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SUMMARY:Knots and links in fluid mechanics - Enciso\, A (Instituto de Cien
 cias Matemticas\; Madrid)
DTSTART:20120724T084500Z
DTEND:20120724T090500Z
UID:TALK39027@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:In this talk I will discuss the existence of steady solutions 
 to the incompressible Euler equations that have stream and vortex lines of
  any prescribed knot (or link) type. More precisely\, I will show that\, g
 iven any locally finite link L in R^3\, one can transform it by a smooth d
 iffeomorphism F\, close to the identity in any C^p norm\, such that F(L) i
 s a set of periodic trajectories of a real analytic steady solution u of t
 he Euler equations in R^3. If the link is finite\, we shall also see that 
 u can be assumed to decay as 1/|x| at infinity\, so that u is in L^p for a
 ll p>3. This problem is motivated by the well-known analysis of the struct
 ure of steady incompressible flows due to V.I. Arnold and K. Moffatt\, amo
 ng others. \n\nTime permitting\, we will also very recent results on the t
 opology of potential flows\, that is\, of steady fluids whose velocity fie
 ld is the gradient of a harmonic function in R^3. These results are closel
 y related to classic questions in potential theory that were first conside
 red by M. Morse and W. Kaplan in the first half of the XX century and have
  been revisited several times after that\, by Rubel\, Shiota and others. \
 n\nThe guiding principle of the talk will be that a strategy of "local\, a
 nalysis-based constructions" + "global approximation methods"\, fitted tog
 ether using ideas from differential topology\, can be used to shed some li
 ght on the qualitative behavior of steady fluid flows. Most of the origina
 l results presented in this talk will be based on the papers: \n\nA. Encis
 o\, D. Peralta-Salas\, Knots and links in steady solutions of the Euler eq
 uation\, Ann. of Math. 175 (2012) 345-367. \n\nA. Enciso\, D. Peralta-Sala
 s\, Submanifolds that are level sets of solutions to a second-order ellipt
 ic PDE\, arXiv:1007.5181. \n\nA. Enciso\, D. Peralta-Salas\, Arnold's stru
 cture theorem revisited\, in preparation.\n
LOCATION:Seminar Room 1\, Newton Institute
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