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The FBI transform, operators with nonsmooth coefficients and the nonlinear wave equation

Daniel Tataru — 1999

Journées équations aux dérivées partielles

The aim of this work is threefold. First we set up a calculus for partial differential operators with nonsmooth coefficients which is based on the FBI (Fourier-Bros-Iagolnitzer) transform. Then, using this calculus, we prove a weaker version of the Strichartz estimates for second order hyperbolic equations with nonsmooth coefficients. Finally, we apply these new Strichartz estimates to second order nonlinear hyperbolic equations and improve the local theory, i.e. prove local well-posedness for initial...

Schrödinger maps

Daniel Tataru — 2012

Journées Équations aux dérivées partielles

The Schrödinger map equation is a geometric Schrödinger model, closely associated to the harmonic heat flow and to the wave map equation. The aim of these notes is to describe recent and ongoing work on this model, as well as a number of related open problems.

Dispersive estimates and absence of embedded eigenvalues

Herbert Koch;  Daniel Tataru — 2005

Journées Équations aux dérivées partielles

In [2] Kenig, Ruiz and Sogge proved ∥ u ∥ L 2 n n - 2 ( ℝ n ) ≲ ∥ L u ∥ L 2 n n + 2 ( ℝ n ) provided n ≥ 3 , u ∈ C 0 ∞ ( ℝ n ) and L is a second order operator with constant coefficients such that the second order coefficients are real and nonsingular. As a consequence of [3] we state local versions of this inequality for operators with C 2 coefficients. In this paper we show how to apply these local versions to the absence of embedded eigenvalues...

Large data local solutions for the derivative NLS equation

Ioan Bejenaru;  Daniel Tataru — 2008

Journal of the European Mathematical Society

We consider the derivative NLS equation with general quadratic nonlinearities. In [2] the first author has proved a sharp small data local well-posedness result in Sobolev spaces with a decay structure at infinity in dimension n = 2 . Here we prove a similar result for large initial data in all dimensions n ≥ 2 .

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