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

By Andrea L. Bertozzi, José Antonio Carrillo, Thomas Laurent. 2009

238 citations recorded by the source corpus.

Subfields: Applied Mathematics, Modeling and Simulation, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/4e0c2d754485e0c78a53f9538753094eafe355e8

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In Applied Mathematics

In Modeling and Simulation

In Public Health, Environmental and Occupational Health

Citations in the corpus

The derivation of swarming models: Mean-field limit and Wasserstein distances (2013, background): “For the local well-posedness of solutions to equation (3.1), we refer to [9, 10, 11, 58].”
Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion (2010, background): “Some recent work on this problem includes [5, 3, 4] for the inviscid case [26, 25, 24] for the problem with fractional diffusion and [27] for the problem in 1D with nonlinear diffusion.”
Stationary states of quadratic diffusion equations with long-range attraction (2011, background): “edium equation, which are nicely collected in the book [53], as well as the papers [45, 19]. In the purely nonlocal case, namely when "= 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 com”
Lp theory for the multidimensional aggregation equation (2011, background): “For the inviscid case, much work has been done recently on the question of finite time blowup in equations of this type, from bounded or smooth initial data [10, 6, 4, 5].”
Recent progress in the theory of Nonlinear Diffusion with Fractional Laplacian Operators (2014, background): “(30) The case m = m1 is borderline and has a logarithmic correction.”
On minimizers of interaction functionals with competing attractive and repulsive potentials (2015, background): “2) These functionals are directly connected to a class of self-assembly/aggregation models which recently have received much attention (see for example, [9, 11, 12, 19, 33, 36, 38, 39, 40, 47, 54, 58, 59]).”
Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations (2011, background): “On the other hand, in the works that study potentials that do produce blow-up [27] [7] [6] [5] the notion of the solution breaks down at the blow-up time.”
Ring patterns and their bifurcations in a nonlocal model of biological swarms (2015, background): “The instabilities of these co-dimension one, radially symmetric solutions nicely characterize the resulting ground state even when the resulting pattern is not co-dimension one.”
Dimensionality of Local Minimizers of the Interaction Energy (2012, background): “ion 5 we will discuss in more detail these questions. Let us nally mention that the gradient ow of the energy E W in the Wasserstein sense d 2 [12, 1, 13] has been extensively studied in recent years [24, 7, 5, 6, 8, 11, 10, 9, 17, 18, 30, 3, 2]. It leads to the nonlocal interaction equation @ @t + ”
Asymptotic Flocking Dynamics for the Kinetic Cucker-Smale Model (2010, result): “Some of the above ideas are reminiscent of arguments used for continuum models of swarming in [13, 3 ].”
A review of the mean field limits for Vlasov equations (2014, background): “Without any particular assumptions, we refer to [17] for a general L theory for aggregation equations, and to [16] for a proof of blow-up for kernels F with singularity.”