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 crosssection, 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.

On Mittag-Leffler Moments for the Boltzmann Equation for Hard Potentials Without Cutoff

“…One example are weak solutions, whose existence was proven by Arkeryd [6] and later extended by Villani [31], under the assumption that initial data has finite mass, energy, entropy and a moment of order 2 + δ, for any δ > 0. Another type of solutions that could be used are measure weak solutions constructed by Lu and Mouhot [25] (see also the result of Morimoto, Wang and Yang [27]). These solutions exist if initial mass and energy are finite, provided that the angular kernel satisfies the following condition π 0 b(cos θ) sin d θ(1 + | log(sin θ)|) < ∞ , which automatically holds for kernels that satisfy condition (2.6) with β < 2.…”

On the Measure Valued Solution to the Inelastic Boltzmann Equation with Soft Potentials

“…Besides, Lu-Mouhot showed existence of weak measure-valued solution without cutoff assumption for hard potential case, having finite mass and energy, in [17,18]. Moreover, Morimoto-Wang-Yang further studied the measure-valued solution in more general non-cutoff case (including finite energy and infinite energy initial datum, hard potential and soft potential), as well as the moments and smoothing property in their series paper [9,23,24].…”

“…In addition, the angular collision part b(cos θ) is an implicitly defined function, asymptotically behaving as, when θ → 0 + , sin θ b(cos θ) θ →0 + ∼ K θ −1−2s , for 0 < s < 1 and K > 0, (1.6) i.e., it has a non-integrable singularity when the deviation angle θ is small. The range of deviation angle θ , namely the angle between pre-and post-collisional velocities, is a full interval [0, π], but it is customary to restrict it to [0, π/2] mathematically, replacing b(cos θ) by its "symmetrized" version [24]:…”

Local well-posedness of the Boltzmann equation with polynomially decaying initial data

“…Even in this case, global-intime existence has only been proven when γ ≥ −2: see [20]. We should also mention measure-valued solutions, which are known to exist globally and regularize depending on the value of γ [36,38]. • Weak solutions.…”