zbMATH — the first resource for mathematics

Long time behaviour of solutions of abstract inequalities. Applications to thermohydraulic and magnetohydrodynamic equations. (English) Zbl 0549.35102
We study some scalar inequalities of parabolic type and we give the leading term of an asymptotic expansion as \(t\to\infty \) for solutions of thermo-hydraulic equations without external excitation. (A phenomenon of resonance is pointed out). We also treat M. H. D. equations, Navier-Stokes equations on a Riemann manifold and scalar inequalities of the type (n(t,x)\(\in {\mathbb{C}}\), \(\Omega\) bounded set in \({\mathbb{R}}^ n):\) \[ |\partial u/\partial t-(\partial /\partial x_ i)(a_{ij}(x,t))\partial u/\partial x_ j|_{L^ 2(\Omega)}\leq n(t)|\nabla u|_{L^ 2(\Omega)^ n} \] where \(a_{ij}(.,t)\) goes to \(a^{\infty}_{ij}(.)\) and n(t) goes to zero in a certain sense when t goes to infinity.
We start with some abstract results on differential inequalities of type: \[ (*)\quad\| (d\Phi /dt)+\nu A\Phi\|_ H\leq n\|\Phi \|_{D(A^{{1\over2}})}\quad (\nu >0), \] where \(\{\) A(t)\(\}\) is a family of self-adjoint unbounded operators on a Hilbert space H. Then these results are applied to the equations and inequalities mentioned previously. The main result for (*) is that the behaviour of \(\Phi\) (t) is characterized by an eigenvalue \(\Lambda^{\infty}\) of the operator \(A^{\infty}=\lim_{t\to +\infty}A(t).\)

35Q30 Navier-Stokes equations
35B40 Asymptotic behavior of solutions to PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
35R45 Partial differential inequalities and systems of partial differential inequalities
76W05 Magnetohydrodynamics and electrohydrodynamics
Full Text: DOI
[1] Bardos, C.; Tartar, L., Sur l’unicité rétrograde des équations paraboliques et quelques questions voisines, Arch. rational mech. anal., 50, 10-25, (1973) · Zbl 0258.35039
[2] Chandrasekhar, S., Hydrodynamic and hydrodynamic stability, (1961), Oxford Univ. Press (Clarendon) London/New York · Zbl 0142.44103
[3] Foias, C.; Saut, J.C., Limite du rapport de l’enstrophie sur l’énergie pour une solution faible des équations de Navier-Stokes, C. R. acad. sci. Paris Sér. 1, 293, 241-244, (1981) · Zbl 0492.35063
[4] Foias, C.; Saut, J.C., Asymptotic behavior, as t → + ∞ of solutions of Navier-Stokes equations and nonlinear spectral manifolds, Indian math. J., 33, 459-477, (1984) · Zbl 0565.35087
[5] Foias, C.; Saut, J.C., Transformation fonctionnelle linéarisant LES équations de Navier-Stokes, C. R. acad. sci. Paris Sér. 1, 295, 325-327, (1982) · Zbl 0545.35074
[6] Ghidaglia, J.M., Étude d’écoulements de fluides visqueux incompressibles: comportement pour LES grands temps et applications aux attracteurs, Thèse de docteur ingénieur, (1984), and On the fractal dimension of attractors for viscous incompressible fluid flows, S.I.A.M. Journal on Mathematical Analysis, in press.
[7] Ghidaglia, J.M., Comportement asymptotique pour une inéquation parabolique. applications à diverses équations de la mécanique des fluides incompressibles, ()
[8] Ghidaglia, J.M., Régularité des solutions de certains problèmes aux limites linéaires liés aux équations d’Euler, Comm. partial differential equations, 9, 13, 1265-1298, (1984) · Zbl 0602.35093
[9] \scJ. M. Ghidaglia, Some backward uniqueness results, Journal of Nonlinear Analysis: Theory Methods and Applications, in press. · Zbl 0622.35029
[10] Guillopé, C., Comportement à l’infini des solutions des équations de Navier-Stokes et propriétés des ensembles fonctionnels invariants (ou attracteurs), Ann. inst. Fourier (Grenoble), 32, 1-37, (1982) · Zbl 0488.35067
[11] Guillopé, C., Remarques à propos du comportement, lorsque t → + ∞, des solutions des équations de Navier-Stokes associées à une force nulle, Bull. soc. math. France, 111, 151-180, (1983) · Zbl 0554.35098
[12] Landau, L.; Lifchitz, E., Electrodynamique des milieux continus, (1969), Editions MIR Moscow
[13] Lax, P.D., A stability theorem for solutions of abstract differential equations, and its application to the study of the local behavior of solutions of elliptic equations, Comm. pure appl. math., 9, 747-766, (1956) · Zbl 0072.33004
[14] Lees, M., Asymptotic behaviour of solutions of parabolic differential inequalities, Canad. J. math., 14, 626-631, (1962) · Zbl 0115.31002
[15] Leray, J., Essai sur le mouvement d’un fluide visqueux emplissant l’espace, Acta math., 63, 193-248, (1934) · JFM 60.0726.05
[16] Leray, J., Essais sur LES mouvements plans d’un liquide visqueux que limitent des parois, J. math. pure appl., 13, 331-419, (1934) · JFM 60.0727.01
[17] Lions, J.L., Quelques méthodes de résolution des problèmes aux limites non linéaires, (1969), Dunod-Gauthier-Villars Paris · Zbl 0189.40603
[18] Lions, J.L.; Malgrange, B., Sur l’unicité rétrograde dans LES problèmes mixtes paraboliques, Math. scand., 8, 277-286, (1960) · Zbl 0126.12202
[19] \scB. Nicolaenko, B. Scheurer, and R. Temam, Some Global Dynamical Properties of the Kuramoto-Sivashinsky Equations: Nonlinear Stability and Attractors, Physica D, in press. · Zbl 0592.35013
[20] Protter, M.H., Properties of solutions of parabolic equations and inequalities, Canad. J. math., 13, 331-345, (1961) · Zbl 0099.30001
[21] Temam, T., Navier-Stokes equations, theory and numerical analysis, (1979), North-Holland Amsterdam · Zbl 0426.35003
[22] Temam, R., Behaviour at time t = 0 of the solutions of semi-linear evolution equations, J. differential equations, 43, 73-92, (1982) · Zbl 0446.35057
[23] Temam, R., ()
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.