ABSTRACTS OF MINISIMPOSIUM

1996

ON A CONSTRUCTION OF DISCONTINUOUS SOLUTIONS FOR EQUATIONS

OF IDEAL LIQUID USING COUPLED VORTICAL AND POTENTIAL FLOW MODELS

1. Stationary equations of ideal liquid in two space dimensions.

(1)

(2)

where P is a pressure, =(u₁,u₂)^T is a velocity vector, =const.

2. Regularity requirement and simplified models. It is well known that under requirements ,P Î C² the system (1),(2) can be reduced to linear equations. Namely in the case of vortical flow, i.e. rot ¹ 0 , (1),(2) can be reduced to the following system

(3)

is stream function,

(4) (5)

or in the case of potential flow, i.e. rot= 0, to the Laplase equation

(6)

is a potential function,

(7)

=grad ,

but Poissons equation for pressure is the same as (5).

3. Formulation of a problem. Let be a domain with sufficiently smooth boundary g . Let a flow be potential in and it be vortical in . Find out interface boundary conditions ensuring the solution constructed by coupling of solutions of corresponding simplified problems to be a weak discontinuous solution of(1),(2) in R².

4. Interface boundary conditions. The system (1),(2) may have discontinuous weak solution