×

zbMATH — the first resource for mathematics

Measure-valued solutions and asymptotic behavior of a multipolar model of a boundary layer. (English) Zbl 0774.76008
The first chapter is devoted to the formulation of the problem for bipolar fluids. In the following two chapters, we deal with the global existence of weak solutions, regularity and smoothing effect. In the fourth chapter, the measure-valued solutions are defined and the behavior of the weak solutions under the vanishing higher viscosity is studied. The limits of strong viscous solutions exhibit the loss of regularity and satisfy equations of motion for monopolar fluids in the sense of regular measures. These conclusions are proved in chapter five. The sixth chapter concerns the uniqueness of weak solutions for bipolar fluids. In the last two chapters, we study the asymptotic behavior. The existence of the universal attractor with a finite Hausdorff dimension is proved.

MSC:
76A05 Non-Newtonian fluids
76D10 Boundary-layer theory, separation and reattachment, higher-order effects
35Q35 PDEs in connection with fluid mechanics
PDF BibTeX XML Cite
Full Text: EuDML
References:
[1] A.V. Babin, M.I. Višik: Attractors of partial differential equations and estimates of their dimensions. Uspekhi Mat. Nauk 38 (1983).
[2] J. Ball: A version of the fundamental theorem for Young measures. PDE’s and continuum models of phase transitions, Lecture Notes in Physics 344 (1989), 207-215. · Zbl 0991.49500
[3] H. Bellout, F. Bloom, J. Nečas: Phenomenological behavior multipolar viscous fluids. · Zbl 0759.76004
[4] R.E. Edwards: Functional analysis. Rinehart and Winston, Holt, 1965. · Zbl 0182.16101
[5] A.E. Green, R.S. Rivlin: Multipolar continuum mechanics. Arch. Rat. Mech. Anal. 16 (1964), 325-353. · Zbl 0133.17604
[6] A.E. Green, R.S. Rivlin: Simple force and stress multipoles. Arch. Rat. Mech. Anal. 17 (1964), 113-147. · Zbl 0244.73005
[7] E. Hewitt, K. Stromberg: Real and abstract analysis. Springer, 1965.
[8] M.A. Krasnoselski, J.B. Ruticki: Convex functions and Orlicz spaces. GITL, Moscow, 1958.
[9] O.A. Ladyženskaya: Mathematical problems of dynamics of viscous incompressible fluids. Nauka, Moscow.
[10] O.A. Ladyženskaya: On the finiteness of the dimension of bounded invariant sets for the Navier-Stokes equations and other related dissipative systems. J. Soviet Math. 28.no.5 (1985), 714-725. · Zbl 0561.76044
[11] J.L. Lions: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris, 1969. · Zbl 0189.40603
[12] J.L. Lions, E. Magenes: Problèmes aux limites non homogènes et applications. Dunod, Paris, 1968. · Zbl 0212.43801
[13] J. Nečas: Sur les normes équivalentes dans \(W^k_p (\Omega )\) et sur la coercivité des formes formellement positives. Les presses de lUnivesité de Montréal (1966).
[14] J. Nečas, A. Novotný: Some qualitative properties of the viscous compressible multipolar heat conductive flow. Commun. in Partial Differential Equations 16(2&3) (1991), 197-220. · Zbl 0777.35061
[15] J. Nečas, A. Novotný, M. Šilhavý: Global solution to the compressible isothermal multipolar fluid. J. Math. Anal. Appl. 162 (1991), 223-241. · Zbl 0757.35060
[16] J. Nečas, A. Novotný, M. Šilhavý: Global solution to the ideal compressible heat-conductive fluid. Comment. Math. Univ. Carolinae 30,3 (1989), 551-564. · Zbl 0702.35205
[17] J. Nečas, A. Novotný, M. Šilhavý: Global solution to the viscous compressible barotropic fluid. · Zbl 0761.76006
[18] J. Nečas, M. Šilhavý: Multipolar viscous fluids. Quart. Appl. Math. 49 (1991), 247-265. · Zbl 0732.76003
[19] A. Novotný: Viscous multipolar fluids-physical background and mathematical theory. Progress in Physics 39 (1991).
[20] Simon J.: Compact sets in the space \(L^p(0,T;B)\). Annali di Mat. Pura ed Applic. 146 (1987), 65-96. · Zbl 0629.46031
[21] R. Temam: Infinite-dimensional dynamical systems in mechanics and physics. Springer-Verlag, New York, 1988. · Zbl 0662.35001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.