Measure valued solutions to the spatially homogeneous Boltzmann equation without angular cutoff

Abstract

A uniform approach is introduced to study the existence of measure valued solutions to the homogeneous Boltzmann equation for both hard potential with finite energy, and soft potential with finite or infinite energy, by using Toscani metric. Under the non-angular cutoff assumption on the cross-section, the solutions obtained are shown to be in the Schwartz space in the velocity variable as long as the initial data is not a single Dirac mass without any extra moment condition for hard potential, and with the boundedness on moments of any order for soft potential.

33 pages

References in corpus (1)

• On the uniqueness for the spatially homogeneous Boltzmann equation with a strong angular singularity

Cited by in corpus (8)

• A fast Fourier spectral method for the homogeneous Boltzmann equation with non-cutoff collision kernels
• The Enskog process for hard and soft potentials
• Self-similar profiles for homoenergetic solutions of the Boltzmann equation for non-cutoff Maxwell molecules
• On the Measure Valued Solution to the Inelastic Boltzmann Equation with Soft Potentials
• Measure Valued Solution to the Spatially Homogeneous Boltzmann Equation with Inelastic Long-Range Interactions
• Frequency multiplier estimates for the linearized relativistic Boltzmann operator without angular cutoff
• Moment estimates and well-posedness of the binary-ternary Boltzmann equation
• Analytic smoothing effect for the nonlinear Landau equation of Maxwellian molecules