Partial differential equations Definition

Strict hyperbolicity

Builds on Hyperbolic partial differential equation Eigenvalue Principal symbol

Definition

Let $\Omega \subseteq \R^n \times \R$ be open, let $N \ge 1$ , and let $A_1, \ldots, A_n \colon \Omega \to \R^{N \times N}$ be continuous. The first-order system $\partial_t u + \sum_{j=1}^{n} A_j(x, t)\, \partial_{x_j} u = 0$ for $u \colon \Omega \to \R^N$ is strictly hyperbolic if for every $(x, t) \in \Omega$ and every $\xi \in \R^n \setminus \set{0}$ , the matrix $A(x, t, \xi) = \sum_{j=1}^{n} \xi_j A_j(x, t)$ has $N$ real and distinct eigenvalues.

Let $P u = 0$ be a scalar linear equation of order $m$ in the variables $(x, t)$ , with principal symbol $p_m(x, t; \xi, \tau)$ . It is strictly hyperbolic with respect to $t$ if, for every $(x, t)$ and every $\xi \ne 0$ , the polynomial $\tau \mapsto p_m(x, t; \xi, \tau)$ has $m$ real and distinct roots.

Remarks

Example: the hyperbolic system $u_t + \begin{pmatrix} 0 & -1 \\ -c^2 & 0 \end{pmatrix} u_x = 0$ with $c > 0$ , obtained from $w_{tt} = c^2 w_{xx}$ by setting $u = (w_x, w_t)$ , has eigenvalues $\pm c$ , which are real and distinct. For the wave operator $\partial_t^2 - c^2 \Delta$ , the principal symbol is $-\tau^2 + c^2 \abs{\xi}^2$ , with the two distinct real roots $\tau = \pm c\abs{\xi}$ when $\xi \ne 0$ . Non-example: the system $u_t + \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} u_x = 0$ has the double eigenvalue $1$ , so it is not strictly hyperbolic; it is not even diagonalizable. Strict hyperbolicity implies that $A(x, t, \xi)$ is diagonalizable over $\R$ , so a strictly hyperbolic system is hyperbolic. The converse fails: $u_t + I u_x = 0$ with $I$ the $2 \times 2$ identity is hyperbolic but not strictly hyperbolic.

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Eigenvalue concept
Hyperbolic partial differential equation concept
Principal symbol concept
Strict hyperbolicity definition

Tag	`TST3NVVB58`
URI	`m2://TST3NVVB58`
Kind	definition
Writing	W0 stub
MSC	35L40, 35L45, 35L65
Defines	Strict hyperbolicity
Uses	Hyperbolic partial differential equation, Eigenvalue, Principal symbol

Strict hyperbolicity

Definition

Remarks

Reading path

Further reading

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