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research [2018/06/07 12:51]
ckoehler
research [2019/02/07 11:19] (current)
ckoehler [A. Analysis of differential and integro-differential equations]
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 ====== Research program ====== ====== Research program ======
  
-The aim of this International Research Training Group is to advance Mathematics in a joint initiative between Bielefeld University and Seoul National University. The focus will be on the mathematical analysis of problems which exhibit singular features or the in uence of randomness. Our research program is structured into the following four research areas:+The aim of this International Research Training Group is to advance Mathematics in a joint initiative between Bielefeld University and Seoul National University. The focus will be on the mathematical analysis of problems which exhibit singular features or the influence ​of randomness. Our research program is structured into the following four research areas:
  
   * **A. Analysis of differential and integro-differential equations** \\ Nonlinear dispersive equations, singular integrals, nonlocal generators of jump processes   * **A. Analysis of differential and integro-differential equations** \\ Nonlinear dispersive equations, singular integrals, nonlocal generators of jump processes
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 **Principal Researchers:​** Sebastian Herr (Bielefeld),​ Moritz Kassmann (Bielefeld)\\ **Principal Researchers:​** Sebastian Herr (Bielefeld),​ Moritz Kassmann (Bielefeld)\\
-**Scientific Partners:​** ​Kyeong-Hun Kim (Korea University), ​Panki Kim (SNU), Soonsik Kwon (KAIST), Ki-Ahm Lee (SNU), Sanghyuk Lee (SNU), Insuk Seo (SNU)\\+**Scientific Partners:​** ​ Panki Kim (SNU), Soonsik Kwon (KAIST), Ki-Ahm Lee (SNU), Sanghyuk Lee (SNU), Insuk Seo (SNU)\\
  
 The aim of this project is to study notoriously difficult Cauchy problems. We investigate nonlinear dispersive equations and systems, as well as integro-differential equations. Concerning nonlinear dispersive equations, such as nonlinear Schrödinger and Wave equations, well-posedness problems for initial data in Sobolev spaces of low regularity will be studied using methods from harmonic analysis. With regard to integro-differential operators, research projects will be offered that extend the theory for the fractional Laplace operator to anisotropic and/or nonlinear nonlocal operators. Characteristic properties of the aforementioned problems can often be described in terms of the underlying scaling behavior, e.g., when studying the interplay between dispersion and nonlinearity or the impact of singularities ​ The aim of this project is to study notoriously difficult Cauchy problems. We investigate nonlinear dispersive equations and systems, as well as integro-differential equations. Concerning nonlinear dispersive equations, such as nonlinear Schrödinger and Wave equations, well-posedness problems for initial data in Sobolev spaces of low regularity will be studied using methods from harmonic analysis. With regard to integro-differential operators, research projects will be offered that extend the theory for the fractional Laplace operator to anisotropic and/or nonlinear nonlocal operators. Characteristic properties of the aforementioned problems can often be described in terms of the underlying scaling behavior, e.g., when studying the interplay between dispersion and nonlinearity or the impact of singularities ​

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