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I'm currently looking through the following paper which examines some dynamics of the Airy$_2$ process: https://arxiv.org/pdf/1106.2717.pdf

On page 2, there appears a PDE of the form

$\partial_t u + Hu = 0, \quad x \leq g(t) $

with boundary conditions

$u(t,x) = 0, \quad x > g(t)$,

$u(l, x) = f(x)$ if $x < g(t)$

where $H = - \partial_x^2 + x$ is the Airy Hamiltonian.

Now, this means that $g$ describes the domain in space - however I don't understand why one might require it to be in $H^1$. The authors point to the proof of Proposition 3.2, but there only a generalisation from a particular solution to the $g\in H^1$ case is done - nothing indicates that that integrability should be necessary for the result to hold.

$H^1$ in one dimension doesn't seem sufficient to get a lot of regularity - if I'm not mistaken, we get that $g\in L^\infty \cap C^{0,\frac{1}{2}}$, but nothing like being locally Lipschitz or other conditions you might want in connection with a PDE that would allow one to, for example, apply divergence theorem.

So, are there situations where this kind of $H^1$ regularity on a domain is important, or was it just some class for which the proof was done in this paper.

An additional doubt I have is that in context, the statement should work even if just $g(l), \ g(r)$ are prescribed, and $g(t) = \infty$ everywhere else, and obviously that lacks any integrability or differentiability whatsoever.

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