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  Physical meaning of elliptic-hyperbolic transitions

+ 4 like - 0 dislike

I am interested in the interpretation of PDE that are not strictly elliptic, parabolic,or hyperbolic: ones that change type in sections of the domain, e.g. the generalised Tricomi equation

$ A(x) u_{,xx} + u_{,yy} = 0$

will be elliptic in regions where $A(x) \geq 0$ and hyperbolic when $A(x) < 0$.

I am aware there is a great deal of literature surrounding the topic (e.g. transonic flows), but mostly regarding well-posedness and such, not so much about the actual physics of what is going on.

For a start, I usually think of hyperbolic equations as having wave-like components in their solutions -- this usually means there is some kind of 'time' involved. What does it mean for a PDE in spatial variables to be hyperbolic? It seems weird to me to think of 'travelling waves' in the $(x,y)$ plane. What about an elliptic PDE involving both spatial and temporal variables?

Can someone explain what is physically happening here? In the non-linear case [$A(x) \rightarrow A(x,u)$] does this kind of elliptic-hyperbolic transition imply something about shocks?

As a simple example, if we have $A(x) =x$ and the domain is something like $D = \{x,y: -a \leq x \leq a, -b \leq y \leq b\}$, what are the 'wave-like' portions doing in the $x < 0$ region? What physics are they conveying?

asked Apr 5, 2016 in General Physics by Blaghol [ no revision ]

Do you know of a case of the generalised Tricomi equation that arises in physics? If so, please point to one. If not, why do you expect a physical interpretation exists?

A fair comment, Arnold. The actual example I am interested in has a somewhat complicated backstory, so I have just written the Tricomi equation as a simple example. Here is a GR example from Stewart, [J. M. Stewart, Classical Quant. Grav. 18 (2001) -- unfortunately paywalled]: Consider the line element:

$ds^2 = -dt^2 + d \rho^2 + \rho^2 \left( d \phi - \Omega d t \right)^{2} + dz^2$, for constant $\Omega$.

Then the Laplace equation for a static variable $u(\rho,\phi,z)$ reads

$\rho^{-1} \left( \rho u_{,\rho} \right)_{,\rho} + \left( \rho^{-2} - \Omega^2 \right) u_{,\phi \phi} + u_{,zz} = 0$,

which is hyperbolic for large $\rho$, and elliptic for small $\rho$. Again -- I don't understand what could be meant by 'wave-like' behaviour in spatial variables.

This particular (simple) example suffers in the respect that there exists a frame which makes the problem purely elliptic -- but you can easily imagine a situation where that is not possible.

Still looking for an answer to this; another example that may provide additional context. If one takes the time independent Euler and Continuity equations of fluid dynamics and linearises about a background which is rigidly rotating, it can be shown that the only remaining equation of motion reads

$u_{,xx} + u_{,yy} + \left( 1 - M^2 \right) u_{,zz} = 0$,

where the velocity field is irrotational and we have written $v^{\alpha} = \nabla^{\alpha} u$, and $M$ is the Mach number (see e.g. Bateman 1929). Such equations are elliptic when $M \leq 1$ and hyperbolic when $M > 1$. Still haven't a clue what it means for the variable $u$ to be 'hyerpbolic'.

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