Sebastian Herr

On the 2D Zakharov system with L2 Schrödinger data

We prove local in time well-posedness for the Zakharov system in two space dimensions with large initial data in L2 × H−1/2 × H−3/2. This is the space of optimal regularity in the sense that the data-to-solution map fails to be smooth at the origin for any rougher pair of spaces in the L2-based Sobolev scale. Moreover, it is a natural space for the Cauchy problem in view of the subsonic limit equation, namely the focusing cubic nonlinear Schrodinger equation. The existence time we obtain depends only upon the corresponding norms of the initial data—a result which is false for the cubic nonlinear Schrodinger equation in dimension two—and it is optimal because Glangetas–Merles solutions blow up at that time.

On the Cauchy problem for the derivative nonlinear Schrödinger equation with periodic boundary condition

It is shown that the Cauchy problem for the DNLS equation in the spatially periodic setting is locally well-posed in Sobolev spaces H^s(T) for s \geq 1/2. Moreover, global well-posedness is shown for s \geq 1 and data with small L^2 norm.

Convolutions of singular measures and applications to the Zakharov system

Abstract Uniform L 2 -estimates for the convolution of singular measures with respect to transversal submanifolds are proved in arbitrary space dimension. The results of Bennett–Bez are used to extend previous work of Bejenaru–Herr–Tataru. As an application, it is shown that the 3D Zakharov system is locally well-posed in the full subcritical regime.

Strichartz estimates for partially periodic solutions to Schrödinger equations in 4d

Author(s): Herr, Sebastian; Tataru, Daniel; Tzvetkov, Nikolay | Abstract: We consider the energy critical nonlinear Schr quot;odinger equation on periodic domains of the form R^m x T^{4-m} with m=0,1,2,3. Assuming that a certain L^4 Strichartz estimate holds for solutions to the corresponding linear Schr quot;odinger equation, we prove that the nonlinear problem is locally well-posed in the energy space. Then we verify that the L^4 estimate holds if m=2,3, leaving open the cases m=0,1.

4d

We establish global well-posedness and scattering for the cubic Dirac equation for small data in the critical space

The Cubic Dirac Equation: Small Initial Data in H-1/2 (R-2)

{H^1(\mathbb{R}^3)}

The Fourier restriction norm method for the Zakharov-Kuznetsov equation

H1(R3). The main ingredient is obtaining a sharp end-point Strichartz estimate for the Klein–Gordon equation. In a classical sense this fails and it is related to the failure of the endpoint Strichartz estimate for the wave equation in space dimension three. In this paper, systems of coordinate frames are constructed in which endpoint Strichartz estimates are recovered and energy estimates are established.

A convolution estimate for two-dimensional hypersurfaces

For α ∈ (1, 2) we prove that the initial-value problem is globally well-posed in the space of real-valued L 2-functions. We use a frequency dependent renormalization method to control the strong low-high frequency interactions.