arXiv subjects

S. Herr

Publications and source records attributed to S. Herr.

3 recordsLinked to original sources

arXiv2007

Low regularity local well-posedness of the Derivative Nonlinear Schrödinger Equation with periodic initial data

The Cauchy problem for the derivative nonlinear Schrödinger equation with periodic boundary condition is considered. Local well-posedness for periodic initial data u_0 in the space ^H^s_r, defined by the norms ||u_0||_{^H^s_r}=|| ^s ^u_0||_{l^r'} is shown in the parameter range s>= 1/2, 2>r>4/3. The proof is based on an adaptation of the gauge transform to the periodic setting and an appropriate variant of the Fourier restriction norm method.

math.AP↗

arXiv2005

On the Cauchy problem for the derivative nonlinear Schroedinger 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.

math.AP↗

arXiv2005

An improved bilinear estimate for Benjamin-Ono type equations

A bilinear estimate in Fourier restriction norm spaces with applications to the Cauchy problem associated to u_t - |D|^αu_x + uu_x =0 is proved, for 1< α<2. As a consequence, local well-posedness in H^s(\R) \cap \dot{H}^{-ω}(\R) follows for s >-{3/4}(α-1) and ω=1/α-1/2. This extends to global well-posedness for all s \geq 0.

math.AP↗