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The tex2html_wrap_inline5897 isentropic equations

We consider the tex2html_wrap_inline5897 system of isentropic equations, governing the density tex2html_wrap_inline6133 and momentum tex2html_wrap_inline6135,

Here tex2html_wrap_inline6137 is the pressure which is assumed to satisfy the (scaled) tex2html_wrap_inline6139 law, tex2html_wrap_inline6141.

The question of existence for this model, depending on the tex2html_wrap_inline6139-law, tex2html_wrap_inline6145, 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 tex2html_wrap_inline6147, consult [23], and is complemented with a new existence proof for tex2html_wrap_inline6145, 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 tex2html_wrap_inline5897 system (2.5.36),

The family of entropy functions associated with (2.5.37) consists of those tex2html_wrap_inline6153's whose Hessians symmetrize the Jacobian, A'(w); the requirement of a symmetric tex2html_wrap_inline6157 yields the Euler-Poisson-Darboux equation, e.g, [6]

Seeking weak entropy functions such that tex2html_wrap_inline6159, leads to the family of weak (entropy, entropy flux) pairs, tex2html_wrap_inline6161, depending on an arbitrary tex2html_wrap_inline6163,

Here, tex2html_wrap_inline6165 is given by

We note that tex2html_wrap_inline6167 is convex iff tex2html_wrap_inline6163 is. Thus by the formal change of variables, tex2html_wrap_inline6171, the weight function tex2html_wrap_inline6165 becomes the 'pseudo-Maxwellian', tex2html_wrap_inline6175,


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


Observe that integration of (2.5.40) against any convex tex2html_wrap_inline6163 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 tex2html_wrap_inline6183 corresponding to tex2html_wrap_inline6185),


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


Finally, we consider the tex2html_wrap_inline5897 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, tex2html_wrap_inline6089, 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, tex2html_wrap_inline6217, are given by
Here and below, tex2html_wrap_inline6219 (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, tex2html_wrap_inline6223.

next up previous contents
Next: References Up: Kinetic Formulations and Regularity Previous: Degenerate parabolic equations

Eitan Tadmor
Mon Dec 8 17:34:34 PST 1997