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11) α1 c2 m1 . 12) u = ϕ(x) + ψ(x). 11) has a solution ϕ(x) = u∗ (x) + ϕ0 (x), ψ(x) = (1 − )u∗ (x) − ϕ0 (x), where = 4α12 c2 /(4α12 c2 − d12 ); ϕ0 an arbitrary harmonic function. 12). 1. Let distribution functions f1 , f2 satisfy condition A . 12); γ1 , λ1 , λ2 are arbitrary constants; ϕ0 (x) an arbitrary harmonic function; (∇u∗ , d1 ) = 0, (∇ϕ0 (x), d1 ) = 0. 12). 12) has a trivial solution u∗ (x) = 0. 12) is a well-known problem about bifurcation point for nonlinear equations. 5). 5), one has ∂ ∂n u∗ (x) + ϕ0 (x) | = 2α1 q1 e0 (x).

The corresponding Lagrangian for the weak relativity is called Darvin Lagrangian . 3. One can obtain Vlasov-Yang-Mills equations in a similar way by exchanging numbers with matrices instead of four potentials Aµ . Conclusion The system of Vlasov-Maxwell equations is obtained by variation of electromagnetic action (Lorentz action) with transition to distribution function. Equation for distribution function is obtained by shifting the equation along particles motion trajectories. 6 Derivation Scheme of Vlasov-Einstein Equation Let us consider the action for the particle in gravitation field and for field : SVE = gµν X˙ µ X˙ ν dqdt + √ −gRd4 x = Sp + Sf .

E. Horst has shown the global existence for classical solution of the modified VM system, where modification means that the current density j is replaced by the convolution j(·, t) ∗ δ, where δ is a member of a canonical sequence of δ-functions. Problem 10. , in the formula for the potential u and in the definition of the local density ρ, the integrals R2 over R3 are replaced by integrals R2 over R2 (over all x or all v, respectively). The question is: Can one obtain the solutions of the flat VP system from the solutions of the VP system (in 3D) by a suitable limit process?

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### A Course in Mathematics Volume II by Frederick S. And Frederick H. Bailey Woods

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