| Abstract: |
| Well-posedness is proved for some classes of singular
second-order SPDEs on a smooth bounded domain
$D$ in $\mathbb{R}^n$ in the form
\[
du(t) - \operatorname{div}\gamma(\nabla u(t))\,dt
+ \beta(u(t))\,dt \ni B(t,u(t))\,dW(t)\,,
\qquad u(0)=u_0\,.
\]
The drift is associated to two maximal monotone
operators $\gamma$ and $\beta$ on $\mathbb{R}^n$
and~$\mathbb{R}$, respectively, on which neither growth
nor smoothness assumptions are imposed.
Moreover, the noise is given by a cylindrical Wiener
process $W$ on a Hilbert space $U$,
with a stochastic integrand $B$ taking values in the
Hilbert-Schmidt operators from $U$ to $L^2(D)$:
classical Lipschitz-continuity hypotheses for the
diffusion coefficient are assumed.
The proof consists in approximating the equation,
finding uniform estimates both pathwise
and in expectation on the approximated solutions,
and then passing to the limit using monotonicity
and compactness arguments.
This study is based on a joint work with
Carlo Marinelli (University College London). |
|