Journal of Mathematical Sciences

, Volume 238, Issue 5, pp 652–657 | Cite as

Weak Solutions of Hopf Type to 2D Maxwell Flows with Infinite Number of Relaxation Times

  • N. A. KarazeevaEmail author
A system of equations describing the motion of fluids of Maxwell type is considered:
$$ \frac{\partial }{\partial t}\upsilon +\upsilon \cdot \nabla \upsilon -\underset{0}{\overset{t}{\int }}K\left(t-\tau \right) d\tau +\nabla p=f\left(x,t\right),\kern0.5em di\upsilon\;\upsilon =0. $$
Here K(t) is an exponential series \( K(t)=\sum \limits_{s=1}^{\infty }{\beta}_s{e}^{-{\alpha}_st} \). The existence of a weak solution for the initial boundary value problem
$$ {\left.\begin{array}{ccc}\upsilon \left(x,0\right)={\upsilon}_0(x),& {\left.\upsilon \cdot n\right|}_{\partial \varOmega }=0,& rot\end{array}\;\upsilon \right|}_{\partial \varOmega }=0 $$

is proved.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach Sc. Pub., New York (1969).Google Scholar
  2. 2.
    A. P. Oskolkov, “On some models of nonstationary systems in the theory of non-Newtonian fluids. IV,” Zap. Nauchn. Semin. POMI, 110, 141–162 (1981).zbMATHGoogle Scholar
  3. 3.
    A. P. Oskolkov, “To the theory of nonstationary flows of the Maxwell fluids and the water solutions of polymers,” Zap. Nauchn. Semin. LOMI, 127, 158–168 (1983).zbMATHGoogle Scholar
  4. 4.
    N. A. Karazeeva, A. A. Cotsiolis, and A. P. Oskolkov, “On dynamical systems generated by initial boundary value problems for the equations of motion of linear viscoelastic fluids,” J. Math. Sci., 3, 73–108 (1991).zbMATHGoogle Scholar
  5. 5.
    E. Hopf, “Über die Anfangswertaufgabe fur die hydrodynamischen Grundgleichungen,” Math. Nachrichten, 4, 213–231 (1950–51).Google Scholar
  6. 6.
    E. Hopf, “Ein allgemeiner Endlichkeitsatz der Hydrodynamik,” Math. Ann., 117, 764–775 (1941).MathSciNetzbMATHGoogle Scholar
  7. 7.
    G. Astarita and G. Marrucci, Principles of non-Newtonian fluid mechanics, Mc-Graw-Hill (1974).Google Scholar
  8. 8.
    V. V. Vlasov and D. A. Medvedev, “Functional differential equations in Sobolev spaces and related problems of spectral theory,” J. Math. Sci., 164, No. 5, 659–841 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    M. E. Gurtin and A. C. Pipkin, “Theory of heat conduction with finite wave speed,” Archive Rat. Mech. Anal., 31, 113–126 (1968).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    S. A. Ivanov and L. Pandolfi, “Heat equations with memory: lack of controllability to the rest,” J. Math. Anal. Appl., 355, 1–11 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    N. A. Karazeeva, “Correct solvability of integro-differential equations in classes of generalized solutions,” Funct. Diff. Equations, 19, No. 1–2, 125–139 (2012).MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.St. Petersburg Department of the Steklov Mathematical Institute, RASSt. PetersburgRussia

Personalised recommendations