Asymptotic Stability of Steady Compressible Flows Abstract Aim of this paper is to acquaint mathematicians and fluid-dynamic researchers with recent results achieved in investigation of nonlinear asymptotic stability with no smallness assumption on initial data, and nonlinear instability of some steady flows of compressible fluids. Here it is exhibited a large class of fluids governed by mixed hyperbolic-parabolic systems for which the time behavior of the velocity is entirely driven by the parabolic part. The following classes of motions are studied: (i) Barotropic viscous gases in rigid domains with compact, either impermeable or porous boundaries, and exterior domains; (ii) Isothermal viscous gases with free boundaries; (iii) Heat conducting viscous polytropic gases. Equations governing non steady flows defined in (i), (ii), (iii) respectively, enter in the following classification: (I) One vector parabolic equation for the velocity, and one scalar hyperbolic equation for the density. (II) One vector parabolic equation for the velocity, and two scalar hyperbolic equations for the density and the height. (III) Two parabolic equations one vector for the velocity, one scalar for the temperature, and one scalar hyperbolic equation for the density. Main goal is to reduce the study of stability, linear and not, to the sign of a suitable functional E called modified energy functional. To this end we employ the classical Lyapunov direct method, also erroneously known as energy method where the Lyapunov functional is identified with a modified energy functional of perturbations E(t), also briefly called Energy! Modified Energy Functional. The modified energy consists in the difference between the total energies in the unsteady E(t), and basic motions Eb, respectively, plus an extra energy term I(t) Em(t) = E(t) -Eb + I. The new tool lies in the construction of a differential equation for I(t), called Free Work Functional which becomes the key ingredient for the proof of uniqueness, asymptotic nonlinear stability, and nonlinear instability. 1
Asymptotic stability of steady compressible fluids
PADULA, Mariarosaria
2011
Abstract
Asymptotic Stability of Steady Compressible Flows Abstract Aim of this paper is to acquaint mathematicians and fluid-dynamic researchers with recent results achieved in investigation of nonlinear asymptotic stability with no smallness assumption on initial data, and nonlinear instability of some steady flows of compressible fluids. Here it is exhibited a large class of fluids governed by mixed hyperbolic-parabolic systems for which the time behavior of the velocity is entirely driven by the parabolic part. The following classes of motions are studied: (i) Barotropic viscous gases in rigid domains with compact, either impermeable or porous boundaries, and exterior domains; (ii) Isothermal viscous gases with free boundaries; (iii) Heat conducting viscous polytropic gases. Equations governing non steady flows defined in (i), (ii), (iii) respectively, enter in the following classification: (I) One vector parabolic equation for the velocity, and one scalar hyperbolic equation for the density. (II) One vector parabolic equation for the velocity, and two scalar hyperbolic equations for the density and the height. (III) Two parabolic equations one vector for the velocity, one scalar for the temperature, and one scalar hyperbolic equation for the density. Main goal is to reduce the study of stability, linear and not, to the sign of a suitable functional E called modified energy functional. To this end we employ the classical Lyapunov direct method, also erroneously known as energy method where the Lyapunov functional is identified with a modified energy functional of perturbations E(t), also briefly called Energy! Modified Energy Functional. The modified energy consists in the difference between the total energies in the unsteady E(t), and basic motions Eb, respectively, plus an extra energy term I(t) Em(t) = E(t) -Eb + I. The new tool lies in the construction of a differential equation for I(t), called Free Work Functional which becomes the key ingredient for the proof of uniqueness, asymptotic nonlinear stability, and nonlinear instability. 1I documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
