Sharp Hypoelliptic Estimates for Some Kinetic Equations
نویسنده
چکیده
We provide a simple overview of some hypoellipticity results with sharp indices for a class of kinetic equations and we outline a general strategy based on some geometrical properties.
منابع مشابه
Hypoelliptic estimates in radiative transfer
We derive the hypoelliptic estimates for a kinetic equation of the form ∂tf + k · ∇xf = (−∆d)h, for (t, x, k) ∈ R× R × S, where d ≥ 1, β > 0, Sd is the unit sphere in Rd+1 and ∆d is the Laplace-Beltrami operator on Sd. Such equations arise in the modeling of high frequency waves in random media with long-range correlations. Assuming some (fractional) Sobolev regularity in the momentum variable ...
متن کاملExponential Return to Equilibrium for Hypoelliptic Quadratic Systems
We study the problem of convergence to equilibrium for evolution equations associated to general quadratic operators. Quadratic operators are nonselfadjoint differential operators with complex-valued quadratic symbols. Under appropriate assumptions, a complete description of the spectrum of such operators is given and the exponential return to equilibrium with sharp estimates on the rate of con...
متن کاملPartial differential equations. -- Gaussian estimates for hypoelliptic operators via optimal control
Partial differential equations. — Gaussian estimates for hypoelliptic operators via optimal control, by UGO BOSCAIN and SERGIO POLIDORO, communicated on 11 May 2007. ABSTRACT. — We obtain Gaussian lower bounds for the fundamental solution of a class of hypoelliptic equations, by using repeatedly an invariant Harnack inequality. Our main result is given in terms of the value function of a suitab...
متن کاملSome Remarks on Degenerate Hypoelliptic Ornstein-uhlenbeck Operators
We study degenerate hypoelliptic Ornstein-Uhlenbeck operators in L spaces with respect to invariant measures. The purpose of this article is to show how recent results on general quadratic operators apply to the study of degenerate hypoelliptic Ornstein-Uhlenbeck operators. We first show that some known results about the spectral and subelliptic properties of Ornstein-Uhlenbeck operators may be...
متن کاملG.H. Hakobyan ESTIMATES OF THE HIGHER ORDER DERIVATIVES OF THE SOLUTIONS OF HYPOELLIPTIC EQUATIONS
In this work we establish a connection between the behaviour of the higher order derivatives of the solutions of the hypoelliptic equation P(D)u = f and the estimates of the derivatives of f (x) in terms of multianisotropic Gevrey classes.
متن کامل