Nahmod, Andrea

Job Title

Professor, Department of Mathematics and Statistics, College of Natural Sciences

Last Name

Nahmod

First Name

Andrea

Discipline

Harmonic Analysis and Representation

Expertise

Harmonic/Nonlinear Fourier Analysis and Partial Differential Equations

Introduction

My research lies at the overlap of Nonlinear Fourier Analysis/Harmonic Analysis and Nonlinear Partial Differential Equations integrating into it tools from geometry, gauge theory and probability. In recent years, its main focus has been to investigate:
(i) the behavior of solutions to nonlinear dispersive equations arising as models both in Physics and in Geometry -both from a deterministic and nondeterministic viewpoint and
(ii) wave-packet analysis techniques and multilinear singular pseudodifferential operators naturally arising in Analysis and PDE.
These are two areas that intimately relate to each other by way of decompositions, frequency interactions analysis and nonlinear estimates.

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Now showing 1 - 10 of 10

On the Well-Posedness of the Wave Map Problem in High Dimensions
(2003-01) Nahmod, Andrea; Stefanov, Atanas; Uhlenbeck, Karen
We construct a gauge theoretic change of variables for the wave map from R × Rn into a compact group or Riemannian symmetric space, prove a new multiplication theorem for mixed Lebesgue-Besov spaces, and show the global well-posedness of a modified wave map equation - n ≥ 4 - for small critical initial data. We obtain global existence and uniqueness for the Cauchy problem of wave maps into compact Lie groups and symmetric spaces with small critical initial data and n ≥ 4.
On Schrodinger Maps
(2003-01) Nahmod, Andrea R; Stefanov, Atanas; Uhlenbeck, Karen
We study the question of well-posedness of the Cauchy problem for Schr¨odinger maps from R 1 ×R 2 to the sphere S 2 or to H2 , the hyperbolic space. The idea is to choose an appropriate gauge change so that the derivatives of the map will satisfy a certain nonlinear Schr¨odinger system of equations and then study this modified Schr¨odinger map system (MSM). We then prove local well posedness of the Cauchy problem for the MSM with minimal regularity assumptions on the data and outline a method to derive well posedness of the Schr¨odinger map itself from it. In proving well posedness of the MSM, the heart of the matter is resolved by considering truly quatrilinear forms of weighted L 2 functions.
Erratum: On Schrödinger Maps
(2004-01) Nahmod, Andrea R; Stefanov, Atanas; Uhlenbeck, Karen
Schrodinger Maps and Their Associated Frame Systems
(2007-01) Nahmod, Andrea; Shatah, Jalal; Vega, Luis; Zeng, Chongchun
In this paper we establish the equivalence of solutions between Schr¨odinger maps into S 2 or H 2 and their associated gauge invariant Schr¨odinger equations. We also establish the existence of global weak solutions into H 2 in two space dimensions. We extend these ideas for maps into compact hermitian symmetric manifolds with trivial first cohomology.
L-p-Boundedness for Time-Frequeny Paraproducts, II
(2002-01) Gilbert, John E; Nahmod, Andrea
This article completes the proof of theLp-boundedness of bilinear operators associated to nonsmooth symbols or multipliers begun in Part I, our companion article [8], by establishing the corresponding Lp-boundedness of time-frequency paraproducts associated with tiles in phase plane. The affine invariant structure of such operators in conjunction with the geometric properties of the associated phase-plane decompositions allow Littlewood–Paley techniques to be applied locally, i. e., on trees. Boundedness of the full time-frequency paraproduct then follows using ‘almost orthogonality’ type arguments relying on estimates for tree-counting functions together with decay estimates.
Bilinear Operators with Non-Smooth Symbol, I
(2001-01) Gilbert, John E; Nahmod, Andrea R
This paper proves the Lp-boundedness of general bilinear operators associated to a symbol or multiplier which need not be smooth. The Main Theorem establishes a general result for multipliers that are allowed to have singularities along the edges of a cone as well as possibly at its vertex. It thus unifies ealier results of Coifman-Meyer for smooth multipliers and ones, such the Bilinear Hilbert transform of Lacey-Thiele, where the multiplier is not smooth. Using a Whitney decomposition in the Fourier plane a general bilinear operator is represented as infinite discrete sums of time-frequency paraproducts obtained by associating wave-packets with tiles in phase-plane. Boundedness for the general bilinear operator then follows once the corresponding Lp-boundedness of time-frequency paraproducts has been established. The latter result is the main theorem proved in Part II, our subsequent paper [11], using phase-plane analysis.
Sobolev Space Estimates and Symbolic Calculus for Bilinear Pseudodifferential Operators
(2006-01) Benyi, Arpad; Nahmod, Andrea R; Torres, Rodolpho H
Bilinear operators are investigated in the context of Sobolev spaces and various techniques useful in the study of their boundedness properties are developed. In particular, several classes of symbols for bilinear operators beyond the so-called Coifman-Meyer class are considered. Some of the Sobolev space estimates obtained apply to both the bilinear Hilbert transform and its singular multipliers generalizations as well as to operators with variable dependent symbols. A symbolic calculus for the transposes of bilinear pseudodifferential operators and for the composition of linear and bilinear pseudodifferential operators is presented too.
Boundedness of bilinear operators with nonsmooth symbols
(2000-01) Gilbert, John; Nahmod, Andrea
We announce the Lp-boundedness of general bilinear operators associated to a symbol or multiplier which need not be smooth. We establish a general result for multipliers that are allowed to have singularities along the edges of a cone as well as possibly at its vertex. It thus unifies ealier results of CoifmanMeyer for smooth multipliers and ones, such the Bilinear Hilbert transform of Lacey-Thiele, where the multiplier is not smooth.
The Cauchy Problem for the Hyperbolic-Elliptic Ishimori System and Schrodinger Maps
(2005-01) Kenig, Carlos; Nahmod, Andrea
We show an improved local in time existence and uniqueness result for Schrödinger maps and for the hyperbolic–elliptic nonlinear system proposed by Ishimori in analogy with the two-dimensional classical continuous isotropic Heisenberg spin (2d-CCIHS) chain. The proof uses fairly standard gauge geometric tools and energy estimates in combination with Kenig's version of the Koch–Tzvetkov method, to obtain a priori estimates for classical solutions to certain dispersive equations.
On Schrodinger and wave maps
(2003-01) Nahmod, Andrea R