STATIC SOLUTIONS TO THE SPHERICALLY SYMMETRIC EINSTEIN-VLASOV SYSTEM: A PARTICLE NUMBER-CASIMIR APPROACH

• The existence of nontrivial static spherically symmetric solutions to the Einstein- Vlasov system is well-known. However, it is an open problem whether or not static solutions arise as minimizers of a variational problem. Apart from being of interest in its own right, it is the connection to nonlinear stability that gives this topic its importance. This problem was considered in [G. Wolansky, Arch. Ration. Mech. Anal., 156 (2001), pp. 205-230], but as has been pointed out in [H. Andr\'easson and M. Kunze, Arch. Ration. Mech. Anal., 235 (2020), pp. 783-791], that paper contained serious flaws. In this work we construct static solutions by solving the Euler- Lagrange equation for the energy density ρ as a fixed point problem. The Euler-Lagrange equation originates from the particle number-Casimir functional introduced in Wolansky (2001). We then define a density function f on phase space which induces the energy density \rho and we show that it constitutes a static solution of the Einstein-Vlasov system. Hence we settle rigorously parts of what Wolansky (2001) attempted to prove.

Subject headings

NATURVETENSKAP  -- Matematik -- Beräkningsmatematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Computational Mathematics (hsv//eng)

NATURVETENSKAP  -- Matematik -- Matematisk analys (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Mathematical Analysis (hsv//eng)

Keyword

variational problem

Einstein-Vlasov system

static solutions

Einstein-Vlasov system

static solutions

variational problem

Publication and Content Type

art (subject category)

ref (subject category)