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SUMMARY:Alexander Shlapunov (Siberian Federal University\, Krasnoyarsk\, R
 ussia)
DTSTART:20201225T110000Z
DTEND:20201225T120000Z
DTSTAMP:20260423T005821Z
UID:mmandim/14
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/mmandim/14/"
 >Existence theorems for regular solutions to the Cauchy problem for the Na
 vier-Stokes equations in R^3</a>\nby Alexander Shlapunov (Siberian Federal
  University\, Krasnoyarsk\, Russia) as part of Mathematical models and int
 egration methods\n\n\nAbstract\nWe consider the Cauchy problem for the Nav
 ier-Stokes equations over ${\\mathbb R}^3 \\times [0\,T]$ with a positive 
 time $T$ over a specially constructed scale of function spaces of Bochner-
 Sobolev type. We prove that the problem induces an open both injective and
  surjective mapping of each space of the scale. In particular\, intersecti
 on of these classes gives a uniqueness and existence theorem for smooth so
 lutions to the Navier-Stokes equations for smooth data with a prescribed a
 symptotic behaviour at the infinity with respect to the time and the space
  variables. Actually\, we propose the following modified scheme of the pro
 of of the existence theorem\, based on apriori estimates and operator appr
 oach in Banach spaces:\n\n1. We prove that the Navier-Stokes equations ind
 uce continuous injective OPEN mapping between the chosen Banach spaces.\n\
 n2. Next\, the standard topological arguments immediately imply that a non
 empty open connected set in a topological vector space coincides with the 
 space itself if and only if the set is closed. This reduces the proof of t
 he existence theorem to an $L^\\mathfrak{s} ([0\,T]\, L^\\mathfrak{r} ({\\
 mathbb R^3}))$ a priori estimate for the INVERSE IMAGE OF PRECOMPACT SETS 
 in the target Banach space where $\\mathfrak{s}$\, $\\mathfrak{s}$ are Lad
 yzhenskaya-Prodi-Serrin numbers satisfying $2/\\mathfrak{s} + 3/\\mathfrak
 {r} = 1$ and $\\mathfrak{r} > 3$. In this way we avoid proving a GLOBAL $L
 ^\\mathfrak{s} ([0\,T]\, L^\\mathfrak{r} ({\\mathbb R^3}))$ a priori estim
 ate.\n\n3. To prove the weak $L^\\mathfrak{s} ([0\,T]\, L^\\mathfrak{r} ({
 \\mathbb R^3}))$ a priori estimate with $\\mathfrak{r} > 3$ we calculate p
 recisely the excess between the left hand side and the right hand side of 
 the corresponding energy inequality\, that equals to $2r$ when expressed i
 n terms of the Lebesgue integrability index $r$. Then we operate with abso
 lutely convergent series involving Lebesgue norms that gives the possibili
 ty to group together summands in a suitable way\, using the energy type in
 equalities\, interpolation inequalities and matching the asymptotic behavi
 our in order to exclude the unbounded sequences in the inverse image of a 
 precompact set.\n\nAn early version of the paper is uploaded on arxiv.org:
  https://arxiv.org/abs/2009.10530\nA similar approach can be used for inve
 stigation of the Navier-Stokes equations in the periodic setting: https://
 arxiv.org/abs/2007.14911\n
LOCATION:https://researchseminars.org/talk/mmandim/14/
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