Airy's equation on $\mathbb R_-$ I am interested in Airy's equation
$$\frac{\partial u}{\partial t}(t,x)=-\frac{\partial^3 u}{\partial x^3}(t,x)$$
on a bounded or semi-bounded domain, e.g. on $(-\infty,0)$. In order to obtain a group of isometries as propagator in $L^2(-\infty,0)$, two boundary conditions have to imposed - e.g., $u'(0)=0$ and either $u(0)=0$ or $u''(0)=0$. But then it seems that the spectrum of the operator is far too large, so I doubt that this operator does not generate a semigroup in $L^2(-\infty,0)$ at all, regardless of the imposed boundary conditions. Is it so? Has anybody ever proved well-posedness of the above equation in some reasonable function or distribution space?
 A: This is a very interesting question and I do not know the answer. I would start at something like
N. A. Larkin, Correct initial boundary value problems for dispersive equations, J. Math. Anal. Appl. 344 (2008), no. 2, 1079--1092.
A: The operators $A=-\,\partial_x^3$ with domain $\mathscr D(A) =
\left(H^3\cap H_0^2\right)(\mathbb R_-)$ and $A^\ast=\partial_x^3$
with domain $\mathscr D(A^\ast) = \left(H^3\cap H_0^1\right)(\mathbb
R_-)$ are $m$-dissipative on $L^2(\mathbb R_-)$. This follows from
$\Re\langle Au,u\rangle =0$ for $u\in\mathscr D(A)$ and $\Re\langle
A^\ast v,v\rangle =-\,\frac12\,|v'(0)|^2\leq 0$ for $v\in\mathscr
D(A^\ast)$. Therefore, $A$ and $A^\ast$ generate $C_0$-contraction
semigroups on $L^2(\mathbb R_-)$, by the Lumer-Phillips
theorem. Moreover, the semigroup $\{e^{tA}\}_{t\geq0}$ consists of
isometries (though not of unitary operators for $t>0$, as $iA$ is not
selfadjoint, see Christian's comment above).
One can also directly solve the initial-boundary value problem
$$
\left\{ \enspace
\begin{aligned}
  & u_t + u_{xxx} = f(t,x), & (t,x)\in \mathbb R_+ \times \mathbb R_-, \\
  & u\bigr|_{x=0} = h_0(t), \enspace u_x\bigr|_{x=0} = h_1(t), \\
  & u\bigr|_{t=0} = u_0(x)
\end{aligned}
\right.
$$
(and similar for the operator $\partial_t-\partial_x^3$ with
one in place of two boundary conditions), e.g., by using the Laplace
transform with respect to $x$. See N. Hayashi, E. Kaikina, Nonlinear
Theory of Pseudodifferential Equations on a Half-line, North-Holland,
2004, which contains a chapter on linear problems.
