Next: References Up: Kinetic Formulations and Regularity Previous: Degenerate parabolic equations

# The isentropic equations

We consider the system of isentropic equations, governing the density and momentum ,

Here is the pressure which is assumed to satisfy the (scaled) law, .

The question of existence for this model, depending on the -law, , was already studied [7],[6] by compensated compactness arguments. Here we revisit this problem with the kinetic formulation presented below which leads to existence result for , consult [23], and is complemented with a new existence proof for , consult [21].

For the derivation of our kinetic formulation of (2.5.36), we start by seeking all weak entropy inequalities associated with the isentropic system (2.5.36),

The family of entropy functions associated with (2.5.37) consists of those 's whose Hessians symmetrize the Jacobian, A'(w); the requirement of a symmetric yields the Euler-Poisson-Darboux equation, e.g, [6]

Seeking weak entropy functions such that , leads to the family of weak (entropy, entropy flux) pairs, , depending on an arbitrary ,

Here, is given by

We note that is convex iff is. Thus by the formal change of variables, , the weight function becomes the 'pseudo-Maxwellian', ,

We arrive at the kinetic formulation of (2.5.36) which reads

Observe that integration of (2.5.40) against any convex recovers all the weak entropy inequalities. Again, as in the scalar case, the nonpositive measure m on the right of (2.5.40), measures the loss of entropy which concentrates along shock discontinuities.

The transport equation (2.5.40) is not purely kinetic due to the dependence on the macroscopic velocity u (unless corresponding to ),

Compensated compactness arguments presented in [23] yield the following compactness result.

Finally, we consider the system

endowed with the pressure law

The system (2.5.41)-(2.5.42) governs the isentropic gas dynamics written in Lagrangian coordinates. In general the equations (2.5.41)-(2.5.42) will be referred to as the p-system (see [20],[30]).

For a kinetic formulation, we first seek the (entropy,entropy flux) pairs, , associated with (2.5.41)-(2.5.42). They are determined by the relations

where F is computed by the compatibility relations

The solutions of (2.5.43) can be expressed in terms of the fundamental solution

where the fundamental solutions, , are given by

Here and below, (rather than v occupied for the specific volume) denotes the kinetic variable. The corresponding kinetic fluxes are then given by

We arrive at the kinetic formulation of (2.5.41)-(2.5.42) which reads, [23]

with macroscopic velocity, .

Next: References Up: Kinetic Formulations and Regularity Previous: Degenerate parabolic equations