Is there a general method for determing the domain of dependence of (higher-order) PDEs?

Question

Is there a general method for determining the domain of dependence of (higher-order) PDEs? I would be plenty happy with a reference to a paper or textbook I could look at; despite much reading around, I couldn't find this problem addressed for anything besides first- and second-order PDEs.

If there's a simple answer to this question, then I probably don't need to write anything else. But for completeness, I'll provide background on the subject and on my particular interest.

Domain of dependence

The domain of dependence of point $(\vec{x}_0,t_0)$ on the solution $U(\vec{x},t)$ of a hyperbolic$^\dagger$ PDE is the subset of the initial conditions which uniquely determine the value $U(\vec{x}_0,t_0)$. For the Advection equation

$U_t + c U_x = 0$

with initial conditions $U(x,0)=f(x)$, the domain of dependence of $(x,t)$ is just the point $x-c t$. For the wave equation

$U_{tt} - c^2 U_{xx} = 0$

the domain of dependence of $(x,t)$ is the line $[x-ct,x+ct]$. In every book I've read, the domain of dependence is only given for PDEs which have been explicitly solved (like these two examples) and no domain is given for a higher-order PDE without explict solutions.

Background for my problem

I'm trying to simulate the evolution of the Wigner function (a pseudo probability distribution over phase space) for a point particle moving in a chaotic potential. The PDE governing the Wigner function $W(x,p,t)$ can be approximated as

$\partial_t W = -\frac{p}{m} \partial_x W + V^\prime (x) \partial_p W - \frac{\hbar^2}{24} V^{\prime \prime \prime} (x) \partial_p^3 W$

or, in the dimensionless PDE notation,

$W_t = -p W_x + f(x) W_p - f^{\prime \prime}(x) W_{ppp} .$

It appears that an instability in my simulation is due to my time steps not satisfying the CFL condition and I'm trying to rigorously derive the CFL condition for this PDE.

This may not be possible, in which case I'll use the CFL condition under the approximation that $f''(x)=0$.

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$^\dagger$ Here I am using "Hyperbolic" in the sense that the Cauchy problem is well defined without restricting to second-order PDEs.

Answer 1

While I am aware of some facts about domain of dependence properties for hyperbolic PDEs, I don't think most of them will be useful for you. The problem is that what you consider as hyperbolic (in your footnote) is too large of a class of equations for the notion to be useful: an illustration is the Heat equation. It is usually classified as a parabolic equation, but it does admit a well-posed initial value problem. So by your definition is hyperbolic. Now it is well known that the heat equation has infinite smoothing properties and infinite speed of propagation.

Now, in the special case of symmetric hyperbolic systems, even in higher orders, one can generally describe the domain of dependence by considering the characteristic cones (see, eg. https://encyclopediaofmath.org/wiki/Characteristic ) of the system. Essentially the "fattest cone" gives you the maximum speed of propagation (which may depend quasilinearly on the solution) and integrating back this cone gets you the domain of dependence.

The domain of dependence properties are really closely associated to a priori energy estimates (see Courant and Hilbert, Methods of Mathematical Physics).

But I don't think this will solve your problem since I don't believe your question can be recast in a form in which such estimates are available.

In particular, looking directly at your equation, on the spatial side it has potentially infinite speed of propagation since the spatial propagation is essentially just a transport equation. So if $W(x,p,t)$ has non-compact support, then the spatial propagation can have arbitrary large speeds. So if your potential vanishes or if your initial data is homogeneous in momentum, your solution will have, as its spatial domain of dependence given by the largest and smallest momentum at which $W$ is supported.

Assuming $f'' = 0$, then you equation can be solved by the method of characteristics: $\partial_t W = v\cdot\nabla W$ where $v(x,p) = (p,f(x))$ is a vector field. The domain of dependence for this problem can be easily found by integrating the vector field independently of the function $W$. I don't know how to deal with your third order term.

Like I said, in general there are only two ways to study domain of dependence properties that are well established, the first is via explicit notion of the Green's function, the second is energy estimates.