Topological Methods in Nonlinear Analysis

Local strong solutions of the nonhomogeneous Navier-Stokes system with control of the interval of existence

Reinhard Farwig, Hermann Sohr, and Werner Varnhorn

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Consider a bounded domain $\Omega\subseteq \mathbb R^3$ with smooth boundary $\partial\Omega$, a time interval $[0,T)$, $0\le T\le \infty$, and in $[0,T) \times\Omega$ the completely nonhomogeneous Navier-Stokes system $u_t - \Delta u+u\cdot \nabla u + \nabla p = f$, $u|_t=0=v_0$, ${\rm div}\,u=k$, $u|_\partial\Omega = g$, with sufficiently smooth data $f,v_0,k,g$. In this general case there are mainly known two classes of weak solutions, the class of global weak solutions, similar as in the well known case $k=0$, $g=0$ which need not be unique, see [R. Farwig, H. Kozono and H. Sohr, Global weak solutions of the Navier-Stokes equations with nonhomogeneous boundary data and divergence, Rend. Sem. Math. Univ. Padova 125 (2011), 51-70], and the class of local very weak solutions, see [H. Amann, Nonhomogeneous Navier-Stokes Equations with Integrable Low-regularity Data, Int. Math. Ser., Kluwer Academic/Plenum Publishing, New York, 2002, 1-26], [H. Amann, Navier-Stokes equations with nonhomogenous Dirichlet data, J. Nonlinear Math. Phys. 10, Suppl. 1 (2003), 1-11], [R. Farwig, G.P. Galdi and H. Sohr, A new class of weak solutions of the Navier-Stokes equations with nonhomogeneous data, J. Math. Fluid Mech. 8 (2006), 423-444], which are uniquely determined but have no differentiability properties and need not satisfy an energy inequality. Our aim is to introduce the new class of local strong solutions in the usual sense for $k\not= 0$, $g\not=0$ satisfying similar regularity and uniqueness properties as in the well known case $k=0$, $g=0$. Further, we obtain precise information through the given data on the interval of existence $[0,T^*)$, $0\le T^*\le T$. Our proof is essentially based on a detailed analysis of the corresponding linear system.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 46, Number 2 (2015), 999-1012.

Dates
First available in Project Euclid: 21 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1458588671

Digital Object Identifier
doi:10.12775/TMNA.2015.074

Mathematical Reviews number (MathSciNet)
MR3494980

Zentralblatt MATH identifier
1362.35207

Citation

Farwig, Reinhard; Sohr, Hermann; Varnhorn, Werner. Local strong solutions of the nonhomogeneous Navier-Stokes system with control of the interval of existence. Topol. Methods Nonlinear Anal. 46 (2015), no. 2, 999--1012. doi:10.12775/TMNA.2015.074. https://projecteuclid.org/euclid.tmna/1458588671


Export citation

References

  • H. Amann, Nonhomogeneous Navier–Stokes Equations with Integrable Low-regularity Data, Int. Math. Ser., Kluwer Academic/Plenum Publishing, New York, 2002, 1–26.
  • ––––, Navier–Stokes equations with nonhomogenous Dirichlet data, J. Nonlinear Math. Phys. 10, Suppl. 1 (2003), 1–11.
  • R. Farwig, G.P. Galdi and H. Sohr, A new class of weak solutions of the Navier–Stokes equations with nonhomogeneous data, J. Math. Fluid Mech. 8 (2006), 423–444.
  • R. Farwig, H. Kozono and H. Sohr, Very weak, weak and strong solutions to the instationary Navier–Stokes system, Topics on Partial Differential Equations, J. Nečas Center for Mathematical Modeling, Lecture Notes, Vol. 2, (P. Kaplický, Š. Nečasová, eds.), pp. 15–68, Prague 2007.
  • ––––, Global weak solutions of the Navier–Stokes equations with nonhomogeneous boundary data and divergence, Rend. Sem. Math. Univ. Padova 125 (2011), 51–70.
  • R. Farwig, H. Sohr and W. Varnhorn, On optimal initial value conditions for local strong solutions of the Navier–Stokes equations, Ann. Univ. Ferrara 55 (2009), 89–110.
  • ––––, Necessary and sufficient conditions on local strong solvability of the Navier–Stokes system, Appl. Anal. 90 (2011), 47–58.
  • ––––, Extensions of Serrin's uniqueness and rgularity conditions for the Navier–Stokes equations, J. Math. Fluid Mech. 14 (2012), 529–540.
  • ––––, Besov space regularity conditions for weak solutions of the Navier-Stokes equations, J. Math. Fluid Mech. 16 (2014), 307–320.
  • H. Kozono and H. Sohr, Remarks on uniqueness of weak solutions of the Navier–Stokes equations, Analysis 16 (1996), 255–271.
  • K. Masuda, Weak solutions of Navier–Stokes equations, Tohoku Math. J. 36 (1984), 623–646.
  • J. Serrin, The initial value problem for the Navier–Stokes equations, Univ. Wisconsin Press, Nonlinear Problems (R.E. Langer, ed.), 1963.
  • H. Sohr, The Navier–Stokes Equations, Birkhäuser–Verlag, Basel–Boston–Berlin, 2001.