×

A variational principle for the nonstationary linear Navier-Stokes equations. (English) Zbl 0795.35083

A boundary-value problem for the linear nonstationary Navier-Stokes system with time-averaged data is considered: \[ u_ t- \nu\Delta u+ \nabla p=F, \quad \text{div } u=0, \qquad x\in\Omega,\;t\in (0,T); \tag{1} \]
\[ (2)\quad u=0,\;x\in \partial\Omega, \qquad (3)\quad \sum_{i=0}^ m \gamma_ i u(x,t_ i)= f(x),\;x\in\Omega. \] Here \(\Omega\subset \mathbb{R}^ n\), \(n\geq 2\) is a bounded domain with a smooth boundary. The functions \(F\) and \(f\) are given. The numbers \(\gamma_ i\), \(t_ i\), \(T\) are prescribed, \(0=t_ 0< t_ 1<\dots <t_ m= T\), \(\gamma_ 0=1\).
The solvability and uniqueness to the problem (1), (2), (3) are established in a space \(H_{x,t}^{2,1}\) if the numbers \(\gamma_ i\) are sufficiently small. A similar result is obtained for a problem for which summation in (3) is exchanged by integration. The classical methods of functional analysis and Galerkin approximation are used.

MSC:

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
46N20 Applications of functional analysis to differential and integral equations
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Belov, P.N.: The numerical methods of weather forecasting, Leningrad: Gidrometeoizdat, 1975 (in Russian)
[2] Filippov, V.M., Scorohodov, A.N.: On quadratic functional for the heat equation. Differential equations, vol. 13, No. 6, 1113–1123 (1977) (in Russian) · Zbl 0355.35039
[3] Fillippov, V.M., Scorohodov, A.N.: The minimum principle of a quadratic functional in the boundary-value problem for the heat equation. Differential equations, vol. 14, No. 11, 1979 (in Russian)
[4] Ladyzhenskaya, O.A.: The mathematical theory of viscous incompressible flow, Gordon and Breach, New York, English translation, Second edition, 1969 · Zbl 0184.52603
[5] Ladyzhenskaya, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and quasi-linear equations of parabolic type. Trans. Math. Monographs 23, Amer. Math. Soc., Providence, R.I., 1968 · Zbl 0174.15403
[6] Mikhlin, S.G.: Linear equations of mathematical physics. Holt, Rinehart and Winston, 1967 · Zbl 0154.10602
[7] Temam, R.: Navier-Stokes equations. Theory and numerical analysis. North Holland Publishing Company, Amsterdam-New York-Oxford, 1977 · Zbl 0383.35057
[8] Shalov, V.M.: A generalization of Friedrichs’ spaces. Doklady of Ac. Sci. USSR, vol. 151, No. 2, 292–294 · Zbl 0199.45302
[9] Shalov, V.M.: Solving the non self-adjoint equations by the variational method. Doklady of Ac. Sci. USSR, vol. 151, No. 3, 511–512 (1963) (in Russian) · Zbl 0211.44503
[10] Shelukhin, V.V.: The problem with time-averaged data for the Navier-Stokes equations. Dinamika Sploshnoi Sredy, vol. 91, 149–167 (1989) (in Russian) · Zbl 0850.76120
[11] Shelukhin, V.V.: The problem of mass transportation for the generalized Burgers equations of gas dynamics, Dinamika sploshnoi sredy, vol. 87, 136–174 (1988) (in Russian) · Zbl 0702.76082
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.