1 Khmelnik S. I. Mathematical Model of Ball Lightning Abstract Based on the Maxwell's equations and on the understanding of the electrical conductivity of the body of ball lightning, a mathematical model of ball lightning is built; the structure of the electromagnetic field and of electric current in it is shown. Next it is shown (as a consequence of this model) that in a ball lightning the flow of electromagnetic energy can circulate and thus the energy obtained by a ball lightning when it occurs can be saved. Sustainability, luminescence, charge, time being, the mechanism of formation and physical modeling of ball lightning are briefly discussed. Content 1. Introduction 2. The solution of Maxwell equations in spherical coordinates 3. Electric currents 4. The Energy Flow 5. About Ball Lightning Stability 6. About Luminescence of the Ball Lightning 7. About the Time of Ball Lightning Existence 8. About a Possible Mechanism of Ball Lightning Formation 9. About the Charge of Ball Lightning 10. About Physical Simulation of the Ball Lightning Appendix 1. Refinement of Solution Appendix 2. "Cubical ball lightning" References 1. Introduction The hypotheses that were made about the nature of ball lightning are unacceptable because they are contrary to the law of energy conservation. This occurs because the luminescence of ball lightning is usually attributed to the energy released in any molecular or chemical transformation, and so it is suggested source of energy, due to which the ball lightning glows is located in it. Kapitsa P.L. 1955 [1]