Topological defects in the abelian Higgs model
Magdalena Czubak Robert L. Jerrard
Discrete & Continuous Dynamical Systems - A 2015, 35(5): 1933-1968 doi: 10.3934/dcds.2015.35.1933
We give a rigorous description of the dynamics of the Nielsen-Olesen vortex line. In particular, given a worldsheet of a string, we construct initial data such that the corresponding solution of the abelian Higgs model will concentrate near the evolution of the string. Moreover, the constructed solution stays close to the Nielsen-Olesen vortex solution.
keywords: Abelian Higgs model minimizers. vortices Nambu-Goto action cosmic strings timelike minimal surface Nielsen-Olesen vortex line
Low regularity well-posedness for the 2D Maxwell-Klein-Gordon equation in the Coulomb gauge
Magdalena Czubak Nina Pikula
Communications on Pure & Applied Analysis 2014, 13(4): 1669-1683 doi: 10.3934/cpaa.2014.13.1669
We consider the Maxwell-Klein-Gordon equation in 2D in the Coulomb gauge. We establish local well-posedness for $s=\frac 14+\epsilon$ for data for the spatial part of the gauge potentials and for $s=\frac 58+\epsilon$ for the solution $\phi$ of the gauged Klein-Gordon equation. The main tool for handling the wave equations is the product estimate established by D'Ancona, Foschi, and Selberg. Due to low regularity, we are unable to use the conventional approaches to handle the elliptic variable $A_0$, so we provide a new approach.
keywords: low regularity. Coulomb gauge null forms local well-posedness Maxwell-Klein-Gordon
Eventual regularization of the slightly supercritical fractional Burgers equation
Chi Hin Chan Magdalena Czubak Luis Silvestre
Discrete & Continuous Dynamical Systems - A 2010, 27(2): 847-861 doi: 10.3934/dcds.2010.27.847
We prove that a weak solution of a slightly supercritical fractional Burgers equation becomes Hölder continuous for large time.
keywords: Holder fractal oscillation supercritical De Giorgi. fractional Burgers

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