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On the Navier-Stokes equations: Existence theorems and energy equalities

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Abstract

Currently available results on the solvability of the Navier-Stokes equations for incompressible non-Newtonian fluids are presented. The order of nonlinearity in the equations may be variable; the only requirement is that it must be a measurable function. Unsteady and steady equations are considered. A lot of attention is paid to the recovery of energy balance, whose violation is theoretically admissible, in particular, in the three-dimensional classical unsteady Navier-Stokes equation. When constructing a weak solution by a limit procedure, a measure arises as a limit of viscous energy densities. Generally speaking, the limit measure contains a nonnegative singular (with respect to the Lebesgue measure) component. It is this singular component that maintains energy balance. Sufficient conditions for the absence of a singular component are studied: in this case, the standard energy equality holds. In many respects, only the regular component of the limit measure is important: in the natural form it is equal to the product of the viscous stress tensor and the gradient of a solution; if this natural form is retained, then the problem is solvable. Conditions are found for the validity of the indicated fundamental representation of the absolutely continuous component of the limit measure.

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Authors and Affiliations

  1. Vladimir State University, ul. Gor’kogo 87, Vladimir, 600000, Russia

    V. V. Zhikov

  2. Moscow State Institute of Radio Engineering, Electronics and Automation (Technical University), pr. Vernadskogo 78, Moscow, 119454, Russia

    S. E. Pastukhova

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  1. V. V. Zhikov
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Correspondence to V. V. Zhikov.

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Original Russian Text © V.V. Zhikov, S.E. Pastukhova, 2012, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2012, Vol. 278, pp. 75–95.

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Zhikov, V.V., Pastukhova, S.E. On the Navier-Stokes equations: Existence theorems and energy equalities. Proc. Steklov Inst. Math. 278, 67–87 (2012). https://doi.org/10.1134/S0081543812060089

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