Abstract
In this paper, we consider distribution solutions to the aggregation equation \({\rho_{t} + \mathrm{div}(\rho \mathbf{u} ) = 0, \; \mathbf{u} = -\nabla V * \rho}\) in \({\mathbb{R}^{d}}\) , where the density ρ concentrates on a co-dimension one manifold. We show that an evolution equation for the manifold itself completely determines the dynamics of such solutions. We refer to such solutions aggregation sheets. When the equation for the sheet is linearly well-posed, we show that the fully non-linear evolution is also well-posed locally in time for the class of bi-Lipschitz surfaces. Moreover, we show that if the initial sheet is C 1 then the solution itself remains C 1 as long as it remains Lipschitz. Lastly, we provide conditions on the kernel \({g(s) = -\frac{\mathrm{d}V}{\mathrm{d}s}}\) that guarantee the solution remains a bi-Lipschitz surface globally in time, and construct explicit solutions that either collapse or blow up in finite time when these conditions fail.
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von Brecht, J.H., Bertozzi, A.L. Well-Posedness Theory for Aggregation Sheets. Commun. Math. Phys. 319, 451–477 (2013). https://doi.org/10.1007/s00220-012-1634-5
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DOI: https://doi.org/10.1007/s00220-012-1634-5