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Looking for critical nonlinearity in the one-dimensional quasilinear Smoluchowski-Poisson system
The surface diffusion flow on rough phase spaces
| 1. | Institute for Applied Mathematics, Leibniz University of Hanover, D-30167 Hanover |
| 2. | Institute of Applied Mathematics and Mechanics, University of Warsaw, 02-097 Warszawa, Poland |
Combining tools from harmonic analysis, such as Besov spaces, multiplier results with abstract results from the theory of maximal regularity we present an analytic framework in which we can investigate weak solutions to the original evolution equation. This approach allows us to prove well-posedness on a large (Besov) space of initial data which is in general larger than $C^2$ (and which is in the distributional sense almost optimal). Our second main result shows that the set of all compact embedded equilibria, i.e. the set of all spheres, is an invariant manifold in this phase space which attracts all solutions which are close enough (which respect to the norm of the phase space) to this manifold. As a consequence we are able to construct non-convex initial data which generate global solutions, converging finally to a sphere.
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