Mathematical entropy, vanishing viscosity, and symmetrizability.

This note explains the relationship between symmetric hyperbolic
form and the existence of an entropy. It
+ defines mathematical entropy,
+ shows that viscosity solution satisfies the entropy inequality, and
+ shows that symmetrizability of a hyperbolic system
  implies the existence of a mathematical entropy.

Below xi:=x^i, fi:=f^i, and ri:=r^i, and the repeated index i
indicates summation from 1 to 3.

We study the conservation law

  u_t + fi_xi = 0.

=== Entropy Framework ===

  To seek an entropy phi, we multiply and dot
  the quasilinear form with phi_u:
  
    phi_u.u_t + phi_u.fi_u.u_xi = 0.
  
  Suppose that we can find phi(u) and Fi(u) such that
  
    Fi_u = phi_u.fi_u.
  
  Then entropy is conserved:
  
    phi_t + Fi_xi = 0.
  
  Observation: in an adiabatic system the value of the "entropy"
    s=log(p\rho^{-\gamma}) is conserved along particle paths:
  
      d_t(s) = 0.
  
    This means that phi:=(\rho s) is entropy in the sense above
    of having an entropy flux:
  
      (\rho s)_t + \Div(\u \rho s) = 0.
  
  If we also assume that phi_uu is positive definite,
  we can show that
  
    phi_t + Fi_xi <= 0 (in the sense of distributions),
  
  and that this picks out the vanishing viscosity solution.
  
  To see this, we consider the viscosity solution

    u_t + fi_xi = \epsilon u_{xi,xi}

  and assume that, as \epsilon goes to zero,
  u converges uniformly to a vanishing viscosity solution.

  As before, we convert the viscosity solution
  to a (viscous) entropy evolution equation by
  taking the dot product with phi_u:

    phi_t + Fi_xi = \epsilon \phi_u.u_{xi,xi}

  Looking for a "chain rule" to simplify the RHS
  (or looking for a product rule to move derivatives from u)
  prompts the observation

    phi_xi = phi_u.u_xi and
    phi_{xi,xi} = (phi_uu.u_xi).u_xi + phi_u.u_{xi,xi},

  so the entropy evolution equation becomes

    phi_t + Fi_xi = \epsilon (phi_{xi,xi} - (phi_uu.u_xi).u_xi).

  Invoking that phi_uu is positive definite,

    (phi_uu.u_xi).u_xi >= 0,

  so we get the viscous entropy evolution inequality

    phi_t + Fi_xi - \epsilon phi_{xi,xi} <= 0.

  Roughly, taking \epsilon -> 0 gives

    phi_t + Fi_xi <= 0.

  More rigorously, to show that this holds in the sense of
  distributions, we multiply the viscous entropy inequality
  by a test function v and integrate (by parts) over time
  T=[0,\infty), and the spatial domain X:

    \int_T \int_X (u v_t + fi v_xi + \epsilon u v_{xi,xi}) >= 0.

  Assuming that u is uniformly bounded and converges as \epsilon
  goes to zero, the dominated convergence theorem allows us
  to bring the limit inside the integral and gives

    \int_T \int_X (u v_t + fi v_xi) >= 0,

  which is what is meant by the statement

    phi_t + Fi_xi <= 0 (in the sense of distributions).

=== symmetric variables framework ===

Given the conservation law

  u_t + fi_xi = 0,

we seek symmetric variables v.
The chain rule says that

  u_v.v_t + fi_v.v_xi = 0.

Suppose that 
  
  u_v and fi_v are symmetric.

Then by Green's theorem path integrals are independent of
(rectilinear) path and can be used to define scalar
potentials q and ri satisfying

  u = q_v and
  fi = ri_v,
  
where we note that q is convex if u_v is positive definite;
in this case u(v) is injective and we can speak of v(u).

The Legendre transform (which is convexity-preserving),

  phi(u) = u.v - q(v),

and the "generalized Legendre transform" (?),

  Fi(u) = fi.v - ri(v),

then allow us to write

  v = phi_u, where phi_uu is positive definite, and
  Fi_u = v.fi_u = phi_u.fi_u,

which satisfies the conditions in the entropy framework
to conclude that

  phi_t + Fi_xi <= 0, as needed.

We remark that the entropy is the Legendre transform of the
potential of the state with respect to the symmetric variables.
(In other words, the derivative of the entropy is the inverse of
the derivative of the potential function of the state variables.)
Similarly, the entropy fluxes are the "generalized Legendre
transform" of the potential of the fluxes with respect to the
symmetric variables.