Search for: All records

We consider solutions of the repulsive Vlasov–Poisson system which are a combination of a point charge and a small gas, i.e., measures of the form\delta_{(\mathcal{X}(t),\mathcal{V}(t))}+\mu^{2}d\mathbf{x}d\mathbf{v}for some(\mathcal{X}, \mathcal{V})\colon \mathbb{R}\to\mathbb{R}^{6}and a small gas distribution\mu\colon \mathbb{R}\to L^{2}_{\mathbf{x},\mathbf{v}}, and study asymptotic dynamics in the associated initial value problem. If initially suitable moments on\mu_{0}=\mu(t=0)are small, we obtain a global solution of the above form, and the electric field generated by the gas distribution \mudecays at an almost optimal rate. Assuming in addition boundedness of suitable derivatives of \mu_{0}, the electric field decays at an optimal rate, and we derive modified scattering dynamics for the motion of the point charge and the gas distribution. Our proof makes crucial use of the Hamiltonian structure. The linearized system is transport by the Kepler ODE, which we integrate exactly through an asymptotic action-angle transformation. Thanks to a precise understanding of the associated kinematics, moment and derivative control is achieved via a bootstrap analysis that relies on the decay of the electric field associated to\mu. The asymptotic behavior can then be deduced from the properties of Poisson brackets in asymptotic action coordinates. 

We prove asymptotic stability of the Poisson homogeneous equilibrium among solu- tions of the Vlasov–Poisson system in the Euclidean space R3. More precisely, we show that small, smooth, and localized perturbations of the Poisson equilibrium lead to global solutions of the Vlasov–Poisson system, which scatter to linear solutions at a polynomial rate as t → ∞. The Euclidean problem we consider here differs signif- icantly from the classical work on Landau damping in the periodic setting, in several ways. Most importantly, the linearized problem cannot satisfy a “Penrose condition”. As a result, our system contains resonances (small divisors) and the electric field is a superposition of an electrostatic component and a larger oscillatory component, both with polynomially decaying rates. 

The goal of this article is twofold. First, we investigate the linearized Vlasov–Poisson system around a family of spatially homogeneous equilibria in the unconfined setting. Our analysis follows classical strategies from physics (Binney and Tremaine 2008, Galactic Dynamics,(Princeton University Press); Landau 1946, Acad. Sci. USSR. J. Phys.10,25–34; Penrose 1960,Phys. Fluids,3,258–65) and their subsequent mathematical extensions (Bedrossian et al 2022, SIAM J. Math. Anal.,54,4379–406; Degond 1986,Trans. Am. Math. Soc., 294,435–53; Glassey and Schaeffer 1994,Transp. Theory Stat. Phys.,23, 411–53; Grenier et al 2021, Math. Res. Lett., 28,1679–702; Han-Kwan et al, 2021, Commun. Math. Phys. 387, 1405–40; Mouhot and Villani 2011, Acta Math., 207, 29–201). The main novelties are a unified treatment of a broad class of analytic equilibria and the study of a class of generalized Poisson equilibria. For the former, this provides a detailed description of the associated Green’s functions, including in particular precise dissipation rates (which appear to be new), whereas for the latter we exhibit explicit formulas. Second, we review the main result and ideas in our recent work (Ionescu et al, 2022 on the full global nonlinear asymptotic stability of the Poisson equilibrium in R3 

We consider the dissipation of the Muskat problem and we give an elementary proof of a surprising inequality of Constantin-Cordoba-Gancedo-Strain [J. Eur. Math. Soc. (JEMS) 15 (2013), pp. 201–227 and Amer. J. Math. 138 (2016), pp. 1455–1494] which holds in greater generality. 

Abstract We construct a class of global, dynamical solutions to the 3 d Euler equations near the stationary state given by uniform “rigid body” rotation. These solutions are axisymmetric, of Sobolev regularity, have non-vanishing swirl and scatter linearly, thanks to the dispersive effect induced by the rotation. To establish this, we introduce a framework that builds on the symmetries of the problem and precisely captures the anisotropic, dispersive mechanism due to rotation. This enables a fine analysis of the geometry of nonlinear interactions and allows us to propagate sharp decay bounds, which is crucial for the construction of global Euler flows.