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Deane Yang
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The system is hyperbolic and has a well-posed Cauchy problem, if you can do the following: Find a linear change of co-ordinates $(z_1, z_2, z_3) = M(t, x, y)$, a linear change of basis for the unknown functions $(a_1, a_2, a_3) = L(u_1, u_2, u_3)$, and a linear change of basis for the eqautions $(b_1, b_2, b_3) = K(c_1, c_2, c_3)$ such that the system can be written in the form $$ \partial_t u + A_1\partial_x u + A_2\partial_y u = c, $$ such that one of the following holds (If so, then the Cauchy problem with initial data specified on the hypersurface $t = 0$ is well-posed):

  1. (strict hyperbolicity) There exists a matrix $Q(\xi,\eta)$ depending smoothly on $(\xi,\eta) \in \mathbb{R}^2$ such that the matrix $Q(\xi,\eta)(\xi A_1 + \eta A_2)Q^{-1}(\xi,\eta)$ is diagonal for all $(\xi,\eta)$.
  2. (symmetric hyperbolicity) The matrices $A_1$ and $A_2$ are symmetric.

A sufficient condition for strict hyperbolicity is the following: The matrix $\xi A_1 + \eta A_2$ has real, distinct eigenvalues for all $(\xi,\eta) \ne (0,0)$.

Deane Yang
  • 27.5k
  • 5
  • 89
  • 180