Abstract
In this paper, we prove the global existence of the weak solution to the mean field kinetic equation derived from the N‐particle Newtonian system. For L1∩L∞ initial data, the solvability of the mean field kinetic equation can be obtained by using uniform estimates and compactness arguments while the difficulties arising from the nonlocal nonlinear interaction are tackled appropriately using the Aubin‐Lions compact embedding theorem.
Highlights
INTRODUCTIONWe investigate a two-dimensional kinetic mean field equation for the mass distribution f(t, x, v) with position x ∈ R2 and velocity v ∈ R2 given by
In this paper, we investigate a two-dimensional kinetic mean field equation for the mass distribution f(t, x, v) with position x ∈ R2 and velocity v ∈ R2 given by∂tf + v · ∇xf + ∇v · [( F ∗ f )f ] + ∇v · (Gf ) = 0. (1.1)Equation (1.1) is motivated by several applications such as crowd dynamics[1,2] or material flow[3] and has been investigated from a numerical and theoretical point of view, see for example previous studies[4,5,6] for a general overview
We prove the global existence of the weak solution to the mean field kinetic equation derived from the N-particle Newtonian system
Summary
We investigate a two-dimensional kinetic mean field equation for the mass distribution f(t, x, v) with position x ∈ R2 and velocity v ∈ R2 given by. We aim to prove the global existence of the weak solution to the mean field kinetic Equation (1.1). In the latter equation, F(x, v) denotes the total interaction force and has the similar structure as x |x|. The proposed model Equation (1.1) involves a singularity comparable with the Coulomb potential in 2-d, resulting from the total interaction force. That means that this singularity, or in other words the nonlocal term, needs extra care in the final limiting process.
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