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UID:7da862c8c2894fff7712a62d111ca33d
CATEGORIES:Hyperbolic & Dispersive PDE Seminar
CREATED:20230417T195925
SUMMARY:Scattering for wave equations with sources and slowly decaying data. 
LOCATION:705
DESCRIPTION:Abstract:  We construct  solutions with prescribed radiation fields for wav
 e equations with polynomially decaying sources close to the lightcone. In t
 his setting, which is motivated by semi-linear wave equations satisfying th
 e weak null condition, solutions to the forward problem have a logarithmic 
 leading order term on the lightcone and non-trivial homogeneous asymptotics
  in the interior of the lightcone. The backward scattering solutions we con
 struct from knowledge of the source and the radiation field at null infinit
 y alone are given to second order  by explicit asymptotic solutions which s
 atisfy novel matching conditions close to the light cone. We also relate th
 e asymptotics of the radiation field towards space-like infinity to explici
 t  homogeneous solutions in the exterior of the light cone for slowly polyn
 omially decaying data corresponding to mass, charge and angular momentum in
  the applications. The somewhat surprising discovery is that these data can
  cause the same logarithmic radiation field as the source term. This requir
 es a delicate analysis of the forward homogeneous solution close to the lig
 ht cone using the invertibility of the Funk transform.This is joint work wi
 th Volker Schlue. 
X-ALT-DESC;FMTTYPE=text/html:<div style="border: 0px; font-style: normal; font-weight: 400; font-size: 1
 6px; line-height: inherit; font-family: Calibri, Arial, Helvetica, sans-ser
 if; margin: 0px; padding: 0px; vertical-align: baseline; color: #000000; le
 tter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text
 -transform: none; white-space: normal; widows: 2; word-spacing: 0px;">Abstr
 act:&nbsp;&nbsp;</div><div style="border: 0px; font-style: normal; font-wei
 ght: 400; font-size: 16px; line-height: inherit; font-family: Calibri, Aria
 l, Helvetica, sans-serif; margin: 0px; padding: 0px; vertical-align: baseli
 ne; color: #000000; letter-spacing: normal; orphans: 2; text-align: start; 
 text-indent: 0px; text-transform: none; white-space: normal; widows: 2; wor
 d-spacing: 0px;">We construct &nbsp;solutions with prescribed radiation fie
 lds for wave equations with polynomially decaying sources close to the ligh
 tcone. In this setting, which is motivated by semi-linear wave equations sa
 tisfying the weak null condition, solutions to the forward problem have a l
 ogarithmic leading order term on the lightcone and non-trivial homogeneous 
 asymptotics in the interior of the lightcone. The backward scattering solut
 ions we construct from knowledge of the source and the radiation field at n
 ull infinity alone are given to second order &nbsp;by explicit asymptotic s
 olutions which satisfy novel matching conditions close to the light cone. W
 e also relate the asymptotics of the radiation field towards space-like inf
 inity to explicit &nbsp;homogeneous solutions in the exterior of the light 
 cone for slowly polynomially decaying data corresponding to mass, charge an
 d angular momentum in the applications. The somewhat surprising discovery i
 s that these data can cause the same logarithmic radiation field as the sou
 rce term. This requires a delicate analysis of the forward homogeneous solu
 tion close to the light cone using the invertibility of the Funk transform.
 </div><div style="border: 0px; font-style: normal; font-weight: 400; font-s
 ize: 16px; line-height: inherit; font-family: Calibri, Arial, Helvetica, sa
 ns-serif; margin: 0px; padding: 0px; vertical-align: baseline; color: #0000
 00; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px
 ; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px;"
 >This is joint work with Volker Schlue.&nbsp;</div>
CONTACT:Hans Lindblad
DTSTAMP:20260827T005744
DTSTART;TZID=America/New_York:20230420T153000
DTEND;TZID=America/New_York:20230420T163000
SEQUENCE:0
TRANSP:OPAQUE
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