Modern Perspectives in Applied Mathematics: Theory and Numerics of PDEs

Global Maxwellians over all space and their relation to conserved quantities of classical kinetic equations

C. Dave Levermore

University of Maryland


When the Cauchy problem for a classical kinetic equation is posed over all D-dimensional Euclidean space, every solution with finite mass, energy, and second spatial moment formally has 4 + 2D + D(D - 1)/2 conserved quantities. The family of global Maxwellians with finite mass over all space has the same number of parameters. They are eternal solutions of the kinetic equation with finite mass, energy, and second spatial moment. We show that a unique such Maxwellian can be associated with each initial data of the Cauchy problem by matching the values of their conserved quantities. This Maxwellian is the minimizer of the Boltzmann entropy over all densities whose conserved quantities have these same values. The set of all such values is characterized by a bound on the trace norm of the angular momentum matrix.