Finite-time blow-up of L∞-weak solutions of an aggregation equation

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Local and global well-posedness for aggregation equations and Patlak–Keller–Segel models with degenerate diffusion (2010, methodology): “This is the approach taken in [4, 8], which we extend to handle singular kernels such as the Newtonian potential, proving uniqueness of weak solutions with no additional assumptions, provided the domain is bounded or d ! 3.”
Lp theory for the multidimensional aggregation equation (2011, methodology): “The results connect recent theory developed for L1 initial data [ 4 , 5, 6] to recent theory for measure solutions [16].”
Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations (2011, methodology): “Hence, for potentials satisfying (NL0)-(NL3) and assuming the same additional conditions of growth at infinity in ∇W as in [27] [7] [5] [20] [8], one can show that the velocity field ∇W * ρ belongs to L 1 ((0, T ); L 2 (ρ(t))) for all 0 < T < T * , see for instance the proofs in [7, Section 3],…”
Stationary states of quadratic diffusion equations with long-range attraction (2011, background): “In the purely nonlocal case, namely when ε = 0, this issue has been studied extensively in many papers [37, 13, 35, 5, 4, 16, 6, 7, 29, 25, 24], combined with the study of the regularity of solutions compared to the attractive singularity of the interaction kernel.”