## Prof. Dr. Elmar Schrohe

### Project leader

Leibniz-Universität Hannover

E-mail: schrohe(at)math.uni-hannover.de

Telephone: +49 511 762-3515

Homepage: http://www2.analysis.uni-hannover.de/~sc...

## Publications within SPP2026

We consider the Cahn-Hilliard equation on manifolds with conical singularities. For appropriate initial data we show that the solution exists in the maximal \(L^q\)-regularity space for all times and becomes instantaneously smooth in space and time, where the maximal \(L^q\)-regularity is obtained in the sense of Mellin-Sobolev spaces. Moreover, we provide precise information concerning the asymptotic behavior of the solution close to the conical tips in terms of the local geometry.

**Related project(s):****30**Nonlinear evolution equations on singular manifolds

We show a Heinz-Kato inequality in Banach spaces for sectorial operators having bounded imaginary powers.

**Related project(s):****30**Nonlinear evolution equations on singular manifolds

We consider the unnormalized Yamabe flow on manifolds with conical singularities. Under certain geometric assumption on the initial cross-section we show well posedness of the short time solution in the \(L^q\)-setting. Moreover, we give a picture of the deformation of the conical tips under the flow by providing an asymptotic expansion of the evolving metric close to the boundary in terms of the initial local geometry. Due to the blow up of the scalar curvature close to the singularities we use maximal \(L^q\)-regularity theory for conically degenerate operators.

**Related project(s):****30**Nonlinear evolution equations on singular manifolds

We study the porous medium equation on manifolds with conical singularities. Given strictly positive initial values, we show that the solution exists in the maximal \(L^q\)-regularity space for all times and is instantaneously smooth in space and time, where the maximal \(L^q\)-regularity is obtained in the sense of Mellin-Sobolev spaces. Moreover, we obtain precise information concerning the asymptotic behavior of the solution close to the singularity. Finally, we show the existence of generalized solutions for non-negative initial data.

Journal | Comm. Partial Differential Equations 43, no 10, 1456-1484 (2018) |

Link to preprint version | |

Link to published version |

**Related project(s):****30**Nonlinear evolution equations on singular manifolds