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SUMMARY:Robert M. Strain (University of Pennsylvania)
DTSTART:20200625T140000Z
DTEND:20200625T145000Z
DTSTAMP:20260423T035631Z
UID:IMS/31
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/IMS/31/">Glo
 bal mild solutions of the Landau and non-cutoff Boltzmann equation</a>\nby
  Robert M. Strain (University of Pennsylvania) as part of PDE seminar via 
 Zoom\n\n\nAbstract\nIn this talk we explain a recent proof of the existenc
 e of small-amplitude global-in-time unique mild solutions to both the Land
 au equation including the Coulomb potential and the Boltzmann equation wit
 hout angular cutoff.  Since the well-known works (Guo\, 2002)  and (Gressm
 an-Strain-2011\, AMUXY-2012) on the construction of classical solutions in
  smooth Sobolev spaces which in particular are regular in the spatial vari
 ables\, it has still remained an open problem to obtain global solutions i
 n an $L^\\infty_{x\,v}$ framework\, similar to that in (Guo-2010)\, for th
 e Boltzmann equation with cutoff in general bounded domains.  \n\n\n\n\nOn
 e main difficulty arises from the interaction between the transport operat
 or and the velocity-diffusion-type collision operator in the non-cutoff Bo
 ltzmann and Landau equations\; another major difficulty is the potential f
 ormation of singularities for solutions to the boundary value problem. \n\
 n\n\n\nIn this work we introduce a new function space with low regularity 
 in the spatial variable to treat the problem in cases when the spatial dom
 ain is either a torus\, or a finite channel with boundary. For the latter 
 case\, either the inflow boundary condition or the specular reflection bou
 ndary condition is considered. An important property of the function space
  is that the $L^\\infty_T L^2_v$ norm\, in velocity and time\, of the dist
 ribution function is in the Wiener algebra  $A(\\Omega)$ in the spatial va
 riables.  \n\n\n\n\nBesides the construction of global solutions in these 
 function spaces\, we additionally study the large-time behavior of solutio
 ns for both hard and soft potentials\, and we further justify the property
  of propagation of regularity of solutions in the spatial variables.  To t
 he best of our knowledge these results may be the first ones to provide an
  elementary understanding of the existence theories for the Landau or non-
 cutoff Boltzmann equations in the situation where the spatial domain has a
  physical boundary.  \n\n\n\n\nThis is a joint work with Renjun Duan (The 
 Chinese University of Hong Kong)\, Shuangqian Liu (Jinan University) and S
 hota Sakamoto (Tohoku University).\n\nhttps://nguyenquochung1241.wixsite.c
 om/qhung/post/pde-seminar-via-zoom\n
LOCATION:https://researchseminars.org/talk/IMS/31/
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