsic fractional function spaces $X^s\subset H^s$ on a noncompact manifold for any $s>\frac{d}{2}$, such that the $X^s$-norm of the second fundamental form can be propagated well along the quasilinear Schr\"odinger flow, (iv) deriving a difference equation to prove the uniqueness result for solutions $F\in C^2$, which is independent in the choices of gauge. Our method turns out to be more robust for large data problem.

math.AP↗

arXiv2025

Large data global well-posedness for the modified Novikov-Veselov system

The modified Novikov-Veselov system (mNV) is a cubic third order dispersive evolution in two space dimensions. It is also completely integrable, belonging to the same hierarchy as the defocusing Davey-Stewartson II (DS II) system. The mNV system is $L^2$ critical. Some time ago, Schottdorf proved that for small $L^2$ initial data, the mNV equation is globally well-posed. In this article, we consider instead the large data problem, using inverse scattering methods. Our main result asserts that the mNV system is globally well-posed for large $L^2$ data, with the solutions scattering as time goes to $\pm \infty$. One key ingredient in the proof, which is of independent interest, is a new nonlinear Gagliardo-Nirenberg inequality for the associated scattering transform. As a byproduct of our main result, we are also able to prove a global well-posedness result for the closely related Novikov-Veselov problem at the critical $\dot H^{-1} + L^1$ level, for a range of data which can heuristically be described as soliton-free. Here we use the associated Miura map to connect the mNV and the NV flows. In order to characterize the range of the Miura map, we prove another result of independent interest, namely a sharp, scale invariant form of the Agmon-Allegretto-Piepenbrink principle in the critical case of two space dimensions.

math.AP↗

arXiv2025

Wave packet decompositions and sharp bilinear estimates for rough Hamiltonian flows

The goal of this paper is to prove bilinear $L^p$ estimates for rough dispersive evolutions satisfying non-degeneracy and transversality assumptions. The estimates generalize the sharp Fourier extension estimates for the cone and the paraboloid. To this end, we require a wave packet decomposition with localization properties in space-time and space-time frequencies. Secondly, we construct a refined wave packet parametrix for dispersive equations with $C^{1,1}$-coefficients by using the FBI transform. As a consequence, we obtain bilinear estimates for solutions to dispersive equations with $C^{1,1}$ coefficients provided that the solutions interact transversely.

math.AP↗

arXiv2025

Global solutions for cubic quasilinear ultrahyperbolic Schr\"odinger flows

In recent work, two of the authors proposed a broad global well-posedness conjecture for cubic quasilinear dispersive equations in two space dimensions, which asserts that global well-posedness and scattering holds for small initial data in Sobolev spaces. As a first validation they proved the conjecture for quasilinear Schr\"odinger flows. In the present article we expand the reach of these ideas and methods to the case of quasilinear ultrahyperbolic Schr\"odinger flows, which is the first example with a nonconvex dispersion relation. The study of local well-posedness for this class of problems, in all dimensions, was initiated in pioneering work of Kenig-Ponce-Vega for localized initial data, and then continued by Marzuola-Metcalfe-Tataru (MMT) and Pineau-Taylor (PT) for initial data in Sobolev spaces in the elliptic and non-elliptic cases, respectively. Our results here mirror the earlier results in the elliptic case: (i) a new, potentially sharp local well-posedness result in low regularity Sobolev spaces, one derivative below MMT and just one-half derivative above scaling, (ii) a small data global well-posedness and scattering result at the same regularity level. One key novelty in this setting is the introduction of a new family of interaction Morawetz functionals which are suitable for obtaining bilinear estimates in the ultrahyperbolic setting. We remark that this method appears to be robust enough to potentially be of use in a large data regime when the metric is not a small perturbation of a Euclidean one.

math.AP↗

arXiv2024

Sharp well-posedness for the free boundary MHD equations

In this article, we provide a definitive well-posedness theory for the free boundary problem in incompressible magnetohyrodynamics. Despite the clear physical interest in this system and the remarkable progress in the study of the free boundary Euler equations in recent decades, the low regularity well-posedness of the free boundary MHD equations has remained completely open. This is due, in large part, to the highly nonlinear wave-type coupling between the velocity, magnetic field and free boundary, which has forced previous works to impose restrictive geometric constraints on the data. To address this problem, we introduce a novel Eulerian approach and an entirely new functional setting, which better captures the wave equation structure of the MHD equations and permits a complete Hadamard well-posedness theory in low-regularity Sobolev spaces. In particular, we give the first proofs of existence, uniqueness and continuous dependence on the data at the sharp $s>\frac{d}{2}+1$ Sobolev regularity, in addition to a blowup criterion for smooth solutions at the same low regularity scale. Moreover, we provide a completely new method for constructing smooth solutions which, to our knowledge, gives the first proof of existence (at any regularity) in our new functional setting. All of our results hold in arbitrary dimensions and in general, not necessarily simply connected, domains. By taking the magnetic field to be zero, they also recover the corresponding sharp well-posedness theorems for the free boundary Euler equations. The methodology and tools that we employ here can likely be fruitfully implemented in other free boundary models.

math.AP↗

arXiv2024

Global solutions for 1D cubic defocusing dispersive equations, Part IV: general dispersion relations

A broad conjecture, formulated by the authors in earlier work, reads as follows: "Cubic defocusing dispersive one dimensional flows with small initial data have global dispersive solutions". Notably, here smallness is only assumed in $H^s$ Sobolev spaces, without any localization assumption. The conjecture was initially proved by the authors first for a class of semilinear Schr\"odinger type models, and then for quasilinear Schr\"odinger flows. In this work we take the next natural step, and prove the above conjecture for a much larger class of one dimensional semilinear dispersive problems with a cubic nonlinearity, where the dispersion relation is no longer of Schr\"odinger type. This result is the first of its kind, for any 1D cubic problem not of Schr\"odinger type. Furthermore, it only requires initial data smallness at critical regularity, a threshold that has never been reached before for any 1D cubic dispersive flow. In terms of dispersive decay, we prove that our global in time solutions satisfy both global $L^6_{t,x}$ Strichartz estimates and bilinear $L^2_{t,x}$ bounds.

math.AP↗

arXiv2024

Nonlinear interpolation and the flow map for quasilinear equations

We prove an interpolation theorem for nonlinear functionals defined on scales of Banach spaces that generalize Besov spaces. It applies to functionals defined only locally, requiring only some weak Lipschitz conditions, extending those introduced by Lions and Peetre. Our analysis is self-contained and independent of any previous results about interpolation theory. It depends solely on the concepts of Friedrichs' mollifiers, seen through the formalism introduced by Hamilton, combined with the frequency envelopes introduced by Tao and used recently by two of the authors and others to study the Cauchy problem for various quasilinear evolutions in partial differential equations. Inspired by this latter work, our main application states that, for an abstract flow map of a quasilinear problem, both the continuity of the flow as a function of time and the continuity of the data to solution map follow automatically from the estimates that are usually proven when establishing the existence of solutions: propagation of regularity via tame a priori estimates for higher regularities and contraction for weaker norms.

math.AP↗

arXiv2024

Near soliton evolution for $2$-equivariant Schr\"odinger Maps in two space dimensions

We consider equivariant solutions for the Schr\"odinger Map equation in $2+1$ dimensions, with values into $\mathbb{S}^2$. Within each equivariance class $m \in \mathbb{Z}$ this admits a lowest energy nontrivial steady state $Q^m$, which extends to a two dimensional family of steady states by scaling and rotation. If $|m| \geq 3$ then these ground states are known to be stable in the energy space $\dot H^1$, whereas instability and even finite time blow-up along the ground state family may occur if $|m| = 1$. In this article we consider the most delicate case $|m| = 2$. Our main result asserts that small $\dot H^1$ perturbations of the ground state $Q^2$ yield global in time solutions, which satisfy global dispersive bounds. Unlike the higher equivariance classes, here we expect solutions to move arbitrarily far along the soliton family; however, we are able to provide a time dependent bound on the growth of the scale modulation parameter. We also show that within the equivariant class the ground state is stable in a slightly stronger topology $X \subset \dot H^1$.

math.AP↗

arXiv2024

Global solutions for cubic quasilinear Schroedinger flows in two and higher dimensions

In recent work the authors proposed a broad global well-posedness conjecture for cubic defocusing dispersive equations in one space dimension, and then proved this conjecture in two cases, namely for one dimensional semilinear and quasilinear Schr\"odinger flows. Inspired by the circle of ideas developed in the proof of the above conjecture, in this paper we expand the reach of these methods to higher dimensional quasilinear cubic Schr\"odinger flows. The study of this class of problems, in all dimensions, was initiated in pioneering work of Kenig-Ponce-Vega for localized initial data, and then continued by Marzuola-Metcalfe-Tataru (MMT) for initial data in Sobolev spaces. The outcomes of this work are (i) a new, potentially sharp local well-posedness result in low regularity Sobolev spaces, one derivative below MMT and just one half derivative above scaling, (ii) a small data global well-posedness and scattering result at the same regularity level, the first result of its kind at least in two space dimensions, and (iii) a new way to think about this class of problems, which, we believe, will become the standard approach in the future.

math.AP↗

arXiv2023

The global well-posedness conjecture for 1D cubic dispersive equations

The goal of this article is to discuss a recent conjecture of the two authors, which aims to describe the long time behavior of solutions to one-dimensional dispersive equations with cubic and higher nonlinearities. These problems arguably represent the single most important example where, even for small initial data, the nonlinear effects are stronger than the dispersive effects. Consequently, the outcome predicted by the conjecture depends essentially on the structure of the nonlinearity, precisely its focusing or defocusing character.

math.AP↗

arXiv2023

Sharp Hadamard local well-posedness, enhanced uniqueness and pointwise continuation criterion for the incompressible free boundary Euler equations

We provide a complete local well-posedness theory in $H^s$ based Sobolev spaces for the free boundary incompressible Euler equations with zero surface tension on a connected fluid domain. Our well-posedness theory includes: (i) Local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data, all in low regularity Sobolev spaces; (ii) Enhanced uniqueness: Our uniqueness result holds at the level of the Lipschitz norm of the velocity and the $C^{1,\frac{1}{2}}$ regularity of the free surface; (iii) Stability bounds: We construct a nonlinear functional which measures, in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) Energy estimates: We prove refined, essentially scale invariant energy estimates for solutions, relying on a newly constructed family of elliptic estimates; (v) Continuation criterion: We give the first proof of a sharp continuation criterion in the physically relevant pointwise norms, at the level of scaling. In essence, we show that solutions can be continued as long as the velocity is in $L_T^1W^{1,\infty}$ and the free surface is in $L_T^1C^{1,\frac{1}{2}}$, which is at the same level as the Beale-Kato-Majda criterion for the boundaryless case; (vi) A novel proof of the construction of regular solutions. Our entire approach is in the Eulerian framework and can be adapted to work in more general fluid domains.

math.AP↗

arXiv2023

A lunar reconnaissance drone for cooperative exploration and high-resolution mapping of extreme locations

An efficient characterization of scientifically significant locations is essential prior to the return of humans to the Moon. The highest resolution imagery acquired from orbit of south-polar shadowed regions and other relevant locations remains, at best, an order of magnitude larger than the characteristic length of most of the robotic systems to be deployed. This hinders the planning and successful implementation of prospecting missions and poses a high risk for the traverse of robots and humans, diminishing the potential overall scientific and commercial return of any mission. We herein present the design of a lightweight, compact, autonomous, and reusable lunar reconnaissance drone capable of assisting other ground-based robotic assets, and eventually humans, in the characterization and high-resolution mapping (~0.1 m/px) of particularly challenging and hard-to-access locations on the lunar surface. The proposed concept consists of two main subsystems: the drone and its service station. With a total combined wet mass of 100 kg, the system is capable of 11 flights without refueling the service station, enabling almost 9 km of accumulated flight distance. The deployment of such a system could significantly impact the efficiency of upcoming exploration missions, increasing the distance covered per day of exploration and significantly reducing the need for recurrent contacts with ground stations on Earth.

cs.RO↗

arXiv2023

Global solutions for 1D cubic dispersive equations, Part III: the quasilinear Schr\"odinger flow

The first target of this article is the local well-posedness question for 1D quasilinear Schr\"odinger equations with cubic nonlinearities. The study of this class of problems, in all dimensions, was initiated in pioneering work of Kenig-Ponce-Vega for localized initial data, and then continued by Marzuola-Metcalfe-Tataru for initial data in Sobolev spaces. Our objective here is to fully redevelop the study of this problem in the 1D case, and to prove a \emph{sharp local well-posedness} result. The second goal of this article is to consider the long time/global existence of solutions for the same problem. This is motivated by a broad conjecture formulated by the authors in earlier work, which reads as follows: ``\emph{Cubic defocusing dispersive one dimensional flows with small initial data have global dispersive solutions}''; the conjecture was initially proved for a well chosen semilinear model of Schr\"odinger type. Our work here establishes the above conjecture for 1D quasilinear Schr\"{o}dinger flows. Precisely, we show that if the problem has \emph{phase rotation symmetry} and is \emph{conservative and defocusing}, then small data in Sobolev spaces yields global, scattering solutions. This is the first result of this type for 1D quasilinear dispersive flows. Furthermore, we prove it at the minimal Sobolev regularity in our local well-posedness result. The defocusing condition is essential in our global result. Without it, the authors have conjectured that \emph{small, $\epsilon$ size data yields long time solutions on the $\epsilon^{-8}$ time-scale}. A third goal of this paper is to also prove this second conjecture for 1D quasilinear Schr\"{o}dinger flows.

math.AP↗

arXiv2022

Long time solutions for 1D cubic dispersive equations, Part II: the focusing case

This article is concerned with one dimensional dispersive flows with cubic nonlinearities on the real line. In a very recent work, the authors have introduced a broad conjecture for such flows, asserting that in the defocusing case, small initial data yields global, scattering solutions. Then this conjecture was proved in the case of a Schr\"odinger dispersion relation. In terms of scattering, our global solutions were proved to satisfy both global $L^6$ Strichartz estimates and bilinear $L^2$ bounds. Notably, no localization assumption is made on the initial data. In this article we consider the focusing scenario. There potentially one may have small solitons, so one cannot hope to have global scattering solutions in general. Instead, we look for long time solutions, and ask what is the time-scale on which the solutions exist and satisfy good dispersive estimates. Our main result, which also applies in the case of the Schr\"odinger dispersion relation, asserts that for initial data of size $\epsilon$, the solutions exist on the time-scale $\epsilon^{-8}$, and satisfy the desired $L^6$ Strichartz estimates and bilinear $L^2$ bounds on the time-scale $\epsilon^{-6}$. To the best of our knowledge, this is the first result to reach such a threshold.

math.AP↗

arXiv2022

Global regularity of Skew mean curvature flow for small data in $d\geq 4$ dimensions

The skew mean curvature flow is an evolution equation for a $d$ dimensional manifold immersed into $\mathbb{R}^{d+2}$, and which moves along the binormal direction with a speed proportional to its mean curvature. In this article, we prove small data global regularity in low-regularity Sobolev spaces for the skew mean curvature flow in dimensions $d\geq 4$. This extends the local well-posedness result in \cite{HT}.

math.AP↗

arXiv2022

Global solutions for 1D cubic defocusing dispersive equations: Part I

This article is devoted to a general class of one dimensional NLS problems with a cubic nonlinearity. The question of obtaining scattering, global in time solutions for such problems has attracted a lot of attention in recent years, and many global well-posedness results have been proved for a number of models under the assumption that the initial data is both \emph{small} and \emph{localized}. However, except for the completely integrable case, no such results have been known for small but non-localized initial data. In this article we introduce a new, nonperturbative method, to prove global well-posedness and scattering for $L^2$ initial data which is \emph{small} but \emph{non-localized}. Our main structural assumption is that our nonlinearity is \emph{defocusing}. However, we do not assume that our problem has any exact conservation laws. Our method is based on a robust reinterpretation of the idea of interaction Morawetz estimates, developed almost 20 years ago by the I-team. In terms of scattering, we prove that our global solutions satisfy both global $L^6$ Strichartz estimates and bilinear $L^2$ bounds. This is a Galilean invariant result, which is new even for the classical defocusing cubic NLS. There, by scaling our result also admits a large data counterpart.

math.AP↗