Searcharxiv⌕ Search

arXiv subjects

Daniel Tataru

Publications and source records attributed to Daniel Tataru.

At least 73 records · Page 4Linked to original sources
arXiv2015

Global well-posedness and scattering of the (4+1)-dimensional Maxwell-Klein-Gordon equation

This article constitutes the final and main part of a three-paper sequence, whose goal is to prove global well-posedness and scattering of the energy critical Maxwell-Klein-Gordon equation (MKG) on $\mathbb{R}^{1+4}$ for arbitrary finite energy initial data. Using the successively stronger continuation/scattering criteria established in the previous two papers, we carry out a blow-up analysis and deduce that the failure of global well-posedness and scattering implies the existence of a nontrivial stationary or self-similar solution to MKG. Then, by establishing that such solutions do not exist, we complete the proof.

math.AP↗
arXiv2014

Two dimensional water waves in holomorphic coordinates

This article is concerned with the infinite depth water wave equation in two space dimensions. We consider this problem expressed in position-velocity potential holomorphic coordinates. Viewing this problem as a quasilinear dispersive equation, we establish two results: (i) local well-posedness in Sobolev spaces, and (ii) almost global solutions for small localized data. Neither of these results are new; they have been recently obtained by Alazard-Burq-Zuily \cite{abz}, respectively by Wu \cite{wu} using different coordinates and methods. Instead our goal is improve the understanding of this problem by providing a single setting for both problems, by proving sharper versions of the above results, as well as presenting new, simpler proofs. This article is self contained.

math.AP↗
arXiv2014

Global bounds for the cubic nonlinear Schrödinger equation (NLS) in one space dimension

This article is concerned with the small data problem for the cubic nonlinear Schrödinger equation (NLS) in one space dimension, and short range modifications of it. We provide a new, simpler approach in order to prove that global solutions exist for data which is small in $H^{0,1}$. In the same setting we also discuss the related problems of obtaining a modified scattering expansion for the solution, as well as asymptotic completeness.

math.AP↗
arXiv2014

Two dimensional water waves in holomorphic coordinates II: global solutions

This article is concerned with the infinite depth water wave equation in two space dimensions. We consider this problem expressed in position-velocity potential holomorphic coordinates,and prove that small localized data leads to global solutions. This article is a continuation of authors' earlier paper arXiv:1401.1252.

math.AP↗
arXiv2014

Semiclassical Lp estimates

The purpose of this paper is to use semiclassical analysis to unify and generalize Lp estimates on high energy eigenfunctions and spectral clusters. In our approach these estimates do not depend on ellipticity and order, and apply to operators which are selfadjoint only at the principal level. They are estimates on weakly approximate solutions to semiclassical pseudodifferential equations. The revision corrects an exponent in the main theorems.

math-ph↗
arXiv2012

Local wellposedness of Chern-Simons-Schrödinger

In this article we consider the initial value problem for the Chern-Simons-Schrodinger model in two space dimensions. This is a covariant NLS type problem which is L^2 critical. For this equation we introduce a so-called heat gauge, and prove that, with respect to this gauge, the problem is locally well-posed for initial data which is small in H^s, s > 0.

math.AP↗
arXiv2012

Quasilinear Schrödinger equations II: Small data and cubic nonlinearities

In part I of this project we examined low regularity local well-posedness for generic quasilinear Schrödinger equations with small data. This improved, in the small data regime, the preceding results of Kenig, Ponce, and Vega as well as Kenig, Ponce, Rolvung, and Vega. In the setting of quadratic interactions, the (translation invariant) function spaces which were utilized incorporated an $l^1$ summability over cubes in order to account for Mizohata's integrability condition, which is a necessary condition for the $L^2$ well-posedness for the linearized equation. For cubic interactions, this integrability condition meshes better with the inherent $L^2$ nature of the Schrödinger equation, and such summability is not required. Thus we are able to prove small data well-posedness in $H^s$ spaces.

math.AP↗
arXiv2012

Low regularity bounds for mKdV

We study the local well-posedness in the Sobolev space H^s for the modified Korteweg-de Vries (mKdV) equation on the real line. Kenig-Ponce-Vega \cite{KPV2} and Christ-Colliander-Tao established that the data-to-solution map fails to be uniformly continuous on a fixed ball in H^s when s<1/4. In spite of this, we establish that for -1/8 < s < 1/4, the solution satisfies global in time H^s(R) bounds which depend only on the time and on the H^s(R) norm of the initial data. This result is weaker than global well-posedness, as we have no control on differences of solutions. Our proof is modeled on recent work by Christ-Colliander-Tao and Koch-Tataru employing a version of Bourgain's Fourier restriction spaces adapted to time intervals whose length depends on the spatial frequency.

math.AP↗
arXiv2012

Sharp L^p bounds on spectral clusters for Lipschitz metrics

We establish L^p bounds on L^2 normalized spectral clusters for self-adjoint elliptic Dirichlet forms with Lipschitz coefficients. In two dimensions we obtain best possible bounds for all p between $2 and infinity, up to logarithmic losses for $6<p\leq 8$. In higher dimensions we obtain best possible bounds for a limited range of p.

math.AP↗
arXiv2012

Uniqueness in Calderon's problem with Lipschitz conductivities

We use X^{s,b}-inspired spaces to prove a uniqueness result for Calderon's problem in a Lipschitz domain under the assumption that the conductivity is Lipschitz. For Lipschitz conductivities, we obtain uniqueness for conductivities close to the identity in a suitable sense. We also prove uniqueness for arbitrary C^1 conductivities.

math.AP↗
arXiv2012

Quasilinear Schrödinger equations I: Small data and quadratic interactions

In this article we prove local well-posedness in low-regularity Sobolev spaces for general quasilinear Schrödinger equations. These results represent improvements of the pioneering works by Kenig-Ponce-Vega and Kenig-Ponce-Rolvung-Vega, where viscosity methods were used to prove existence of solutions in very high regularity spaces. Our arguments here are purely dispersive. The function spaces in which we show existence are constructed in ways motivated by the results of Mizohata, Ichinose, Doi, and others, including the authors.

math.AP↗
arXiv2011

Price's Law on Nonstationary Spacetimes

In this article we study the pointwise decay properties of solutions to the wave equation on a class of nonstationary asymptotically flat backgrounds in three space dimensions. Under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we establish a $t^{-3}$ local uniform decay rate (Price's law \cite{MR0376103}) for linear waves. As a corollary, we also prove Price's law for certain small perturbations of the Kerr metric. This result was previously established by the second author in \cite{Tat} on stationary backgrounds. The present work was motivated by the problem of nonlinear stability of the Kerr/Schwarzschild solutions for the vacuum Einstein equations, which seems to require a more robust approach to proving linear decay estimates.

math.AP↗