2019/10/15 by Christopher Henderson, Henderson, Christopher, Stanley Snelson +3
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.1910.07138
openalex publication_date 2019/10/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the Cauchy problem for the spatially inhomogeneous non-cutoff\nBoltzmann equation with polynomially decaying initial data in the velocity\nvariable. We establish short-time existence for any initial data with this\ndecay in a fifth order Sobolev space by working in a mixed L2 and L^\∞\nspace that allows to compensate for potential moment generation and obtaining\nnew estimates on the collision operator that are well-adapted to this space.\nOur results improve the range of parameters for which the Boltzmann equation is\nwell-posed in this decay regime, as well as relax the restrictions on the\ninitial regularity. As an application, we can combine our existence result with\nthe recent conditional regularity estimates of Imbert-Silvestre\n(arXiv:1909.12729 [math.AP]) to conclude solutions can be continued for as long\nas the mass, energy, and entropy densities remain under control. This\ncontinuation criterion was previously only available in the restricted range of\nparameters of previous well-posedness results for polynomially decaying initial\ndata.\n