Abstract
We study a linearly transformed particle method for the aggregation equation with smooth or singular interaction forces. For the smooth interaction forces, we provide convergence estimates in L^1 and L^infty norms depending on the regularity of the initial data. Moreover, we give convergence estimates in bounded Lipschitz distance for measure valued solutions. For singular interaction forces, we establish the convergence of the error between the approximated and exact flows up to the existence time of the solutions in L^1 cap L^p norm.
Highlights
In this work, we are interested in showing the convergence of approximated particle schemes to the Cauchy problem for the so-called aggregation equation
Proof Using (3.12), (3.13) and the fact that eC t + C t ≤ e2C t, we find eF n+1 ≤ e2C t eF n + C t (h2 + t + hθn), eF n+1 ≤ e2C N t (eF 0 + N t (h2 + t + hθn)) ≤ C(h2 + t + hθn), n ≤ N − 1, follows by a summation using eF 0 = 0
Since h ≤ 1, we conclude that θn ≤ eC N t θ0 + eC N t h+ t h
Summary
We are interested in showing the convergence of approximated particle schemes to the Cauchy problem for the so-called aggregation equation. Global in time unique weak measure solutions can be constructed for any probability measure as initial data under suitable smoothness assumptions on the interaction potential. Under the above assumptions of either smooth or singular potentials, the proofs of the global-in-time well-posedness of weak measure solutions and the local-intime well-posedness of weak solutions for initial data in (L1 ∩ L p)(Rd ) spaces are essentially based on the fact that the velocity field is regular enough to have meaningful characteristics. Certain Sobolev regularity is needed on the initial data to obtain convergence of the LTP method in Lebesgue spaces for both smooth and singular potentials. Finite volume and finite difference schemes have been shown to be convergent to weak measure solutions of the aggregation equations for mildly singular potentials in [31,58].
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