We study the inhomogeneous B\'enard equations on the infinite layer, $\Omega={\mathbb R}^2\times (-\frac12,\frac12)$, provided with an exterior force $f=f(z)$, depending only on the bounded variable $z\in[-\frac12,\frac12]$. There is a unique equilibrium solution $v=v(z)$ depending only on $z$. We study the stability of small $v(z)$, once under $L$-periodic perturbations, and once under spatially localized perturbations, i.e., perturbations in ${\mathcal L}^2(\Omega)$. Loss of stability may occur in the neighbourhood of the critical Rayleigh numbers $\lambda_L$ and $\lambda_\omega$, where $\lambda_L$ refers to the $L$-periodic setting, $\lambda_\omega$ to the ${\mathcal L}^2(\Omega)$ setting. Among others we give a characterization of $\lambda_\omega$ in terms of Orr-Sommerfeld theory. It is shown that if $\lambda_L\ne\lambda_\omega$ then $v(z)$ may be stable under $L$-periodic perturbations, but is necessarily unstable under ${\mathcal L}^2(\Omega)$ perturbations. The proofs are based on energy methods and on Bloch space theory.

Motivated by the classical model for the binary alloy solidification (crystallization) problem, we show the local in time existence and uniqueness of solutions to a parabolic system strongly coupled through free boundary conditions of Stefan type. Using a modification of the standard change of variables method and coercive estimates in a weighted Hölder space (the weight being a power of $t$) we obtain solutions with maximal global regularity (having at least equal regularity for $t>0$ as at the initial moment).

We study the quadratic optimal control problem over a finite time horizon for a class of abstract systems with analytic underlying semigroup $e^{tA}$ and unbounded control operator $B$. It is assumed that a suitable decomposition of the operator $B^*e^{tA^*}$ is valid, where only one component satisfies a `singular estimate', whereas for the other component specific regularity properties hold. Under these conditions, we prove well posedness of the associated differential Riccati equation, and in particular that the gain operator is bounded on a dense set. In spite of the unifying abstract framework used, the prime motivation (and application) of the resulting theory of linear-quadratic problems comes from optimal boundary control of a thermoelastic system with clamped boundary conditions. The non-trivial trace regularity estimate showing that this PDE mixed problem fits into the distinct class of models under examination---for which we have developed the present, novel optimal control theory---is established, as well.