,Vo,,,v,ear Analyris. Theory. Methods & Apphcorrons. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Vol. 8. No. 1, pp. 17-38. 1984. 036?-546)(/U %3.00+ .OO Prmted in Great Britam 0 1984 Pergamon Press Ltd. zyxwvutsrqpo ON THE MATHEMATICAL PROBLEM OF LINEAR AND NON- LINEAR HYDRODYNAMIC STABILITY WITH COMPLETELY PERTURBED DATA* FABIO Rosso Istituto di Matematica “R. Caccioppoli”. Universita di Napoli, Via Mezzocannone 8, 80134 Napoli. Italia zyxwvutsrqponml (Receiued 10 December 1982) Key words and phrases: Hydrodynamic stability, boundary disturbances. method of linear stability. Serrin’s method. 1. INTRODUCTION AN INCREDIBLE number of papers have been devoted so far to the problem of stability of solutions to the Navier-Stokes equations when disturbances to the initial data only are supposed to cause instabilities. On the contrary, as far as we know, only two papers (see [l]. [2]) have dealt with the same problem when also disturbances both to the boundary data and to the body force are taken into account; the results of these latter papers are confined, however, to laminar flows and to particular spatial geometries. Within any typical experiment it is practically impossible to control all the data with arbitrary precision; therefore it is of both theoretical and practical interest to investigate the general problem of linear and nonlinear hydrodynamic stability with completely perturbed data. In particular it is extremely interesting to see whether or not the range of stability? becomes, in the general case, smaller than that determined by disturbances to the initial data only. Although one intuitively expects an affirmative answer, we shall show that the range remains the same, independently of the size of the data in the case of linear stability; the same holds in the fully nonlinear case provided only that either boundary disturbances are not “too large” or they are sufficiently regular. Our approach to the mathematical problem is the natural and direct generalization (at least in principle) both of the so-called “method of linear stability” and of the variational method, suggested for the first time by Serrin [3], for studying nonlinear hydrodynamic stability. Both methods have been not only fruitfully and extensively employed in applications, but are also deeply related to each other [4], [5], [6], [7]. To be a little more precise, let us denote with u the velocity of a (steady or unsteady) basic motion zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA m occurring in a two or three-dimensional bounded region, and with Re > 0 the Reynolds number of m. Let moreover uo, ur, andfbe, respectively, disturbances to the initial and boundary value of u and to the body force. Let also 9(q) be the real part of the first eigenvalue q of the linear operator related to the problem of linear stability (see Section 4) * Preliminary drafts of this research were communicated by the author during the Meeting “Waves and Stability in Continuous Media” (Catania, Italy, November 1981) and at the “Fifth Canadian Symposium on Fluid Dynamics” (Windsor, Canada, June 1982). within the framework of activities of the G.N.F.M. of the Italian C.N.R. t That is, the interval of the real positive line allowed to the Reynolds number of the basic flow for this remains stable. 17