BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Alexander Shlapunov (Siberian Federal University\, Krasnoyarsk\, R ussia) DTSTART:20201009T110000Z DTEND:20201009T120000Z DTSTAMP:20260423T040740Z UID:mmandim/8 DESCRIPTION:Title: Existence Theorems for Regular Spatially Periodic Solutions to the Navier –Stokes Equations in R^3\nby Alexander Shlapunov (Siberian Federal U niversity\, Krasnoyarsk\, Russia) as part of Mathematical models and integ ration methods\n\n\nAbstract\nWe consider the initial problem for the Navi er–Stokes equations over ${\\mathbb R}^3 \\times [0\,T]$ with a positive time $T$ in the spatially periodic setting. Identifying periodic vector-v alued functions on ${\\mathbb R}^3$ with functions on the $3\\\,$-dimensio nal torus ${\\mathbb T}^3$\, we prove that the problem induces an open bot h injective and surjective mapping of specially constructed scale of funct ion spaces of Bochner–Sobolev type parametrised with the smoothness inde x $s\\in \\mathbb{N}$. The intersection of these classes with respect $s$ gives a uniqueness and existence theorem for smooth solutions to the Navie r–Stokes equations for each finite $T>0$. Then additional intersection w ith respect to $T\\in (0\, +\\infty)$ leads to a uniqueness and existence theorem for smooth solutions and data having prescribed asymptotic behavio ur at the infinity with respect to the time variable. Actually\, we propos e the following modified scheme of the proof of the existence theorem\, ba sed on apriori estimates and operator approach in Banach spaces:\n\n1. We prove that the Navier–Stokes equations induce continuous injective OPEN mapping between the chosen Banach spaces.\n\n2. Next\, the standard topolo gical arguments immediately imply that a nonempty open connected set in a topological vector space coincides with the space itself if and only if th e set is closed. This reduces the proof of the existence theorem to an $L^ \\mathfrak{s} ([0\,T]\, L^\\mathfrak{r} ({\\mathbb R^3}))$ a priori estima te for the INVERSE IMAGE OF PRECOMPACT SETS in the target Banach space whe re $\\mathfrak{s}$\, $\\mathfrak{r}$ are Ladyzhenskaya–Prodi–Serrin nu mbers 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 estimate.\n\n3. To prove the we ak $L^\\mathfrak{s} ([0\,T]\, L^\\mathfrak{r} ({\\mathbb R^3}))$ a priori estimate with $\\mathfrak{r} > 3$ we calculate precisely the excess betwee n the left hand side and the right hand side of the corresponding energy i nequality\, that equals to $2r$ when expressed in terms of the Lebesgue in tegrability index $r$. Then we operate with absolutely convergent series i nvolving Lebesgue norms that gives the possibility to group together summa nds in a suitable way\, using the energy type inequalities\, interpolation inequalities and matching the asymptotic behaviour in order to exclude th e unbounded sequences in the inverse image of a precompact set.\n LOCATION:https://researchseminars.org/talk/mmandim/8/ END:VEVENT END:VCALENDAR