Small Data Global Well-Posedness for a Boltzmann Equation via Bilinear Spacetime Estimates

Well/ill-posedness bifurcation for the Boltzmann equation with constant collision kernel

“…Of course, the Boltzmann equation is certainly very different from the nonlinear dispersive PDEs and the nuts and bolts designed for one, so far, do not fit the other. Interestingly, the recent series of papers [11][12][13] by T. Chen, Denlinger, and Pavlović suggests that a systematic study of the Boltzmann equation using tools built for the dispersive PDEs might indeed be possible.…”

“…Technical problems remain (if they do not get worse) though, since the form of the collision operators does not improve and derivatives are complicated under the Wigner transform while the analysis of (1.1) was not very developed for a long time. In [11][12][13], incorporating the new developments [14-37, 44, 46, 47, 52, 55, 63-65] on the quantum many-body hierarchy dynamics that rely on the analysis of (1.1), with some highly nontrivial technical improvisions, T. Chen, Denlinger, and Pavlović provided an alternate dispersive PDE based route for proving the local well-posedness of the Boltzmann equation. The solutions provided in [11][12][13] differ from previous work in the sense that they solve the Boltzmann equation a.e.…”

“…In [11][12][13], incorporating the new developments [14-37, 44, 46, 47, 52, 55, 63-65] on the quantum many-body hierarchy dynamics that rely on the analysis of (1.1), with some highly nontrivial technical improvisions, T. Chen, Denlinger, and Pavlović provided an alternate dispersive PDE based route for proving the local well-posedness of the Boltzmann equation. The solutions provided in [11][12][13] differ from previous work in the sense that they solve the Boltzmann equation a.e. instead of everywhere in time.…”

“…We follow the lead of [11][12][13], but we study ill-posedness instead of well-posedness in this paper (and we are not using the Wigner transform). It is of mathematical and physical interest to prove well-posedness at the "optimal" regularity as it would mean all the nonlinear interactions of the complicated underlying nonlinear equation have been analyzed and accounted for.…”

On the emergence of quantum Boltzmann fluctuation dynamics near a Bose-Einstein Condensate

“…The well-posedness problem for the Boltzmann equation is a large and important research area; we will not address questions of this type in this work, but refer, for instance, to [4,5,6,7,8,9,10,14,19,34,35,25,50,52,53,54,78,81,85,86,105,107,110,108,116,117,149,155,156,157,158,165,202,201,203,207,208,209,211] and references therein.…”