On nonlinear integro-differential Oskolkov-Stokes system with variable exponent of nonlinearity and small parameter
Анотація
Some nonlinear parabolic integro-differential system with the variable expo\-nent of the nonlinearity is considered. The initial-boundary value problem for this system is investigated and the existence and uniqueness theorems for the problem are proved. Some convergence results if parameter tends to zero are also obtained.
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R. Temam, Nivier-Stokes equations. Theory and numerical. North-Holland Publishing Company, 1979.
J. Simon, Nonhomogeneous viscous incompressible fluids: existence of velocity, density and preassure, SIAM J. Math. Anal. 21 (1990), no. 5, 1093--1117. DOI: 10.1137/0521061
J. A. Langa, J. Real, and J. Simon, Existence and regularity of the pressure for the stochastic Navier-Stokes equations, Appl. Math. Optimization 48 (2003), no. 3, 195--210. DOI: 10.1007/s00245-003-0773-7
L. Diening, P. Harjulehto, P. Hasto, and M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents, Springer, Heidelberg, 2011. DOI: 10.1007/978-3-642-18363-8
S. Antontsev and S. Shmarev, Evolution PDEs with nonstandard growth conditions. Existence, uniqueness, localization, blow-up, Atlantis Studies in Diff. Eq., Vol. 4, Paris: Atlantis Press, 2015. DOI: 10.2991/978-94-6239-112-3
V. Radulescu and D. Repovs, Partial differential equations with variable exponents: variational methods and qualitative analysis,
CRC Press, Boca Raton, London, New York, 2015.
O. Buhrii and N. Buhrii, Nonlocal in time problem for anisotropic parabolic equations with variable exponents of nonlinearities, J. Math. Anal. Appl. 473 (2019), no. 2, 695-711. DOI: 10.1016/j.jmaa.2018.12.058
O. M. Buhrii and N. V. Buhrii, Doubly nonlinear elliptic-parabolic variational inequalities with variable exponents of nonlinearities, Advances in Nonlin. Variat. Ineq. 22 (2019), no.~2, 1--22.
M. Bokalo, O. Buhrii, and N. Hryadil, Initial-boundary value problems for nonlinear elliptic-parabolic equations with variable exponents of nonlinearity in unbounded domains without conditions at infinity, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 192 (2020), Article ID 111700, 17 p.
DOI: 10.1016/j.na.2019.111700
O. Buhrii and N. Buhrii, Integro-differential systems with variable exponents of nonlinearity, Open Math. 15 (2017), 859-883. DOI: 10.1515/math-2017-0069
O. Buhrii and N. Buhrii, On initial-boundary value problem for nonlinear integro-differential equations with variable exponents of nonlinearity, NTMSCI 5 (2017), no. 3, 128-153. DOI: 10.20852/ntmsci.2017.191
M. M.Bokalo and O. V. Ilnytska, Classical solutions of problems for parabolic equations with variable integral delay, Bukovyn. Mat. Zh. 5 (2017), no. 1--2, 18--36 (in Ukrainian).
M. Bokalo and I. Skira, Almost periodic solutions for nonlinear integro-differential elliptic-parabolic equations with variable exponents of nonlinearty, Int. J. Evol. Equ. 10 (2017), no. 3-4, 297-314.
M. M. Bokalo and I. V. Skira, The Fourier problem for weakly nonlinear integro-differential elliptic-parabolic systems, Mat. Stud. 51 (2019), no. 1, 59--73. DOI: 10.15330/ms.51.1.59-73
A. P. Oskolkov, A small-parameter quasilinear parabolic system approximating the Navier–Stokes system, J. Sov. Math. 1 (1973), no. 4, 452-470. DOI: 10.1007/BF01084587
O. M. Buhrii, Visco-plastic, newtonian, and dilatant fluids: Stokes equations with variable exponent of nonlinearity, Mat. Stud. 49 (2018), no. 2, 165--180. DOI: 10.15330/ms.49.2.165-180
O. Buhrii and M. Khoma, On initial-boundary value problem for nonlinear integro-differential Stokes system, Visn. L’viv. Univ., Ser. Mekh.-Mat. 85 (2018), 107--119. DOI: 10.30970/vmm.2018.85.107-119
R. A. Adams, Sobolev spaces, Academic Press, New York, San Francisco, London, 1975.
H. Gajewski, K. Groger, und K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1974.
L. C. Evans, Partial differential equations, Graduate Studies in Mathematics,
Amer. Math. Soc., Providence, RI, 1998.
O. M. Buhrii, Finiteness of time vanishing of the solution of a nonlinear parabolic variational inequality with variable exponent of nonlinearity,
Mat. Stud. 24 (2005), no. 2, 167-172 (in Ukrainian).
T. Roubicek, Nonlinear partial differential equations with applications, Birkhauser Verlag, Basel, Boston, Berlin, 2005. DOI: 10.1007/978-3-0348-0513-1
E. Suhubi, Functional analysis, Kluwer Acad. Publ., Dordrecht, Boston, London, 2003.
J. W. Lee and D. O'Regan, Existence results for differential equations in Banach spaces, Comment. Math. Univ. Carolin. 34 (1993), no. 2, 239--251.
H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Springer, New York, Dordrecht, Heidelberg, London, 2011. DOI: 10.1007/978-0-387-70914-7
J.-P. Aubin, Un theoreme de compacite, C. R. Acad. Sci., Paris 256 (1963), no. 24, 5042--5044.
F. Bernis, Existence results for doubly nonlinear higher order parabolic equations on unbounded domains, Math. Ann. 279 (1988), 373-394.
DOI: 10.1007/BF01456275
O. Buhrii, G. Domans'ka, and N. Protsakh, Initial boundary value problem for nonlinear differential equation of the third order in generalized Sobolev spaces, Visn. L’viv. Univ., Ser. Mekh.-Mat. 64 (2005), 44-61.
J.-L. Lions, Quelques methodes de resolution des problemes aux limites non lineaire, Dunod, Gauthier-Villars, Paris, 1969.
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