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| * **D. Heat semigroups and Dirichlet forms on manifolds and metric spaces** \\ Stochastic differential equations and Sobolev regularity on infinite-dimensional and fractal state spaces, singular drifts, propagation speed on manifolds and metric spaces, heat kernel estimates for operators with singular coefficients and magnetic energy forms | * **D. Heat semigroups and Dirichlet forms on manifolds and metric spaces** \\ Stochastic differential equations and Sobolev regularity on infinite-dimensional and fractal state spaces, singular drifts, propagation speed on manifolds and metric spaces, heat kernel estimates for operators with singular coefficients and magnetic energy forms | ||
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| + | ===== A. Analysis of differential and integro-differential equations ===== | ||
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| + | **Principal Researchers: | ||
| + | **Scientific Partners:** Kyeong-Hun Kim (Korea University), | ||
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| + | 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 | ||