Opis

• Title:
  A note on the $3$-D Navier-Stokes equations
• Creator:
  Cholewa, Jan W.;
  Dłotko, Tomasz
• Description:
  We consider the Navier-Stokes model in a bounded smooth domain $\Omega\subset \mathbb R^3$. Assuming a smallness condition on the external force $f$, which does not necessitate smallness of $\| f\|_{[L^2(\Omega)]^3}$-norm, we show that for any smooth divergence free initial data $u_0$ there exists ${\mathcal T}={\mathcal T}(\|u_0\|_{[L^2(\Omega)]^3})$ satisfying $$ {\mathcal T} \to 0 \quad \text{as } \|u_0\|_{[L^2(\Omega)]^3}\to 0 \quad \text{and} \quad {\mathcal T} \to \infty \quad \text{as } \|u_0\|_{[L^2(\Omega)]^3}\to \infty, $$% and such that either a corresponding regular solution ceases to exist until $\mathcal T$ or, otherwise, it is globally defined and approaches a maximal compact invariant set $\mathbb A$. The latter set $\mathbb A$ is a global attractor for the semigroup restricted to initial velocities $u_0$ in a certain ball of fractional power space $X^{1/4}$ associated with the Stokes operator, which in turn does not necessitate smallness of the gradient norm $\|\nabla u_0\|_{[L^2(\Omega)]^3}$. Moreover, $\mathbb A$ attracts orbits of bounded sets in $X$ through Leray-Hopf type solutions obtained as limits of viscous parabolic approximations.
• Object availability:
  Dostęp otwarty
• Rights:
  Prawa autorskie (c) 2018 Topological Methods in Nonlinear Analysis
• Date:
  2018.03.18
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  angielski
• Source:
  Topological Methods in Nonlinear Analysis; Vol 52, No 1 (September 2018); 195 - 212;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2017.049/14346 ;https://apcz.umk.pl/TMNA/article/view/TMNA.2017.049/14347
• Subject:
  Navier-Stokes equations;
  global solutions;
  small data;
  asymptotic behavior;
  35Q30;
  35B40
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2017.049
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  poproś o pozwolenie
• Type:
  pozostałe