arXiv subjects

E. Compaan

Publications and source records attributed to E. Compaan.

4 recordsLinked to original sources

arXiv2019

Pointwise Convergence of the Schr\"odinger Flow

In this paper we address the question of the pointwise almost everywhere limit of nonlinear Schr\"odinger flows to the initial data, in both the continuous and the periodic settings. Then we show how, in some cases, certain smoothing effects for the non-homogeneous part of the solution can be used to upgrade to a uniform convergence to zero of this part, and we discuss the sharpness of the results obtained. We also use randomization techniques to prove that with much less regularity of the initial data, both in continuous and the periodic settings, almost surely one obtains uniform convergence of the nonlinear solution to the initial data, hence showing how more generic results can be obtained.

math.AP↗

arXiv2018

A note on global existence for the Zakharov system on $\mathbb{T}$

We show that the one-dimensional periodic Zakharov system is globally well-posed in a class of low-regularity Fourier-Lebesgue spaces. The result is obtained by combining the I-method with Bourgain's high-low decomposition method. As a corollary, we obtain probabilistic global existence results in $L^2$-based Sobolev spaces. We also obtain global well-posedness in $H^{\frac12+} \times L^2$, which is sharp (up to endpoints) in the class of $L^2$-based Sobolev spaces.

math.AP↗

arXiv2018

Low-regularity global well-posedness for the Klein-Gordon-Schr\"odinger system on $\mathbb R^{+}$

In this paper we establish an almost optimal well-posedness and regularity theory for the Klein-Gordon-Schr\"odinger system on the half line. In particular we prove local-in-time well-posedness for rough initial data in Sobolev spaces of negative indices. Our results are consistent with the sharp well-posedness results that exist in the full line case and in this sense appear to be sharp. Finally we prove a global well-posedness result by combining the $L^2$ conservation law of the Schr\"odinger part with a careful iteration of the rough wave part in lower order Sobolev norms.

math.AP↗

arXiv2016

Smoothing for the Zakharov & Klein-Gordon-Schrödinger Systems on Euclidean Spaces

This paper studies the regularity of solutions to the Zakharov and Klein-Gordon-Schrödinger systems at low regularity levels. The main result is that the nonlinear part of the solution flow falls in a smoother space than the initial data. This relies on a new bilinear $X^{s,b}$ estimate, which is proved using delicate dyadic and angular decompositions of the frequency domain. Such smoothing estimates have a number of implications for the long-term dynamics of the system. In this work, we give a simplified proof of the existence of global attractors for the Klein-Gordon-Schrödinger flow in the energy space for dimensions $d = 2,3$. Secondly, we use smoothing in conjunction with a high-low decomposition to show global well-posedness of the Klein-Gordon-Schrödinger evolution on $\mathbb{R}^4$ below the energy space for sufficiently small initial data.

math.AP↗