On Direct Lyapunov Method in Continuum Theories
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.
KeywordsViscous Fluid Continuum Theory Rest State Dissipative Term Helmholtz Free Energy
Unable to display preview. Download preview PDF.
- 1.M. Padula, On the exponential stability of the rest of a viscous compressible fluid, Proc. Soc. Trends Appl. Math. Mech., Nice 25–29 May, 1998, pp. 317–326.Google Scholar
- 2.M. Padula and V. A. Solonnikov, On the global existence of nonsteady motions of a fluid drop and their exponential decay to a uniform rigid rotation, Proc. Meeting Rolduc, 2001.Google Scholar
- 8.L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon Press, 1959.Google Scholar
- 10.E. B. Dussan V., Hydrodynamic stability and instability of fluid systems with interfaces, Arch. Ration. Mech. Anal. 57 (1975), 364–379.Google Scholar
- 11.B. J. Jin and M. Padula, On existence of nonsteady compressible viscous flows in a horizontal layer with free upper surface, J. Math. Anal. Appl. [To appear]Google Scholar
- 12.V. Pukhnachov and V. A. Solonnikov, On the problem of dynamic contact angle, PMM USSR 46 (1983), 771–779.Google Scholar
- 13.G. Guidoboni and B. J. Jin, On the onset of convection for a horizontal layer of incompressible fluid with upper free boundary, and Marangoni effect in the Boussinesq approximation. [To appear]Google Scholar