### Abstract

We study the nonlinear stability of hydrostatic equilibria of an ideal incompressible stratified fluid. We obtain a new a priori estimate for finite-amplitude perturbation of the basic equilibrium state. The main idea of our approach is based on a special decomposition of the density perturbation, namely, we split the density perturbation in two parts, the first part depends on time but has zero initial value, the second one is in some sense time-independent (its L_{2}-norm is time-independent). This decomposition allows us to obtain the a priori estimate for the time-dependent part of the perturbation and hence for the total perturbation. In our approach we avoid the problem of a smooth extension of a locally convex function beyond its initial domian of definition that arises in applications of Arnold's method. Taking advantage of this fact, we consider the nonlinear stability of equilibrium states of stratified fluid endowed with two densities. Such a kind of problem appears, e.g., in atmospheric physics when symmetric flows of a stratified fluid are considered. As a result, we obtain a sufficient condition for nonlinear stability of a general equilibrium state of such a doubly stratified fluid as well as an a priori estimate for perturbations of arbitrary amplitude.

Original language | English |
---|---|

Pages (from-to) | 123-133 |

Number of pages | 11 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 118 |

Issue number | 1-2 |

Publication status | Published - 1998 |

### Fingerprint

### ASJC Scopus subject areas

- Applied Mathematics
- Statistical and Nonlinear Physics

### Cite this

*Physica D: Nonlinear Phenomena*,

*118*(1-2), 123-133.

**Note on the nonlinear stability of stratified fluid equilibria.** / Vladimirov, V. A.; Ilin, K. I.

Research output: Contribution to journal › Article

*Physica D: Nonlinear Phenomena*, vol. 118, no. 1-2, pp. 123-133.

}

TY - JOUR

T1 - Note on the nonlinear stability of stratified fluid equilibria

AU - Vladimirov, V. A.

AU - Ilin, K. I.

PY - 1998

Y1 - 1998

N2 - We study the nonlinear stability of hydrostatic equilibria of an ideal incompressible stratified fluid. We obtain a new a priori estimate for finite-amplitude perturbation of the basic equilibrium state. The main idea of our approach is based on a special decomposition of the density perturbation, namely, we split the density perturbation in two parts, the first part depends on time but has zero initial value, the second one is in some sense time-independent (its L2-norm is time-independent). This decomposition allows us to obtain the a priori estimate for the time-dependent part of the perturbation and hence for the total perturbation. In our approach we avoid the problem of a smooth extension of a locally convex function beyond its initial domian of definition that arises in applications of Arnold's method. Taking advantage of this fact, we consider the nonlinear stability of equilibrium states of stratified fluid endowed with two densities. Such a kind of problem appears, e.g., in atmospheric physics when symmetric flows of a stratified fluid are considered. As a result, we obtain a sufficient condition for nonlinear stability of a general equilibrium state of such a doubly stratified fluid as well as an a priori estimate for perturbations of arbitrary amplitude.

AB - We study the nonlinear stability of hydrostatic equilibria of an ideal incompressible stratified fluid. We obtain a new a priori estimate for finite-amplitude perturbation of the basic equilibrium state. The main idea of our approach is based on a special decomposition of the density perturbation, namely, we split the density perturbation in two parts, the first part depends on time but has zero initial value, the second one is in some sense time-independent (its L2-norm is time-independent). This decomposition allows us to obtain the a priori estimate for the time-dependent part of the perturbation and hence for the total perturbation. In our approach we avoid the problem of a smooth extension of a locally convex function beyond its initial domian of definition that arises in applications of Arnold's method. Taking advantage of this fact, we consider the nonlinear stability of equilibrium states of stratified fluid endowed with two densities. Such a kind of problem appears, e.g., in atmospheric physics when symmetric flows of a stratified fluid are considered. As a result, we obtain a sufficient condition for nonlinear stability of a general equilibrium state of such a doubly stratified fluid as well as an a priori estimate for perturbations of arbitrary amplitude.

UR - http://www.scopus.com/inward/record.url?scp=22244442159&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=22244442159&partnerID=8YFLogxK

M3 - Article

VL - 118

SP - 123

EP - 133

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 1-2

ER -