© Hindawi Publishing Corp. ON LOCAL SMOOTH SOLUTIONS FOR THE VLASOV EQUATION WITH THE POTENTIAL OF INTERACTIONS ±r−2

For the initial value problem for the Vlasov equation with the potential of interactions±r−2, we prove the existence and uniqueness of a local solution with values in the Schwartz space S of infinitely differentiable functions rapidly decaying at infinity. 2000 Mathematics Subject Classification: 35Q72, 82C05, 82C40. 1. Introduction, notation, and results. Vlasov equation and its various modifica-tions are classical equations of physics. They appear in the mean-field approximations of the dynamics of a large number of interacting classical particles (molecules). Cur-rently, there is a numerous literature devoted to its mathematical treatments. In par-ticular, in [2, 5, 7, 12] a well-posedness for this equation supplied with initial data an