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Dynamics of Inviscid Fluids 55

2. Some Simple Existence and Uniqueness

Results

In what follows we will present, successively, some existence and
uniqueness results for the solutions of the Euler system (equations). A

special accent is put on the uniqueness results because, in fluid dynam-
ics, there is a large variety of methods, not necessarily direct (i.e., they
could also be inverse, semi-inverse, etc.), which enable us to construct
a solution fulfilling the given requirements and which, if a uniqueness
result already exists, will be the right solution we were looking for.

At the same time we will limit our considerations to the “strong”
solutions, i.e., the solutions associated to the continuous flows, while the
other solutions (weak, etc.) will be considered within a more general
frame, in the next chapter.

We will start by focussing on additional requirements concerning the
associated boundary conditions. The slip-conditions on a rigid wall —

which are necessary conditions for any deformable continuum and which,
in the particular case of the inviscid fluids are proved to be also sufficient
for the mathematical coherence of the joined model — take the known
form v-n = O or, when the wall is moving, v,-n = 0 (v, being the
relative velocity of the fluid versus the wall).

If our fluid is in contact with another ideal fluid, the contact surface
(interface) is obviously a material surface whose shape is not “a priori”
known. But we know that across such an interface the stress should

be continuous. As in the case of the ideal fluid the stress comes to
the pressure, we will have that across this contact surface of (unknown)
equation F = 0, there are both F = 0 (the Euler-Lagrange criterion for
material surfaces) and p; = pa (p, and pe being the limit values of the
pressure at the same point of the interface, a point which is “approached”
from the fluid (1), and from the fluid (2) respectively). The existence

of two conditions, the kinematic condition (F = 0) and the dynamic
condition (pi = p2|r—o) does not lead to an over-determined problem
because this time, we should not determine only the solution of the
respective equation but also the shape of the boundary F = O, the
boundary which carries the last data. In other words, in this case, we
deal with an inverse problem.

If the flow is not adiabatic we will have to know either the temperature
T(r,t) or the vector q on the boundary of the flow domain.

If the flow is adiabatic, from the energy equation we will have s = 0
and, if the fluid is also perfect s = c, In i + so, the Euler system will

have five equations with five (scalar) unknowns vy, p, p. If, additionally,