On improvement of summability properties in nonautonomous Kolmogorov equations

Under suitable conditions, we obtain some characterization of supercontractivity, ultraboundedness and ultracontractivity of the evolution operator $G(t,s)$ associated to a class of nonautonomous second order parabolic equations with unbounded coefficients defined in $I\times\R^d$, where $I$ is a right-halfline. For this purpose, we establish an Harnack type estimate for $G(t,s)$ and a family of logarithmic Sobolev inequalities with respect to the unique tight evolution system of measures $\{\mu_t: t \in I\}$ associated to $G(t,s)$. Sufficient conditions for the supercontractivity, ultraboundedness and ultracontractivity to hold are also provided

Non autonomous parabolic problems with unbounded coefficients in unbounded domains

Given a class of nonautonomous elliptic operators \A(t) with unbounded coefficients, defined in \overline{I \times \Om} (where $I$ is a right-halfline or $I=\R$ and \Om\subset \Rd is possibly unbounded), we prove existence and uniqueness of the evolution operator associated to \A(t) in the space of bounded and continuous functions, under Dirichlet and first order, non tangential homogeneous boundary conditions. Some qualitative properties of the solutions, the compactness of the evolution operator and some uniform gradient estimates are then proved

LpL^p-estimates for parabolic systems with unbounded coefficients coupled at zero and first order

We consider a class of nonautonomous parabolic first-order coupled systems in the Lebesgue space $L^p({\mathbb R}^d;{\mathbb R}^m)$, $(d,m \ge 1)$ with $p\in [1,+\infty)$. Sufficient conditions for the associated evolution operator ${\bf G}(t,s)$ in $C_b({\mathbb R}^d;{\mathbb R}^m)$ to extend to a strongly continuous operator in $L^p({\mathbb R}^d;{\mathbb R}^m)$ are given. Some $L^p$-$L^q$ estimates are also established together with $L^p$ gradient estimates

Invariant measures for systems of Kolmogorov equations

In this paper we provide sufficient conditions which guarantee the existence of a system of invariant measures for semigroups associated to systems of parabolic differential equations with unbounded coefficients. We prove that these measures are absolutely continuous with respect to the Lebesgue measure and study some of their main properties. Finally, we show that they characterize the asymptotic behaviour of the semigroup at infinity

Strong convergence of solutions to nonautonomous Kolmogorov equations

We study a class of nonautonomous, linear, parabolic equations with unbounded coefficients on $\mathbb R^{d}$ which admit an evolution system of measures. It is shown that the solutions of these equations converge to constant functions as $t\to+\infty$. We further establish the uniqueness of the tight evolution system of measures and treat the case of converging coefficients

Rigorous derivation of the Kuramoto-Sivashinsky equation in a 2D weakly nonlinear Stefan problem

In this paper we are interested in a rigorous derivation of the Kuramoto-Sivashinsky equation (K--S) in a Free Boundary Problem. As a paradigm, we consider a two-dimensional Stefan problem in a strip, a simplified version of a solid-liquid interface model. Near the instability threshold, we introduce a small parameter $\varepsilon$ and define rescaled variables accordingly. At fixed $\varepsilon$, our method is based on: definition of a suitable linear 1D operator, projection with respect to the longitudinal coordinate only, Lyapunov-Schmidt method. As a solvability condition, we derive a self-consistent parabolic equation for the front. We prove that, starting from the same configuration, the latter remains close to the solution of K--S on a fixed time interval, uniformly in $\varepsilon$ sufficiently small