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J. Colliander

Publications and source records attributed to J. Colliander.

23 records · Page 2Linked to original sources
arXiv2002

A refined global well-posedness result for Schrodinger equations with derivative

In this paper we prove that the 1D Schrödinger equation with derivative in the nonlinear term is globally well-posed in $H^{s}$, for $s>\frac12$ for data small in $L^{2}$. To understand the strength of this result one should recall that for $s<\frac12$ the Cauchy problem is ill-posed, in the sense that uniform continuity with respect to the initial data fails. The result follows from the method of almost conserved energies, an evolution of the ``I-method'' used by the same authors to obtain global well-posedness for $s>\frac23$. The same argument can be used to prove that any quintic nonlinear defocusing Schrödinger equation on the line is globally well-posed for large data in $H^{s}$, for $s>\frac12$.

math.AP↗
arXiv2001

On solutions for the Kadomtsev-Petviashvili I equation

Oscillatory integral techniques are used to study the well-posedness of the KP-I equation for initial data that are small with respect to the norm of a weighted Sobolev space involving derivatives of total order no larger than 2.

math.AP↗
arXiv2001

Sharp Global well-posedness for KdV and modified KdV on $\R$ and $\T$

The initial value problems for the Korteweg-de Vries (KdV) and modified KdV (mKdV) equations under periodic and decaying boundary conditions are considered. These initial value problems are shown to be globally well-posed in all $L^2$-based Sobolev spaces $H^s$ where local well-posedness is presently known, apart from the $H^{1/4} (\R)$ endpoint for mKdV. The result for KdV relies on a new method for constructing almost conserved quantities using multilinear harmonic analysis and the available local-in-time theory. Miura's transformation is used to show that global well-posedness of modified KdV is implied by global well-posedness of the standard KdV equation.

math.AP↗
arXiv2001

Global well-posedness for Schrödinger equations with derivative

We prove that the 1D Schrödinger equation with derivative in the nonlinear term is globally well-posed in $H^{s}$, for $s>2/3$ for small $L^{2}$ data. The result follows from an application of the ``I-method''. This method allows to define a modification of the energy norm $H^{1}$ that is ``almost conserved'' and can be used to perform an iteration argument. We also remark that the same argument can be used to prove that any quintic nonlinear defocusing Schrödinger equation on the line is globally well-posed for large data in $H^{s}$, for $s>2/3$ .

math.AP↗