TY - JOUR

T1 - Asymptotics of the spectrum of variational problems arising in the theory of fluid oscillations

AU - Суслина, Татьяна Александровна

PY - 2024/1/1

Y1 - 2024/1/1

N2 - This work is a survey of results on the spectral asymptotics of variational problems arising in the theory of small oscillations of a fluid in a vessel near the equilibrium position. These problems were posed by Kopachevsky in the late 1970s and cover various fluid models. The statements of the problems are given both in the form of boundary-value problems in the domain Ω ⊂ R3 occupied by the fluid in the equilibrium state and in the form of variational problems on the spectrum of the ratio of quadratic forms. The common features of all the problems under consideration are the presence of an “elliptic” constraint (the Laplace equation for an ideal fluid or a homogeneous Stokes system for a viscous fluid), as well as the occurrence of the spectral parameter in the boundary condition on the free (equilibrium) surface Γ. The spectrum in the considered problems is discrete; the spectral counting functions have power-law asymptotics.

AB - This work is a survey of results on the spectral asymptotics of variational problems arising in the theory of small oscillations of a fluid in a vessel near the equilibrium position. These problems were posed by Kopachevsky in the late 1970s and cover various fluid models. The statements of the problems are given both in the form of boundary-value problems in the domain Ω ⊂ R3 occupied by the fluid in the equilibrium state and in the form of variational problems on the spectrum of the ratio of quadratic forms. The common features of all the problems under consideration are the presence of an “elliptic” constraint (the Laplace equation for an ideal fluid or a homogeneous Stokes system for a viscous fluid), as well as the occurrence of the spectral parameter in the boundary condition on the free (equilibrium) surface Γ. The spectrum in the considered problems is discrete; the spectral counting functions have power-law asymptotics.

KW - boundary-value problem

KW - fluid oscillation

KW - small oscillation

KW - spectral asymptotics

KW - variational problem

UR - https://www.mendeley.com/catalogue/50cd05a9-4054-306f-84ee-7e4be5be671a/

U2 - 10.1007/s10958-024-06912-9

DO - 10.1007/s10958-024-06912-9

M3 - Article

VL - 278

SP - 152

EP - 193

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 1

ER -