We consider the compressible Euler equation with a Coriolis term and prove a lower bound on the time of existence of solutions in terms of the speed of rotation, sound speed, and size of the initial data. Along the way, we obtain precise dispersive decay estimates for the linearized equation. In the incompressible limit, this improves current bounds for the incompressible Euler-Coriolis system as well.

We consider the Vlasov–Poisson system in a C^3 convex domain D with a perfectly conducting wall. We introduce the asymptotic domain D_∞ for the domain D. Then under acceptable assumptions on D, we show that for localized initial data, the velocity of particles is asymptotically supported in the (closure of the) asymptotic domain D_∞ and the solutions exhibit the asymptotics of modified scattering.

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 $\delta_{(\mathcal{X}(t),\mathcal{V}(t))}+\mu^2d{\bf x}d{\bf v}$ for some $(\mathcal{X}, \mathcal{V}):\mathbb{R}\to\mathbb{R}^6$ and a small gas distribution $\mu:\mathbb{R}\to L^2_{{\bf x},{\bf v}}$, 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 $\mu$ decays 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 a 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 nonlinear Landau damping in optimal weighted Gevrey-3 spaces for solutions of the confined Vlasov-Poisson system on ^d×^d which are small perturbations of homogeneous Penrose-stable equilibria. We also prove the existence of nonlinear scattering operators associated to the confined Vlasov-Poisson evolution, as well as suitable injectivity properties and Lipschitz estimates (also in weighted Gevrey-3 spaces) on these operators. Our results give definitive answers to two well-known open problems in the field, both of them stated in the recent review of Bedrossian [4, Section 6].

In this paper, we investigate the dynamics of solutions of the Muskat equation with initial interface consisting of multiple corners allowing for linear growth at infinity. Specifically, we prove that if the initial data contains a finite set of small corners then we can find a precise description of the solution showing how these corners desingularize and move at the same time. At the analytical level, we are solving a small data critical problem which requires renormalization. This is accomplished using a nonlinear change of variables which serves as a logarithmic correction and accurately describes the motion of the corners during the evolution.

We prove asymptotic stability of the Poisson homogeneous equilibrium among solutions of the Vlasov–Poisson system in the Euclidean space ℝ^3 . 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 significantly 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.

Computational and Experimental Research in Mathematics (ICERM). This once fledgling center has now grown into a mature institute, and this article aims to reintroduce ICERM to the international mathematical community, provide an update on its achievements over the last decade, and encourage mathematical scientists of all stages to participate in its future.

While it is well known that constant rotation induces linear dispersive effects in various fluid models, we study here its effect on long time nonlinear dynamics in the inviscid setting. More precisely, we investigate stability in the 3d rotating Euler equations in R3 with a fixed speed of rotation. We show that for any M>0, axisymmetric initial data of sufficiently small size epsilon lead to solutions that exist for a long time at least epsilon-M and disperse. This is a manifestation of the stabilizing effect of rotation, regardless of its speed. To achieve this we develop an anisotropic framework that naturally builds on the available symmetries. This allows for a precise quantification and control of the geometry of nonlinear interactions, while at the same time giving enough information to obtain dispersive decay via adapted linear dispersive estimates.

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.

The goal of this article is twofold. First, we investigate the linearized Vlasov–Poisson system around a family of spatially homogeneous equilibria in R3 (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 (arXiv: 2205.04540 )) on the full global nonlinear asymptotic stability of the Poisson equilibrium in R3 .

We show the existence of self-similar solutions for the Muskat equation. These solutions are parameterized by $0 0$.

We construct a class of global, dynamical solutions to the 3d 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.

We prove small data modified scattering for the Vlasov-Poisson system in dimension $d=3$ using a method inspired from dispersive analysis. In particular, we identify a simple asymptotic dynamic related to the scattering mass.

This book provides a definitive proof of global nonlinear stability of Minkowski spacetime as a solution of the Einstein-Klein-Gordon equations of general relativity. Along the way, a novel robust analytical framework is developed, which extends to more general matter models. The book proves global regularity at an appropriate level of generality of the initial data, and then proves several important asymptotic properties of the resulting spacetime, such as future geodesic completeness, peeling estimates of the Riemann curvature tensor, conservation laws for the ADM tensor, and Bondi energy identities and inequalities. The book is self-contained, providing complete proofs and precise statements, which develop a refined theory for solutions of quasilinear Klein-Gordon and wave equations, including novel linear and bilinear estimates. Only mild decay assumptions are made on the scalar field and the initial metric is allowed to have nonisotropic decay consistent with the positive mass theorem. The framework incorporates analysis both in physical and Fourier space, and is compatible with previous results on other physical models such as water waves and plasma physics.

We prove global in time Strichartz estimates on the semi-periodic cylinder with initial data in L-2. This extends the local results of Takaoka-Tzvetkov [13] and the global estimates with loss of derivatives of Hani-Pausader [9] and Barron [1]. (c) 2020 Published by Elsevier Ltd.

In this note, we prove the profile decomposition for hyperbolic Schr\odinger (or mixed signature) equations on $\mathbb{R}^2$ in two cases, one mass-supercritical and one mass-critical. First, as a warm up, we show that the profile decomposition works for the ${\dot H}^{\frac12}$ critical problem, which gives a simple generalization of for instance one of the results in Fanelli-Visciglia (2013). Then, we give the derivation of the profile decomposition in the mass-critical case by proving an improved Strichartz estimate. We will use a very similar approach to that laid out in the notes of Killip-Visan (2008), but we are forced to do a double Whitney decomposition to accommodate an extra scaling symmetry that arises in the problem with mixed signature.

We consider the Vlasov–Poisson system with repulsive interactions. For initial data a small, radial, absolutely continuous perturbation of a point charge, we show that the solution is global and disperses to infinity via a modified scattering along trajectories of the linearized flow. This is done by an exact integration of the linearized equation, followed by the analysis of the perturbed Hamiltonian equation in action-angle coordinates.

We construct (modified) scattering operators for the Vlasov-Poisson system in three dimensions, mapping small asymptotic dynamics as $t\to -\infty$ to asymptotic dynamics as $t\to +\infty$. The main novelty is the construction of modified wave operators, but we also obtain a new simple proof of modified scattering. Our analysis is guided by the Hamiltonian structure of the Vlasov-Poisson system. Via a pseudo-conformal inversion we recast the question of asymptotic behavior in terms of local in time dynamics of a new equation with singular coefficients which is approximately integrated using a generating function.