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DTSTART;TZID=America/New_York:20211202T130000
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DTSTAMP:20240304T053731Z
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UID:10002571-1638450000-1638453600@cmsa.fas.harvard.edu
SUMMARY:Kerr Geodesics and Self-consistent match between Inspiral and Transition-to-merger
DESCRIPTION:Abstract: The two-body motion in General Relativity can be solved perturbatively in the small mass ratio expansion. Kerr geodesics describe the leading order motion. After a short summary of the classification of polar and radial Kerr geodesic motion\, I will consider the inspiral motion of a point particle around the Kerr black hole subjected to the self-force. I will describe its quasi-circular inspiral motion in the radiation timescale expansion. I will describe in parallel the transition-to-merger motion around the last stable circular orbit and prove that it is controlled by the Painlevé transcendental equation of the first kind. I will then prove that one can consistently match the two motions using the method of asymptotically matched expansions.
URL:https://cmsa.fas.harvard.edu/event/12-2-2021-general-relativity-seminar/
LOCATION:MA
CATEGORIES:General Relativity Seminar
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DTSTART;TZID=America/New_York:20211216T130000
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DTSTAMP:20240304T052540Z
CREATED:20240214T075612Z
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UID:10002573-1639659600-1639663200@cmsa.fas.harvard.edu
SUMMARY:Low regularity ill-posedness for 3D elastic waves and for 3D ideal compressible MHD driven by shock formation
DESCRIPTION:Abstract: We construct counterexamples to the local existence of low-regularity solutions to elastic wave equations and to the ideal compressible magnetohydrodynamics (MHD) system in three spatial dimensions (3D). Inspired by the recent works of Christodoulou\, we generalize Lindblad’s classic results on the scalar wave equation by showing that the Cauchy problems for 3D elastic waves and for 3D MHD system are ill-posed in $H^3(R^3)$ and $H^2(R^3)$\, respectively. Both elastic waves and MHD are physical systems with multiple wave speeds.  We further prove that the ill-posedness is caused by instantaneous shock formation\, which is characterized by the vanishing of the inverse foliation density. In particular\, when the magnetic field is absent in MHD\, we also provide a desired low-regularity ill-posedness result for the 3D compressible Euler equations\, and it is sharp with respect to the regularity of the fluid velocity.  Our proofs for elastic waves and for MHD are based on a coalition of a carefully designed algebraic approach and a geometric approach. To trace the nonlinear interactions of various waves\, we algebraically decompose the 3D elastic waves and the 3D ideal MHD equations into $6\times 6$ and $7\times 7$ non-strictly hyperbolic systems. Via detailed calculations\, we reveal their hidden subtle structures. With them\, we give a complete description of solutions’ dynamics up to the earliest singular event\, when a shock forms. This talk is based on joint works with Haoyang Chen and Silu Yin.
URL:https://cmsa.fas.harvard.edu/event/12-16-2021-general-relativity-seminar/
LOCATION:MA
CATEGORIES:General Relativity Seminar
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