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On direct Lyapunov method in continuum theories. (English) Zbl 1046.35089

Birman, Michael Sh. (ed.) et al., Nonlinear problems in mathematical physics and related topics I. In honor of Professor O. A. Ladyzhenskaya. New York, NY: Kluwer Academic/Plenum Publishers (ISBN 0-306-47333-X). Int. Math. Ser., N.Y. 1, 289-302 (2002).
Summary: Let \(S_b\) be a basic motion. We consider two aspects of the direct Lyapunov method of stability theory. The first one is related to the control of perturbations of \(S_b\) in terms of the data (stability in mean), and the second one is related to an asymptotic decay to zero for perturbation. First, for a Lyapunov functional we take the difference between the total energy of a given flow and that of the basic flow. An algorithm for computing the norm of perturbation (in a certain space) is demonstrated by three examples. We also propose the useful technique based on the general variational formulation. The algorithm consists in the choice of a test function. Precisely, we note that different test functions can be used for the same formulation and provide us with different informations. We show how to choose the test function in three examples.
For the entire collection see [Zbl 0998.00008].

MSC:

35Q35 PDEs in connection with fluid mechanics
76E19 Compressibility effects in hydrodynamic stability
80A20 Heat and mass transfer, heat flow (MSC2010)
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