A variational principle for linear evolution problems nonlocal in time. (English. Russian original) Zbl 0835.34078

Sib. Math. J. 34, No. 2, 369-384 (1993); translation from Sib. Mat. Zh. 34, No. 2, 191-207 (1993).
Summary: For a differential operator equation, problems nonlocal in time are considered: a data average in time is prescribed instead of initial data. Solving the problem is reduced to the problem of minimizing a quadratic functional. As an example, the linearized Navier-Stokes system is considered.


34G10 Linear differential equations in abstract spaces
49R50 Variational methods for eigenvalues of operators (MSC2000)
35Q30 Navier-Stokes equations
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[1] P. N. Belov, Numerical Methods for Weather Forecast [in Russian], Gidrometeoizdat, Leningrad (1975).
[2] V. V. Shelukhin, ?The problem of particle movement in the model Burgers system of a viscous gas,? Dinamika Sploshn. Sredy (Novosibirsk),87, 136-154 (1988). · Zbl 0702.76082
[3] V. V. Shelukhin, ?A problem with time-averaged data for Navier-Stokes equations,? Dinamika Sploshn. Sredy (Novosibirsk),91, 149-167 (1989). · Zbl 0850.76120
[4] V. V. Shelukhin, ?A problem with time-averaged data for monotone parabolic equations,? Dinamika Sploshn. Sredy (Novosibirsk),93/94, 186-189 (1989).
[5] V. M. Shalov, ?Some generalization of Friedrichs spaces,? Dokl. Akad. Nauk SSSR,151, No. 2, 292-294 (1963). · Zbl 0199.45302
[6] V. M. Shalov, ?Solution of nonselfadjoint equations by variational method,? Dokl. Akad. Nauk SSSR,151, No. 3, 511-512 (1963). · Zbl 0211.44503
[7] V. M. Shalov, ?The minimum principle for the quadratic functional related to a hyperbolic equation,? Differentsial’nye Uravneniya,1, No. 10, 1338-1365 (1965).
[8] V. M. Filippov and A. N. Skorokhodov, ?On the quadratic functional related to the heat equation,? Differentsial’nye Uravneniya,13, No. 6, 1113-1123 (1977). · Zbl 0355.35039
[9] V. M. Filippov and A. N. Skorokhodov, ?The minimum principle for the quadratic functional related to the boundary value problem of heat conduction,? Differentsial’nye Uravneniya,13, No. 8, 1434-1445 (1977).
[10] J.-P. Aubin, Approximate Solution of Elliptic Boundary Value Problems [Russian translation], Nauka, Moscow (1969).
[11] H. Gajewski, K. Gröger, and K. Zacharias, Nonlinear Operator Equations and Operator Differential Equations [Russian translation], Mir, Moscow (1978).
[12] F. Riesz and B. Sz.-Nagy, Lectures on Functional Analysis [Russian translation], Izdat. Inostr. Lit., Moscow (1954).
[13] S. G. Mikhlin, Linear Partial Differential Equations [in Russian], Vyssh. Shkola, Moscow (1977). · Zbl 0378.45003
[14] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis [Russian translation], Mir, Moscow (1981). · Zbl 0529.35002
[15] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type [in Russian], Nauka, Moscow (1967). · Zbl 0164.12302
[16] O. A. Ladyzhenskaya, Mathematical Problems of Dynamics of a Viscous Incompressible Fluid [in Russian], Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow (1961). · Zbl 0131.09402
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