Lp theory for the multidimensional aggregation equation

By Andrea L. Bertozzi, Thomas Laurent, Jesús Rosado. 2011

173 citations recorded by the source corpus.

We consider well‐posedness of the aggregation equation ∂tu + div(uv) = 0, v = −▿K * u with initial data in \input amssym ${\cal P}_2 {\rm (\Bbb R}^d {\rm )} \cap L^p ({\Bbb R}^d )$ in dimensions 2 and higher. We consider radially symmetric kernels where the singularity at the origin is of order |x|α, α > 2 − d, and prove local well‐posedness in \input amssym ${\cal P}_2 { (\Bbb R}^d {\rm )} \cap L^p ({\Bbb R}^d )$ for sufficiently large p < ps. In the special case of K(x) = |x|, the exponent ps = d/(d = 1) is sharp for local well‐posedness in that solutions can instantaneously concentrate mass for initial data in \input amssym ${\cal P}_2 { (\Bbb R}^d {\rm )} \cap L^p ({\Bbb R}^d )$ with p < ps. We also give an Osgood condition on the potential K(x) that guarantees global existence and uniqueness in \input amssym ${\cal P}_2 { (\Bbb R}^d {\rm )} \cap L^p ({\Bbb R}^d )$. © 2010 Wiley Periodicals, Inc.

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

View Paper: https://www.semanticscholar.org/paper/ff90cf197f0e840dc427282fd5c9fcd2e75f0de9

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Citations in the corpus

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.”
Dimensionality of Local Minimizers of the Interaction Energy (2012, background): “Well-posedness theories for these repulsiveattractive potentials in various functional settings have been provided in [24, 1, 8, 11, 9, 3].”
Swarm dynamics and equilibria for a nonlocal aggregation model (2011, background): “A significant component of this research has dealt with the well-posedness of the initial-value problem for (1) [9, 10, 11, 12, 13].”
The derivation of swarming models: Mean-field limit and Wasserstein distances (2013, background): “More precisely, Bertozzi et al. [11, Theorem 1.1] showed that if ∇ W ∈ W 1 ,p ′ ( R d ) and ρ 0 ∈ L p ( R d ) ∩ P 2 ( R d ), then there exists T ∗ > 0 and a unique nonnegative solution to (3.1) satisfying Unfortunately, one can not directly apply those results for potentials satisfying assumptions…”
Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations (2011, methodology): “Let us also remark that in the works [27] [7] [5] [20] [8] the energy identity (2.39) was used as a tool for proving blow-up of the L p -norm in finite time.”
Uniqueness for Keller-Segel-type chemotaxis models (2012, background): “ic properties that imply the uniqueness for bounded solutions. Let us ﬁnally mention that part of the strategy is related to the uniqueness of solutions to ﬂuid and aggregation equations developed in [34, 27, 5, 28, 19, 7, 29]. The main novelty here is the interplay between the diﬀusive and the aggr”
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”