Mittag-Leffler moments and weighted L∞ estimates for solutions to the Boltzmann equation for hard potentials without cutoff
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In this thesis we study analytic properties of solutions to the spatially homogeneous Boltzmann equation for collision kernels corresponding to hard potentials without the angular cutoff assumption, i.e. the angular part of the kernel is non-integrable with prescribed singularity rate. We study behavior in time of such solutions for large velocities i.e. their tails. We do this in two settings - L¹ and L∞. In the L¹ setting, we study Mittag-Leffler moments of solutions of the Cauchy problem under consideration. These moments, obtained by integrating the solution against a Mittag-Leffler function, are a generalization of exponential moments since Mittag-Leffler functions asymptotically behave like exponential functions. Mittag-Leffler moments can be also represented as infinite sums of renormalized polynomial moments. However, instead of considering renormaliztion by integer factorials that would lead to classical exponential moments, we renormalize by Gamma functions with non-integer arguments. By analyzing the convergence of partial sums sequences of these infinite sums, we prove the propagation and generation in time of Mittag-Leffler moments. In the case of propagation, orders of these moments depend on the singularity rate of the angular collision kernel. In the case of generation, the orders depend on the potential rate of the kernel. The proof uses a subtle combination of angular averaging and angular singularity cancellation, to show that partial sums satisfy an ordinary differential inequality with a negative term of the highest order while controlling all positive terms, whose solutions are uniformly bounded in time and number of terms. These techniques apply to both generation and propagation of Mittag-Leffler moments, with some variations depending on the case. In the L∞ setting, we prove that solutions to the Boltzmann equation that satisfy propagation in time of weightedL¹ bounds also satisfy propagation in time of weighted L∞ bounds. To emphasize that the propagation in time of weighted L∞ bounds relies on the propagation in time of weighted L¹ bounds, we express our main result using certain general weights. Consequently we apply the main result to cases of exponential and Mittag-Leffler weights, for which propagation in time of weighted L¹ bounds holds. Hence we obtain propagation in time of exponentially or Mittag-Leffler weighted L∞ bounds on the solution.