Hyperbolic Partial Differential Equations - Hyperbolic System of Partial Differential Equations

Hyperbolic System of Partial Differential Equations

Consider the following system of first order partial differential equations for unknown functions, where

(*) \quad \frac{\partial \vec u}{\partial t} + \sum_{j=1}^d \frac{\partial}{\partial x_j} \vec {f^j} (\vec u) = 0,

are once continuously differentiable functions, nonlinear in general.

Now define for each a matrix

A^j:= \begin{pmatrix} \frac{\partial f_1^j}{\partial u_1} & \cdots & \frac{\partial f_1^j}{\partial u_s} \\ \vdots & \ddots & \vdots \\ \frac{\partial f_s^j}{\partial u_1} & \cdots & \frac{\partial f_s^j}{\partial u_s} \end{pmatrix} ,\text{ for }j = 1, \ldots, d.

We say that the system is hyperbolic if for all the matrix has only real eigenvalues and is diagonalizable.

If the matrix has distinct real eigenvalues, it follows that it's diagonalizable. In this case the system is called strictly hyperbolic.

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