Geodesic
The weak solutions of Hopf type to 2D Maxwell flows with infinite number of relaxation times
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 47, Tome 461 (2017), pp. 140-147 
N. A. Karazeeva. The weak solutions of Hopf type to 2D Maxwell flows with infinite number of relaxation times. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 47, Tome 461 (2017), pp. 140-147. http://geodesic.mathdoc.fr/item/ZNSL_2017_461_a7/
@article{ZNSL_2017_461_a7,
     author = {N. A. Karazeeva},
     title = {The weak solutions of {Hopf} type to {2D} {Maxwell} flows with infinite number of relaxation times},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {140--147},
     year = {2017},
     volume = {461},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2017_461_a7/}
}
TY  - JOUR
AU  - N. A. Karazeeva
TI  - The weak solutions of Hopf type to 2D Maxwell flows with infinite number of relaxation times
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2017
SP  - 140
EP  - 147
VL  - 461
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2017_461_a7/
LA  - en
ID  - ZNSL_2017_461_a7
ER  - 
%0 Journal Article
%A N. A. Karazeeva
%T The weak solutions of Hopf type to 2D Maxwell flows with infinite number of relaxation times
%J Zapiski Nauchnykh Seminarov POMI
%D 2017
%P 140-147
%V 461
%U http://geodesic.mathdoc.fr/item/ZNSL_2017_461_a7/
%G en
%F ZNSL_2017_461_a7

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

The system of equations, describing motion of fluids of Maxwell type is considered $$ \frac\partial{\partial t}v+v\cdot\nabla v-\int_0^t K(t-\tau)\Delta v(x,\tau)\,d\tau+\nabla p=f(x,t),\quad\operatorname{div}v=0. $$ Here $K(t)$ is exponential series $K(t)=\sum_{s=1}^\infty\beta_se ^{-\alpha_st}$. The existence of weak solution for initial boundary value problem $$ v(x,0)=v_0(x),\quad v\cdot n|_{\partial\Omega}=0,\quad\operatorname{rot}v|_{\partial\Omega}=0 $$ is proved.

[1] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Math. Appl., 2, Gordon and Breach Sc. Pub., N.Y., 1969 | MR

[2] A. P. Oskolkov, “On some models of nonstationary systems in the theory of non-Newtonian fluids. IV”, Zap. Nauchn. Semin. POMI, 110, 1981, 141–162 | MR | Zbl

[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, 1983, 158–168 | MR | Zbl

[4] N. A. Karazeeva, A. A. Cotsiolis, A. P. Oskolkov, “On dynamical systems generated by initial boundary value problems for the equations of motion of linear viscoelastic fluids”, Proc. Steklov Inst. Math., 188 (1991), 73–108 | MR | Zbl

[5] E. Hopf, “Über die Anfangswertaufgabe fur die hydrodynamischen Grundgleichungen”, Math. Nachrichten, 4 (1950–51), 213–231 | DOI | MR

[6] E. Hopf, “Ein allgemeiner Endlichkeitsatz der Hydrodynamik”, Math. Ann., 117 (1941), 764–775 | MR | Zbl

[7] G. Astarita, G. Marrucci, Principles of non-Newtonian fluid mechanics, Mc-Graw-Hill, 1974

[8] V. V. Vlasov, D. A. Medvedev, “Functional differential equations in Sobolev spaces and related problems of spectral theory”, J. Math. Sci., 164:5 (2010), 659–841 | DOI | MR | Zbl

[9] M. E. Gurtin, A. C. Pipkin, “Theory of heat condaction with finite wave speed”, Archive Rat. Mech. Anal., 31 (1968), 113–126 | DOI | MR | Zbl

[10] S. A. Ivanov, L. Pandolfi, “Heat equations with memory: lack of controllability to the rest”, J. Math. Anal. Appl., 355 (2009), 1–11 | DOI | MR | Zbl

[11] N. A. Karazeeva, “Correct solvability of integro-differential equations in classes of generalized solutions”, Funct. Diff. Equations, 19:1–2 (2012), 125–139 | MR