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SUMMARY:Singular Limits and Long-Time Behaviour in Fluid Mechanics Models 
 via the Relative Entropy Method - Agnieszka Świerczewska-Gwiazda\, Univer
 sity of Warsaw
DTSTART:20251023T140000Z
DTEND:20251023T150000Z
UID:TALK239263@talks.cam.ac.uk
CONTACT:Matthew Colbrook
DESCRIPTION:Singular limits in partial differential equations occur when c
 ertain parameters reach extreme regimes\, leading to changes in regularity
 \, the appearance of singularities\, or transitions between different phys
 ical behaviors. Understanding such limits is essential for linking mathema
 tical models across scales and for describing complex phenomena in areas s
 uch as fluid dynamics\, materials science\, and astrophysics.\n\nIn this t
 alk\, I will present recent results concerning the high-friction limit for
  systems arising in fluid mechanics. Following this approach\, we rigorous
 ly derive the nonlocal Cahn–Hilliard equation as a limit of the nonlocal
  Euler–Korteweg equation using the relative entropy method. \nBy applyin
 g recent results on the connection between nonlocal and local Cahn–Hilli
 ard models\, we also rigorously obtain the large-friction nonlocal-to-loca
 l limit. The analysis is carried out for dissipative measure-valued soluti
 ons of the nonlocal Euler-Korteweg equation\, which are known to exist glo
 bally in time.\n\nThis framework provides a novel way to derive equations 
 that may lack classical solutions by introducing nonlocal effects in the f
 luid system and employing the relative entropy method. I will also discuss
  the high-friction limit of the Euler-Poisson system and various applicati
 ons of the relative entropy method in fluid mechanics\, including weak-str
 ong uniqueness results and asymptotic limits.\n\nFinally\, I will focus on
  a recent result concerning the unconditional stability of certain radiall
 y symmetric steady states of compressible viscous fluids in domains with i
 nflow/outflow boundary conditions\, showing that any (not necessarily symm
 etric) solution of the corresponding evolutionary problem converges to a s
 ingle radially symmetric steady state.\n
LOCATION:Centre for Mathematical Sciences\, MR14
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