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Register a new userHypercontractivity and asymptotic behaviour in nonautonomous Kolmogorov equations
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We consider a class of nonautonomous second order parabolic equations with unbounded coefficients defined in $ItimesR^d$, where $I$ is a right-halfline. We prove logarithmic Sobolev and Poincare inequalities with respect to an associated evolution system of measures ${mu_t: t in I}$, and we deduce hypercontractivity and asymptotic behaviour results for the evolution operator $G(t,s)$.
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Using time dependent Lyapunov functions, we prove pointwise upper bounds for the heat kernels of some nonautonomous Kolmogorov operators with possibly unbounded drift and diffusion coefficients.
We consider a class of nonautonomous elliptic operators ${mathscr A}$ with unbounded coefficients defined in $[0,T]timesR^N$ and we prove optimal Schauder estimates for the solution to the parabolic Cauchy problem $D_tu={mathscr A}u+f$, $u(0,cdot)=g$.
We study a class of elliptic operators $A$ with unbounded coefficients defined in $ItimesCR^d$ for some unbounded interval $IsubsetCR$. We prove that, for any $sin I$, the Cauchy problem $u(s,cdot)=fin C_b(CR^d)$ for the parabolic equation $D_tu=Au$ admits a unique bounded classical solution $u$. This allows to associate an evolution family ${G(t,s)}$ with $A$, in a natural way. We study the main properties of this evolution family and prove gradient estimates for the function $G(t,s)f$. Under suitable assumptions, we show that there exists an evolution system of measures for ${G(t,s)}$ and we study the first properties of the extension of $G(t,s)$ to the $L^p$-spaces with respect to such measures.
We study Liouville-type theorems and the asymptotic behaviour of positive solutions near an isolated singular point $zetainpartialOmegacup{infty}$ of the quasilinear elliptic equations $$-text{div}(| abla u|_A^{p-2}A abla u)+V|u|^{p-2}u =0quadtext{in } Omegasetminus{zeta},$$ where $Omega$ is a domain in $mathbb{R}^d$ ($dgeq 2$), and $A=(a_{ij})in L_{rm loc}^{infty}(Omega;mathbb{R}^{dtimes d})$ is a symmetric and locally uniformly positive definite matrix. The potential $V$ lies in a certain local Morrey space (depending on $p$) and has a Fuchsian-type isolated singularity at $zeta$.
We study the behavior of solutions to the incompressible $2d$ Euler equations near two canonical shear flows with critical points, the Kolmogorov and Poiseuille flows, with consequences for the associated Navier-Stokes problems. We exhibit a large family of new, non-trivial stationary states of analytic regularity, that are arbitrarily close to the Kolmogorov flow on the square torus $mathbb{T}^2$. This situation contrasts strongly with the setting of some monotone shear flows, such as the Couette flow: in both cases the linearized problem exhibits an inviscid damping mechanism that leads to relaxation of perturbations of the base flows back to nearby shear flows. While this effect persists nonlinearly for suitably small and regular perturbations of some monotone shear flows, for the Kolmogorov flow our result shows that this is not possible. Our construction of these stationary states builds on a degeneracy in the global structure of the Kolmogorov flow on $mathbb{T}^2$. In this regard both the Kolmogorov flow on a rectangular torus and the Poiseuille flow in a channel are very different, and we show that the only stationary states near them must indeed be shears, even in relatively low regularity $H^3$ resp. $H^{5+}$. In addition, we show that this behavior is mirrored closely in the related Navier-Stokes settings: the linearized problems near the Poiseuille and Kolmogorov flows both exhibit an enhanced rate of dissipation. Previous work by us and others shows that this effect survives in the full, nonlinear problem near the Poiseuille flow and near the Kolmogorov flow on rectangular tori, provided that the perturbations lie below a certain threshold. However, we show here that the corresponding result cannot hold near the Kolmogorov flow on $mathbb{T}^2$.
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