TY - JOUR

T1 - Blow-up in multidimensional aggregation equations with mildly singular interaction kernels

AU - Bertozzi, Andrea L.

AU - Carrillo, José A.

AU - Laurent, Thomas

PY - 2009/4/27

Y1 - 2009/4/27

N2 - We consider the multidimensional aggregation equation ut - ∇Δ (u∇ΔK * u) = 0 in which the radially symmetric attractive interaction kernel has a mild singularity at the origin (Lipschitz or better). In the case of bounded initial data, finite time singularity has been proved for kernels with a Lipschitz point at the origin (Bertozzi and Laurent 2007 Commun. Math. Sci. 274 717-35), whereas for C2 kernels there is no finite-time blow-up. We prove, under mild monotonicity assumptions on the kernel K, that the Osgood condition for well-posedness of the ODE characteristics determines global in time well-posedness of the PDE with compactly supported bounded nonnegative initial data. When the Osgood condition is violated, we present a new proof of finite time blow-up that extends previous results, requiring radially symmetric data, to general bounded, compactly supported nonnegative initial data without symmetry. We also present a new analysis of radially symmetric solutions under less strict monotonicity conditions. Finally, we conclude with a discussion of similarity solutions for the case K(x) = |x| and some open problems. © 2009 IOP Publishing Ltd and London Mathematical Society.

AB - We consider the multidimensional aggregation equation ut - ∇Δ (u∇ΔK * u) = 0 in which the radially symmetric attractive interaction kernel has a mild singularity at the origin (Lipschitz or better). In the case of bounded initial data, finite time singularity has been proved for kernels with a Lipschitz point at the origin (Bertozzi and Laurent 2007 Commun. Math. Sci. 274 717-35), whereas for C2 kernels there is no finite-time blow-up. We prove, under mild monotonicity assumptions on the kernel K, that the Osgood condition for well-posedness of the ODE characteristics determines global in time well-posedness of the PDE with compactly supported bounded nonnegative initial data. When the Osgood condition is violated, we present a new proof of finite time blow-up that extends previous results, requiring radially symmetric data, to general bounded, compactly supported nonnegative initial data without symmetry. We also present a new analysis of radially symmetric solutions under less strict monotonicity conditions. Finally, we conclude with a discussion of similarity solutions for the case K(x) = |x| and some open problems. © 2009 IOP Publishing Ltd and London Mathematical Society.

U2 - 10.1088/0951-7715/22/3/009

DO - 10.1088/0951-7715/22/3/009

M3 - Article

VL - 22

SP - 683

EP - 710

IS - 3

ER -