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Nonlinear evolution equations on singular manifolds

This project concerns quasilinear parabolic problems on conic manifolds both with and without boundary and on edge manifolds via a maximal $$L^p$$-regularity approach. We are interested in results on the existence, uniqueness and regularity of solutions close to the singular points and in detecting effects that do not appear in the absence of singularities. The main examples we will consider are the porous medium equation, the Cahn-Hillard equation and the Yamabe flow

The objectives of this project can be summarized in four major parts:

1. Extend the results for cone differential operators on manifolds with boundary in the direction of maximal $$L^p$$-regularity theory for nonlinear PDEs.
2. Apply this to nonlinear problems on manifolds with conical singularities and boundary.
3. Show existence of a long time solution together with its asymptotics for conic manifolds with and without boundary.
4. Show existence, uniqueness and regularity results for the above-mentioned problems in the more complicated case of manifolds with edges.

## Publications

We consider the long-time behaviour of the mean curvature flow of spacelike hypersurfaces in the Lorentzian product manifold M×R, where M is asymptotically flat. If the initial hypersurface F0⊂M×R is uniformly spacelike and asymptotic to M×{s} for some s∈R at infinity, we show that the mean curvature flow starting at F0 exists for all times and converges uniformly to M×{s} as t→∞.

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):
30Nonlinear evolution equations on singular manifolds

In this paper we discuss Perelman's Lambda-functional, Perelman's Ricci shrinker entropy as well as the Ricci expander entropy on a class of manifolds with isolated conical singularities. On such manifolds, a singular Ricci de Turck flow preserving the isolated conical singularities exists by our previous work. We prove that the entropies are monotone along the singular Ricci de Turck flow. We employ these entropies to show that in the singular setting, Ricci solitons are gradient and that steady or expanding Ricci solitons are Einstein.

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

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

We decompose locally in time maximal $$L^{q}$$-regular solutions of abstract quasilinear parabolic equations as a sum of a smooth term and an arbitrary small−with respect to the maximal $$L^{q}$$-regularity space norm−remainder. In view of this observation, we next consider the porous medium equation and the Swift-Hohenberg equation on manifolds with conical singularities. We write locally in time each solution as a sum of three terms, namely a term that near the singularity is expressed as a linear combination of complex powers and logarithmic integer powers of the singular variable, a term that decays to zero close to the singularity faster than each of the non-constant summands of the previous term and a remainder that can be chosen arbitrary small with respect e.g. to the $$C^{0}$$-norm. The powers in the first term are time independent and determined explicitly by the local geometry around the singularity, e.g. by the spectrum of the boundary Laplacian in the situation of straight conical tips. The case of the above two problems on closed manifolds is also considered and local space asymptotics for the solutions are provided.

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

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## Team Members

Dr. Nikolaos Roidos
Researcher
Leibniz-Universität Hannover
roidos(at)math.uni-hannover.de

Prof. Dr. Elmar Schrohe